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arXiv:cond-mat/9402108 · Physical Review Letters

Randomness at the edge: Theory of quantum Hall transport at filling ν=2/3

C. L. Kane, Matthew P. A. Fisher, Joseph Polchinski

Published Jun 27, 1994• 316 citations
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Abstract

Current Luttinger liquid edge state theories for filling \ensuremath{\nu}=2/3 predict a nonuniversal Hall conductance, in disagreement with experiment. Upon inclusion of random edge tunneling we find a phase transition into a new disorder-dominated edge phase. An exact solution of the random model in this phase gives a quantized Hall conductance of 2/3 and a neutral mode propagating upstream. The presence of the neutral mode changes the predicted temperature dependence for tunneling through a point contact from ${\mathit{T}}^{2/\ensuremath{\nu}\mathrm{\ensuremath{-}}2}$ to ${\mathit{T}}^{2}$.

DOI
10.1103/physrevlett.72.4129

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Current Luttinger liquid edge state theories for filling \ensuremath{\nu}=2/3 predict a nonuniversal Hall conductance, in disagreement with experiment. Upon inclusion of random edge tunneling we find a phase transition into a new disorder-dominated edge phase. An exact solution of the random model in this phase gives a quantized Hall conductance of 2/3 and a neutral mode propagating upstream. The presence of the neutral mode changes the predicted temperature dependence for tunneling through a point contact from T2/\ensuremathν\ensuremath−2{\mathit{T}}^{2/\ensuremath{\nu}\mathrm{\ensuremath{-}}2}T2/\ensuremathν\ensuremath−2 to T2{\mathit{T}}^{2}T2.
The conductance for tunneling through an impurity in a Luttinger liquid is described by a universal scaling function. We compute this scaling function exactly, by using the thermodynamic Bethe ansatz and a kinetic (Boltzmann) equation. This model has been proposed to describe resonant tunneling through a point contact between two $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/3$ quantum Hall edges. Recent experiments on quantum Hall devices agree well with our exact results. We also derive the exact conductance and I(V)I\left(V\right)I(V) curve, out of equilibrium, in this fully interacting system.
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.
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.
The effects of impurity scattering on a general Abelian fractional quantum Hall (FQH) edge state are analyzed within the chiral-Luttinger-liquid model of low-energy edge dynamics. We find that some disordered edges can have several different phases characterized by different symmetries. The stable impurity edge phases are in general more symmetric than the original clean system and demonstrate the phenomenon of dynamical symmetry restoration at low energies and long length scales. The phase transitions between different disordered phases are characterized by broken symmetries and obey Landau's symmetry-breaking principle for continuous phase transitions. Phase diagrams for various edges are found using a system of coordinates for the interactions between modes in a quantum Hall edge. The temperature dependence of tunneling through a point contact is calculated and is found to be able to distinguish different impurity edge phases of the same FQH state.
For fractional quantum Hall states on a finite disk, we study the thermoelectric transport properties under the influence of the presence of an edge and its reconstruction. In a recent study on a torus [Phys. Rev. B 101, 241101(R) (2020)], Sheng and Fu found a universal non-Fermi liquid power-law scaling of the thermoelectric conductivity ${\ensuremath{\alpha}}_{xy}\ensuremath{\propto}{T}^{\ensuremath{\eta}}$ for the gapless composite Fermi liquid state. The exponent $\ensuremath{\eta}\ensuremath{\sim}0.5$ appears an independence of the filling factors and the details of the interactions. In the presence of an edge, we find the properties of the edge spectrum dominants the low-temperature behaviors and a nonuniversal scaling behavior in low temperature is observed. To consider individually the effect of the edge states, the real space entanglement spectrum of the model wave function, which contains only the spectrum of edge excitation is employed. Its spectrum energies could be tuned by varying the radius of the subsystem. In the non-Abelian Moore-Read state, the Majorana neutral edge mode is found to have more significant effects than that of the charge mode in the low temperature.