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Comptes Rendus Physique

Edge states tunneling in the fractional quantum Hall effect: integrability and transport

Hubert Saleur

Published Jan 1, 2002• 2 citations
Publisher

Abstract

This is a short review of nonperturbative techniques that have been used in the past 5 years to study transport out of equilibrium in low dimensional, strongly interacting systems of condensed matter physics. These techniques include massless factorized scattering, the generalization of the Landauer Büttiker approach to integrable quaisparticles, and duality. The case of tunneling between edges in the fractional quantum Hall effect is discussed in details.

DOI
10.1016/s1631-0705(02)01366-x

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This is a short review of nonperturbative techniques that have been used in the past 5 years to study transport out of equilibrium in low dimensional, strongly interacting systems of condensed matter physics. These techniques include massless factorized scattering, the generalization of the Landauer Büttiker approach to integrable quaisparticles, and duality. The case of tunneling between edges in the fractional quantum Hall effect is discussed in details.
This thesis deals with the study of transport properties of integer and fractional QH edge states and it is based on the work I performed during my Ph.D. studies. The focus of this thesis is on Luttinger liquids far from equilibrium and their relaxation dynamics. Since Boltzmann, a fundamental aspect of statistical mechanics has been the understanding of the emergence of an equilibrium state. Interactions play a crucial role in the thermalization process that drives a system through states described by the Gibbs equilibrium ensemble. Therefore, it seems counterintuitive that a strongly interacting system, such as the Luttinger liquid, should not present any relaxation dynamics. This peculiar fact is due to the integrability of the Luttinger model, i.e. the existence of an infinite number of conserved quantities that precludes the equilibration process. However, in the past few years it has become clear that integrable systems can present some kind of relaxation, even though not towards the Gibbs equilibrium ensemble. Remarkably, the necessity of correctly taking into account some particular non-equilibrium configurations, also revealed the necessity of modifying bosonization, a technique widely used to study strongly interacting systems in one dimension. In this work we focus on three different cases: • Relaxation of high energy electrons injected in a ν = 1/3 chiral Luttinger liquid and in a standard Luttinger liquid. • Heating and the emergence of effective temperatures in a Quantum Hall system at fractional filling fraction ν = 2/3 partitioned by a Quantum Point Contact. • Effect of relaxation on shot-noise measurement of the quasi-particle charge in a ν = 2 QH state.
We have recently calculated exact nonequilibrium quantum transport properties through a point contact in a Luttinger liquid. Using a particular quasiparticle basis of the Hilbert space dictated by integrability, we here compute explicitly the exact I(V) characteristic and conductance out of equilibrium as a function of driving voltage V and temperature T. These are described by universal scaling functions of two variables, the scaled point-contact interaction strength, and V/T. The differential-conductance curve as a function of the interaction strength broadens significantly as V/T is increased, and develops a pronounced maximum at a (universal) critical value (eV/kB{\mathit{k}}_{\mathit{B}}kB​T)=7.188 68. . .. In addition, we derive an exact duality between strong and weak backscattering. The theory presented here has recently been realized experimentally in resonant tunneling-transport experiments between edge states in fractional quantum Hall effect devices. In this context the exact duality is between electron tunneling and Laughlin-quasiparticle tunneling.
In fractional quantum Hall systems, quasiparticles of fractional charge can tunnel between the edges at a quantum point contact. Such tunneling (or backscattering) processes contribute to charge transport and provide information on both the charge and statistics of the quasiparticles involved. Here, we study quasiparticle tunneling in the Moore-Read state, in which quasiparticles of charges e/4e/4e/4 (non-Abelian) and e/2e/2e/2 (Abelian) may coexist and both contribute to edge transport. On a disk geometry, we calculate the matrix elements for e/2e/2e/2 and e/4e/4e/4 quasiholes to tunnel through the bulk of the Moore-Read state, in an attempt to understand their relative importance. We find that the tunneling amplitude for charge e/2e/2e/2 quasihole is exponentially smaller than that for charge e/4e/4e/4 quasihole, and the ratio between them can be (partially) attributed to their charge difference. We find that including long-range Coulomb interaction only has a weak effect on the ratio. We discuss briefly the relevance of these results to recent tunneling and interferometry experiments at filling factor $\ensuremath{\nu}=5/2$.
A general description of weak and strong tunneling fixed points is developed in the chiral-Luttinger-liquid model of quantum Hall edge states. Tunneling fixed points are a subset of ``termination'' fixed points, which describe boundary conditions on a multicomponent edge. The requirement of unitary time evolution for bounded edges gives a nontrivial consistency condition for possible low-energy boundary conditions. The effects of interactions and random hopping on fixed points are studied through a perturbative renormalization-group (RG) approach which generalizes the Giamarchi-Schulz RG for disordered Luttinger liquids to broken left-right symmetry and multiple modes. We apply our approach to a number of examples, such as tunneling between a quantum Hall edge and a superconductor and tunneling between two quantum Hall edges in the presence of interactions. Interactions are shown to induce a continuous renormalization of effective tunneling charge for the integrable case of tunneling between two Laughlin states.
In contemporary physics, especially in condensed-matter physics, fermionic topological order and its protected edge modes is one of the most important objects. In this work, we propose a systematic construction of the cylinder partition corresponding to the fermionic fractional quantum Hall effect and a general mechanism for obtaining the candidates of the protected edge modes. In our construction, when the underlying conformal field theory has Z2{Z}_{2}Z2​ duality defects corresponding to the fermionic Z2{Z}_{2}Z2​ electric particle, we show that the fractional quantum Hall partition function has a fermionic T duality. This duality is analogous to (hopefully the same as) the dualities in the dual resonance models, typically known as supersymmetry, and it provides a renormalization-group (RG) theoretical understanding of the topological phases. We also introduce a modern understanding of bulk topological degeneracies and topological entanglement entropy. This understanding is based on the traditional tunnel problem and the recent conjecture of correspondence between bulk RG flow and boundary conformal field theory. Our formalism provides an intuitive and general understanding of the modern physics of topologically ordered systems in the traditional language of RG and fermionization, and it may serve as a complement to more mathematical physical frameworks such as fermionic category theories.
The quantum Hall effect is one of the most challenging problems in condensed matter physics in which many rare physical phenomena like fractional charges and generally the fractional statistics are involved. In the integer quantum Hall effect the role of the disorder is more than others. The important point in describing these systems is that their bulk states should be gapful and in this sense the system should be incompressible. For describing the electron condensates in an arbitrary filling fraction with the odd denominator one needs the topological U(1) Chern-Simons effective field theory. Our motivation for using these field theories is the emergence of quantized conductivity. This effective field theory also has the ability of constructing the hierarchical fractional condensates with various filling factors. The excitations of these theories have fractional statistics which are investigated in this review. The non-Abelian Chern-Simons for quantum Hall systems will be also investigated. The interesting point is that the edge states of these systems are gapless and their excitations are equipped with the Kac-Moody algebra. The connection to the conformal field theories can be established using this point. The investigation of this correspondence is one of the aims of the present thesis. The connection to the logarithmic conformal field theories (e.g. for the Haldane-Rezaei wave function) is also reported in some works which will be addressed in this thesis.