Transport in Low Dimensional Strongly Correlated Ele troni System by Feifei Li B. S., Nanjing University, 2001 S . M., Nanjing University, 2004 Thesis Submitted in partial fulllment of the requirements for the degree of Do tor of Philosophy in the Department of Physi s at Brown University Providen e, Rhode Island May, 2010 Abstra t of Transport in Low Dimensional Strongly Correlated Ele troni System by Feifei Li, Ph.D., Brown University, May, 2010. This thesis presents two theoreti al investigations on transport in low dimensional strongly orrelated ele troni systems. In the study of one dimensional system, we demonstrate that spin urrent an be gen- erated by an a voltage in a one- hannel quantum wire with strong repulsive ele tron intera tions in the presen e of a non-magneti impurity and uniform stati magneti eld. We show that in a ertain range of voltages, the spin urrent an exhibit a power dependen e on the a voltage bias with a negative exponent. The spin urrent expressed in units of ~/2 per se ond an be ome mu h larger than the harge urrent in units of the ele tron harge per se ond. The spin urrent generation requires nei- ther spin-polarized parti le inje tion nor time-dependent magneti elds. In the study of non-Abelian statisti s in two dimensional quantum Hall system, we suggest an experiment whi h an determine the physi al state for the ν = 5/2 quan- tum Hall plateau. The proposal involves transport measurements in the geometry with three quantum Hall edges onne ted by two quantum point onta ts. In on- trast to interferen e experiments, this approa h an distinguish the Pfaan and anti- Pfaan states as well as dierent states with identi al Pfaan or anti-Pfaan statis- ti s. In addition, the transport is not sensitive to the u tuations of the number of the quasiparti les trapped in the system. © Copyright 2009 by Feifei Li This dissertation by Feifei Li is a epted in its present form by the Department of Physi s as satisfying the dissertation requirement for the degree of Do tor of Philosophy. Date Prof. Dima Feldman, Dire tor Re ommended to the Graduate Coun il Date Prof. Brad Marston, Reader Date Prof. James Valles, Jr., Reader Approved by the Graduate Coun il Date Sheila Bonde Dean of the Graduate S hool iii Vita Edu ation ˆ Brown University, Providen e, RI 02912 2004-2009  Ph.D. in Physi s. Department of Physi s. July 2009 ˆ Nanjing University, Nanjing, Jiangsu Provin e, China 1997-2004  M.S . in Physi s. Department of Physi s. June 2004  B.S . in Physi s. Department of Physi s. July 2001 Publi ations ˆ D. E. Feldman, Feifei Li, Charge-statisti s separation and probing non-Abelian states, Phys. Rev. B 78, 161304(R) (2008) (Editors' suggestion). iv v ˆ Bernd Braune ker, D. E. Feldman, and Feifei Li, Spin urrent and re ti ation in one-dimensional ele troni systems, Phys. Rev. B 76, 085119 (2007). ˆ Li Fei-Fei, Li Zheng-Zhong and Xiao Ming-Wen, Ee ts of temperature and ele tron ee tive mass on bias-dependent tunnelling magnetoresistan e, Chinese Phys. 14, 1025 (2005). ˆ Fei-fei Li, Zheng-zhong Li, Ming-wen Xiao, Jun Du, Wang Xu and An Hu, Bias dependen e and inversion of the tunneling magnetoresistan e in ferromagneti jun tions, Phys. Rev. B 69, 054410 (2004). ˆ Fei-fei Li, Zheng-zhong Li, Ming-wen Xiao, Jun Du, Wang Xu, An Hu, and John Q. Xiao, Bias dependent tunneling in ferromagneti jun tions and inversion of the tunneling magnetoresistan e from a quantum me hani al point of view, J. Appl. Phys. 95, 7243 (2004). ˆ Li Fei-Fei, Xiao Ming-Wen, Li Zheng-Zhong, Hu An and Xu Wang, Ee t of Barrier Width on Bias-Dependent Tunnelling in Ferromagneti Jun tions, Chi- nese Phys. Lett. 21, 2271 (2004). ˆ Wen-Ting Sheng, W. G. Wang, X. H. Xiang, F. Shen, Fei-Fei Li, T. Zhu, Z. Zhang, Zheng-Zhong Li, Jun Du, An Hu and John Q. Xiao, Probing Tunnel Barrier Shape and Its Ee ts on Inversed Tunneling Magnetoresistan e at High Bias, J. Ele . Mat. 33, 1274 (2004). Prefa e This thesis provides a theoreti al investigation of transport in strongly intera ting system in one and two spa ial dimension. Chp. 1 provides an introdu tion to Luttinger liquid theory and quantum Hall liq- uid theory. A review of bosonization te hnique is presented in Se . 1.1.2, followed by some appli ations in 1D spinless and spinful system where the on ept of spin- harge separation is introdu ed. A brief intro ution to quantum Hall (QH) liquid in lud- ing fra tional harge, edge ex itation and abelian/non-Abelian fra tional statisti s is presented in Se . 1.2. The rest of the two hapters are devoted to two proje ts that I have worked on during my Ph.D. years. In Chapter 2, we present a theoreti al investigation on harge and spin transport in 1D strongly intera ting system where we nd an interesting possibility to generate pure spin urrent in asymmetri quantum wires using only an AC voltage and spa ially uniform magneti eld. In Chapter 3, we present an experimental proposal that an be used to probe the state of quantum Hall liquids at 5/2 lling. We demonstrate that transport measurements in a two-point onta t vi vii geometry is enough to distingish six of the ompeting andidates for 5/2 state. In ontrast to the interferen e experiments proposed for the same purpose, our proposal is not sensitive to the quasi-parti le u tuations in the system. Finally, a summary of the thesis is given in Chp. 4. A knowledgments I would like to express tremendous thanks to my thesis advisor Professor Dima Feld- man for his invaluable guidan e, inspiration and en ouragement. I owe so mu h to Dima, way more than I an express in words. The last ve years have been unexpe t- edly hallenging to me. There were those tough times when I was onfused, lost and depressed. It was Dima's patien e, en ouragement and are that helped me eventu- ally get out of the di ulty. Today, I am able to present my Ph.D. thesis as well as ondently look forward to my a ademi areer. All these would be impossible without Dima! I would also like to thank Professor Brad Marston and Professor James Valles, Jr. for being in my dissertation defense ommittee and reading this thesis. In parti ular, I am thankful to Professor Brad Marston for his wonderful tea hing and for intro- du ing to me Python whi h has be ome my favorate programming language. I am grateful to Professor J. Mi hael Kosterlitz and my o emates Dina Obeid, Chenjie Wang and Pengyu Liu for the weekly group study of the Renormalization Group. My viii ix understanding of RG is so mu h rened with the help of Professor Kosterlitz's inspir- ing questions and our group dis ussion. I also want to re organize my ollabarator Dr. Bernd Braune ker and olleagues K. T. Law, Alnaz Alipour-Assiabi, Andrew Callan-Jones, Ookie Ma for all the valuable dis ussions and suggestions. I am deeply thankful to my master thesis advisor Professor Zheng-zhong Li who al- ways guides me like a wise father, and Professor Gang Xiao for all the en ouragement and help. Finally I would like to thank my parents Wanli Li, Guiyu Tang and my wife Hongyuan Fu for their immeasurable love and support. Thank you for bringing so mu h happi- ness to me! This work was supported in part by the NSF under grant numbers DMR-0213818, DMR-0544116, and PHY99-07949, and by BSF grant 2006371 as well as Salomon Resear h Award. Contents Vita iv Prefa e vi A knowledgments viii List of Figures xiii 1 Introdu tion 3 1.1 Luttinger Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.1 The breakdown of Landau's Fermi liquid theory in 1D . . . . 3 1.1.2 Bosonization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.1.2.1 Tomanaga-Luttinger model . . . . . . . . . . . . . . 11 1.1.2.2 Bosonization of free hiral Hamiltonian . . . . . . . . 15 1.1.3 Exa t solution of 1D intera ting Fermi system . . . . . . . . . 25 1.1.3.1 Spinless system . . . . . . . . . . . . . . . . . . . . . 25 1.1.3.2 Model with spin: harge-spin separation . . . . . . . 29 x 1.2 Quantum Hall Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . 35 1.2.1 Integer Quantum Hall Ee t . . . . . . . . . . . . . . . . . . . 36 1.2.2 Fra tional Quantum Hall Ee t . . . . . . . . . . . . . . . . . 43 1.2.3 Edge ex itations in Quantum Hall Liquids . . . . . . . . . . . 46 1.2.4 Fra tional Charge and Fran tional Statisti s . . . . . . . . . . 50 1.2.5 Non-Abelian Statisti s and ν = 5/2 state . . . . . . . . . . . . 54 2 Generating Spin Current in Asymmetri Quantum Wire 59 2.1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.2 Model and Physi s of the problem . . . . . . . . . . . . . . . . . . . . 61 2.3 Bosonization and Keldysh Te hnique . . . . . . . . . . . . . . . . . . 71 2.4 Re ti ation urrents . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 2.5 Con lusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3 Probing non-Abelian Statisti s in quantum Hall Liquids at 5/2 ll- ing 85 3.1 Introdu tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.2 The two-point- onta t Geometry and Model Hamiltonian . . . . . . . 88 3.3 Analyti al Cal ulation . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.3.1 General analyti al formula for Current and Short noise . . . . 91 3.3.2 Analyti al results for Ea h States . . . . . . . . . . . . . . . . 97 3.4 Con lusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 xi 4 Summary 117 A Cal ulation of the Correlation Fun tions in 1D 118 B Te hni al details in Chapter 2 124 B.1 High potential barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 B.1.1 Model without intera tion . . . . . . . . . . . . . . . . . . . . 124 B.1.2 Model with intera tion . . . . . . . . . . . . . . . . . . . . . . 125 B.2 Estimation of higher perturbative orders . . . . . . . . . . . . . . . . 126 B.2.1 Parametrization of s aling dimensions . . . . . . . . . . . . . . 128 B.2.2 Most relevant operators . . . . . . . . . . . . . . . . . . . . . 128 B.2.3 Se ond order ontribution to the urrent . . . . . . . . . . . . 130 B.2.4 Third order ontribution to the urrent . . . . . . . . . . . . . 131 B.2.5 Numeri al estimates . . . . . . . . . . . . . . . . . . . . . . . 134 B.3 Expli it evaluation of the third order urrents . . . . . . . . . . . . . 135 Bibliography 138 xii List of Figures 1.1 O upation number versus momentum for free ele tron gas. . . . . . . 4 1.2 Phase spa e argument for a general s attering pro ess for ele trons lose to the Fermi surfa e. The onservation of momentum and energy signi antly limits the phase spa e(k, θ ) that supports the s attering. 6 1.3 O upation number versus momentum for Fermi liquids. The dis on- tinuity at Fermi surfa e Z is smaller than one as a result of intera tion. 8 1.4 Real spa e pi ture showing the ee t of intera tion in Fermi Liquids. Ele tron an always move without signi antly pushing all of the other ele trons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.5 Phase spa e for s attering in 1D. In ontrast to higher dimension, the phase spa e is not limited be ause simultaneous onservation of mo- mentum and energy is guaranteed. . . . . . . . . . . . . . . . . . . . 10 1.6 Energy spe trums of the Hamiltonians Eq. (1.18) and (1.19). The Tomanaga-Luttinger model (blue urves) and a realisti  Hamiltonian have the same low energy physi s (q ≪ kF ). . . . . . . . . . . . . . . 14 xiii 1.7 S attering pro esses in 1D, (a) g4 pro ess between ele trons near the same Fermi point. (b) g2 pro ess between ele trons sitting on opposite Fermi points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.8 O upation number nR (k) in a Tomanaga-Luttinger model to be om- pared with the Fermi Liquid ase in Fig. 1.3. The dis ontinuity disap- pears as a result of intera tion. . . . . . . . . . . . . . . . . . . . . . 29 1.9 g1 pro ess in a model with spin. It represents the s attering between ele trons of opposite spins sitting at opposite Fermi points. . . . . . . 31 1.10 Classi al Hall ee t. . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 1.11 The degenerate angular momentum states in zeroth Landau Level . . 41 1.12 (a) DOS of a translational invariant sample. (b) DOS of a system with disorder where the shaded region onsists lo alized states. . . . . . . . 42 1.13 Potential prole of an IQH liquid with boundary. The bulk ex itation has a gap ~ωc while the edge ex itation is gapless. l labels the angular momentum state in symmetri gauge. . . . . . . . . . . . . . . . . . . 47 2.1 Sket h of the one-dimensional ondu tor onne ted to two ele trodes on both ends. Currents are driven through a voltage bias V that is applied to the left ele trode while the right ele trode is kept at ground. The system is magnetized by the eld H. Ele trons are ba ks attered o the asymmetri potential U (x). U (x) 6= 0 in the region of size aU ∼ 1/kF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 xiv 2.2 Double-well potential with quasistationary levels. The transmission oe ient is maximal in the shaded regions. The narrow potentials u1 (x) and u2 (x) are entered at the positions x = 0 and x = a (a < kF−1 ), respe tively, and are modeled by δ -fun tions in Eq. (B.1). . . . 64 2.3 Normalized harge re ti ation urrent Ic /Ic0 and spin re ti ation urrent Is /Is0 versus applied voltage V /V0 for non-intera ting ele trons with EF = 400ǫ0 , µH = 75ǫ0 , u1 = 50ǫ0 a and u2 = −50ǫ0 a, where ǫ0 = ~2 /ma2 (see Fig. 2.2 and Appendix B.1). Ic0 = 50eǫ0 /~, Is0 = 25ǫ0 and V0 = 50ǫ0 /e are arbitrary referen e urrents and voltage. . . . . 68 2.4 Normalized harge re ti ation urrent Ic /Ic0 and spin re ti ation urrent Is /Is0 versus applied voltage V /V0 for intera ting ele trons with EF = 100ǫ0 , µH = 25ǫ0 , γ = 12.6ǫ0 a/e, u1 = 25ǫ0 a and u2 = 50ǫ0 a, where ǫ0 = ~2 /ma2 (see Fig. 2.2 and Appendix B.1). Ic0 = 50eǫ0 /~, Is0 = 25ǫ0 , and V0 = 50ǫ0 /e are arbitrary referen e urrents and volt- age. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 2.5 Qualitative representation of the spin re ti ation urrent. The spin urrent ex eeds the harge urrent and follows a power-law dependen e on the voltage with a negative exponent in the interval of voltages V ∗ < V < V ∗∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.1 Setup with two quantum point onta ts. Arrows show the propagation dire tion of harged modes. . . . . . . . . . . . . . . . . . . . . . . . 88 xv 3.2 The Keldysh ontour. O1 (t1 ), O2 (t2 ) an lie on the forward bran h (σ = +) or ba kward bran h (σ = −). . . . . . . . . . . . . . . . . . . 93 3.3 K=8 state, only harged boson φ propagate on edge 3. Hen e S2 /I2 = 2q = 4e. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 3.4 Pfaan state, harged boson φc ontributes to both urrent and shot noise while Majorana fermion λ ontributes to noise only. Hen e there is ex essive noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 3.5 Edge-re onstru ted Pfaan state with voltage applied on sour e 2. Only left moving Majorana fermion λ ontributes to the urrent and shot noise in drain 1. Hen e I1 = 0 and S1 6= 0. . . . . . . . . . . . . 112 xvi 1 Intera tions among parti les are what make the physi s of ele troni systems so ri h. But the way how intera tions hange the system from a non-intera ting one depends on dimensionality. Indeed, ele trons in three spa ial dimension (3D) form Fermi liquids[1℄ whose low energy physi s is essentially similar to non-intera ting Fermi gases, even though the intera tion may be strong. The little dieren e between Fermi gases and 3D Fermi liquid is due to the fa t that intera tions among ele trons lose to the Fermi surfa e are very ine ient in 3D. On the ontrary, in one dimen- sion (1D) systems, e.g., quantum wires, intera tions are extremely e ient and the systems form Luttinger liquids[2℄. Unlike Fermi liquids whose low energy ex itations are fermioni quasi-parti les, Luttinger liquids have separte harge and spin ex ita- tions that are both Bosons. The most exoti situation appears in two spa ial dimension (2D) with high magneti eld, where ele trons form quantum Hall liquids whose Hall ondu tan e versus magneti eld exhibit series of plateaus. These plateaus are ex- a tly quantized at integer multiples (integer quantum Hall ee t) or ertain fra tions (fra tional quantum Hall ee t) of a universial onstant e2 /h[3, 4℄. In ontrast to 1D and 3D, the elemenary low energy ex itations in 2D quantum Hall uids are neither Fermions nor Bosons, but anyons that arry fra tional statisti s.[5, 6℄ Interestingly, the physi s of Fermi liquids and Luttinger liquids meet on the edge of the quantum Hall liquid. It turns out that the edge of an integer quantum Hall liquid is a Fermi liquid[7℄ and the edge of a fra tional quantum Hall liquid is a Luttinger liquid.[8℄ In 2 this thesis, we fo us on the transport properties of two types of Luttinger liquids, quan- tum wires and edges of fra tional quantum Hall liquids. We begin with introdu tions to 1D and 2D ele troni systems. Chapter 1 Introdu tion 1.1 Luttinger Liquids 1.1.1 The breakdown of Landau's Fermi liquid theory in 1D The simplest Fermi system is non-intera ting ele tron gas. Consider a model with non-intera ting spinless ele trons in a box. The Hamiltonian of the system is 2 ˆ = − ~ ∇2 . H (1.1) 2m The eigenstates of this Hamiltonian are plan waves 1 ik·r |ki = e (1.2) L3/2 with energy ~2 k 2 ǫ(k) = . (1.3) 2m 3 4 Here L is the size of the box and k ≡ |k|. Due to Pauli Ex lusion Prin iple, ea h k state an only a omondate one spinless ele tron. As a result, the ground state at T =0 is formed by lling N ele trons up to the Fermi energy EF ≡ ~2 kF2 /2m, where the Fermi wavenumber kF is determined by µ ¶3 µ ¶ L 3 3 πk = N. (1.4) 2π 4 F If one plots the o upation number as a fun tion of momentum, it has a dis ontinuity Z = 1 at the Fermi surfa e (see Fig. 1.1). Using the language of se ond quantization, the ground state of the system an be written as Y |GSi = Ck† |0i. (1.5) k kF , |ESi = Ck†′ Ck |GSi, k < kf < k ′ . (1.6) Su h ex itation is alled parti le-hole ex itation. Obviously the ex ited state is still an eigenstate of the free Hamiltonian Eq. (1.1). It has innite lifetime. 5 The fermioni nature of the ele trons plays an essential role in the thermodynam- i al behavior of the system. Be ause of the Pauli Ex lusion, ele trons are for ed to piled up to a very high energy (Fermi energy EF ). For a normal metal, EF is of the order of 105 K . As a result, thermal u tuation at room temperature (∼ 300K ) an only ex ite a small fra tion of ele trons lose to the Fermi surfa e. In a real system, ele trons are not free but an be s attered by ea h other through Coulomb for e. The Hamiltonian has an additional potential term, ˆ ~2 2 ˆ H=− ∇ +V. (1.7) 2m Be ause of Vˆ , Eq. (1.6) is no longer an eigenstate of the system. The presen e of the intera tion term in the Hamiltonian makes our life very di ult. In general we an only treat intera tion perturbatively. However, intera tion energy in a typi al system is usually of the same order ompared to the kineti energy. Stri tly speak- ing, perturbation theory an not be applied. Given all these ompli ations, one may doubt any of the ni e features in the non-intera ting Fermi gas pi ture still survives. Surprisingly, thanks to Landau's Fermi Liquid theory, our life is mu h simpler in two or three dimension if we are interested only in the low temperature physi s of the system. Landau remarkably showed that an intera ting fermion system is essentially ∗ similar to a free fermion system . The genius of Landau's theory omes from the observation that ele tron-ele tron s attering near the Fermi surfa e is ine ient due ∗ There are ex eptions, e.g., super ondu tors. 6 to Pauli Ex lusion Prin iple and onservation laws, regardless of the strength of inter- a tion. This an be understood through a simple phase spa e argument. Consider a s attering pro ess in Fig. (1.2) where two ele trons with initial momentum k1 and k2 are s attered into momentum states k1′ and k2′ . Most states below the Fermi surfa e k1 k2 O1 O2 k1′ k2′ Figure 1.2: Phase spa e argument for a general s attering pro ess for ele trons lose to the Fermi surfa e. The onservation of momentum and energy signi antly limits the phase spa e(k, θ ) that supports the s attering. are o upied. So a low energy pro ess must have k1 , k2 , k1′ , k2′ ∼ kF . In addition, the s attering pro ess must simultaneously onserves both momentum and energy, k1 + k2 = k1′ + k2′ , (1.8) k12 + k22 = (k1′ )2 + (k2′ )2 . (1.9) It then follows from geometry that only the states within two small shaded region in Fig. ( 1.2) an be s attered into. The width of the region is roughly ∆k = k1 − kF . It an be shown[1℄ that the probability of s attering is proportional to (∆k)2 in 3D. 7 Therefore the lifetime τ of the ex itation diverges as τ ∼ (∆k)−2 ∼ (E − EF )−2 . (1.10) Note that the power dependen e of τ in Eq. (1.10) is ru ial for the parti le-hole ex itation's longevity. Consider the wavefun tion with a nite lifetime τ e−i(E−EF )t e−t/τ . (1.11) It has a period of 2π/(E − EF ). If the damping time τ is smaller than one os illating period, the wave will be overdamped. Eq. (1.10) ensures that the ex itation be omes more and more well-dened for energy loser and loser to the Fermi surfa e. Now we have the following pi ture for an intera ting ele tron system. Starting from a free ele tron gas, we ex ite on ele tron under the Fermi sea (k < kF ) to just above the Fermi surfa e. Su h ex itation has a momentum q = k′ − k and energy E = (k ′2 − k 2 )/2m. Now imagine we turn on the intera tion, the ex ited ele tron will start to intera t therefore a quiring potential energy. So the energy of the ex itation be omes E ′ 6= E . Its momentum remains q be ause inner for e onserves total momentum. Following the above phase spa e argument, if the ex ited ele tron lies just above the Fermi surfa e, it will live for a long time before it de ays into the ba kground. Therefore, upon the intera tion is fully turned on, a bare ele tron be omes a so alled quasi-parti le whi h has denite momentum q and energy E ′. It has nite but very long lifetime. The quasi-parti le is in one-to-one orresponden e to the bare ele tron in a non-intera ting system. One an think of it as an ele tron 8 surrounded by louds of parti le-hole ex itations. Be ause the parti le-hole ex itation is bosoni , the quasi-parti le maintains the bare ele tron's fermioni nature. As a result, the intera ting system still have a sharp Fermi surfa e but with a redu ed dis ontinuity Z < 1 as shown in Fig. ( 1.3). Z < 1 ree ts the fa t that only a portion of the ele tron remains in the quasi-parti le state. The `rest' of the ele tron ontributes to the parti le-hole loud. Obviously, the stronger is the intera tion, the less is the portion of the ele tron that remains in the quasi-parti le state, hen e the smaller is the fermioni dis ontinuity Z. Having a nite dis ontinuity 0 δ terms in Eq. (1.50), we an interpret it as the ee tive band width of the system. But stri tly speaking, it is merely a mathemati al helper and an not be interpreted as any ` ut-o ' length as emphasized by Haldane[2℄. Eq. (1.45) now takes an elegant form ˆ ψˆR (x) = ηˆR eikF R x+iφR (x) . (1.51) Using Eq. (1.47), the parti le density is related to φˆR (x) as 1 ρˆR (x) = (kF + ∂x φˆR (x)) (1.52) 2π Eq. (1.51) is alled bosonization identity. It is the entral formula in bosonization te hnique. Another important formula we must have is the ommutator for ˆ . φ(x) It 23 an be al ulated using the master ommutator Eq. (1.36) whi h yields ′ ˆ ˆ ′ 2π X eip(x−x )−δ|p| [φR (x), φR (x )] = L p6=0 p X 2π ′ X 2π ′ = eip(x−x +iδ) − e−ip(x−x −iδ) . (1.53) p>0 Lp p>0 Lp Re all that p takes dis rete value p = n2π/L, n = 1, 2, 3, · · · , and using ln(1 − y) = P∞ − n=1 y n /n, 2π 2π [φˆR (x), φˆR (x′ )] = − ln[1 − ei L (x−x +iδ) ] + ln[1 − e−i L (x−x −iδ) ] ′ ′ 2π ′ 2π ′ L −−→ −−∞ → − ln[ L (δ − i(x − x ))] + ln[ L (δ + i(x − x ))] = −i arg(δ − i(x − x′ )) + i arg(δ + i(x − x′ )) = iπ Sgn(x − x′ ) (1.54) where Sgn(x − x′ ) is the sign fun tion. We an bosonize a hiral left Hamiltonian in a similar way. The dieren e between right moving and left moving lies in the sign of the master ommutator qL ρL (q), ρˆL (q ′ )] = δq,−q′ [ˆ , (1.55) 2π This is be ause left moving ele trons ll the Fermi sea from k = +∞ to k = −∞ whi h is in the reverse order of right moving ones hen e there is a sign dieren e P P between k (hnL,k+q i0 − hnL,k i0 ) and k (hnR,k+q i0 − hnR,k i0 ). As a result, there are some extra minus signs here and there in the bosonization di tionary for left movers. We will not bother to go over all the details again and simply summarize them as 24 follows. Denote R/L by r = +/−, The free hiral Hamiltonian is Z ˆr = H dx ψˆr† (−ir~vF r ∂x )ψˆr (1.56) ∞ q Cˆr,q † ˆ X = r~vF Cr,q . (1.57) q=−∞ The density operator ρˆr (q) satises qL ρr (q), ρˆr (q ′ )] = −rδq,−q′ [ˆ . (1.58) 2π This allows us to bosonize the Hamiltonian as ˆ r = π ~vF X H ρˆr (p)ˆ ρr (−p). (1.59) L p By dening a hiral boson eld X 2π φˆr (x) = −i eipx−δ|p|/2 ρˆr (−p), (1.60) p6=0 pL one an bosonize the fermion operator ψˆr (x) and density ρˆR (x) as ˆ ψˆr (x) = ηˆr eir(kF r x+φr (x)) (1.61) 1 ρˆr (x) = (kF r + ∂x φˆr (x)) (1.62) 2π φˆr (x) does not ommute with itself: [φˆr (x), φˆr′ (x′ )] = δr,r′ iπrSgn(x − x′ ) (1.63) Finally, using L 2π 2 L Z Z dx(∂x φˆr (x))2 = X ′ ( )( dxei(p+p )x )ˆ ρr (−p′ ) ρr (−p)ˆ 0 p6=0,p′ 6=0 L 0 (2π)2 X = ρˆr (p)ˆ ρr (−p), (1.64) L p6=0 25 the bosonized Hamiltonian an also be written in terms of φˆr (x) L ˆ r = 2π N Z H ˆr2 ~vF r + ~vF r dx(∂x φˆr (x))2 . (1.65) L 4π 0 1.1.3 Exa t solution of 1D intera ting Fermi system 1.1.3.1 Spinless system In se 1.1.2.1, we have obtained the following full Hamiltonian for a spinless Tomanaga- Luttinger model assuming short range intera tion, Z ˆ= H dx ψR† (−i~vF ∂x )ψR + ψL† (+i~vF ∂x )ψL (1.66) g4 Z + dx ρR (x)2 + ρˆL (x)2 ) + g2 ρˆR (x)ˆ (ˆ ρL (x). (1.67) 2 This Hamiltonian an not be diagonalized in ψˆr (x) basis. But after bosonization, it be omes quadrati 2π Z ~vF ˆ H= ˆ 2 ˆ ~vF (NR + NL ) + dx[(∂x φˆR (x))2 + (∂x φˆL (x))2 ] L 4π 1 g4 ˆR (x))2 + g4 (∂x φˆL (x))2 + g2 ∂x φˆR (x)∂x φˆL (x)]. Z + dx[ (∂ x φ (1.68) (2π)2 2 2 where we have used Eqs. (1.65), (1.62) and dx∂x φˆr (x) = φˆr (L)− φˆr (0) = 0 (periodi R boundary onditioin). We now pro eed to diagonalize Eq. (1.68). Introdu ing two boson elds φˆR (x) + φˆL (x) ˆ φ(x) ≡ , (1.69) 2 φˆL (x) − φˆR (x) ˆ θ(x) ≡ . (1.70) 2 26 Unlike the hiral eld φˆr (x), φ(x) ˆ and θ(x) ˆ now ommute with themselves, ˆ [φ(x), ˆ ′ )] = [θ(x), θ(x ˆ ˆ ′ )] = 0. θ(x (1.71) But φ(x) ˆ and θ(x ˆ ′ ) does not ommute, ˆ [φ(x), ˆ ′ )] = −i π Sgn(x − x′ ). θ(x (1.72) 2 The physi al meaning of φ(x) ˆ and θ(x) ˆ an be understood though their gradient, ˆ ∂x φ(x) = π(ˆ ρR (x) + ρˆL (x)) − kF , (1.73) ˆ ∂x φ(x) = π(ˆ ρL (x) − ρˆR (x)). (1.74) Thus ∂x φ(x) ˆ is the density u tuation at point x. ∂x θ(x) ˆ is simply the urrent operator sin e it ounts the dieren e between right and left moving ele trons. In terms of ˆ φ(x) and θ(x) ˆ , (∂x φˆR (x))2 + (∂x φˆL (x))2 = 2(∂x φ(x)) ˆ 2 ˆ + 2(∂x θ(x))2 , (1.75) ∂x φˆR (x)∂x φˆL (x) = (∂x φ(x)) ˆ 2 ˆ − (∂x θ(x))2 . (1.76) Hen e Hˆ is diagonal g4 g2 g4 g2 Z ˆ= ~ H dx[(vF + + ˆ )(∂x φ(x))2 + (vF + − ˆ )(∂x θ(x))2 ] 2π 2~π 2~π 2~π 2~π Z ~u = ˆ dx[g −1 (∂x φ(x))2 ˆ + g(∂x θ(x))2 ], (1.77) 2π where g4 g2 ug −1 = (vF + + ), (1.78) 2π 2π g4 g2 ug = (vF + − ). (1.79) 2π 2π 27 whi h an be solved as r g4 2 g2 u = (vF + ) − ( )2 , (1.80) 2π 2π s vF + g4 /2π − g2 /2π g= . (1.81) vF + g4 /2π + g2 /2π Obviously for repulsive intera tion (g2 > 0), g < 1. For attra tive intera tion (g2 < 0), g > 1. For non-intera ting system (g4 = g2 = 0), g = 1. u has the dimension of velo ity, it is the speed of the density wave in the system. Eqs. (1.77), (1.80) and (1.81) omplete the exa t solution to spinless Tomanaga-Luttinger model. Bosonization enable us to al ulate several physi al properties with ease. Let us use it to see how 1D intera ting system diers from a Fermi liquid. We al ulate the o upation number n(k). We know that there is a dis ontinuity at k = kF in a Fermi liquid. In Se . 1.1.1, we argue that in 1D, this dis ontinuity should be zero due to the ee tiveness of ele tron-ele tron s attering. Now we an see whether this is the ase. To al ulate o upation number (in fa t to al ulate anything), it is onvenient to use orrelation fun tion sin e they are related by Z nR (k) = dxeikx hψˆR† (x, 0+ )ψˆR (0, 0)i. (1.82) Using bosonization di tionary Eq. (1.61) and Eqs. (1.69), (1.70), nR (k) an be 28 expressed as Z + nR (k) = dxei(k−kF )x he−iφR (x,0 ) eiφR (0,0) i Z dxei(k−kF )x he−i[φ(x,0 ] ei[φ(x,0)+θ(x,0)] i + )+θ(x,0+ ) = Z dxei(k−kF )x he−i[(φ(x,0 )+(θ(x,0+ )−θ(0,0))] i + )−φ(0,0) = Z 2 dxei(k−kF )x e− 2 h[(φ(x,0 )+(θ(x,0+ )−θ(0,0))] (1.83) 1 + )−φ(0,0) i = Z + )−φ(0,0) 2 i− 1 h 2 dxei(k−kF )x e− 2 h[φ(x,0 ] [θ(x,0+ )−θ(0,0)] i−h[φ(x,0+ )−φ(0,0)][θ(x,0+ )−θ(0,0)]i 1 = 2 (1.84) where in line (1.83), we have used (1.85) 1 2 heA i = e 2 hA i . It is orre t as long as A is quadrati in φ and θ. In order to al ulate Eq. (1.84), we need the following three orrelation fun tions whose al ulation an be found in Appendix. A: (δ + u|τ |)2 + x2 · ¸ g2 h[φ(x, τ ) − φ(0, 0)] i = ln , (1.86) 2 δ2 (δ + u|τ |)2 + x2 · ¸ 1 2 h[θ(x, τ ) − θ(0, 0)] i = ln , (1.87) 2g δ2 i hφ(x, τ )θ(0, 0)i = − Arg(Sgn(τ )δ + uτ + ix). (1.88) 2 Substituting Eqs. (1.86), (1.87), (1.88) into Eq. (1.84), we have ]−iArg(δ+ix) Z dxei(k−kF )x e− 4 (g+1/g) ln[(δ 1 2 +x2 )/δ 2 nR (k) = ¸ 41 (g+1/g) δ2 · e−iArg(δ+ix) . Z = dxe i(k−kF )x (1.89) δ 2 + x2 29 The pre ise form of the integration is a little hard to al ulate but the essential physi s an be easily obtained using dimensional analysis, whi h gives (1.90) 1 nR (k) ∼ |k − kF | 2 (g+1/g)−1 Thus nR (k) has a positive power dependen e on |k − kF |. A plot of nR (k) is show Fig. 1.8 where we see the dis ontinuity at Fermi surfa e is indeed 0 in agreement with our intuition. n(k) 1 |k − kF | 2 (g+1/g)−1 0 k kF Figure 1.8: O upation number nR (k) in a Tomanaga-Luttinger model to be ompared with the Fermi Liquid ase in Fig. 1.3. The dis ontinuity disappears as a result of intera tion. 1.1.3.2 Model with spin: harge-spin separation So far only spinless model is treated. The model with spin is easily generalized from the spinless one. Instead of having two hiral eld φR and φL , we now have φR↑ , φR↓ , φL,↑ and φL,↓ . Denoting r = R/L = ± and σ =↑ / ↓= ±, the generi Hamiltonian 30 has the form ˆ=H H ˆ0 + H ˆ int , (1.91) where Hˆ 0 is the kineti energy Z ˆ0 = (1.92) X † H dx ψr,σ (−ir∂x )ψr,σ . r=±,σ=± The intera tion Hˆ int onsists of three terms: ˆ int = H H ˆ4 + H ˆ2 + H ˆ 1. (1.93) ˆ 4 and H H ˆ 2 des ribes g4 and g2 pro ess explained in Se . 1.1.2.1 and Fig. 1.7. Sin e now dierent spins an s atter, the kinds of s attering in both pro esses doubled, i.e., one has g4k , g4⊥ , and g2k , g2⊥ , where k denotes s attering between the same spins and ⊥ denotes s attering between opposite spins. In addition, there exists another pro ess alled g1 whi h is absent in the spinless model. This pro ess represents s attering between opposite spins at opposite Fermi points as shown in Fig. 1.9. Note that only opposite spins should be onsidered in g1 pro ess, be ause if the ele trons have the same spin, it is equivalent to a g2k pro ess by a permutation of Fermi operators. Physi ally, this is due to the fa t that ele trons with the same spin are identi al parti les hen e one an freely ex hange the initial labeling of the two ele trons whi h turns a g1 graph into to a g2k one. This is also the reason why we do not have to onsider g1 pro ess in the spinless model. 31 E(k) EF k kF −kF g1 process Figure 1.9: g1 pro ess in a model with spin. It represents the s attering between ele trons of opposite spins sitting at opposite Fermi points. ˆ 4, H H ˆ 2 and H ˆ 1 have the following generi form Z hg g4⊥ i ˆ4 = 4k (1.94) X H dx ρˆr,σ (x)ˆ ρr,σ (x) + ρˆr,σ ρˆr,−σ (x) , r=±,σ=± 2 2 XZ ˆ2 = (1.95) £ ¤ H dx g2k ρˆR,σ (x)ˆ ρL,σ (x) + g2⊥ ρˆR,σ (x)ˆ ρL,−σ (x) , σ=± XZ ˆ1 = H dx g1⊥ ψˆL,σ † (x)ψˆR,−σ † (x)ψˆL,−σ (x)ψˆR,σ (x). (1.96) σ=± The bosonization s heme is essentially the same as the spinless one ex ept being more ompli ated. One introdu e the boson eld X 2π φˆr,σ (x) = −i eipx−δ|p|/2 ρˆr,σ (−p), (1.97) p6=0 pL 32 whi h leads to the following mapping: ˆ ψˆr,σ (x) = ηˆr eir(kF x+φr (x)) , (1.98) 1 ρˆr,σ (x) = (kF + ∂x φˆr,σ (x)), (1.99) 2π Z L 2π ˆ 2 ~vF L Z † dx ψr,σ (−ir∂x )ψr,σ = Nr ~vF + dx(∂x φˆr,σ (x))2 . (1.100) o L 4π 0 Thus Eqs. (1.92), (1.94), (1.95) and (1.96) are mapped into 2π X ˆ 2 ~vF L X ˆ 2 Z ˆ H0 = N ~vF + dx (∂x φr,σ ) , (1.101) L r,σ r 4π 0 r,σ " # g ´2 g X ³ Z ³ ´³ ´ ˆ 4 = dx 4k ∂x φˆr,σ + 4k ∂x φˆr,σ ∂x φˆr,−σ , (1.102) X H 2 r,σ 2 r,σ Z " # X³ ´³ ´ X³ ´³ ´ ˆ 2 = dx g2k H ∂x φˆR,σ ∂x φˆL,σ + g2⊥ ∂x φˆR,σ ∂x φˆL,−σ , (1.103) σ σ Z ˆ ˆ ˆ ˆ ˆ1 = (1.104) X H dx g2⊥ ei(φL,σ +φR,σ −φR,−σ −φL,−σ ) . σ At this stage, we an introdu e two elds to representing the total harge degree of freedom φˆρ and total spin degree of freedom φˆσ , 1 Xˆ φˆρ (x) = √ φr,σ (x), (1.105) 2 r,σ 1 X ˆ φˆσ (x) = √ σ φr,σ . (1.106) 2 r,σ They have the gradient, 1 ˆ ∂x φρ (x) = (ˆ ρR,↑ + ρˆL,↑ ) + (ˆ ρR,↓ + ρˆL,↓ ) − 2kF /π, (1.107) 2π 1 ˆ ∂x φσ (x) = (ˆ ρR,↑ + ρˆL,↑ ) − (ˆ ρR,↓ + ρˆL,↓ ) − 2kF /π. (1.108) 2π 33 So ∂x φˆρ measures the total harge u tuation, ∂x φˆσ measures the total spin u tua- tion. The elds onjugate to them are dened as 1 X ˆ θˆρ (x) = √ rφr,σ (x), (1.109) 2 r,σ 1 X ˆ θˆσ (x) = √ rσ φr,σ (x). (1.110) 2 r,σ It is not hard to show that the ommutators between them satisfy π [φˆρ (x), θˆρ (x′ )] = [φˆσ (x), θˆσ (x′ )] = −i Sgn(x − x′ ), (1.111) 2 others = 0. (1.112) Eqs. (1.105), (1.106), (1.109), (1.110) an be inverted as 1 ³ ´ φˆr,σ (x) = √ φˆρ (x) + σ φˆσ (x) + rθˆρ + rσ θˆσ . (1.113) 2 This allows us to easily write the Hamitonian in terms of harge and spin elds, ˆ4 = 1 Z h ³ ´ ³ ´i H dx (g4k + g4⊥ ) (∂x φˆρ )2 + ∂x θˆρ )2 + (g4k − g4⊥ ) (∂x φˆσ )2 + ∂x θˆσ )2 , 4π 2 (1.114) ˆ2 = 1 Z h ³ ´ ³ ´i H dx (g2k + g2⊥ ) (∂x φˆρ )2 − ∂x θˆρ )2 + (g2k − g2⊥ ) (∂x φˆσ )2 − ∂x θˆσ )2 , 4π 2 (1.115) Z √ ˆ1 = H dx 2g2⊥ cos(2 2φˆσ (x)). (1.116) 34 Putting everything together, the full Hamiltonian reads ˆ=H H ˆ0 + H ˆ4 + Hˆ2 + H ˆ1 (1.117) g4k + g4⊥ g2k + g2⊥ µ ¶ hvF Z = dx + 2 + 2 [∂x φρ (x)]2 2π 4π 4π g4k + g4⊥ g2k + g2⊥ µ ¶ hvF + + 2 − 2 [∂x θρ (x)]2 2π 4π 4π g4k − g4⊥ g2k − g2⊥ µ ¶ hvF + + + [∂x φσ (x)]2 2π 4π 2 4π 2 g4k − g4⊥ g2k − g2⊥ µ ¶ hvF + + − [∂x θσ (x)]2 2π 4π 2 4π 2 ˆ1 +H ˆρ + H ≡H ˆσ. (1.118) where Z ˆ ρ = ~uρ H dx[gρ−1 (∂x φˆρ (x))2 + gρ (∂x θˆρ (x))2 ], (1.119) 2π Z ˆ σ = ~uσ H dx[gσ−1 (∂x φˆσ (x))2 + gσ (∂x θˆσ (x))2 ] + H ˆ 1, (1.120) 2π r g4k + g4⊥ 2 g2k + g2⊥ 2 uρ = (vF + ) −( ), (1.121) 2π 2π s vF + (g4k + g4⊥ )/2π − (g2k + g2⊥ )/2π gρ = , (1.122) vF + (g4k + g4⊥ )/2π + (g2k + g2⊥ )/2π r g4k − g4⊥ 2 g2k − g2⊥ 2 uσ = (vF + ) −( ), (1.123) 2π 2π s vF + (g4k − g4⊥ )/2π − (g2k − g2⊥ )/2π gσ = . (1.124) vF + (g4k − g4⊥ )/2π + (g2k − g2⊥ )/2π The above result learly shows that the harge and spin degree of freedom are om- pletely de oupled. Remarkably, they travel with dierent velo ities uρ 6= uσ . This 35 phenomena is alled spin- harge separation. It is an essential feature of one dimen- sional ele troni system. As a result of the separation, the harge urrent and spin urrent in a 1D system , e.g, quantum wire, an be tuned as mu h as possible. It turns out that in 1D asymmetri quantum wire, a nearly pure d spin urrent without harge an be generated by an a voltage. We will explain the details of this result in Chapter. 2. 1.2 Quantum Hall Liquids The integer quantum Hall ee t (IQHE) was dis overed by Klauss von Klitzing in 1980[3℄. What Klitzing found is that at low temperature (a few K s above absolute zero) and strong magneti eld (or the order of Tesla), the Hall- ondu tivity GH versus magneti eld B urves in some two dimensional ele tron systems exhibit plateaus. What is remarkable about the plateaus of GH is that they o ur at integer multiples of a universal onstant e2 /h. The a ura y of the quantization is better than 10−8 and is independent of materials. It is so a urate that IQHE be ame the basis for the US standard of resistan e in January 1990. The quantization learly indi ates something fundamental in this phenomena. But the surprise has not ended. Two years later, Daniel Tsui, Horst Stormer and Arthur Gossard dis overed that in some higher mobility samples and under stronger magneti eld, the quantization an o ur at a fra tion of e2 /h [4℄. The fra tional number v was rst found to be 1/3 but was qui kly extended to a long list of odd denominators e.g., 2/7, 2/5, 2/3, 36 et . Theoreti al understanding of the fra tional quantum Hall Ee t ame on 1983 when Laughlin published his famous variational wavefun tion [10℄. Laughlin also predi ted that the elementary ex itation in FQHE is quasi-parti le/quasi-hole with harge being only a fra tion of ele tron harge e. The idea was extended further to fra tional statisti s whi h states that the quasi-parti le in a FQH liquid is neither a Fermion nor Boson, but an anyon whose wavefun tion a quires a phase eiθ whi h is dierent from 1 or −1. Both fra tional harge and fra tional statisti s were observed in experiments[11, 12, 13℄. Interestingly, it was dis overed that even denominator FQHE like 5/2 is also possible. And it is proposed that the quasi-parti le in 5/2 FQHE an have exoti non-abelian statisti s[14℄. A potential appli ation of non- Abelian statisti s in ludes Fault-tolerant quantum omputing[14℄. 1.2.1 Integer Quantum Hall Ee t Classi al Hall ee t was dis overed by Edwin Hall in 1879. Hall found that a potential dieren e will o ur a ross a ondu ting metal in magneti eld. The potential is traverse to the ele tri urrent and the magneti eld. The physi s of Hall ee t an be easily understood. Consider a 2D ondu tor under a perpendi ular magneti eld ~ = B zˆ shown in Fig. (1.10). The urrent ows in y dire tion ~j = nev yˆ, where n is B the area density of ele trons, e < 0 and v are the harge and speed of the ele trons. To maintain a stati longitudinal urrent, the ele tri for e must be balan ed by Lorenz 37 for e, eE~ = e ~v × B ~ c v j =⇒ E = B = B, ~ ⊥B E ~ ⊥ ~j. (1.125) c nec Thus the Hall ondu tan e dened as B~z + + + + + + ~j − − − − − − Figure 1.10: Classi al Hall ee t. I jw nec GH ≡ = = . (1.126) VH Ew B is proportional to the magneti eld. Here w is the traverse width of the sample. So lassi ally, one would expe t GH vs B is a straight line whi h is obviously not the urve observed by Klitzing. In order to understand the plateaus, one must go to quantum me hani s. The quantum me hani al solution of this problem was solved long time ago. The basi result is that 2D ele trons gas under magneti eld onsists of quantized states En = ~ωc (n + 12 ) with ωc = eB/mc the y lotron frequen y. Ea h energy level is highly degenerated with a degenera y eBA/hc, where A is the area of the system. The solution using Landau gauge an be found in most quantum me hani s books. 38 Here we present a solution using symmetri gauge whi h is more onvenient when we deal with fra tional quantum Hall ee t. The Hamiltonian is 2 2 Hˆ = − ~ (∂x − i e Ax )2 − ~ (∂y − i e Ay )2 (1.127) 2m ~c 2m ~c B (Ax , Ay ) = (−y, x), symmetri gauge. (1.128) 2 Introdu ing omplex oordinate z = x + iy , and using the identity ˆ 2 = 1 (Aˆ − iB)( h i Aˆ2 + B ˆ Aˆ + iB) ˆ + (Aˆ + iB)( ˆ Aˆ − iB) ˆ , (1.129) 2 Eq. (1.127) an be written as ~2 ³ ˆ ˆ ´ ˆ H=− ˆ ˆ Dz Dz¯ + Dz¯Dz , (1.130) m with ˆ z = 1 ∂x − i e Ax − i ∂y − i e Ay , h³ ´ ³ ´i D (1.131) 2 ~c ~c ˆ z¯ = 1 ∂x − i e Ax + i ∂y − i e Ay . h³ ´ ³ ´i D (1.132) 2 ~c ~c using 1 1 ∂z = (∂x − i∂y ), ∂z¯ = (∂x + i∂y ). (1.133) 2 2 and dening 1 B Az ≡ (Ax − iAy ) = −i z¯, (1.134) 2 4 1 B Az¯ ≡ (Ax + iAy ) = i z. (1.135) 2 4 39 ˆ Z and D D ˆ z¯ an be expressed in terms of z and z¯, ˆ z = ∂z − i e Az , D ˆ z¯ = ∂z¯ − i e Az¯. D (1.136) ~c ~c It is onvenient to dene the magneti length lB c~ 2 lB ≡ , (1.137) eB where we have assumed eB > 0. Thus ˆ z = ∂z − z¯ , D ˆ z¯ = ∂z¯ + z . D (1.138) 2 2 4lB 4lB Eq. (1.130) an be further simplied through a transformation ezz¯/4lB as 2 ~2 µ ¶h i ˆ = e−zz¯/4lB2 H − ˆb†ˆb + ˆbˆb† ezz¯/4lB2 , (1.139) m where ˆb† =ezz¯/4lB2 D ˆ z e−zz¯/4lB2 2 z¯ −zz¯/4lB2 =ezz¯/4lB (∂z − 2 )e 4lB z¯ (1.140) 2 2 =ezz¯/4lB e−zz¯/4lB (∂z + ∂z (−z z¯/4lB 2 )− 2 ) 4lB z¯ =∂z − 2 , 2lB ˆb =ezz¯/4lB2 D ˆ z¯e−zz¯/4lB2 =∂z¯. (1.141) It is easily veried that 1 [ˆb, ˆb† ] = − 2 . (1.142) 2lB 40 Hen e ˆb† and ˆb are boson operators up to a onstant. Thus Eq. (1.139) des ribes a harmoni os illator 2~2 µ ¶³ ´ ˆ = e−zz¯/4lB2 H − ˆb†ˆb ezz¯/4lB2 + 1 ~ωc , (1.143) m 2 with the famous harmoni energy spe trum 1 En = (n + )~ωc . (1.144) 2 The ground state Φ0 satises ³ ´ ˆb ezz¯/4lB2 Φ0 (z) = 0 (1.145) 2 /4l2 =⇒ Φ0 (z, z¯) = e−|z| B f (z), where f (z) is an arbitrary fun tion of z . The ex ited states an be onstru ted as ³ ´n ³ ´n ezz¯/4lB Φn = ˆb† ezz¯/4lB Φ0 ˆb† f (z). (1.146) 2 2 2 /4l2 =⇒ Φn (z, z¯) = e−|z| B Sin e f (z) is an analyti fun tion, it an be expanded as a series of z . Thus the wavefun tion of zeroth Landau Level Φ0 an be expanded as ∞ X 2 /4l2 Φ0 = cl z l e−|z| B l=0 ∞ (1.147) X ≡ cl φl , l=0 where the basis φl = z l e−|z| represents a state with denite angular momentum 2 /4l2 B l. The probability distribution is |φl |2 = |z|2l e−|z| . It has a maximum at 2 /2l2 B √ |zl | = lB 2l, l = 0, 1, 2, · · · . (1.148) 41 Be ause of the exponential e−|z| , |φl |2 qui kly dies out away from the maximum 2 /2l2 B √ zl . So ea h angular momentum state has the shape of a ring with radius lB 2l. Ea h state o upies equal area 2πlB2 . These orbitals are depi ted in Fig. (1.11). Ea h orbital in the same Landau Level has the same energy hen e they are highly Figure 1.11: The degenerate angular momentum states in zeroth Landau Level degenerated. The degenera y D is determined by the quantum number of the largest orbital. For a system with area A, A eBA D= 2 = . (1.149) 2πlB hc The degenera y leads us to dene a very useful quantity, the lling fa tor ν ≡ N/D, (1.150) with N the number of ele trons in the system. Thus, when ν = 1, all states in the lowest Landau Level that lie within the area A are lled. If ν = 1/3, only 1/3 of the states are lled whi h is the ase for fra tional quantum Hall liquid at 1/3 plateau. 42 Another way to look at the lling fa tor is that its inverse is a measure of the number of ux quanta per ele tron. The ux quanta for the system is Φ0 = hc/e. The number of ux quanta is NΦ0 = BA/Φ0 = D. Thus the number of ux quanta per ele tron is NΦ0 /N = D/N = 1/ν . The lling fa tor is very useful when we deal with FQHE. Fig. (1.12a) shows the density of states of the non-intera ting system we have just E(~ωc ) E 5 5 4 4 3 3 2 2 1 1 DOS DOS (a) (b) Figure 1.12: (a) DOS of a translational invariant sample. (b) DOS of a system with disorder where the shaded region onsists lo alized states. solved. It onsists of a series of δ -fun tions with weight A/2πlB2 . As we will see, the quantum Hall Ee t an be understood if the system is in ompressible and if there is disorder. The nite energy gap ~ωc between the Landau Levels guarantees that the system is in ompressible when the Landau Level is fully lled.¶ But nite energy ¶ This is be ause upon all the degenerated states in one energy level is fully lled, ompressing the system just a little bit would redu e the area A hen e the degenera y D, therefore pushing some ele trons up to a higher energy level whi h would ost energy or the order ~ωc . 43 gap alone is not enough to explain the the plateaus in IQHE. In fa t, one an show using Lorenz invarian e, that the Hall ondu tan e is always trivially linear in B if the system is translationally invariant[15℄. Therefore plateaus an only o ur when there is disorder. In the presen e of disorder, some ele trons are pinned to lo alized states, as a result, ea h Landau Level is broaden into bands in Fig. ( 1.12b). It turns out that only a narrow region of the states in the enter of ea h Landau Level remain extended. All states away from the enters are lo alized states. Therefore, the gap in the disordered system is a mobility gap, i.e., gap between extended states. Now we an understand the plateaus in IQHE. Imagine we de rease the magneti eld B ontinuously, the Fermi level swipes the DOS ontinuously from the bottom to the top. When the Fermi level lies inside the region of extended states, the system shows linear in rease of Hall ondu tan e with magneti eld. However, only extended states an respond to magneti eld hen e ontribute to the Hall ondu tan e. When the Fermi level swipes a ross the region of lo alized states, the Hall ondu tan e remains un hanged until the Fermi level nds the next extended states. The repeated pattern of extended states and lo alized states leads to the repeated plateaus in GH vs B urve. 1.2.2 Fra tional Quantum Hall Ee t As we have seen, plateaus in IQHE an be understood through an essentially non- intera ting pi ture. The two ingredients are are gaps in the energy spe trum and 44 disorder. At ea h plateau, the system is an in ompressible liquid with all extended states in one Landau Level lled. What we learned from the fra tional quantum Hall ee t is rather astonishing. The quantization o urs when the system is still lling ele trons into one Landau Level. This ee t must ome from intera tion. In the pi ture of IQHE we have ignored the oulomb intera tion between ele trons. This may be valid if there is no ground state degenera y (all states in one level are just lled). But in the partially lled FQH liquid, there is an enormous degenera y ∼ D! (νD)!(D−νD)! , with D in Eq. (1.149). Theses degenerate states are not stable under intera tion. k Hen e to understand FQHE one must solve the intera ting problem. Unfortunately, no method known solves the intera ting Hamiltonian dire tly. Fortunately however, the ee t an be essentially understood through Laughlin's variational fun tion[10℄. Laughlin suggested that for v = 1/m, |zi |2 /4lB (1.151) 2 P Ψm = Πi0 where we have used Eq. (1.157) and n = νeB/2πc~. ρ(k) is the Fourier transform of 49 ρ(x), 1 X ikx ρ(x) = e ρ(k). (1.159) L k Note that the edge wave is hiral (right moving) and is propagating with velo ity v, hen e the equation of motion for ρ(x) must be ∂t ρ(x, t) + v∂x ρ(x, t) = 0 (1.160) It is not hard to verify that if we identify the oordinate q = ρ(k) and anoni al momentum p = i2π~ρ(−k)/νLk , then the Hamiltonian equations yield ∂H q˙ = ⇒ ρ(k) ˙ = −ivkρ(k), ∂ρ ∂H p˙ = − ⇒ ρ(−k) ˙ = ivkρ(−k), (1.161) ∂q whi h exa tly reprodu e the equation of motion (1.160). Hen e ρ(k) and i2πρ(−k)/νLk are anoni al onjugated  oordinates and momentum. The lassi al Hamiltonian an then be easily quantized by demanding [ρ(k), i2π~ρ(−k ′ )/νLk ′ ] = i~δk,−k′ , (1.162) or Lp [ρ(p), ρ(p′ )] = ν δp,−p′ . (1.163) 2π This has the same stru ture as the one we obtained in a Luttinger liquid (see Eq. (1.58)). Thus the quantized Hamitonian des ribes a hiral Luttinger liquid. The 50 ele tron operator an be obtained following the same steps as we derive Eq. (1.45) in Se . 1.1.2.2 P 2π ipx 1 p νp e ρ(−p) ψ(x) = ηe = ηeikF x+i ν φ(x) (1.164) with X 2π φ(x) = eipx ρ(−p) (1.165) p6=0 pL and [φ(x), φ(x′ )] = iπ Sgn(x − x′ ) (1.166) 1.2.4 Fra tional Charge and Fran tional Statisti s The elementary ex itations from the Laughlin state have the remarkable property that they arry fra tional harge and fra tional statisti s. This does not mean ele - trons are split into pie es. It is a lo al de it or ex ess of harge resulting from the ompli ated and orrelated motion of the ele trons. Quasi-holes are vorti es in the fra tional quantum Hall liquid, whi h an be understood as the quantum in arnation of a lassi al whirl pool. To understand the quasi-hole in fra tional quantum Hall liquids, let us onsider rst the response of a lled Landau Level ν = 1 to a magneti ux tube arrying a total ux Φ inserted adiabati ally into the enter of the system. The ux adds an Aharonov-Bohm phase of θ = 2πΦ/Φ0 to the wavefun tion, so the lth angular momentum state in the symmetri gauge be omes θ 2 /4l2 φl = z l+ 2π e−|z| B (1.167) 51 If the Φ is exa t one ux quanta Φ0 , the lth orbital be omes the (l + 1)th orbital. Hen e, by inserting one unit of ux quanta, all single ele tron states are pushed one orbital away from the enter. As a result, pre isely one ele tron is expelled from the enter of the system whi h reates a lo al harge de it of e. If we apply the same idea to the Laughlin state Eq. (1.151), inserting one ux quanta into the enter of the system adds a phase fa tor θ = 2π to ea h ele tron so the wavefun tion be omes Ψ+ m (ξ) =Πi (zi − ξ)Ψm |zi |2 /4lB 2 P =Πi (zi − ξ)Πi R 53 ontributes zero. Hen e Z γ=i dxdyρ+ (z)2πi r I(eV, −1/2) and |I(−eV, 1/2)| < |I(−eV, −1/2)|. Hen e, Isr > ~ r I 2e c . By an appropriate hoi e of parameters, one an produ e any ratio of the spin and harge re ti ation urrents. Note that the re ti ation ee t for non-intera ting ele trons is possible even if the potential U (x) is symmetri . The asymmetry of the system, ne essary for re ti ation, is introdu ed by the applied voltage bias. The harge density inje ted into the wire from the leads is proportional to µL + µR and hen e is dierent for the opposite signs of the voltage. If the inje ted harge density were independent of the voltage sign, i.e. µR os illated between eV /2 and −eV /2 and µL = −µR os illated between −eV /2 and eV /2, then the re ti ation ee t would be impossible for non-intera ting ele trons. This follows from Eqs. (2.1-2.3) and the fa t that the transmission oe ient T (E) is independent of the dire tion of the in oming wave for non-intera ting parti les[51℄. In the presen e of ele tron repulsion both the asymmetry of the potential and the 66 voltage dependen e of the inje ted harge ontribute to the re ti ation urrent. It turns out, that in the ase of strong ele tron intera tion, the re ti ation ee t due to the asymmetry of the potential barrier dominates. The above example is based on a spe ial form of the potential barrier in the wire and assumes that the magneti eld and hemi al potentials are tuned in order to obtain the desired ee t. As shown below, in the presen e of strong repulsive ele tron intera tion no tuning is ne essary and no quasistationary states are needed to obtain the spin urrent whi h is greater than the harge urrent. In fa t, the spin re ti ation ee t is possible even for weak asymmetri potentials U (x). This an be understood from the following toy model (a related model for re ti ation in a two-dimensional ele tron gas was studied in Ref. [52℄): Let there be no uniform magneti eld H and no asymmetri potential U (x). Instead, both right↔left and spin-up↔spin-down symmetries are broken by a weak oordinate-dependent magneti eld Bz (x) 6= Bz (−x), whi h is lo alized in a small of size ∼ 1/kF (we do not in lude the omponents Bx,y in the toy model). Let us also assume that the spin-up and -down ele trons do not intera t with the ele trons of the opposite spin. Then the system an be des ribed as the ombination of two spin-polarized one- hannel wires with opposite spin-dependent potentials ±µBz (x), where µ is the ele tron magneti moment. A ording to Ref. [36℄ an a bias generates a re ti ation urrent in ea h of those two systems and the urrents are proportional to the ubes of the potentials (±µBz )3 . Thus, I↑r = −I↓r . Hen e, no net harge urrent I r = I↑r + I↓r is generated in 67 the leading order. At the same time, there is a nonzero spin urrent in the third order in Bz . A similar ee t is present in a more realisti Luttinger liquid model onsidered below. The main fo us of this thesis is on the ase of weak asymmetri potentials. A simple model with strong impurities is studied in Appendix B.1. In Figs. 2.3 and 2.4 we have represented the results from a numeri al evaluation for the spin and harge urrents Is,c r for the potential shown in Fig. 2.2. Fig. 2.3 shows the non- intera ting ase. In Fig. 2.4 we represent the ase of strong ele tron intera tion. We have hosen parameters (explained in the gure aptions) su h that Icr is smaller than Isr for a range of the applied voltage. Further information on the numeri al approa h is given in Appendix B.1. Transport in a strongly intera ting system in the presen e of a strong asymmetri potential U (x) is a di ult problem whi h annot be solved analyti ally and is sensitive to a parti ular hoi e of the potential. As we have mentioned, Appendix B.1 ontains the numeri al analysis of a simple model of intera ting ele trons with a strong potential barrier. On the other hand, the intera ting problem an be solved analyti ally in the limit of a weak potential U (x) with the help of the bosonization and Keldysh te hniques (Se . 2.3). We will see that the re ti ation urrent exhibits a number of universal features, independent of the form of the potential U (x). In parti ular, in a wide interval of intera tion strength, the spin re ti ation urrent an ex eed the harge re ti ation urrent for an arbitrary shape of the asymmetri potential barrier. Re ti ation is a nonlinear 68 0.15 I / I rc c0 I / I rs s0 0.00 0 I / I r -0.15 -0.30 0.0 0.2 0.4 0.6 V/V 0 Figure 2.3: Normalized harge re ti ation urrent Ic /Ic0 and spin re ti ation ur- rent Is /Is0 versus applied voltage V /V0 for non-intera ting ele trons with EF = 400ǫ0 , µH = 75ǫ0 , u1 = 50ǫ0 a and u2 = −50ǫ0 a, where ǫ0 = ~2 /ma2 (see Fig. 2.2 and Ap- pendix B.1). Ic0 = 50eǫ0 /~, Is0 = 25ǫ0 and V0 = 50ǫ0 /e are arbitrary referen e urrents and voltage. 0.004 I / I rc c0 I / I rs s0 0.002 0 I / I r 0.000 -0.002 0.00 0.05 0.10 0.15 0.20 V/V 0 Figure 2.4: Normalized harge re ti ation urrent Ic /Ic0 and spin re ti ation ur- rent Is /Is0 versus applied voltage V /V0 for intera ting ele trons with EF = 100ǫ0 , µH = 25ǫ0 , γ = 12.6ǫ0 a/e, u1 = 25ǫ0 a and u2 = 50ǫ0 a, where ǫ0 = ~2 /ma2 (see Fig. 2.2 and Appendix B.1). Ic0 = 50eǫ0 /~, Is0 = 25ǫ0 , and V0 = 50ǫ0 /e are arbitrary referen e urrents and voltage. 69 transport phenomenon. Thus, it annot be observed at low voltages at whi h the I − V urve is linear and hen e symmetri . In Luttinger liquids the I − V urve is nonlinear at eV > kB T , where T is the temperature [53, 54℄. We will on entrate on the limit of the zero temperature whi h orresponds to the strongest re ti ation. We expe t qualitatively the same behavior at T ∼ V . At higher temperatures the harge and spin re ti ation ee ts disappear. Sin e the temperatures of the order of millikelvins an be a hieved with dilution refrigeration, the re ti ation ee t is possible even for the voltages as low as V . 1 µV. In this thesis we fo us on the low-frequen y a bias. We dene the re ti ation urrent as the d response to a low-frequen y square voltage wave of amplitude V : Isr (V ) = [Is (V ) + Is (−V )]/2, (2.4) Icr (V ) = [Ic (V ) + Ic (−V )]/2. (2.5) The above d urrents express via the urrents of spin-up and -down ele trons: Icr = I↑r + I↓r , Isr = (~/2e)[I↑r − I↓r ]. The spin urrent ex eeds the harge urrent if the signs of I↑r and I↓r are opposite. Equations (2.4,2.5) for the d - urrents do not ontain the frequen y ω of the a -bias. They are valid as long as the frequen y ω < eV /~. (2.6) Indeed, as shown below, the re ti ation urrent is determined by ele tron ba ks at- tering o the asymmetri potential. Hen e, one an negle t the time-dependen e of the a voltage in Eqs. (2.4,2.5) if the period of the a bias ex eeds the duration τ 70 of one ba ks attering event. The time τ ∼ τtravel + τuncertainty in ludes two ontribu- tions. τtravel is the time of the ele tron travel a ross the potential barrier. τuncertainty omes from the un ertainty of the energy of the ba ks attered parti le. If the bar- rier amplitude U (x) < EF and the barrier o upies a region of size aU ∼ 1/kF then τtravel ∼ 1/kF v ∼ ~/EF , where v ∼ ~kF /m is the ele tron velo ity. The energy un er- tainty ∼ eV translates into τuncertainty ∼ ~/eV . Thus, for eV < EF one obtains the ondition (2.6). The same ondition an be derived with the approa h of Appendix A of Ref. [55℄ and emerges in a related problem [56℄. Note that for realisti voltages the low-frequen y ondition (2.6) allows rather high frequen ies. Even for V ∼ 1 µV the maximal ω ∼ 1 GHz. There remains the question of the asymmetri impurity: We require a potential U (x) that is lo alized within ∼ 1/kF . A possible realization is to generate two dierent (symmetri ) lo al potentials by two gates within a distan e ∼ 1/kF or an ele tri potential reated by an asymmetri gate of size ∼ 1/kF pla ed at the distan e ∼ 1/kF from the wire. Ele tron densities of ρ ∼ 1011 cm−2 are possible nowadays in 2-dimensional ele tron gases, yielding 1/kF up to several 10 nm. Connement in a one-dimensional wire will redu e the ele tron density further so that this number may in rease further. Modern te hniques allow pla ing ele tri gates of widths of ∼ 20 nm at distan es of ∼ 20 − 50 nm. A realization of an asymmetri potential in this way is, therefore, within the rea h. Alternatively, in the ase of shorter ele tron wave-length, it should be possible to pla e an asymmetri ally shaped STM tip lose to the wire. 71 An applied bias would yield an asymmetri s attering potential. With su h a tip the asymmetry annot be dire tly tuned, but most of our predi tions are not sensitive to the pre ise shape of the potential. Certainly, an asymmetri potential may simply emerge by han e due to the presen e of two point impurities of unequal strength at the distan e ∼ 1/kF . 2.3 Bosonization and Keldysh Te hnique At ω < V , the al ulation of the re ti ation urrents redu es to the al ulation of the stationary ontributions to the d I − V urves Is (V ) and Ic (V ) that are even in the voltage V . We assume that the Coulomb intera tion between distant harges is s reened by the gates. This will allow us to use the standard Tomonaga-Luttinger model with short range intera tions. [53, 54℄ Ele tri elds of external harges are also assumed to be s reened. Thus, the applied voltage reveals itself only as the dieren e of the ele tro hemi al potentials E1 and E2 of the parti les inje ted from the left and right reservoirs. We assume that one lead is onne ted to the ground so that its ele tro hemi al potential E2 = EF is xed. The ele tro hemi al potential of the se ond lead E1 = EF + eV is ontrolled by the voltage sour e (see Fig. 2.1). Sin e the Tomonaga-Luttinger model aptures only low-energy physi s, we assume that eV < EF , where EF is of the order of the bandwidth. Re ti ation o urs due to ba ks attering o the asymmetri potential U (x). We will assume that the asymmetri potential is weak, U (x) < EF . This will enable us to use perturbation 72 theory. We assume that the magneti eld H ouples only to the ele tron spin and we negle t the orre tion −eA/c to the momentum in the ele tron kineti energy. Indeed, for a uniform eld one an hoose A ∼ y , where the y -axis is orthogonal to the wire, and y is small inside a narrow wire. As shown in Ref. [48℄, su h a system allows a formulation within the bosonization language and, in the absen e of the asymmetri potential, an be des ribed by a quadrati bosoni Hamiltonian X Z (2.7) X ¡ ¢ ¡ ¢ H0 = dx ∂x φνσ Hνσ,ν ′ σ′ ∂x φν ′ σ′ , ν,ν ′ =L,R σ,σ ′ =↑,↓ where σ is the spin proje tion and ν = R, L labels the left and right moving ele trons, whi h are related to the boson elds φνσ as ψνσ † † ±i(kF νσ x+φνσ (x)) (x) ∼ ηνσ e with ± for ν = R, L. The operators ηνσ † are the Klein fa tors adding a parti le of type (ν, σ) to the system, and kF νσ /π is the density of (ν, σ) parti les in the system. The densities of the spin-up and -down ele trons are dierent sin e the system is polarized by the external magneti eld. The 4×4 matrix H des ribes the ele tron-ele tron intera tions. In the absen e of spin-orbit intera tions, L ↔ R parity is onserved and we an introdu e the quantities φσ = φLσ + φRσ and Πσ = φLσ − φRσ su h that the Hamiltonian de ouples into two terms depending on φσ and Πσ only. In the absen e of the external eld, this Hamiltonian would further be diagonalized by the ombinations φc,s ∝ φ↑ ± φ↓ , and similarly for Πc,s , expressing the spin and harge separation. This is here no longer the ase be ause of the external magneti eld. If we fo us on the φ elds only (as Π will not appear in the operators des ribing ba ks attering o U (x)), the elds 73 diagonalizing the Hamiltonian, φ˜c,s , have a more ompli ated linear relation to φ↑,↓ , whi h we an write as      √ √ φ↑   gc [1 + α] gs [1 + β]  φ˜c   =   √  , (2.8) √   φ↓ gc [1 − α] − gs [1 − β] φ˜s and whi h orresponds to the matrix AˆT of Ref. [48℄. The normalization has been hosen su h that the propagator of the φ˜ elds with respe t to the Hamiltonian (2.7) evaluates to hφ˜c,s (t1 )φ˜c,s (t2 )i = −2 ln(i(t1 − t2 )/τc + δ), where δ > 0 is an innitesimal quantity and τc ∼ ~/EF the ultraviolet uto time. For non-intera ting ele trons without a magneti eld, gc = gs = 1/2. gc < 1/2 (> 1/2) for repulsive (attra tive) intera tions. The intera tion onstants depend on mi ros opi details and the magneti eld. The dimensionless parameter whi h ontrols the intera tion strength is the ratio of the potential and kineti energies of the ele trons. This ratio grows as the harge density de reases and hen e lower ele tron densities orrespond to stronger repulsive intera tion. In the absen e of the magneti eld, terms in (2.7) in √ the form of exp(±2i gs φs ) may be ome relevant and open a spin gap for gs < 1/2. In our model they an be negle ted sin e they are suppressed by the rapidly os illating fa tors exp(±2i[kF ↑ − kF ↓ ]x). It is onvenient to model the leads as the regions near the right and left ends of the wire without ele tron intera tion [45, 46, 47℄. Ba ks attering o the impurity potential U (x) is des ribed by the following on- tribution to the Hamiltonian[53, 54℄ H = H0 + H ′ : (2.9) X H′ = U (n↑ , n↓ )ein↑ φ↑ (0)+in↓ φ↓ (0) , n↑ ,n↓ 74 where the elds are evaluated at the impurity position x = 0 and U (n↑ , n↓ ) = U ∗ (−n↑ , −n↓ ) sin e the Hamiltonian is Hermitian. The elds Π do not enter the above equation due to the onservation of the ele tri harge and the z -proje tion of the spin. The Klein fa tors are not written be ause they drop out in the perturbative expansion. U (n↑ , n↓ ) are the amplitudes of ba ks attering of n↑ spin-up and n↓ spin- down parti les with nσ > 0 for L → R and nσ < 0 for R → L s attering. U (n↑ , n↓ ) an be estimated as[53, 54℄ U (n↑ , n↓ ) ∼ kF dxU (x)ein↑ 2kF ↑ x+in↓ 2kF ↓ x ∼ U , where U R is the maximum of U (x). In the ase of a symmetri potential, U (x) = U (−x), the oe ients U (n↑ , n↓ ) are real. The spin and harge urrent an be expressed as Is,c = L1s,c + Rs,c 1 = L2s,c + Rs,c 2 , (2.10) where Lis,c and Rs,c i denote the urrent of the left- and right-movers near ele trode i, respe tively (see Fig. 2.2). For a lean system (U (x) = 0), the urrents obey[45, 46, 47℄ 1 Rs,c 2 = Rs,c , L1s,c = L2s,c and Ic = 2e2 V /h, Is = 0. With ba ks attering o U (x), parti les are transferred between L and R in the wire, and hen e Rs2 = Rs1 + dSR /dt, Rc2 = Rc1 + dQR /dt, where QR and SR denote the total harge and the z -proje tion of the spin of the right-moving ele trons [56℄. The urrents L2s,c and Rs,c 1 are determined by the leads (i.e. the regions without ele tron intera tion in our model [45, 46, 47℄) and remain the same as in the absen e of the asymmetri potential. Thus, the spin and harge urrent an be represented as Ic = 2e2 V /h + Icbs and Is = Isbs , where the 75 ba ks attering urrent operators are [36, 37, 53, 54℄ Iˆcbs = dQ ˆ R /dt = i[H, Q ˆ R ]/~ −ie X = (n↑ + n↓ )U (n↑ , n↓ )ein↑ φ↑ (0)+in↓ φ↓ (0) , (2.11) ~ n ,n ↑ ↓ Iˆsbs = dSˆR /dt i X =− (n↑ − n↓ )U (n↑ , n↓ )ein↑ φ↑ (0)+in↓ φ↓ (0) . (2.12) 2 n ,n ↑ ↓ The al ulation of the re ti ation urrents redu es to the al ulation of the urrents (2.11), (2.12) at two opposite values of the d voltage. To nd the ba ks attered urrent we use the Keldysh te hnique [57, 58℄. We assume that at t = −∞ there is no ba ks attering in the Hamiltonian (U (x) = 0), and then the ba ks attering is gradually turned on. Thus, at t = −∞, the numbers NL and NR of the left- and right-moving ele trons onserve separately: The system an be des ribed by a partition fun tion with two hemi al potentials E1 = EF + eV and E2 = EF onjugated with the parti le numbers NR and NL . This initial state determines the bare Keldysh Green fun tions. We will onsider only the zero temperature limit. It is onvenient to swit h [56℄ to the intera tion representation H0 → H0 − E1 NR − E2 NL . This transformation indu es a time dependen e in the ele tron reation and annihilation operators. As the result ea h exponent in Eq. (2.9) is multiplied by exp(ieV t[n↑ + n↓ ]/~). In the Keldysh formulation [57, 58℄ the ba ks attering urrents (2.11), (2.12) are 76 evaluated as bs Ic,s = h0|S(−∞, 0)Iˆc,s bs S(0, −∞)|0i, (2.13) where |0i is the ground state for the Hamiltonian H0 , Eq. (2.7), and S(t, t′ ) the evolution operator for H ′ from t′ to t in the intera tion representation with respe t to H0 . The result of this al ulation depends on the elements of the matrix (2.8), whi h des ribe the low-energy degrees of freedom and depend on the mi ros opi details. Several regimes are possible [37℄ at dierent values of the parameters gs > 0, gc > 0, α and β . In this thesis we fo us on one parti ular regime, in whi h the main ontribution to the re ti ation urrent omes from ba ks attering operators U (1, 0), U (0, −1) and U (−1, 1). 2.4 Re ti ation urrents In order to al ulate the urrent (2.13) we will expand the evolution operator in powers of U (n, m). Su h perturbative approa h is valid only if U < EF . The details of the perturbative al ulation are dis ussed in Appendix B.3. The urrents (2.13) an be estimated using a renormalization group pro edure [53, 54℄. As we hange the energy s ale E , the ba ks attering amplitudes U (n↑ , n↓ ) s ale as U (n, m; E) ∼ U (n, m)(E/EF )z(n,m) , (2.14) 77 where the s aling dimensions are z(n, m) = n2 [gc (1 + α)2 + gs (1 + β)2 ] + m2 [gc (1 − α)2 +gs (1 − β)2 ] + 2nm[gc (1 − α2 ) − gs (1 − β 2 )] − 1 = n2 A + m2 B + 2nmC − 1. (2.15) The renormalization group (RG) stops at the s ale of the order E ∼ eV . At this s ale the ba ks attering urrent an be represented as Ic,s bs = V rc,s (V ), where the ee tive ree tion oe ient rc,s (V ) is given by the sum of ontributions of the form[36, 53, 54℄ (const)U (n1 , m1 ; E = eV )U (n2 , m2 ; eV ) . . . U (np , mp ; eV ). (2.16) Su h a perturbative expansion an be used as long as U (n, m; E = eV ) < EF (2.17) for every (n, m). This ondition denes the RG uto voltage V ∗ su h that U (n0 , m0 ; E = eV ∗ ) = EF (2.18) for the most relevant operator U (n0 , m0 ). The RG pro edure annot be ontinued to lower energy s ales E < V ∗ . One expe ts that the leading ontribution to the ba ks attering urrent emerges in the se ond order in U (n, m), if the above ondition is satised. The leading on- tribution to the re ti ation urrent may however emerge in the third order. Indeed, the se ond order ontributions to the harge urrent were omputed in Ref. [53, 54℄. 78 The spin urrent an be found in exa tly the same way. The result is (2.19) X bs(2) Ic,s (V ) ∼ (const)|U (n, m)|2 |V |2z(n,m)+1 sign(V ). If the (unrenormalized) U (n, m) were independent of the voltage, the above ur- rent would be an odd fun tion of the bias and hen e would not ontribute to the re ti ation urrent. The ba ks attering amplitudes depend[53, 54℄ on the harge densities kF νσ though, whi h in turn depend on the voltage in our model [36℄. The voltage-dependent orre tions to the amplitudes are linear in the voltage at low bias eV ≪ EF . Hen e, the se ond order ontributions to the re ti ation urrents s ale as U 2 |V |2z(n,m)+2 (see Appendix B.2). The additional fa tor of V makes the se ond order ontribution smaller than the leading third order ontribution (2.24) at su iently high impurity strength U ≪ EF (as shown in Appendix B.2, U/EF must ex eed [V /EF ]1+z(1,0)−z(0,1)−z(1,−1) ). Note that the se ond order ontribution to the re ti a- tion urrent is nonzero even for a symmetri potential U (x) and emerges solely due to the voltage dependen e of the inje ted harge density ( f. Se . 2.2). The leading third order ontribution emerges solely due to the asymmetry of the s atterer. The main third order ontribution omes from the three ba ks attering operators, most relevant in the renormalization group sense (small z(n, m), Eq. (2.15)). They are identied in Appendix B.2. Under onditions (B.8,B.9,B.10,B.14,B.16,B.17), the most relevant operator is U (1, 0), the se ond most relevant U (0, −1), and the third most relevant U (−1, 1). The uto voltage V ∗ is determined by the s aling dimension z(1, 0), eV ∗ ∼ EF (U/EF )1/[1−A] . The leading non-zero third order ontributions to 79 the spin and harge urrents ome from the produ t of the above three operators in the Keldysh perturbation theory (see Appendix B). This leads to bs Ic,s ∼ U 3 V 2(A+B−C−1) . (2.20) This ontribution dominates the spin re ti ation urrent at EF (U/EF )1/[2+2C−2B] ≡ eV ∗∗ > eV > eV ∗ (2.21) as is lear from the omparison with the leading se ond order ontribution Is,2 bs ∼ U 2 V 2A , Eq. (B.12). Interestingly, the urrent (2.20) grows as the voltage de reases in the regime (B.8,B.9,B.10,B.14,B.16,B.17). However, does the urrent (2.20) a tually ontribute to the re ti ation ee t? In general, (2.20) is the sum of odd and even fun tions of the voltage and only the even part is important for us. One might naively expe t that su h a ontribution has the same order of magnitude for the spin and harge urrents. A dire t al ulation shows, however, that this is not the ase and the spin re ti ation urrent is mu h greater than the harge re ti ation urrent. In order to al ulate the prefa tors in the right hand side of Eq. (2.20) one has to employ the Keldysh formalism. The details are explained in Appendix B.3. Here let us shortly summarize the essential steps: The third order Keldysh ontribution redu es to the integral of P (t1 , t2 , t3 ) = hTc exp(iφ↑ (t1 ) + ieV t1 /~) exp(−iφ↓ (t2 ) − ieV t2 /~) × exp(i[−φ↑ (t3 ) + φ↓ (t3 )])i (2.22) 80 over (t1 − t3 ) and (t2 − t3 ), where Tc denotes time ordering along the Keldysh on- tour −∞ → 0 → −∞ and the angular bra kets denote the average with respe t to the ground state of the non-intera ting Hamiltonian (2.7). The integration an be performed analyti ally as dis ussed in Appendix B.3. One nds 16eτc2 ¯ eV τc ¯a+b+c−2 ¯ ¯ Icbs = sign(eV ) ¯¯ ¯ Γ(1 − a)Γ(1 − b) π~3 ~ ¯ πa × Γ(2 − a − b − c)Γ(a + b − 1) sin 2 πb π(a + b) × sin sin sin π(a + b + c) 2 2 × Re[U (1, 0)U (−1, 1)U (0, −1)], (2.23) ¯a+b+c−2 16τc2 ¯ πa πb ¯¯ eV τc ¯¯ Isbs = 2 sin sin π~ 2 2 ¯ ~ ¯ × Γ(a + b − 1)Γ(2 − a − b − c)Γ(1 − a)Γ(1 − b) n π(a + b + c) × Im[U (1, 0)U (−1, 1)U (0, −1)] cos 2 πc π(a − b) π(a + b + c) π(a + b) ×[sin + cos sin + sin 2 2 2 2 π(a + b + c) 1 × cos ] + Re[U (1, 0)U (−1, 1)U (0, −1)] 2 2 π(a − b) o × sin sin π(a + b + c)sign(eV ) , (2.24) 2 where a = 2A − 2C , b = 2B − 2C , c = 2C and τc ∼ ~/EF is the ultraviolet uto time. The harge urrent (2.23) is an odd fun tion of the voltage and hen e does not ontribute to the re ti ation ee t. The spin urrent (2.24) is a sum of an even and 81 odd fun tions and hen e determines the spin re ti ation urrent ¯a+b+c−2 16τc2 ¯ πa πb π(a + b + c) ¯¯ eV τc ¯¯ Isr = 2 sin sin cos π~ 2 2 2 ¯ ~ ¯ × Γ(a + b − 1)Γ(2 − a − b − c)Γ(1 − a)Γ(1 − b) π(a − b) π(a + b + c) × Im[U (1, 0)U (−1, 1)U (0, −1)][cos sin 2 2 πc π(a + b) π(a + b + c) + sin + sin cos ]. (2.25) 2 2 2 It is non-zero if Im[U (1, 0)U (−1, 1)U (0, −1)] 6= 0, whi h is satised for asymmet- ri potentials. The leading ontribution to the harge re ti ation urrents omes from other terms in the perturbation expansion. Thus, we expe t that in the region (B.8,B.9,B.10,B.14,B.16,B.17), the spin re ti ation urrent ex eeds the harge re ti-  ation urrent in an appropriate interval of voltages (2.21). The dieren e between the spin and harge re ti ation urrent an be easily understood from the limit A = B . In that ase the harge urrent hanges its sign under the transformation U (1, 0) ↔ U (0, −1), V → −V . Sin e U (1, 0) and U (0, −1) enter the urrent only in the ombination U (1, 0)U (0, −1), this means that the harge urrent must be an odd fun tion of the voltage bias. A similar argument shows that at A = B the spin re ti ation urrent is an even fun tion of the voltage in agreement with Eq. (2.24). The voltage dependen e of the spin re ti ation urrent is illustrated in Fig. 2.5. The expression (2.20) des ribes the urrent in the voltage interval V ∗∗ > V > V ∗ . In this interval the urrent in reases as the voltage de reases in the regime (B.8,B.9,B.10,B.14,B.16,B.17). At lower voltages the perturbation theory breaks 82 Isr V∗ V ∗∗ V Figure 2.5: Qualitative representation of the spin re ti ation urrent. The spin urrent ex eeds the harge urrent and follows a power-law dependen e on the voltage with a negative exponent in the interval of voltages V ∗ < V < V ∗∗ . down. The urrent must de rease as the voltage de reases below V ∗ and eventu- ally rea h 0 at V = 0. At higher voltages, EF > eV > eV ∗∗ , the se ond or- der re ti ation urrent (2.19) dominates. The leading se ond order ontribution Isr ∼ |U (1, 0)|2 V 2z(1,0)+2 grows as the voltage in reases. The harge re ti ation ur- rent has the same order of magnitude as the spin urrent. The Tomonaga-Luttinger model annot be used for the highest voltage region EF ∼ eV . It is easier to dete t harge urrents than spin urrents. However, the measurement of the spin urrent an be redu ed to the measurement of harge urrents: Let us split the right end of the wire into two bran hes and pla e them in opposite strong magneti elds so that only ele trons with one spin orientation an propagate in ea h bran h. If both bran hes are grounded, they still inje t exa tly the same harge and spin urrents into the wire as one unpolarized lead. However, the urrent generated 83 in the wire will split between two bran hes into the urrents of spin-up and spin-down ele trons. If they are opposite then pure spin urrent is generated. 84 2.5 Con lusions In Se . 2, we have shown that re ti ation in quantum wires in a uniform magneti eld an lead to a spin urrent that largely ex eeds the harge urrent. The thesis fo uses on the regime of low voltages and weak asymmetri potentials in whi h the perturbation theory provides quantitatively exa t predi tions. Qualitatively the same behavior is expe ted up to eV, U ∼ EF . The spin re ti ation ee t is solely due to the properties of the wire and does not require time-dependent magneti elds or spin polarized inje tion as from magneti ele trodes. The urrents are driven by the voltage sour e only. In an interval of low voltages the spin urrent grows as the voltage de reases. In ontrast to some other situations, the z - omponent of the total spin onserves and hen e the d spin urrent is onstant throughout the system. Chapter 3 Probing non-Abelian Statisti s in quantum Hall Liquids at 5/2 lling Several states were proposed as andidates for the ν = 5/2 quantum Hall plateau. We suggest an experiment whi h an determine the physi al state. The proposal involves transport measurements in the geometry with three quantum Hall edges onne ted by two quantum point onta ts. In ontrast to interferen e experiments, this approa h an distinguish the Pfaan and anti-Pfaan states as well as dierent states with identi al Pfaan or anti-Pfaan statisti s. The transport is not sensitive to the u tuations of the number of the quasiparti les trapped in the system. 85 86 3.1 Introdu tion The Pauli spin-statisti s theorem does not apply in two dimensions and anyons an exist in addition to usual fermions and bosons. Dierent types of anyoni statisti s an be lassied as Abelian and non-Abelian. In the former ase, the wave fun tion of an anyon system a quires a phase fa tor after one parti le makes a ir le around other anyons. In the latter ase not only the wave fun tion but also the quantum state hanges after a parti le moves along a losed loop. Gauge invarian e guarantees that fra tionally harged ex itations of ele troni systems must be anyons. Hen e, the observation of fra tional harges in quantum Hall systems [11, 12, 13℄ proves the existen e of fra tional statisti s. However, the experiments [11, 12, 13, 59℄ have provided little information about the details of the quasiparti le statisti s in the quantum Hall ee t (QHE). In parti ular, the existen e of non-Abelian anyons [14℄ remains an open problem. A promising pla e to look for non-Abelian statisti s is the QHE plateau at the lling fa tor ν = 5/2. The nature of the 5/2 QHE state has not been understood yet and most theoreti al proposals involve non-Abelian statisti s [17, 18, 19, 20℄. However, a simpler Abelian state is also a possibility [60, 20℄. Six simplest theoreti al proposals are des ribed in Table. (3.1) (see also Ref. [20℄). Several interferometry experiments were proposed [21, 22, 23, 24, 25, 26, 27, 28, 29, 30℄ for probing the 5/2 state. The simplest approa h is based on the Fabry-Perot geometry [21, 22, 23, 24, 25℄. This approa h an however work only if the number 87 state modes statisti s K=8 1R Abelian 331 2R Abelian Pfaan 3/2R Non-Abelian edge-re onstru ted Pfaan 2R+1/2L Non-Abelian non-equilibrated anti-Pfaan 1R+3/2L Non-Abelian disorder-dominated anti-Pfaan 1R+3/2L Non-Abelian Table 3.1: Proposed 5/2 states. The 2nd olumn shows the numbers of the right- and left-moving modes (R and L), Majorana fermions being ounted as 1/2 of a mode. of the quasiparti les trapped in lo alized states inside the interferometer does not u tuate on the time s ale of the experiment [61℄. Sin e the energy gap for neutral ex itations is likely to be low, this ondition may not be easy to satisfy at realis- ti temperatures. The Ma h-Zehnder interferometry is free from this limitation but shares another limitation with the Fabry-Perot approa h: it annot distinguish any of the non-Abelian states listed in Table. (3.1) from ea h other. So far interferome- try [62℄ and other approa hes [63, 64℄ allowed the measurement of the quasiparti le harge q = e/4 in the 5/2 state. This is not su ient for the determination of the physi al 5/2 state sin e q = e/4 is predi ted by all theories of the 5/2 plateau listed in Table. (3.1). Thus, some method other than interferometry is desirable. One idea onsists in he king s aling relations [20, 65℄ su h as the power dependen e of the urrent on the voltage I ∼ V s in a quantum point onta t [20℄. Unfortunately, even in the simplest ase of the Laughlin states the theory has not been re on iled with the measurements of the I − V urve [66℄. Besides, this approa h is not expe ted to distinguish the anti-Pfaan and edge-re onstru ted Pfaan states [20℄. In this thesis, we suggest a 88 two-point- onta t geometry whi h involves only transport experiment. We show that this approa h leads to qualitatively dierent results for all states listed in Table (3.1) and is not sensitive to the number of the trapped quasi-parti les. The reset of this se tion is organized as follows. In Se . 3.2, we explain the two-point- onta t geometry and present the model Hamiltonian that des ribe the system. In Se . 3.3, we present analyti al al ulation for six of the theoreti al andidate states respe tively. Our result is summarized in Se . 3.4. 3.2 The two-point- onta t Geometry and Model Hamil- tonian Fig. ( 3.1) shows our two-point- onta t geometry. ν = 5/2 Γ1 Γ2 S3 D3 ν=2 S1 D1 S2 D2 Figure 3.1: Setup with two quantum point onta ts. Arrows show the propagation dire tion of harged modes. This geometry is similar to the one used in experiments with Laughlin states [67℄. 89 Edge 3 onne ts sourse S3 with drain D3 and separates regions with lling fa tors 5/2 and 2. Edges 1 and 2 onne t sour e S1 with drain D1 and sour e S2 with drain D2 and separate the region with lling fa tor 2 from the region with lling fa tor 0. Ele trons an tunnel a ross the integer QHE region through quantum point onta ts QPC1 and QPC2 with the tunneling amplitudes Γ1 and Γ2 (Fig. 1), respe tively, at the distan e a from ea h other. Sour e S3 is maintained at zero voltage. We onsider two situations for the ele tri potentials of sour es S1 and S2: 1) V (S1) = V , V (S2) = 0 and 2) V (S1) = 0, V (S2) = V . In the rst ase we al ulate the urrent I2 and noise S2 in drain D2. In the se ond ase we nd the urrent I1 and noise S1 in drain D1. Nonzero I1 is possible only in some states due to the presen e of ontra-propagating edge modes on the boundary between the ν = 5/2 and ν = 2 regions. We will see below that dierent states an be distinguished by zero/nonzero I1 and/or S1 and universal or nonuniversal Fano fa tors F1 = S1 /I1 and F2 = S2 /I2 . The out omes of the proposed experiment for dierent states are summarized in Table. ( 3.2) in Se . (3.4). The physi s is analogous in a similar geometry with all three edges separating a QHE liquid with ν = 5/2 from regions with ν = 0. However, an analyti al ulation is impossible in that setup and its numeri al analysis will be dis ussed elsewhere. The system in Fig. 1 has the following Lagrangian 3 Z 2 (3.1) X X L= Lk − dt [Γk Tk + h.c.], k=1 k=1 where Lk are the Lagrangians of the three edges, Γk the tunneling amplitudes at 90 the two QPC's, Tk = ǫ†k ψ(xk )ηk the tunneling operators, ǫk the ele tron annihilation operators on edges 1 and 2, ψ(xk ) the ele tron annihilation operators on edge 3 (x1 = 0, x2 = a) and the Klein fa tors ηk make sure that the tunneling operators Tk ommute. The operator ψ(x) and a tion L3 of fra tional QHE edge 3 depend on the model (Table. (3.1)). L1,2 des ribe hiral Fermi-liquid systems. Edges 1 and 2 have two hannels with spin up and down but it is su ient to in lude only one of them in L1,2 be ause of the spin onservation at the tunneling events. The zero-temperature orrelation fun tions of the elds ǫ assume the Fermi-liquid form hǫ†k (t1 )ǫk (t2 )i ∼ 1/[i(t1 − t2 ) + δ]. Below we use the perturbation theory to al ulate the urrent and noise in the order Γ21 Γ22 . The perturbative al ulation is legitimate if Γ2 (eV )2s+1 /Ec2s+2 < eV , where s is the s aling dimension of the operators Tk and Ec the ut-o energy of the order of the QHE energy gap. We negle t the thermal noise ∼ Γ2 T 2s+1 . For Γ ∼ Ecs+1 /(eV )s this ondition redu es to eV > T . Thus, we on entrate on the low-temperature limit T = 0. We assume that the distan e a between the point onta ts is su iently large, a ≫ aV = hv/(eV ), where v is of the order of the edge mode velo ity. This will allow us to treat QPC1 and QPC2 as independent. At the same time we negle t equilibration between dierent o- and ontra-propagating edge modes on the s ale a. Indeed, if the Lagrangian of any model in Table. (3.1) ontains a large ontribution, responsible for edge equilibration, or su h ontribution renormalizes to a large value at the s ale aV then it opens a gap and/or modies the 91 hara ter of soft modes and hen e hanges the model. 3.3 Analyti al Cal ulation 3.3.1 General analyti al formula for Current and Short noise The purpose of our perturbation al ulation is to nd out two quantities, urrent I2 and shot noise S2 at drain 2 (see Fig. (3.1). Here we present a general formula due ? to Kane and Fisher[ ℄ to al ulate I2 and S2 . We assume that qV (S1 ) < 0, i.e., the hemi al potential of edge 1 is lower than the potentials of edges 2 and 3 when voltage is applied on Sour e 1. The full Lagrangian has the form L = L1 + L2 + L3 + O 1 + O 2 (3.2) where L1 and L2 des ribes hiral Fermi-liquid systems in whi h ele trons have the following fermion orrelation fun tion 1 hǫk (t)ǫk (0)i ∼ (3.3) δ + it O1 and O2 represents the tunneling terms O1 = Γ1 (T1+ + T1− ) (3.4) O2 = Γ2 (T2+ + T2− ) (3.5) where we have introdu ed ± to denote the dagger or non-dagger of an operator. The spe i expression for L3 and Ti depends on the a tual state of 5/2 region. This 92 will in term determine the urrent and shot noise at drain 2. For the general formula that we are going to derive in this subse tion, the expression of Ti± need not to be spe ied. Spe i forms for Ti± in dierent model will be presented in Se . 3.3, where we present al ulation for dierent L3 . In order to treat the applied voltage, it is onvenient to swit h to the intera tion representation su h that all three hemi al potentials be ome 0. This introdu es a time-dependen e into the operator T1 ∼ exp(iqV t/~). The operator of the urrent through QPC2 an be found as the time derivative of the harge on edge 2, −iq I2 = Γ2 (T2+ − T2 ) (3.6) ~ where q is the arrier harge. The shot noise through QPC2 is dened as Z S2 (t0 ) = dt1 [I2 (t0 )I2 (t1 ) + I2 (t1 )I2 (t0 )] (3.7) Using Keldysh te hnique[57, 58℄ The expe tation value of I2 under Keldysh te hnique is · µ ¶¸ i Z hI2 (t0 )i = hTˆc I2 (t0 ) exp − dt [O1 (t) + O2 (t)] i (3.8) ~ c Here h...i is the thermal expe tation value averaged under the system in the innite past when Γ1 = Γ2 = 0. O1± , O2± are operators in intera tion presentation. c denotes Keldysh ontour, whi h runs from −∞ → ∞ → −∞. Tˆc spe ies ordering along Keldysh ontour shown in Fig. ( 3.2). The time t0 is arbitrary and an be hosen to lie on the forward Keldysh ontour (−∞ → ∞). 93 O1 (t1 ) σ=+ O2 (t2 ) −∞ +∞ t O2 (t4 ) σ=− O1 (t3 ) Figure 3.2: The Keldysh ontour. O1 (t1 ), O2 (t2 ) an lie on the forward bran h (σ = +) or ba kward bran h (σ = −). Eq. (3.8) is al ulated perturbatively by expanding the exponential fa tor as µ ¶ X ∞ i 1 Z Z Z Z exp − dt [O1 (t) + O2 (t)] = dt1 dt2 · · · dtn ~ c n=0 n! c c c ·µ ¶µ ¶ µ ¶¸ ˆ i i i × Tc − [O1 (t1 ) + O2 (t1 )] − [O1 (t2 ) + O2 (t2 )] · · · − [O1 (tn ) + O2 (tn )] ~ ~ ~ (3.9) Every nonzero ontribution to I2 in ludes all four operators T1 , T1† , T2 , T2† . Hen e the leading ontribution for I2 is fourth order∼ Γ21 Γ22 and havs the following form ¶3 Z Γ 2 Γ2 µ −i hI2 (t0 )i = 1 dt1 dt2 dt3 3! ~ c × hTˆc [I2 (t0 ) [O1 (t1 ) + O2 (t1 )] [O1 (t2 ) + O2 (t2 )] [O1 (t3 ) + O2 (t3 )]]i (3.10) Be ause the operator I2 (t0 ) in the integrand already ontains one Γ2 O2± , we need to sele t another Γ2 O2± from the three [O1 + O2 ] bra kets. There are 3 ways of doing this whi h an els the fa tor 1/3! to 1/2!. After the se ond Γ2 O2± is hosen, we left with no hoi e but to sele t two T1 s from the rest of the two [O1 + O2 ]bra kets. Hen e ¶3 Z Γ 2 Γ2 µ −i hI2 (t0 )i = 1 dt1 dt2 dt3 hTˆc I(t0 )O2 (t1 )O1 (t2 )O1 (t3 )i (3.11) 2! ~ c 94 In the same spirit, we an derive the expression for the shot noise ·Z µ ¶¸ −i Z hS2 (t0 ) = hTˆc dt1 (I2 (t0 )I2 (t1 ) + I2 (t1 )I2 (t0 )) exp − dt [O1 (t) + O2 (t)] i ~ c Z ≈ hTˆc dt1 (I2 (t0 )I2 (t1 ) + I2 (t1 )I2 (t0 )) µ ¶2 Γ1 Γ2 −i × [O1 (t2 ) + O2 (t2 )] [O1 (t3 ) + O2 (t3 )]i 2! ~ µ ¶2 Z −i Z = Γ1 Γ2 dt1 dt2 dt3 hTˆc I(t0 )I(t1 )O1 (t2 )O1 (t3 )i. (3.12) ~ c Eqs. (3.11), (3.12) an be further simplied to an expression involving only Ti± s. Introdu e σ = ± denoting forward/ba kward path, the integration over Keldysh ontour an be written as Z X Z ∞ dt → σ dt (3.13) c σ=± −∞ Next, make the following substitution, Oi± (t) → Oi± (σ, t) (3.14) where σ indi ate whi h bran h the operator lies on. With these onvention, one has ¶3 X Z Z Z Γ 2 Γ2 µ −i hI2 (t0 )i = 1 dt1 dt2 dt3 2 ~ σ1 ,σ2 ,σ3 × hTˆc [I2 (+, t0 )σ1 O2 (σ1 , t1 )σ2 O1 (σ2 , t2 )σ3 O1 (σ3 , t3 )]i µ ¶3 X Γ21 Γ2 −i Z Z Z = σ1 σ2 σ3 dt1 dt2 dt3 4 ~ σ ,σ ,σ ,σ , 0 1 2 3 × hTˆc [I2 (σ0 , t0 )O2 (σ1 , t1 )O1 (σ2 , t2 )O1 (σ3 , t3 )]i (3.15) The extra 1/2 fa tor omes from the following argument. hI2 (t0 )i does not depend on whi h bran h t0 lies on, so equal ontribution of integral an be obtained when t0 95 is on lower bran h σ0 = −. Thus summing over σ0 yields twi e of hI2 (t0 )i. For the same reason, hS2 (t0 )i an be simplied as µ ¶2 X Z Z Z −i hS2 (t0 )i = Γ1 Γ2 dt1 dt2 dt3 ~ σ2 ,σ3 × hTˆc [I2 (t0 )I2 (t1 )σ2 O1 (t2 )σ3 O1 (t3 )]i µ ¶2 X Γ1 Γ2 −i Z Z Z = σ2 σ3 dt1 dt2 dt3 4 ~ σ ,σ ,σ ,σ 0 1 2 3 × hTˆc [I2 (σ0 , t0 )I2 (σ1 , t1 )O1 (σ2 , t2 )O1 (σ3 , t3 )]i. (3.16) Next, introdu e s = ± to denote the dagger or non-dagger of an operator, one has −iq X s I2 = Γ2 sT2 (3.17) ~ s X Oi = Tis (3.18) s This allows us to further simplify Eqs. (3.15), (3.16) into qΓ2 Γ2 Z d3 thTˆc T2s0 (σ0 , t0 )T2s1 (σ1 , t1 )T1s2 (σ2 , t2 )T1s3 (σ3 , t0 )i X hI2 (t0 )i = 1 4 2 s0 σ1 σ2 σ3 4~ {σk },{sk } (3.19) q 2 Γ21 Γ22 Z d3 thTˆc T2s0 (σ0 , t0 )T2s1 (σ1 , t1 )T1s2 (σ2 , t2 )T1s3 (σ3 , t0 )i X hS2 (t0 )i = s0 s1 σ2 σ3 4~4 {σi },{sk } (3.20) Clearly, only the term with s0 + s1 = s2 + s3 = 0 has non zero orrelation fun tion, 96 therefore qΓ2 Γ2 Z d3 thTˆc T2s0 (σ0 , t0 )T2−s0 (σ1 , t1 )T1s2 (σ2 , t2 )T1−s2 (σ3 , t0 )i X hI2 (t0 )i = 1 4 2 s0 σ1 σ2 σ3 4~ {σk },s0 ,s2 (3.21) q 2 Γ21 Γ22 Z d3 thTˆc T2s0 (σ0 , t0 )T2−s0 (σ1 , t1 )T1s2 (σ2 , t2 )T1−s2 (σ3 , t0 )i X hS2 (t0 )i = s0 s1 σ2 σ3 4~4 {σi },s0 ,s2 (3.22) There is one symmetry that an bring the above expressions into an even simpler form. Due to translational invarian e of the orrelation fun tion, one an show that the value of the integral does not hange under (s0 ↔ −s0 , σ0 ↔ σ1 ) and (s2 ↔ −s2 , σ2 ↔ σ3 ). Applying these symmetries, we obtain qΓ21 Γ22 X Z hI2 (t0 )i = 4 (σ1 − σ0 )σ2 σ3 d3 thTˆc T2+ (σ0 , t0 )T2− (σ1 , t1 )T1+ (σ2 , t2 )T1− (σ3 , t3 )i 2~ {σk } (3.23) q 2 Γ21 Γ22 X Z hS2 (t0 )i = − 4 σ2 σ3 d3 thTˆc T2+ (σ0 , t0 )T2− (σ1 , t1 )T1+ (σ2 , t2 )T1− (σ3 , t3 )i ~ {σk } (3.24) d3 t = R R Finally, we emphasize that dt1 dt2 dt3 , but be ause the orrelation fun - tion depends only on the time dieren es, d3 t an be integration over any three d3 t(· · · ) = R R R of the four time variables, e.g., dt1 dt2 dt3 (· · · ) = dt0 dt1 dt2 (· · · ) = R dt3 dt1 dt2 (· · · ). 97 3.3.2 Analyti al results for Ea h States K=8 State We begin with the simplest ase of the K = 8 state [60, 20℄. Cal ulations are similar but longer for the other models. The simpli ations is due to the existen e of only one edge mode on edge 3 in the K = 8 model whose Lagrangian is Z L3 /~ = −(2/π) dtdx[∂t φ∂x φ + v(∂x φ)2 ]. (3.25) In this model, we have T1 (t) = ǫ†1 (t)ei8φ(0,t) , (3.26) T2 (t) = ǫ†2 (t)ei8φ(a,t) , (3.27) where in ontrast to all other states in Table. ( 3.1), only ele tron pairs but not ele trons an tunnel into the K = 8 edge ∗ . Thus, ψ = exp(i8φ) and ǫk should be understood as pair annihilation. Lagrangian Eq. (3.25) gives the orrelation fun tion 1 hφ(x, t)φ(0, 0)i = hφ(t − x/v)φ(0)i = − ln [δ + i(t − x/v)] . (3.28) 8 And the pair annihilation operator ǫk is has orrelation fun tion hǫ†k (t)ǫk (0)i ∼ 1/(δ + it)4 (3.29) ∗ The K=8 state is formed by tightly bound Cooper pairs and hen e ele trons are gapped. 98 From Eqs. (3.23), (3.24), the essential quantity in our al ulation is the integration of the four point orrelation fun tion hTˆc T2+ (σ0 , t0 )T2− (σ1 , t1 )T1+ (σ2 , t2 )T1− (σ3 , t3 )i = hTˆc ǫ2 (σ0 , t0 − a/v)ǫ†2 (σ1 , t1 − a/v)ihTˆc ǫ1 (σ2 , t2 )ǫ†1 (σ3 , t3 )i × hTˆc e−i8φ(σ0 ,t0 −a/v) e+i8φ(σ1 ,t1 −a/v) e−i8φ(0,σ2 ,t2 ) ei8φ(0,σ3 ,t3 ) i σ01 σ23 = (δ + iσ01 t01 ) (δ + iσ23 t23 )4 4 eiωt23 1 (δ + iσ02 (t02 − a/v))8 (δ + iσ13 (t13 − a/v))8 × , (3.30) (δ + iσ01 t01 )8 (δ + iσ23 t23 )8 (δ + iσ12 (t12 − a/v))8 (δ + iσ03 (t03 − a/v))8 where w ≡= −qV /~ > 0 is the applied voltage and σij arises from time ordering along Keldysh ontour whi h an be expli itly expressed as 1 σij = [(σj − σi ) + Sgn(ti − tj )(σi + σj )]. (3.31) 2 In the limit of innite distan e between two QPCs a → ∞, the only times to on- tribute will be those with t01 , t23 ∼ 0 and t12 , t03 ∼ a/v . Consequently, σ02 = σ12 = σ2 , σ03 = σ13 = σ3 by Eq. (3.31). Therefore Z d3 thTˆc T2+ (σ0 , t0 )T2− (σ1 , t1 )T1+ (σ2 , t2 )T1− (σ3 , t3 )i σ01 σ23 Z = d3 t eiωt23 (δ + iσ01 t01 ) (δ + iσ23 t23 )12 12 (δ + iσ2 (t02 − a/v))8 (δ + iσ3 (t13 − a/v))8 × (δ + iσ2 (t12 − a/v))8 (δ + iσ3 (t03 − a/v))8 ≡ F (σ01 , σ23 , σ2 , σ3 ) (3.32) 99 Substitute Eq. (3.32) into Eq. (3.23) and using the fa t that   σi Sgn(tij ),   σi = σj σij = (3.33)   −σi ,  σi = −σj the urrent and shot noise at QPC2 an be expressed as qΓ21 Γ22 X I2 = σ0 σ2 σ3 F (σ0 , σ23 , σ2 , σ3 ), (3.34) ~4 σ ,σ ,σ 0 2 3 q 2 Γ21 Γ22 X S2 = − σ2 σ3 [F (σ0 Sgn(t01 ), σ23 , σ2 , σ3 ) + F (−σ0 , σ23 , σ2 , σ3 )] ~4 σ ,σ ,σ 0 2 3 2q 2 Γ21 Γ22 (3.35) X = − σ2 σ3 F (σ0 , σ23 , σ2 , σ3 ). ~4 σ0 ,σ2 ,σ3 Note that in deriving Eq. (3.35), one needs the following identity F (σ0 Sgn(t01 ), σ23 , σ2 , σ3 ) = (3.36) X X F (σ0 , σ23 , σ2 , σ3 ) σ0 σ0 whi h an be proved by ombing the dt1 part of F (σ0 Sgn(t01 ), σ23 , σ2 , σ3 ) with R t0 −∞ the dt1 part of F (−σ0 Sgn(t01 ), σ23 , σ2 , σ3 ) as follows, R∞ t0 F (σ0 Sgn(t01 ), σ23 , σ2 , σ3 ) X σ0 XZ σ0 Sgn(t01 ) ∞ Z = dt2 dt3 dt1 (· · · ) σ0 −∞ (δ + iσ0 Sgn (t 01 )t01 )12 ·Z t0 Z ∞ ¸ XZ σ0 −σ0 = dt2 dt3 dt1 (· · · ) + dt1 (· · · ) σ0 −∞ (δ + iσ0 t01 )12 t0 (δ − iσ0 t01 )12 ·Z t0 Z ∞ ¸ XZ σ0 σ0 = dt2 dt3 dt1 (· · · ) + dt1 (· · · ) σ0 −∞ (δ + iσ0 t01 )12 t0 (δ + iσ0 t01 )12 Z t0 XZ σ0 = dt2 dt3 dt1 dt1 (· · · ) σ0 −∞ (δ + iσ0 t01 )12 (3.37) X = F (σ0 , σ23 , σ2 , σ3 ) σ0 100 Next, we al ulate the integral F (σ, σ23 , σ2 , σ3 ) in Eq. (3.32). By looking at the poles, one dis overs Z σ0 = σ3 ⇒ dt0 (· · · ) = 0 Z σ0 = −σ2 ⇒ dt1 (· · · ) = 0 Hen e only the term with σ0 = σ2 = −σ3 is non-zero and we have qΓ21 Γ22 X I2 = − σ0 F (σ0 , −σ0 , σ0 , −σ0 ). (3.38) ~4 σ 0 Moreover, the t2 integral in F (σ0 , −σ0 , σ0 , −σ0 ) with ω > 0 is non-zero only if σ0 = −1. Therefore, in the K = 8 ase, we have only one non-zero integral F (−, +, −, +) and the urrent and shot noise are qΓ21 Γ22 I2 = F (−, +, −, +), (3.39) ~4 2q 2 Γ21 Γ22 S2 = F (−, +, −, +). (3.40) ~4 The integral an be al ulated analyti ally but its value is unimportant for the al- ulation of the Fano fa tor F2 = S2 /I2 =2q=4e. The result is easy to understand from Fig. ( 3.3). Proportionality to S2 to I2 ree ts the fa t that the urrent I2 is reated by a random ux of harge-2e parti les. Certainly, the same Fano fa tor would be seen in a simpler geometry with one tunneling onta t. The situation is more interesting in the remaining models from Table. In all of them q = e and the Fano fa tor in a single-QPC geometry is the same, F = 2e. However, in the two-QPC geometry (Fig. 3.1), their transport properties are onsiderably dierent. 101 K=8 φ edge 3 S3 D3 edge 1 ν=2 edge 2 V I2 , S 2 Figure 3.3: K=8 state, only harged boson φ propagate on edge 3. Hen e S2 /I2 = 2q = 4e. Pfaan State The Pfaan state [17℄ has two edge modes [14℄: a harged boson φc and neutral Ma- jorana fermion λ whi h propagate with dierent velo ities vc and vn . The Majorana mode ontains information about non-Abelian statisti s and thus the Pfaan state exhibits harge-statisti s separation. The Lagrangian Z L3 /~ = dtdx[−∂x φc (∂t + vc ∂x )φc /(2π) + iλ(∂t + vn ∂x )λ], (3.41) In this model, the ele tron operator is ψ = λ exp(−2iφ), so T1 (t) = ǫ†1 (t)λ(t)ei2φc (0,t) , (3.42) T2 (t) = ǫ†2 (t)λ(t)ei2φc (a,t) . (3.43) The orrelation fun tion for φ(x, t) is easily determined from Lagrangian (3.41) 1 hφc (x, t)φc (0, 0)i = hφc (t − x/vc )φc (0)i = − ln [δ + i(t − x/vc )] , (3.44) 2 102 and the orrelation fun tion for ǫk (x, t) is hǫ†k (x, t)ǫk (0, 0)i ∼ 1/(δ + i(t − x/v)). (3.45) We will also need the four-point orrelation fun tion for Majorana fermions, i.e., hλ(1)λ(2)λ(3)λ(4)i =hλ(1)λ(2)ihλ(3)λ(4)i − hλ(1)λ(3)ihλ(2)λ(4)i + hλ(1)λ(4)ihλ(2)λ(3)i (3.46) where hλ(x, t)λ(0, 0)i = hλ(t − x/vn )λ(0)i = 1/[δ + i(t − x/vn )] (3.47) The al ulations follow the same line as above. We obtain the four point orrelation fun tion hTˆc T2+ (σ0 , t0 )T2− (σ1 , t1 )T1+ (σ2 , t2 )T1− (σ3 , t3 )i = hTˆc ǫ2 (σ0 , t0 − a/vc )ǫ†2 (σ1 , t1 − a/vc )ihTˆc ǫ1 (σ2 , t2 )ǫ†1 (σ3 , t3 )i × hTˆc e−i2φ(σ0 ,t0 −a/v) e+i2φ(σ1 ,t1 −a/v) e−i2φ(0,σ2 ,t2 ) ei2φ(0,σ3 ,t3 ) i × hTˆc λ(σ0 , t0 − a/vn )λ(σ1 , t1 − a/vn )λ(σ2 , t2 )λ(σ3 , t3 )i σ01 σ23 = (δ + iσ01 t01 ) (δ + iσ23 t23 ) eiωt23 1 (δ + iσ02 (t02 − a/vc ))2 (δ + iσ13 (t13 − a/vc ))2 × (δ + iσ01 t01 )2 (δ + iσ23 t23 )2 (δ + iσ12 (t12 − a/vc ))2 (δ + iσ03 (t03 − a/vc ))2 · σ01 σ23 σ12 σ03 × + δ + iσ01 t01 δ + iσ23 t23 δ + iσ12 (t12 − a/vn ) δ + iσ03 (t03 − a/vn ) ¸ σ02 σ13 − . (3.48) δ + iσ02 (t02 − a/vn ) δ + iσ13 (t13 − a/vn ) 103 As we an see from Eq. (3.48), the four point orrelation fun tion in Pfaan state is onsiderably dierent from the one obtained in the K = 8 (see Eq. (3.30)). Physi ally, the dieren e is due to the presen e of the Majorana fermion λ whi h travels with velo ity vn 6= vc . As a result, one needs to take into a ount the ontributions to the urrent and shot noise from both t12 , t03 ≈ a/vc and t12 , t03 ≈ −a/vn . Moreover the urrent and noise omes from three ontributions, Z d3 thTˆc T2+ (σ0 , t0 )T2− (σ1 , t1 )T1+ (σ2 , t2 )T1− (σ3 , t3 )i = FA (σ01 , σ23 , σ2 , σ3 ) + FB (σ01 , σ23 , σ2 , σ3 ) + FC (σ01 , σ23 , σ2 , σ3 ) (3.49) with eiωt23 (σ01 σ23 )2 Z FA (σ01 , σ23 , σ2 , σ3 ) = d3 t (δ + iσ01 t01 )3 (δ + iσ23 t23 )3 (δ + iσ2 (t02 − a/vc ))2 (δ + iσ3 (t13 − a/vc ))2 × , (3.50) (δ + iσ2 (t12 − a/vc ))2 (δ + iσ3 (t03 − a/vc ))2 eiωt23 σ01 σ23 σ2 σ3 Z 3 FB (σ01 , σ23 , σ2 , σ3 ) = dt 3 3 (δ + iσ01 t01 ) (δ + iσ23 t23 ) δ + iσ2 (t12 − a/vn ) δ + iσ3 (t03 − a/vn ) (δ + iσ2 (t02 − a/vc ))2 (δ + iσ3 (t13 − a/vc ))2 × , (3.51) (δ + iσ2 (t12 − a/vc ))2 (δ + iσ3 (t03 − a/vc ))2 eiωt23 σ01 σ23 −σ2 σ3 Z FC (σ01 , σ23 , σ2 , σ3 ) = d3 t 3 3 (δ + iσ01 t01 ) (δ + iσ23 t23 ) δ + iσ2 (t02 − a/vn ) δ + iσ3 (t13 − a/vn ) (δ + iσ2 (t02 − a/vc ))2 (δ + iσ3 (t13 − a/vc ))2 × , (3.52) (δ + iσ2 (t12 − a/vc ))2 (δ + iσ3 (t03 − a/vc ))2 where we have used the approximation σ02 = σ12 = σ2 and σ03 = σ13 = σ3 whi h follows from the same argument between Eq. (3.31) and Eq. (3.32). Ea h type of 104 integral ontributes independently to the urrent and shot noise at QPC2, thus I2 = IA + IB + IC , (3.53) S2 = SA + SB + SC , (3.54) where qΓ21 Γ22 X Ii = σ0 σ2 σ3 Fi (σ0 , σ23 , σ2 , σ3 ), (3.55) ~4 σ ,σ ,σ 0 2 3 2qΓ21 Γ22 (3.56) X Si = − σ2 σ3 Fi (σ0 , σ23 , σ2 , σ3 ), i = A, B, C. ~4 σ0 ,σ2 ,σ3 We now pro eed to al ulate IA , IB and IC . 1. IA FA (σ01 , σ23 , σ2 , σ3 ) has the same stru ture as Eq. (3.32) ex ept dierent powers. Just like the K = 8 ase, we have only one non-zero integral FA (−, +, −, +), hen e qΓ21 Γ22 IA = FA (−, +, −, +) ~4 qΓ21 Γ22 eiωt23 1 (δ − i(t02 − a/vc ))2 (δ + i(t13 − a/vc ))2 Z 3 = d t ~4 (δ − it01 )3 (δ + it23 )3 (δ − i(t12 − a/vc ))2 (δ + i(t3 − a/vc ))2 qΓ21 Γ22 64π 3 5 = × ω (3.57) ~4 5! 2qΓ21 Γ22 SA = FA (−, +, −, +) ~4 2qΓ21 Γ22 64π 3 5 = × ω . (3.58) ~4 5! 105 2. IB One performs a hange of variables t2 → t2 + a/vn t3 → t3 + a/vn , eiωt23 σ0 σ23 σ2 σ3 Z FB (σ, σ23 , σ2 , σ3 ) = d3 t 2 2 (δ + iσt01 ) (δ + iσ23 t23 ) δ + iσ2 t12 δ + iσ3 t03 (δ + iσ2 (t02 − ∆t))2 (δ + iσ3 (t13 − ∆t))2 × , (3.59) (δ + iσ2 (t12 − ∆t))2 (δ + iσ3 (t03 − ∆t))2 where a a ∆t = − . (3.60) vn vc We only onsider the limit of a → ∞, Under this approximation, the ontribu- tion from the poles at t12 ∼ ∆t, t03 ∼ ∆t are of the order (∆t)− 2 ∼ a−2 → 0. Hen e we only need to al ulate the ontributions from t12 ∼ 0 and t03 ∼ 0, in whi h ase, (δ + iσ2 (t02 − ∆t))2 (δ + iσ3 (t13 − ∆t))2 → 1. (3.61) (δ + iσ2 (t12 − ∆t))2 (δ + iσ3 (t03 − ∆t))2 Thus FB an be approximated as eiωt23 σ0 σ23 σ2 σ3 Z FB (σ, σ23 , σ2 , σ3 ) = d3 t 3 3 . (δ + iσt01 ) (δ + iσ23 t23 ) δ + iσ2 t12 δ + iσ3 t03 (3.62) By looking at the poles, one nds Z σ = σ3 ⇒ dt0 (· · · ) = 0 (3.63) Z σ = −σ2 ⇒ dt1 (· · · ) = 0 (3.64) 106 Hen e only the term with σ = σ2 = −σ3 is non-zero, (3.65) X IBσ = − F (σ, −σ, σ, −σ). σ Moreover, the t2 integral of FB (σ, −σ, σ, −σ) with ω > 0 is non-zero only if σ = −1. So we got qΓ21 Γ22 IB = FB (−, +, −, +) ~4 qΓ21 Γ22 eiωt23 (−1)(1) (−1) (1) Z 3 = 4 d t 3 3 ~ (δ − it01 ) (δ + it23 ) δ − it12 δ + it03 qΓ21 Γ22 8π 3 5 = × ω (3.66) ~4 5! 2qΓ21 Γ22 SB = FB (−, +, −, +) ~4 2qΓ21 Γ22 8π 3 5 = × ω . (3.67) ~4 5! 3. IC Similarly to FB (σ, σ23 , σ2 , σ3 ), FC (σ, σ23 , σ2 , σ3 ) an be approximated as eiωt23 σ0 σ23 −σ2 σ3 Z FC (σ, σ23 , σ2 , σ3 ) = d3 t . (δ + iσt01 )3 (δ + iσ23 t23 )3 δ + iσ2 t02 δ + iσ3 t13 (3.68) By looking at the poles, one nds Z σ = σ2 ⇒ dt0 (· · · ) = 0 (3.69) Z σ = −σ3 ⇒ dt1 (· · · ) = 0 (3.70) Hen e only the term with σ = −σ2 = σ3 is non-zero, (3.71) X IBσ = − F (σ, σ, −σ, σ). σ 107 Moreover, the t2 integral of FB (σ, σ, −σ, σ) with ω > 0 is non-zero only if σ = +1. So we nally have qΓ21 Γ22 IC = − FB (+, +, −, +) ~4 qΓ21 Γ22 eiωt23 (1)(1) (1) (1) Z =− 4 d3 t 3 3 ~ (δ + it01 ) (δ + iσ23 t23 ) δ + iσ2 t02 δ + iσ3 t13 qΓ21 Γ22 8π 3 5 =− × ω (3.72) ~4 5! 2qΓ21 Γ22 SC = 2 FB (+, +, −, +) ~4 2qΓ21 Γ22 8π 3 5 = × ω . (3.73) ~4 5! Substituting Eqs. (3.57),(3.58),(3.66),(3.67),(3.72),(3.73) into Eqs. (3.85),(3.86), we have qΓ2 Γ2 64π 3 5 8π 3 5 8π 3 5 qΓ2 Γ2 64π 3 5 µ ¶ I2 = 14 2 × ω + ω − ω = 14 2 × ω , (3.74) ~ 5! 5! 5! ~ 5! 2q 2 Γ21 Γ22 64π 3 5 8π 3 5 8π 3 5 q 2 Γ21 Γ22 160π 3 5 µ ¶ S2 = × ω + ω + ω = × ω . (3.75) ~4 5! 5! 5! ~4 5! Thus, the Fano fa tor 5 5 F2 = q = e (3.76) 2 2 is universal and ex eeds the double arrier harge. The above result has a simple explanation. After tunneling to edge 3 at QPC1, a harge-e hole splits into harged and neutral ex itations whi h propagate towards QPC2 with dierent velo ities. When the harged ex itations arrives to QPC2, its energy an be used for the tunneling of the harge e into edge 2. This pro ess is responsible for IA and SA . When a neutral ex itation arrives to QPC2 its energy 108 Pfaan λ edge 3 S3 φc D3 edge 1 ν=2 edge 2 V I2 , S 2 Figure 3.4: Pfaan state, harged boson φc ontributes to both urrent and shot noise while Majorana fermion λ ontributes to noise only. Hen e there is ex essive noise. an also be used for a tunneling event. However, sin e the reation and annihilation operators of the Majorana fermion λ are the same, harge an tunnel both from and to edge 3. This explains why IB = −IC . On the other hand, both tunneling dire tions ontribute to the ex essive noise Sn = SB + SC and in rease the Fano fa tor in omparison with a single-mode system. Edge-re onstru ted Pfaan State In the edge-re onstru ted Pfaan state [20℄ there are three modes: right-moving harged and neutral Bose-modes φc and φn and a left-moving Majorana fermion λ: Z L3 /~ =1/(4π) dtdx[−2∂x φc (∂t + vc ∂x )φc − ∂x φn (∂t + vn ∂x )φn + w∂x φc ∂x φn + 4πiλ(∂t − vλ )λ]. (3.77) Due to the left-moving mode, a non-zero S1 be omes possible in ontrast to the non- re onstru ted Pfaan state. The theory has three most relevant ele tron reation 109 operators on edge 3: λ exp(2iφc ) and exp(2iφc ± iφn ). Thus, one needs to introdu e three pairs of tunneling onstants Γ(λ) k , Γk (+) and Γ(−) k , where k = 1, 2 labels QPC's. The intera tion between the two Bose-modes ae t the Fano fa tor F2 whi h depends on all six tunneling onstants. We fo us instead on the urrent and noise at QPC1 when V (S1) = 0, V (S2) 6= 0. Non-zero S1 be omes possible due to the ontra- propagating Majorana mode and hen e only the following tunneling operators should be taken into a ount T1 (t) = ǫ†1 (t)λ(t)ei2φc (0,t) , (3.78) T2 (t) = ǫ†2 (t)λ(t)ei2φc (a,t) . (3.79) T1 and T2 above are almost the same as the ones in the Pfaan ase ex ept this time λ and φc are moving in opposite dire tions. By symmetry of the geometry (see Fig. 3.1), the al ulation of I1 and S1 in the present ase is equivalent to the al ulation of I2 and S2 in the Pfaan ase with the following repla ement vc → −vc . (3.80) 110 Hen e just like the Pfaan ase, the integral of the four point orrelation fun tion is the sum of three types of integrals, Z d3 t hTˆc T2+ (σ0 , t0 )T2− (σ1 , t1 )T1+ (σ2 , t2 )T1− (σ3 , t3 )i σ01 σ23 = (δ + iσ01 t01 ) (δ + iσ23 t23 ) eiωt23 1 (δ + iσ02 (t02 + a/vc ))2 (δ + iσ13 (t13 + a/vc ))2 × (δ + iσ01 t01 )2 (δ + iσ23 t23 )2 (δ + iσ12 (t12 + a/vc ))2 (δ + iσ03 (t03 + a/vc ))2 · σ01 σ23 σ12 σ03 × + δ + iσ01 t01 δ + iσ23 t23 δ + iσ12 (t12 − a/vn ) δ + iσ03 (t03 − a/vn ) ¸ σ02 σ13 − . δ + iσ02 (t02 − a/vn ) δ + iσ13 (t13 − a/vn ) ≈ GA (σ01 , σ23 , σ0 , σ1 ) + GB (σ01 , σ23 , σ2 , σ3 ) + GC (σ01 , σ23 , σ2 , σ3 ), (3.81) where eiωt23 (σ01 σ23 )2 Z 3 GA (σ01 , σ23 , σ0 , σ1 ) = dt (δ + iσ01 t01 )3 (δ + iσ23 t23 )3 (δ − iσ0 (t02 + a/vc ))2 (δ − iσ1 (t13 + a/vc ))2 × , (3.82) (δ − iσ0 (t03 + a/vc ))2 (δ − iσ1 (t12 + a/vc ))2 eiωt23 σ01 σ23 σ2 σ3 Z GB (σ01 , σ23 , σ2 , σ3 ) = d3 t 3 3 , (3.83) (δ + iσ01 t01 ) (δ + iσ23 t23 ) δ + iσ2 t12 δ + iσ3 t03 eiωt23 σ01 σ23 −σ2 σ3 Z GC (σ01 , σ23 , σ2 , σ3 ) = d3 t 3 3 . (3.84) (δ + iσ01 t01 ) (δ + iσ23 t23 ) δ + iσ2 t02 δ + iσ3 t13 Note that in obtaining Eq. (3.82), we have used similar onsideration to what was did between Eq. (3.31) and Eq. (3.32).† And in obtaining Eqs. (3.83),(3.84), we have used the same argument that leads to Eq. (3.62). Thus the urrent and shot noise at † Ex ept that this time the only times that ontributes are t01 , t23 ∼ 0 and t03 , t12 ∼ −a/v whi h leads to σ03 = σ02 = σ0 and σ12 = σ13 = σ1 111 QPC2 ome from three ontributions I1 = IA + IB + IC , (3.85) S1 = SA + SB + SC , (3.86) where IA needs not expli it al ulation: q|Γλ1 Γλ2 |2 X IA = (σ1 − σ0 )σ2 σ3 GA (σ0 , σ23 , σ0 , −σ0 ) 2~4 σ0 ,σ1 ,σ2 ,σ3 q|Γλ1 Γλ2 |2 (σ1 − σ0 ) (GA (σ0 , σ2 Sgn(t23 ), σ0 , −σ1 ) − GA (σ0 , −σ2 , σ0 , σ1 )) X = 2~4 σ0 ,σ1 ,σ2 = 0, (3.87) q|Γλ1 Γλ2 |2 X SA = − σ2 σ3 GA (σ0 , σ23 , σ2 , σ3 ) ~4 σ0 ,σ1 ,σ2 ,σ3 q|Γλ1 Γλ2 |2 (GA (σ01 , σ2 Sgn(t23 ), σ0 , σ1 ) − GA (σ01 , −σ3 , σ0 , σ1 )) X = − ~4 σ0 ,σ1 ,σ2 = 0, (3.88) and q|Γλ1 Γλ2 |2 X IB,C = σ0 σ2 σ3 GB,C (σ0 , σ23 , σ2 , σ3 ), (3.89) ~4 σ ,σ ,σ 0 2 3 2q|Γλ1 Γλ2 |2 (3.90) X SB,C = − σ2 σ3 GB,C (σ0 , σ23 , σ2 , σ3 ). ~4 σ0 ,σ2 ,σ3 The al ulation of IB and IC are exa tly the same as the Pfaan ase, one nds q|Γλ1 Γλ2 |2 8π 3 5 IB = −IC = × ω , (3.91) ~4 5! SB = SC = 2eIB . (3.92) 112 Thus in edge-re onstru ted Pfaan state, we nd I1 = 0 (3.93) q|Γλ1 Γλ2 |2 32π 3 5 S1 = × ω . (3.94) ~4 5! The physi s of the above result an be easily understood from Fig. 3.5. The only left moving mode is Majorana fermion whi h an not arry any harge hen e I1 = 0. On the other hand, the tunneling of Majorana fermion reates nite noise S1 6= 0 just like the Pfaan ase. Edge-re onstru ted Pfaan λ edge 3 S3 D3 ν=2 I1 , S 1 V Figure 3.5: Edge-re onstru ted Pfaan state with voltage applied on sour e 2. Only left moving Majorana fermion λ ontributes to the urrent and shot noise in drain 1. Hen e I1 = 0 and S1 6= 0. Non-equilibrated anti-Pfaan State We now onsider the anti-Pfaan state [18, 19℄. We start with the simpler non- equilibrated version of that state. It has two ontra-propagating harged modes φ1,2 113 and a Majorana fermion: Z L3 /~ =1/(4π) dtdx[−∂x φ1 (∂t + v1 ∂x )φ1 + 2∂x φ2 (∂t − v2 ∂x )φ2 + w∂x φ1 ∂x φ2 + 4πiλ(∂t − vλ ∂x )λ] (3.95) The model has many independent ele tron operators but only one most relevant tunneling operator dominates the transport. Thus, it is su ient to in lude only two tunneling onstants Γ1,2 in the model. Due to the presen e of ontra-propagating harged modes both I1 , S1 and I2 , S2 are nonzero, if V (S2) or V (S1) 6= 0 respe tively, in ontrast to all previous models. The urrent and noise I1 , S1 are proportional to |Γ1 Γ2 |2 . Hen e, the Fano fa tor F1 is independent of Γk . Disorder-dominated anti-Pfaan State The disorder-dominated anti-Pfaan state [18, 19℄ has one harged mode and three ontra-propagating Majorana modes λn with the same velo ity vλ , Z 3 (3.96) X L3 /~ = dtdx[−∂x φc (∂t + vc ∂x )φc ]/(2π) + i λn (∂t − vλ ∂x )λn ] n=1 The tunneling operators be ome (Γk~λ) exp(−2iφc )ǫ†k ηk , where the tunneling ampli- tudes (1) (2) Γk = (Γk , Γk , Γk ), (3) k = 1, 2 (3.97) are three- omponent ve tors. Just like in the non-equilibrated anti-Pfaan model, I1 , S1 6= 0. They are propor- tional to Γ4 . Only ontributions with two Γ(n) k (with the same or dierent n) and 114 two omplex onjugate Γl(m)∗ (with the same or dierent m) are allowed. Ea h power of Γ(n) 1 or Γ(n)∗ 1 must be a ompanied by the same power of Γ(n) 2 or its onjugate. Besides, the a tion is invariant with respe t to orthogonal transformations ~λ → Oˆ~λ, ˆ k and hen e so are the urrent and noise. Hen e, Γk → OΓ q|Γλ1 Γλ2 |2 8π 3 5 I I1 = 4 × ω [c1 |Γ1 Γ2 |2 + cI2 |Γ1 Γ∗2 |2 ], (3.98) ~ 5! 2 λ λ 2 2q |Γ1 Γ2 | 8π 3 5 S S1 = 4 × ω [c1 |Γ1 Γ2 |2 + cS2 |Γ1 Γ∗2 |2 ], (3.99) ~ 5! where cIl , cSl are onstants. They an be determined by the following onsiderations. If only one omponent of ea h of the ve tors Γk is nonzero then the problem redu es to the edge-re onstru ted Pfaan model. From the omparison with the results for that model Eqs. (3.93),(3.94) one nds cI1 + cI2 = 0, (3.100) cS1 + cS2 = 1. (3.101) Finally, the analysis of the same type as in the K = 8 model shows that cS1 = cI1 , (3.102) cS2 = −cI2 . (3.103) Thus |Γ1 Γ2 |2 + |Γ1 Γ∗2 |2 F1 = 2e . (3.104) |Γ1 Γ2 |2 − |Γ1 Γ∗2 |2 From this result one an see a drasti dieren e between two versions of the anti- Pfaan model. Let a gate ele trode modify the shape of edge 3. This hanges the edge 115 disorder ontribution to the a tion. In the non-equilibrated model, it in ludes terms, linear in φ1,2 , like dxdtu(x)∂x φ1 , where u(x) is random. The disorder ontribution R to the a tion of the disorder-dominated model is quadrati in Majorana fermions. We ignored su h ontributions so far sin e they an be gauged out [18, 19℄ from the a tion by a linear transformation of the elds at the expense of hanging Γ's. In the non-equilibrated model this does not ae t the Fano fa tor, independent of Γk . The Fano fa tor depends on the edge disorder and hen e the edge shape in the disorder- dominated state ‡ . 331 State The last state is the Abelian (331) state [60, 20℄ with the a tion Z L3 /~ = −1/(4π) dtdx[3∂t φ1 ∂x φ1 − 4∂t φ1 ∂x φ2 + 4∂t φ2 ∂x φ2 + wn,m ∂x φn ∂x φm ] (3.105) The two most (and equally) relevant ele tron reation operators are exp(3iφ1 − 2iφ2 ) and exp(iφ1 + 2iφ2 ). Hen e, two pairs of Γ's must be in luded in the model. Just like in the Pfaan state, I1 , S1 = 0. At the same time, one an easily see that F2 depends on the intera tion strength wn,m and thus is nonuniversal in ontrast to the Pfaan and K = 8 ases. ‡ One an dete t the non-equilibrated anti-Pfaan state from transport measurements in a bar geometry sin e the quantum Hall ondu tan e σ in that state is 7e2 /(2h). Indeed, the left ele trode inje ts harge into the φ1- hannel and two integer QHE edges on the lower edge. This ontributes 3e2 /h to σ. It also inje ts additional urrent from the left to right into the φ2- hannel on the upper edge. 116 state modes statisti s signature K=8 1R A F2 = 4e; I1 , S1 = 0 Pfaan 3/2R N F2 = 5/2e; I1 , S1 = 0 331 2R A Nonuniversal F2 ; I1 , S1 = 0 edge-re onstru ted 2R+1/2L N S1 6= 0, I1 = 0 Pfaan non-equilibrated 1R+3/2L N F1 independent of the edge shape anti-Pfaan disorder-dominated 1R+3/2L N F1 depends on the edge shape anti-Pfaan Table 3.2: Signature from proposed two-point- onta t geometry. The 2nd olumn shows the numbers of the right- and left-moving modes (R and L), Majorana fermions being ounted as 1/2 of a mode. 'A' and 'N' denote Abelian and non-Abelian statis- ti s. Ii , Si denote the urrent and shot noise measured in Drain i. Fi is the Fano fa tor Fi = Si /Ii 3.4 Con lusion In on lusion, we suggest an experiment whi h an distinguish six andidate states for the 5/2 QHE plateau. The harge-statisti s separation leads to dierent transport properties in the two-QPC geometry. The signatures of all states are summarized in Table (3.2). nnn Chapter 4 Summary In summary, we have presented two theoreti al investigations for transport in low di- mensional ele troni systems with strong intera tion. In the study of one dimensional system, we demonstrate that a nearly pure spin urrent without harge an be gen- erated by an a voltage in a quantum wire with asymmetri non-magneti potential. The spin urrent generation requires no magneti leads nor time-dependent magneti elds. In the study of non-Abelian statisti s, we present an experimental proposal that an determine the a tual state of quantum Hall liquid at 5/2 lling. In parti ular, we show that urrent and shot noise measurement in a two-point- onta t geometry distinguish K = 8, 331, Pfaan, non-equilibrated anti-Pfaan, edge-re onstru ted Pfaan and disorder-dominated anti-Pfaan states. The proposed experiment is not sensitive to the quasi-parti le u tuations in the system. 117 Appendix A Cal ulation of the Correlation Fun tions in 1D This is easier to be done using the Lagrangian language and path integral formalism. From Eq. (1.72), one has ˆ ~ ˆ [θ(x), ∂x φ(x)] = i~δ(x − x′ ). (A.1) π Thus ˆ ∂x θ(x)~/π is the anoni al momentum onjugate to ˆ , φ(x) i.e., ˆ Π(x) ˆ = ∂x θ(x)~/π (A.2) So the a tion is Z ~ ~u −1 S= dt dx [ ∂x θ(x, t)∂t φ(x, t) − [g (∂x φ(x, t))2 + g(∂x θ(x, t))2 ]] (A.3) π 2π 118 119 By going into imaginary time, the partition fun tion an be expressed in terms of fun tional integral as Z β~ Z ~ ~u −1 −S = dτ dx[ ∂x θ(x, τ )i∂τ φ(x, τ ) − [g (∂x φ(x, τ ))2 + g(∂x θ(x, τ ))2 ]] 0 π 2π (A.4) The partition fun tion is Z Z= Dφ(x, τ )Dθ(x, τ )e−S/~ (A.5) And the time-ordered orrelation fun tion an be expressed as ˆ 0)i = 1 Z ˆ τ )φ(0, hTτ φ(x, Dφ(x, τ )Dθ(x, τ )φ(x, τ )φ(0, 0)e−S/~ (A.6) Z Be ause the boson eld satises φ(x, τ + ~β) = ψ(x, τ ) and φ(x + L, τ ) = φ(x, τ ), we an expand the elds as Fourier sum 1 X1X φ(x, τ ) = φ(km , ωn )eikm +iωn τ , ωn = 2nπ/~β, km = 2mπ/L (A.7) ~β ω L k n m Z ~β/2 Z L/2 φ(km , ωn ) = dτ dxφ(x, τ )e−ikm −iωn τ (A.8) −~β/2 −L/2 120 and similarly for θ(x, τ ) and θ(km , ωn ). The a tion in terms of Fourier transform has the following quadrati form −S 1 X = [2ikm ωn φ(km , ωn )θ(−km , −ωn ) ~ 2π~βL ω ,k n m u 2 2 − km φ(km , ωn )φ(−km , −ωn ) − ugkm θ(km , ωn )θ(−km , −ωn )] g    2 −1 X  ukm /g −ikm ωn   φ(q)  = (φ(−q), θ(−q))     (A.9) 2π~βL q 2   −ikm ωn ugkm θ(q)   1X  φ(q)  =− (φ(−q), θ(−q)) M −1   2 q   θ(q) with q ≡ (km , ωn ) and   2 1  ukm /g −ikm ωn  M −1 =   (A.10) π~βL  2  −ikm ωn ugkm whi h an be solved as   π~βL  gu iωn /km  M=   (A.11) km u2 + ωn2  2  iωn /km u/g The onvenien e of the path integral formulism for this parti ular quadrati a tion is that the orrelation fun tion is dire tly related to the matrix M as a result of the Gaussian integral 1 P ∗ Dφ[q]φ∗ (q)φ(q ′)e− 2 q φ (q)M (q)φ(q) R ∗ ′ hφ (q)φ(q )i = R − 21 q φ∗ (q)M (q)φ(q) P = M −1 (q)δq,q′ , (A.12) Dφ[q]e where φ(x, τ ) is real: φ∗ (q) = φ(−q) (A.13) 121 Therefore, we immediately have    hφ(q)φ(q ′)i hφ(q)θ(q ′)i    = δq,−q′ M   hθ(q)φ(q )i hθ(q)θ(q )i ′ ′   δq,−q′ π~βL  gu iωn /km  =  2 u2 + ω 2   (A.14) km n  iωn /km u/g Using Eq. (A.14), one an obtain the orrelation fun tion in real spa e (x, τ ). For example, à !2 1 X 1 X h[φ(x, τ ) − φ(0, 0)]2 i = h φ(km , ωn )(eikm x+iωn τ − 1) i ~β ω L k n m 1 XX hφ(q)φ(q ′)i eikm x+iωn τ − 1) eikm′ x+iωn′ τ − 1) ¡ ¢¡ ¢ = (~βL)2 q q′ πgu X 2 − ei(km x+ωτ ) − e−i(km x+ωτ ) = 2 u2 + ω 2 (A.15) ~βL q km For zero temperature limit β → 0 and innite system size L → ∞, the above sum- mation an be expressed as integral ∞ Z ∞ πgu dk dω 2 − ei(km x+ωτ ) − e−i(km x+ωτ ) Z 2 h[φ(x, τ ) − φ(0, 0)] i = 2 u2 + ω 2 ~βL−∞ 2π/L −∞ 2π/~β km Z ∞ Z ∞ g 2 − ei(km x+ωuτ ) − e−i(km x+ωuτ ) = dk dω 2 + ω2 (A.16) 4π −∞ −∞ km The integral over ω an be easily done using residue theorem, and we arrive at ∞ g 2 − e−u|k||τ |+ikx − e−u|k||τ |−ikx Z 2 h[φ(x, τ ) − φ(0, 0)] i = (A.17) 4 −∞ |k| 122 Now, the integral diverges at |k| → ∞ as a result of innity bandwidth in the model. To mimi nite bandwidth, one an introdu e a onvergen e fa tor e−δ|k| with δ = 0+ . ∞ g 2 − e−u|k||τ |+ikx − e−u|k||τ |−ikx −δ|k| Z 2 h[φ(x, τ ) − φ(0, 0)] i = e 4 −∞ |k| ∞ g 2 − e−uk|τ |+ikx − e−uk|τ |−ikx −δk Z = e (A.18) 2 0 |k| Integral of the above an be evaluated using the following tri k, dene ∞ e−(δ+u|τ |±ix)k Z I± (x, τ ) ≡ dk (A.19) 0 k Sin e ∞ dI± (x, τ ) ±i Z = ∓i dk e−(δ+u|τ |±ix)k = − (A.20) dx 0 δ + u|τ | ± ix Hen e x dI(x, τ ) Z I± (x, τ ) = I± (0, τ ) + dx 0 dx · ¸ δ + u|τ | ± ix = I± (0, τ ) − ln (A.21) δ + u|τ | R∞ Now I± (0, τ ) = 0 e−(δ+u|τ |)k /k an be evaluated using the same tri k whi h gives ∞ e−δk · ¸ δ + u|τ | Z I± (0, τ ) = dk − ln (A.22) 0 k δ The rst term in above formula is diverging, but they are exa tly an eled by the 2 in the numerator in Eq. (A.18). Substitute Eq. (A.22) into Eq. (A.18), we nally have (δ + u|τ |)2 + x2 · ¸ g h[φ(x, τ ) − φ(0, 0)] i = ln2 . (A.23) 2 δ2 123 We have the orrelation fun tion for φ(x, τ ). Using essential the same method, we an obtained (δ + u|τ |)2 + x2 · ¸ 1 2 h[θ(x, τ ) − θ(0, 0)] i = ln , (A.24) 2g δ2 i hφ(x, τ )θ(0, 0)i = − Arg(Sgn(τ )δ + uτ + ix). (A.25) 2 Appendix B Te hni al details in Chapter 2 B.1 High potential barrier In this appendix we rst briey onsider the model of non-intera ting ele trons, Se . 2.2, and then a simple Hartree-type model for strongly intera ting ele trons. B.1.1 Model without intera tion We onsider non-intera ting ele trons in the presen e of the potential U (x) = u1 δ(x) + u2 δ(x − a). (B.1) The transmission oe ient an be found from elementary quantum me hani s, 1 T (E) = 2 , (B.2) (1 − 2s1 s2 sin ka)2 + (s1 + s2 + s1 s2 sin 2ka)2 where E = ~2 k 2 /2m and si = mui /k~2 . The spin and harge re ti ation urrents an be omputed from Eqs. (2.1,2.2). Fig. 2.3 shows their voltage dependen e for a 124 125 ertain hoi e of u1 , u2 , the voltage bias V and the magneti eld H . B.1.2 Model with intera tion It is di ult to nd a general analyti expression for the urrent in the regime when both the ele tron intera tion and potential barrier are strong. If all hara teristi energies, U , eV , ~2 /[ma2 ] and the typi al potential energy of an ele tron EP , are of the order of EF then one an estimate the spin and harge re ti ation urrents with dimensional analysis: Icr ∼ eEF /~, Isr ∼ EF . To obtain a qualitative pi ture of the intera tion ee ts in the ase of a high potential barrier (B.1), we restri t our dis ussion to a simple model in the spirit of the zero-mode approximation [68℄. We assume that ele trons move in a self- onsistent Hartree-type eld. In our ansatz the self- onsistent eld takes three dierent onstant values VL , VM and VR on the left of the potential barrier, between two δ -fun tion s atterers and on the right of the potential barrier. In the spirit of the Luttinger liquid model, we assume that the onstants VL , VM and VR are proportional to the average harge density in the respe tive regions, e.g., VM = dxρ(x), where γ is γ Ra a 0 the intera tion onstant. A result is shown in Fig. 2.4. We see that the voltage dependen e of the spin and harge re ti ation urrent exhibits a behavior similar to the non-intera ting ase. 126 B.2 Estimation of higher perturbative orders In this appendix we ompare ontributions to the re ti ation urrents from dierent orders of perturbation theory. We fo us on the regime when the third order ontribu- tion dominates. The appendix ontains 5 subse tions and has the following stru ture: 1) We introdu e a parametrization for the s aling dimensions (2.15). 2) We dis uss the operators most relevant in the RG sense. 3) We determine at what onditions the se ond order ontribution to the re ti ation urrent dominates. Subse tion 3 also ontains a lemma whi h is important in subse tion 4. 4) We determine at what onditions the third order ontribution to the urrent dominates. 5) We estimate the voltages and urrents at whi h the spin re ti ation urrent an ex eed the harge re ti ation urrent in realisti systems. As shown in Refs. [36℄ and [37℄, there are two ee ts leading to re ti ation in Luttinger liquids, whi h are here very shortly summarized: The density-driven and the asymmetry-driven re ti ation ee ts. The former appears at se ond order in U . It appears be ause the ba ks attering potential depends on the parti le densities in the system, whi h in turn are modied by the external voltage bias. The leading order ba ks attering urrents are of the form[53, 54℄ I bs (V ) ∼ sign(V )U 2 |V |α so that the re ti ation urrents, I r = [I bs (V )+I bs (−V )] vanish. Due to the density dependen e, however, an expansion of U to linear order in V an els the sign(V ), and we obtain a re ti ation urrent I r ∼ U 2 |V |α+1 . The asymmetry-driven re ti ation ee t appears at third order in U . It is due 127 solely to the spatial asymmetry of the potential U (x): Due to ba ks attering o U , s reening harges a umulate lose to the impurity. Those reate an ele trostati nonequilibrium ba ks attering potential W (x) for in ident parti les, leading to an ee tive potential U¯ (x) = U (x) + W (x). The spatial distribution of harges follows from the shape of U (x) and the applied voltage bias. An asymmetri U (x) leads to dierent ele trostati potentials for positive or negative bias, and hen e to re - ti ation. If we expand the urrent, I bs ∼ U¯ 2 ∼ U 2 + U W + . . . , the asymmetry appears rst at order U W . Sin e the harge density in the vi inity of the impurity is modied by the modi ation of the parti le urrent through ba ks attering, W itself is (self- onsistently) related to the ba ks attering urrent as W ∼ I bs . Hen e W ∼ U 2 , so that the asymmetri re ti ation ee t appears rst at third order in U , I r ∼ U W ∼ U 3. The main result of this paper are expressions for the urrents that result from the perturbation theory at third order in the impurity potential U . In this appendix we show that the onsidered ontribution indeed dominates the se ond and other third order expressions in the region dened by Eq. (B.8,B.9,B.10,B.14,B.16,B.17). In addition, we give the proof that higher perturbative orders N ≥ 4 annot ex eed these values in the onsidered range of the system parameters gc , gs , α and β . Unless we want to emphasize the orre t dimensions, we set EF = 1, e = 1 and ~ = 1 in this appendix. We assume that U < EF and eV < EF . An important observation is the following: In Eq. (2.16), mi = 0. P P i ni = i 128 This follows from the fa t that in the absen e of ba ks attering the numbers of right- and left-movers with dierent spin orientations onserve. B.2.1 Parametrization of s aling dimensions A ording to Eq. (2.15) z(n, m) = n2 A + m2 B + 2nmC − 1, (B.3) where A = [gc (1 + α)2 + gs (1 + β)2 ] (B.4) B = [gc (1 − α)2 + gs (1 − β)2 ] (B.5) C = [gc (1 − α2 ) − gs (1 − β 2 )] (B.6) Sin e gc and gs are positive, A and B are also positive. C an have any sign. It satises the inequality √ |C| < AB. (B.7) √ Indeed, AB − C 2 = 4gc gs (1 − αβ)2 > 0. Any values of A, B > 0 and − AB < C < √ √ √ √ √ AB are possible. For example, one an set α = β = ( A − B)/( A + B), √ √ √ √ √ √ gc = ( A + B)2 [1 + C/ AB]/8, gs = ( A + B)2 [1 − C/ AB]/8. B.2.2 Most relevant operators Depending on the values of A, B and C many dierent possibilities for relative impor- tan e of dierent ba ks attering operators U (m, n) exist. In the paper we fo us on the 129 situation when the most relevant operator is U (1, 0), the se ond most relevant opera- tor is U (0, −1) and the third most relevant operator is U (−1, 1) ( ertainly, the s aling dimensions of the operators U (n, m) and U (−n, −m) are always the same). We will also assume that the operator U (1, 0) is relevant in the RG sense, i.e., z(1, 0) < 0. The analysis of the situation in whi h U (0, −1) is the most relevant operator, U (1, 0) is the se ond most relevant and U (−1, 1) is the third most relevant follows exa tly the same lines. Similarly, little hanges if U (1, 1) is the third most relevant operator. The s aling dimensions of the three aforementioned operators are A−1, B −1 and A+B −2C −1. The following inequality must be satised in order for these operators to be most relevant ba ks attering operators: A − 1 < B − 1 < A + B − 2C − 1 < [all other scaling dimensions ]. Hen e B > A > 2C. (B.8) Sin e z(1, 0) < 0, A < 1. (B.9) When are all other operators less relevant? We must onsider three lasses of opera- tors: 1) U (1, 1); 2) U (n, 0) and U (0, n) with |n| > 1; 3) all other operators. 1) Sin e z(1, 1) = A + B + 2C − 1, one nds C > 0. (B.10) 2) z(0, n) = Bn2 − 1 > z(n, 0) = An2 − 1 ≥ 4A − 1 > A + B − 2C − 1. Thus, 3A + 2C > B. (B.11) 130 3) z(n, m) − (A + B − 2C − 1) = An2 + Bm2 + 2Cnm − (A + B − 2C) ≥ An2 + Bm2 − C(n2 + m2 ) − A − B + 2C = (A − C)(n2 − 1) + (B − C)(m2 − 1) > 0 sin e B − C > A − C > 0 in a ordan e with Eq. (B.8), |n|, |m| ≥ 1 and either |n| or |m| ex eeds 1. Thus, ase 3) gives no new restri tion on A, B and C . B.2.3 Se ond order ontribution to the urrent When is the se ond order ontribution to the re ti ation urrent dominant? Any operator U (n, m) an be represented as U˜ (n, m) + V U1 (n, m) + . . . , where U1 ∼ U˜ /EF . Any se ond order ontribution to the urrent whi h ontains U˜ only is an odd fun tion of the voltage bias. Indeed, any su h ontribution is proportional to U˜ (n, m)U˜ ∗ (n, m) = U˜ (n, m)U˜ (−n, −m). The transformation U˜ ↔ U˜ ∗ , V → −V hanges the sign of the urrent. At the same time, the transformation U˜ ↔ U˜ ∗ annot hange the se ond order urrent at all. Hen e, it is odd in the voltage. The same argument applies to any perturbative ontribution whi h ontains only U˜ , if every operator U˜ (n, m) enters in the same power as U˜ ∗ (n, m). In parti ular, if only two operators U˜ (n, 0) and U˜ (0, m) and their onjugate enter then the resulting urrent ontribution is odd. Thus, all se ond order ontributions to the re ti ation urrent must ontain U1 . As is lear from Eq. (2.19), the leading se ond order ontribution is proportional to the square of the most relevant operator, |U (1, 0)|2 . It s ales as I2 ∼ V U 2 V 2z(1,0)+1 ∼ U 2 V 2A . (B.12) 131 In this subse tion we dis uss at what onditions this ontribution dominates for all V > V ∗ (see Eq. (2.18)). Sin e U (1, 0) is the most relevant operator, its renormalized amplitude U (1, 0; E = V ) ex eeds the renormalized amplitude of all other operators on every energy s ale. At the same time it remains lower than 1 (i.e. EF ) for V > V ∗ . This ertainly means that the renormalized amplitudes are smaller than 1 for all other operators too. Hen e, the produ t of any operators is smaller than the produ t of any two of them and that produ t annot ex eed U 2 (1, 0; E). This guarantees that the se ond order urrent (B.12) ex eeds any se ond or higher order ontribution whi h ontains any operator V U1 (n, m). Thus, we have to ompare I2 with higher order ontributions to the re ti ation urrent whi h ontain U˜ only. Every su h ontribution is at least third order and ontains at least one operator less relevant than U (0, −1) [if it ontains U (±1, 0) and U (0, ±1) only then it must ontain V U1 as dis ussed above℄. Thus, any re ti ation urrent ontribution with U˜ only annot ex eed U 3 V 2z(1,0)+z(1,−1)+1 . Comparison with (B.12) at V ∼ V ∗ leads to the ondition B > 2C + 1. (B.13) B.2.4 Third order ontribution to the urrent The most interesting question is dierent. When does the third order ontribution dominate the re ti ation urrent? We will fo us on the third order ontribution I3 proportional to U (1, 0)U (0, −1)U (−1, 1) at V ∼ V ∗ . Note that this ontribution is 132 proportional to ∼ V 2(A+B−C−1) and hen e s ales as a negative power of the voltage, if A + B < C + 1. (B.14) At V ∼ V ∗ , U ∼ V 1−A . Thus, I3 (V = V ∗ ) ∼ V 2B−A−2C+1 . (B.15) We need to ompare I3 , Eq. (B.15), with the following types of ontributions: 1) those ontaining at least three dierent operators [we treat a pair of U (n, m) and U (−n, −m) = U ∗ (n, m) as one operator℄; 2) those ontaining only one type of oper- ators; 3) those ontaining two types of operators. Cases 1) and 2) are easy. 1) I3 ontains the produ t of the three most relevant operators and hen e always ex eeds the produ t of any other three dierent operators at any energy s ale EF > V > V ∗ . Any ontribution with three dierent operators is the produ t of three dierent operators times perhaps some other ombination of operators whi h annot ex eed 1 at EF > V > V ∗ . Hen e it is smaller than I3 . 2) Any ontribution to the re ti ation urrent with only one type of operators must ontain V U1 . As dis ussed in the previous subse tion, the leading ontribution of su h type emerges in the se ond order. It is I2 , Eq. (B.12). At V ∼ V ∗ , I2 (V = V ∗ ) ∼ V 2 . The ondition I2 (V ∗ ) < I3 (V ∗ ) means that 2B < A + 2C + 1. (B.16) 133 3) We have to onsider three possibilities: 3.1) one operator has the form U (n, 0) and the se ond operator has the form U (k, m), m 6= 0 or one operator has the form U (0, m) and the other one has the form U (n, k), n 6= 0; 3.2) both operators have the form U (ni , 0) or both operators have the form U (0, mi ); 3.3) both operators have the form U (ni , mi ) with ni , mi 6= 0. 3.1). Let us assume that one operator has the form U (n, 0) and the se ond one is U (k, m). The ase of the operators U (0, m) and U (n, k) an be onsidered in exa tly the same way. We must have the same number of operators U (k, m) and U (−k, −m) in the perturbative ontribution sin e the sum of the se ond indexes ±m must be 0. [The other ases are overed in 3.3).℄ From the analysis of the sum of the rst indexes one on ludes that the operators U (n, 0) and U (−n, 0) also enter in the same power. It follows from the previous subse tion that the perturbative ontribution must ontain at least one U1 operator and hen e is smaller than I2 . Hen e, it is also smaller than I3 . 3.2) We will fo us on the ase when both operators have the form U (ni , 0). The ase when both operators have the form U (0, mi ) is very similar and does not lead to a new restri tion on A, B and C . The s aling dimensions of the operators U (n, 0) are An2 − 1. Operators with greater n are less relevant. Sin e the on- tribution ontains two dierent operators, it must be at least third order [we treat U (n, 0) and U (−n, 0) as the same operator!℄. At least one of the two operators must have |ni | > 1 (otherwise all operators are U (±1, 0)). Thus, the ontribution 134 annot ex eed U 2 (1, 0; E = V )U (2, 0; E = V ) ∼ U 3 V 6A−2 . The omparison with I3 ∼ U 3 V 2A+2B−2C−2 at EF > V > V ∗ , yields: B < 2A + C. (B.17) Note that the above ondition is stronger than (B.11). 3.3) This ase is easy: the ontribution must be at least third order again. Both operators U (ni , mi ) are less relevant than U (1, 0) and U (0, −1) and no more relevant than U (1, −1). Thus, the ontribution is automati ally smaller than I3 at any energy s ale EF > V > V ∗ . We now have a full set of onditions at whi h the third order ontribution domi- nates at V ∼ V ∗ and the spin re ti ation urrent s ales as a negative power of the voltage. These are equations (B.8,B.9,B.10,B.14,B.16,B.17). The above analysis shows that I3 ex eeds any ontribution to the spin re ti ation urrent whi h does not ontain V U1 in the whole region EF > V > V ∗ . I2 dominates the remaining ontributions for any V > V ∗ . The ontributions be ome equal, I2 = I3 , at V = V ∗∗ = U 1/[2+2C−2B] . In the interval of voltages V ∗∗ > V > V ∗ , the spin re ti ation urrent is dominated by I3 . At V > V ∗∗ , the spin and harge re ti ation urrents are dominated by I2 . B.2.5 Numeri al estimates In order to get a feeling about the magnitude of the ee t, let us onsider a parti ular hoi e of parameters A = B = 7/12, C = 7/24, eV ∼ 0.01EF , eV ∗ ∼ 10−4 EF . 135 For su h A, B and C the s aling dimensions of the three most relevant operators are the same. The inequalities (B.9,B.10,B.14,B.16,B.17) are satised. The equality A = B = 2C orresponds to a limiting ase of (B.8). One nds that U ∼ 0.01EF and eV ∗∗ ∼ 0.1EF . Repeating the arguments of the previous se tion one an estimate the leading orre tion to I3 as δI ∼ (eV /EF )7/12 I3 ≪ I3 . The spin re ti ation urrent is the dieren e of two opposite ele tri urrents of the spin-up and -down ele trons times ~/[2e]. Even if EF is as low as ∼ 0.1 meV, this still orresponds to the voltage V of the order of mi rovolts and the urrents∗ (2.25) of spin-up and -down ele trons of the order of pi oamperes, i.e. within the ranges probed in experiments with semi ondu tor heterostru tures. Certainly, the urrent in reases, if EF or V ∗ is in reased. B.3 Expli it evaluation of the third order urrents The harge or spin urrents in the third order in the potentials U are evaluated from the following perturbative expression: (−i)3 X dt1 dt2 Z bs(3) Ic,s (V )= (n↑ ± n↓ ) 2! CK ~2 × hTc Uˆ (n↑ , n↓ ; 0)Uˆ (m↑ , m↓ ; t1 )Uˆ (l↑ , l↓ ; t2 )i, (B.18) ∗ Note a large numeri al fa tor in Eq. (2.25). 136 where the sum runs over indi es satisfying nσ + mσ + lσ = 0 for σ =↑, ↓, CK is the Keldysh ontour −∞ → 0 → ∞, Tc the time order on CK , and we omitted a onstant prefa tor. The operators Uˆ are given by Uˆ (n↑ , n↓ ; t) = U (n↑ , n↓ )ei(n↑ +n↓ )teV /~ein↑ φ↑ (t)+in↓ φ↓ (t) . (B.19) The most relevant expressions are those arising from the ombinations U (1, 0)U (0, −1)U (−1, 1) and U (−1, 0)U (0, 1)U (1, −1) (see Appendix B.2). The third order ontributions to the urrent ontain orrelation fun tions of the form P (t1 , t2 , t3 ) = hTc e±i[φ↑ (t1 )−φ↓ (t2 )−φ↑ (t3 )+φ↓ (t3 )] i (B.20) We evaluate the orrelation fun tions within the quadrati model des ribed by Eq. (2.7) and use the relations (2.8) and hφ˜c,s (t1 )φ˜c,s (t1 )i = −2 ln(i(t1 − t2 )/τc + δ), with an innitesimal δ > 0 and τc ∼ ~/EF the ultraviolet uto time. This leads to ¡ ¢2C−2A P (t1 , t2 , t3 ) = iTc (t1 − t3 )/τc + δ ¢2C−2B ¡ ¢−2C , (B.21) ¡ × iTc (t2 − t3 )/τc + δ iTc (t1 − t2 )/τc + δ where Tc (ti − tj ) = (ti − tj ), if time ti stays later than tj on the Keldysh ontour, and otherwise Tc (ti − tj ) = (tj − ti ). The expression (B.21) is independent of the ± signs in Eq. (B.20). The spin and harge urrent ontributions, proportional to U (1, 0)U (0, −1)U (−1, 1), are omplex onjugate to those proportional to U (−1, 0)U (0, 1)U (1, −1) = U ∗ (1, 0)U ∗ (0, −1)U ∗ (− Thus, it is su ient to al ulate only the ontributions of the rst type. In the ase 137 of the harge urrent, their al ulation redu es to the al ulation of the following two integrals over the Keldysh ontour: Z dt1 dt3 P (t1 , 0, t3 ) exp(ieV t1 /~) (B.22) and Z dt2 dt3 P (0, t2 , t3 ) exp(−ieV t2 /~). (B.23) One of the times t1 and t2 is zero sin e the urrent operator is taken at t = 0 in Eq. (2.13). The two integrals an be evaluated in exa tly the same way. We will onsider only the rst integral. We nd 8 integration regions. They orrespond to 2 × 2 = 4 possibilities for the bran hes of the Keldysh ontour on whi h t1 and t3 are lo ated and two possible relations |t1 | > |t3 | or |t3 | > |t1 |. In all 8 ases, we rst integrate over t3 . The integral redu es to the Euler B -fun tion. Then we integrate over t1 . This yields a Γ-fun tion. Finally, we obtain Eq. (2.23). The spin urrent ontains three ontributions proportional to U (1, 0)U (0, −1)U (−1, 1). Two of them redu e to the integrals (B.22) and (B.23). The third ontribution is pro- portional to Z dt1 dt2 P (t1 , t2 , 0) exp(ieV [t1 − t2 ]/~). (B.24) Again we have eight integration regions determined by the hoi e of the bran hes of the Keldysh ontour and the relations |t1 | > |t2 | and |t2 | > |t1 |. In ea h region it is onvenient to introdu e new integration variables: τ = |t1 − t2 | and t = min(t1 , t2 ). The integration over t redu es to a B -fun tion. The integration over τ produ es an additional Γ-fun tion fa tor. Finally, one obtains Eq. (2.24). Bibliography [1℄ A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinskii, Methods of Quantum Field Theory in Statisti al Physi s (Dover, New York, 1963). [2℄ F. D. M. Haldane. `Luttinger liquid theory' of one-dimensional quantum uids. I. Properties of the Luttinger model and their extension to the general 1D inter- a ting spinless Fermi gas, J. Phys. C. 14, 2585 (1981). [3℄ K. v. Klitzing, G. Dorda, and M. Pepper. New method for high-a ura y deter- mination of the ne-stru ture onstant based on quantized Hall resistan e, Phys. Rev. Lett. 45, 494 (1980). [4℄ D. C. Tsui, H. L. Stormer, and A. C. Gossard. Two-dimensional magnetotrans- port in the extreme quantum limit, Phys. Rev. Lett. 48, 1559 (1982). [5℄ F. Wil zek. Quantum me hani s of fra tional-spin parti les, Phys. Rev. Lett. 49, 957 (1982). [6℄ X. Wen, Quantum Field Theory of Many-body Systems: From the Origin of Sound to an Origin of Light and Ele trons (Oxford University Press, 2004). 138 139 [7℄ B. I. Halperin. Quantized Hall ondu tan e, urrent- arrying edge states, and the existen e of extended states in a two-dimensional disordered potential, Phys. Rev. B 25, 2185 (1982). [8℄ X.-G. WEN. Theory of the edge states in fra tional quantum Hall ee t, Int. J. Mod. Phys. B 6, 1711 (1992). [9℄ T. Giamar hi, Quantum Physi s in One Dimension (Oxford University, Oxford, 2004). [10℄ R. B. Laughlin. Anomalous quantum Hall ee t: An in ompressible quantum uid with fra tionally harged ex itations, Phys. Rev. Lett. 50, 1395 (1983). [11℄ R. de Pi iotto, , M. Reznikov, M. Heiblum, V. Umansky, G. Bunin, and D. Ma- halu. Dire t observation of a fra tional harge, Nature 389, 162 (1997). [12℄ L. Saminadayar, D. C. Glattli, Y. Jin, and B. Etienne. Observation of the e/3 fra tionally harged Laughlin quasiparti le, Phys. Rev. Lett. 79, 2526 (1997). [13℄ V. J. Goldman and B. Su. Resonant tunneling in the quantum Hall regime: Mea- surement of fra tional harge, S ien e 267, 1010 (1995). [14℄ C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. D. Sarma. Non-Abelian anyons and topologi al quantum omputation, Rev. Mod. Phys. 80, 1083 (2008). [15℄ S. M. Girvin, The quantum hall ee t: Novel ex itations and broken symmetries, arxiv.org: ond-mat/9907002 (2000). 140 [16℄ J. M. Caillol, D. Levesque, J. J. Weis, and J. P. Hansen. A Monte Carlo study of the lassi al two-dimensional one- omponent plasma, J. Stat. Phys. 28, 325 (1982). [17℄ G. Moore and N. Read. Nonabelions in the fra tional quantum Hall ee t, Nu l. Phys. B 360, 362 (1991). [18℄ M. Levin, B. I. Halperin, and B. Rosenow. Parti le-Hole symmetry and the Pfaf- an state, Phys. Rev. Lett. 99, 236806 (2007). [19℄ S.-S. Lee, S. Ryu, C. Nayak, and M. P. A. Fisher. Parti le-Hole symmetry and the ν = 5/2 quantum Hall state, Phys. Rev. Lett. (2007). [20℄ B. J. Overbos h and X.-G. Wen, Phase transitions on the edge of the ν = 5/2 Pfaan and anti-Pfaan quantum Hall state, arXiv.org:0804.2087 (2008). [21℄ C. de C. Chamon, D. E. Freed, S. A. Kivelson, S. L. Sondhi, and X. G. Wen. Two point- onta t interferometer for quantum Hall systems , Phys. Rev. B 55, 2331 (1997). [22℄ E. Fradkin, C. Nayak, A. Tsvelik, and F. Wil zek. A Chern-Simons ee tive eld theory for the Pfaan quantum Hall state, Nu l. Phys. B 516, 704 (1998). [23℄ S. Das Sarma, M. Freedman, and C. Nayak. Topologi ally prote ted qubits from a possible non-Abelian fra tional quantum Hall state, Phys. Rev. Lett. 94, 166802 (2005). 141 [24℄ A. Stern and B. I. Halperin. Proposed experiments to probe the non-Abelian ν= 5/2 quantum Hall state, Phys. Rev. Lett. 96, 016802 (2006). [25℄ P. Bonderson, A. Kitaev, and K. Shtengel. Dete ting non-Abelian statisti s in the ν = 5/2 fra tional quantum Hall state, Phys. Rev. Lett. 96, 016803 (2006). [26℄ Y. Ji, Y. Chung, D. Sprinzak, M. Heiblum, D. Mahalu, and H. Shtrikman. An ele troni Ma h-Zehnder interferometer, Nature 422, 415 (2003). [27℄ K. T. Law, D. E. Feldman, and Y. Gefen. Ele troni Ma h-Zehnder interferom- eter as a tool to probe fra tional statisti s, Phys. Rev. B 74, 045319 (2006). [28℄ D. E. Feldman and A. Kitaev. Dete ting non-Abelian statisti s with an ele troni Ma h-Zehnder interferometer, Phys. Rev. Lett. 97, 186803 (2006). [29℄ D. E. Feldman, Y. Gefen, A. Kitaev, K. T. Law, and A. Stern. Shot noise in an anyoni Ma h-Zehnder interferometer, Phys. Rev. B 76, 085333 (2007). [30℄ K. T. Law. Probing non-Abelian statisti s in nu = 12/5 quantum Hall state, Phys. Rev. B 77, 205310 (2008). [31℄ T. Christen and M. Buttiker. Gauge-invariant nonlinear ele tri transport in mesos opi ondu tors, Europhysi s Lett. 35, 523 (1996). [32℄ P. Reimann, M. Grifoni, and P. Hänggi. Quantum rat hets, Phys. Rev. Lett. 79, 10 (1997). 142 [33℄ J. Lehmann, S. Kohler, P. Hänggi, and A. Nitzan. Mole ular wires a ting as oherent quantum rat hets, Phys. Rev. Lett. 88, 228305 (2002). [34℄ D. Sán hez and M. Büttiker. Magneti -eld asymmetry of nonlinear mesos opi transport, Phys. Rev. Lett. 93, 106802 (2004). [35℄ B. Spivak and A. Zyuzin. Signature of the ele tron-ele tron intera tion in the magneti -eld dependen e of nonlinear I − V hara teristi s in mesos opi sys- tems, Phys. Rev. Lett. 93, 226801 (2004). [36℄ D. E. Feldman, S. S heidl, and V. M. Vinokur. Re ti ation in Luttinger Liquids, Phys. Rev. Lett. 94, 186809 (2005). [37℄ B. Braune ker, D. E. Feldman, and J. B. Marston. Re ti ation in one- dimensional ele troni systems, Phys. Rev. B 72, 125311 (2005). [38℄ V. Krsti , S. Roth, M. Burghard, K. Kern, and G. L. J. A. Rikken. Magneto- hiral anisotropy in harge transport through single-walled arbon nanotubes, J. Chem. Phys. 117, 11315 (2002). [39℄ J. Wei, M. Shimogawa, Z. Wang, I. Radu, R. Dormaier, and D. H. Cobden. Magneti -eld asymmetry of nonlinear transport in arbon nanotubes, Phys. Rev. Lett. 95, 256601 (2005). [40℄ A. D. Martino, R. Egger, and A. M. Tsvelik. Nonlinear magnetotransport in intera ting hiral nanotubes, Phys. Rev. Lett. 97, 076402 (2006). 143 [41℄ P. Sharma and C. Chamon. Quantum pump for spin and harge transport in a Luttinger Liquid, Phys. Rev. Lett. 87, 096401 (2001). [42℄ P. Sharma. How to reate a spin urrent, S ien e 307, 531 (2005). [43℄ D. A. Abanin, P. A. Lee, and L. S. Levitov. Spin-Filtered edge states and quantum Hall ee t in graphene, Phys. Rev. Lett. 96, 176803 (2006). [44℄ M. Pustilnik, E. G. Mish henko, and O. A. Starykh. Generation of spin Current by Coulomb drag, Phys. Rev. Lett. 97, 246803 (2006). [45℄ D. L. Maslov and M. Stone. Landauer ondu tan e of Luttinger liquids with leads, Phys. Rev. B 52, R5539 (1995). [46℄ V. V. Ponomarenko. Renormalization of the one-dimensional ondu tan e in the Luttinger-liquid model, Phys. Rev. B 52, R8666 (1995). [47℄ I. Sa and H. J. S hulz. Transport in an inhomogeneous intera ting one- dimensional system, Phys. Rev. B 52, R17040 (1995). [48℄ T. Hikihara, A. Furusaki, and K. A. Matveev. Renormalization of impurity s at- tering in one-dimensional intera ting ele tron systems in magneti eld, Phys. Rev. B 72, 035301 (2005). [49℄ P. Bruno and J. Wunderli h. Resonant tunneling spin valve: A novel magneto- ele troni s devi e, J. Appl. Phys. 84, 978 (1998). 144 [50℄ A. Slobodskyy, C. Gould, T. Slobodskyy, C. R. Be ker, G. S hmidt, and L. W. Molenkamp. Voltage- ontrolled spin sele tion in a Magneti Resonant Tunneling diode, Phys. Rev. Lett. 90, 246601 (2003). [51℄ L. D. Landau and E. M. Lifshitz, Quantum Me hani s (Butterworth-Heinemann, Oxford, 1977). [52℄ M. S heid, M. Wimmer, D. Ber ioux, and K. Ri hter. Zeeman rat hets for bal- listi spin urrents, Phys. Status Solidi C 3, 4235 (2006). [53℄ C. L. Kane and M. P. A. Fisher. Transmission through barriers and resonant tunneling in an intera ting one-dimensional ele tron gas, Phys. Rev. B 46, 15233 (1992). [54℄ A. Furusaki and N. Nagaosa. Single-barrier problem and Anderson lo alization in a one-dimensional intera ting ele tron system, Phys. Rev. B 47, 4631 (1993). [55℄ F. Dol ini, B. Trauzettel, I. Sa, and H. Grabert. Transport properties of single- hannel quantum wires with an impurity: Inuen e of nite length and temper- ature on average urrent and noise, Phys. Rev. B 71, 165309 (2005). [56℄ D. E. Feldman and Y. Gefen. Ba ks attering o a point impurity: Current en- han ement and ondu tan e greater than e2/h per hannel, Phys. Rev. B 67, 115337 (2003). [57℄ L. V. Keldysh. Sov. Phys. JETP 20, 1018 (1965). 145 [58℄ J. Rammer and H. Smith. Quantum eld-theoreti al methods in transport theory of metals, Rev. Mod. Phys. 58, 323 (1986). [59℄ F. E. Camino, W. Zhou, and V. J. Goldman. Realization of a Laughlin quasi- parti le interferometer: Observation of fra tional statisti s, Phys. Rev. B 72, 075342 (2005). [60℄ X. G. Wen and A. Zee. Classi ation of Abelian quantum Hall states and matrix formulation of topologi al uids, Phys. Rev. B 46, 2290 (1992). [61℄ C. L. Kane. Telegraph noise and fra tional statisti s in the quantum Hall ee t, Phys. Rev. Lett. 90, 226802 (2003). [62℄ R. L. Willett, L. N. Pfeier, and K. W. West. Measurement of lling fa tor 5/2 quasiparti le interferen e with observation of harge e/4 and e/2 period os illa- tions, PNAS 106, 8853 (2009). [63℄ M. Dolev, M. Heiblum, V. Umansky, A. Stern, and D. Mahalu. Observation of a quarter of an ele tron harge at the ν = 5/2 quantum Hall state, Nature 452, 829 (2008). [64℄ I. P. Radu, J. B. Miller, C. M. Mar us, M. A. Kastner, L. N. Pfeier, and K. W. West. Quasi-parti le properties from tunneling in the formula fra tional quantum Hall state, S ien e 320, 899 (2008). 146 [65℄ G. A. Fiete, W. Bishara, and C. Nayak. Multi hannel Kondo models in non- Abelian quantum Hall droplets, Phys. Rev. Lett. 101, 176801 (2008). [66℄ A. M. Chang. Chiral Luttinger liquids at the fra tional quantum Hall edge, Rev. Mod. Phys. 75, 1449 (2003). [67℄ E. Comforti, Y. C. Chung, M. Heiblum, and V. Umansky. Multiple s attering of fra tionally harged quasiparti les, Phys. Rev. Lett. 89, 066803 (2002). [68℄ I. L. Aleiner, P. W. Brouwer, and L. I. Glazman. Quantum ee ts in Coulomb blo kade, Phys. Rep. 358, 309 (2002).