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arXiv:cond-mat/0304271 · Physical review. B, Condensed matter

Charge and current in the quantum Hall matrix model

T. H. Hansson, Janik Kailasvuori, A. Karlhede

Published Jul 29, 2003• 19 citations
arXivHTMLPDF

Abstract

We extend the quantum Hall matrix model to include couplings to external electric and magnetic fields. The associated current suffers from matrix ordering ambiguities even in the classical model. We calculate the linear response at low momenta---this is unambigously defined. In particular, we obtain the correct fractional quantum Hall conductivity, and the expected density modulations in response to a weak and slowly varying magnetic field. In the quantum theory the ordering ambiguities are more severe, but we are able to construct a good density operator, satisfying the pertinent lowest Landau level commutator algebra, for the special case of $2\ifmmode\times\else\texttimes\fi{}2$ matrices. Our results support the conjecture that the quantum Hall matrix models do describe important aspects of the dynamics of electrons in the lowest Landau level.

DOI
10.1103/physrevb.68.035327

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We extend the quantum Hall matrix model to include couplings to external electric and magnetic fields. The associated current suffers from matrix ordering ambiguities even in the classical model. We calculate the linear response at low momenta---this is unambigously defined. In particular, we obtain the correct fractional quantum Hall conductivity, and the expected density modulations in response to a weak and slowly varying magnetic field. In the quantum theory the ordering ambiguities are more severe, but we are able to construct a good density operator, satisfying the pertinent lowest Landau level commutator algebra, for the special case of 2\ifmmode×\else\texttimes\fi22\ifmmode\times\else\texttimes\fi{}22\ifmmode×\else\texttimes\fi2 matrices. Our results support the conjecture that the quantum Hall matrix models do describe important aspects of the dynamics of electrons in the lowest Landau level.
We study three diffent approximations of the usual many-body model for the fractional quantum Hall effect (FQHE) consisting in a two-dimensional electron gas (2DEG) in a uniform electric and magnetic field.For each of them we consider the stochastic limit and we find the quantum Langevin equation and the generators of the master equation.This allows us to calculate the explicit form of the conductivity and the resistivity tensors.The relevance of the levels beyond the lowest Landau's one is explained, and a fine tuning relation between the electric and the magnetic fields is obtained from the stochastic resonance principle.It is shown that this relation could be at the origin of the quantization of the resistivity tensor.
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We investigate the theory of the fractional quantum Hall effect (QHE) proposed a long time ago by Lopez and Fradkin \cite{Fradkin1991chern} to describe the principal Jain series. The magnetic fluxes of the statistical gauge field attached to electrons remain at rest in the reference frame moving together with the electron liquid. In the laboratory reference frame the electric field of the statistical gauge field forms and screens the external electric field. The fractional QHE conductivity appears as a consequence of this screening already on the mean field theory level. We consider a relativistic extension of the model, and propose an alternative description of the fractional QHE based on macroscopic motion of the electron liquid within the Zubarev statistical operator approach. It is this macroscopic motion of electrons which in this pattern gives rise to the fractional QHE. Within this approach we propose the proof to all orders of perturbation theory that the interaction corrections cannot change the above mentioned mean field theory result for the QHE conductivity.
In a recent paper by S. Moroz, C. Hoyos, and L. Radzihovsky [Phys. Rev. B 91, 195409 (2015)], it is claimed that the conductivity at low frequency $\ensuremath{\omega}$ and small wave vector qqq along the edge of a quantum Hall system (that possesses Galilean invariance along the edge) contains a universal contribution of order q2{q}^{2}q2 that is determined by the orbital spin per particle in the bulk of the system, or alternatively by the shift of the ground state. (These quantities are known to be related to the Hall viscosity of the bulk.) In this Comment we calculate the real part of the conductivity, integrated over $\ensuremath{\omega}$, in this regime for the edge of a system of noninteracting electrons filling either the lowest, or the lowest $\ensuremath{\nu} (\ensuremath{\nu}=1,\phantom{\rule{0.16em}{0ex}}2,...)$, Landau level(s), and show that the q2{q}^{2}q2 term is nonuniversal and depends on details of the confining potential at the edge. In the special case of a linear potential, a form similar to the prediction is obtained; it is possible that this corrected form of the prediction may also hold for fractional quantum Hall states in systems with special forms of interactions between electrons.
We use the dynamical structure factors of the quantum Hall states at $\ensuremath{\nu}=1/3$ and 1/21/21/2 in the lowest Landau level to study their excitation spectrum. Using the density matrix renormalization group in combination with the time-dependent variational principle on an infinite cylinder geometry, we extract the low-energy properties. At $\ensuremath{\nu}=1/3$, a sharp magnetoroton mode and the two-roton continuum are present and the finite-size effects can be understood using the fractional charge of the quasiparticle. At $\ensuremath{\nu}=1/2$, we find low-energy modes with linear dispersion and the static structure factor $\overline{s}(q)\ensuremath{\sim}{(q\ensuremath{\ell})}^{3}$ in the limit $q\ensuremath{\ell}\ensuremath{\rightarrow}0$. The properties of these modes agree quantitatively with the predictions of the composite-fermion theory placed on the infinite cylinder.
Abstract We consider an infinitely extended system of fermions on a d-dimensional lattice with (magnetic) translation-invariant short-range interactions. We further assume that the system has a gapped ground state. Physically, this is a model for the bulk of a generic topological insulator at zero temperature, and we are interested in the current response of such a system to a constant external electric field. Using the non-equilibrium almost-stationary states approach, we prove that the longitudinal current density induced by a constant electric field of strength ε\varepsilon ε ε is of order O(ε∞)\mathcal {O}(\varepsilon ^\infty )O(ε∞) O ( ε ∞ ) , i.e. the system is an insulator in the usual sense. For the Hall current density we show instead that it is linear in ε\varepsilon ε ε up to terms of order O(ε∞)\mathcal {O}(\varepsilon ^\infty )O(ε∞) O ( ε ∞ ) . The proportionality factor σH\sigma _\textrm{H}σH​ σ H is by definition the Hall conductivity, and we show that it is given by a generalization of the well known double commutator formula to interacting systems. As a by-product of our results, we find that the Hall conductivity is constant within gapped phases, and that for d=2d=2d=2 d = 2 the relevant observable that “measures” the Hall conductivity in experiments, the Hall conductance, not only agrees with σH \sigma _{\textrm{H}}σH​ σ H in expectation up to O(ε∞)\mathcal {O}(\varepsilon ^\infty )O(ε∞) O ( ε ∞ ) , but also has vanishing variance. A notable difference to several existing results on the current response in interacting fermion systems is that we consider a macroscopic system exposed to a small constant electric field, rather than to a small voltage drop.