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arXiv:cond-mat/9408035 · Europhysics Letters (EPL)

Integer Quantum Hall Effect for Lattice Fermions

K. Ziegler

Published Oct 1, 1994• 10 citations
arXivHTMLPDF

Abstract

A two-dimensional lattice model for non-interacting fermions in a magnetic field with half a flux quantum per plaquette and N levels per site is considered. This is a model which exhibits the integer quantum Hall effect (IQHE) in the presence of disorder. It presents an alternative to the continuous picture for the IQHE with Landau levels. The large- N limit can be solved: two Hall transitions appear and there is an interpolating behaviour between the two Hall plateaux. Although this approach to the IQHE is different from the traditional one with Landau levels because of different symmetries (continuous for Landau levels and discrete here), some characteristic features are reproduced. For instance, the slope of the Hall conductivity is infinite at the transition points and the electronic states are delocalized only at the transitions.

DOI
10.1209/0295-5075/28/1/009

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A two-dimensional lattice model for non-interacting fermions in a magnetic field with half a flux quantum per plaquette and N levels per site is considered. This is a model which exhibits the integer quantum Hall effect (IQHE) in the presence of disorder. It presents an alternative to the continuous picture for the IQHE with Landau levels. The large- N limit can be solved: two Hall transitions appear and there is an interpolating behaviour between the two Hall plateaux. Although this approach to the IQHE is different from the traditional one with Landau levels because of different symmetries (continuous for Landau levels and discrete here), some characteristic features are reproduced. For instance, the slope of the Hall conductivity is infinite at the transition points and the electronic states are delocalized only at the transitions.
We introduce and analyze a class of model systems to study transitions in the integer quantum Hall effect (IQHE). Even without disorder our model exhibits an IQHE transition as a control parameter is varied. We find that the transition is in the two-dimensional Ising universality class and compute all associated exponents and critical transport properties. The fixed point has time-reversal, particle-hole, and parity invariance. We then consider the effect of quenched disorder on the IQHE transition and find the following. (i) Randomness in the control parameter (which breaks all the above symmetries) translates into bond randomness in the Ising model and is hence marginally irrelevant. The transition may equally well be viewed as a quantum percolation of edge states localized on equipotentials. The absence of random-phase factors for the edge states is responsible for the nongeneric (Ising) critical properties. (ii) For a random magnetic field (which preserves particle-hole symmetry in every realization) the model exhibits an exactly solvable fixed line, described in terms of a product of a Luttinger liquid and an SU(n) spin chain. While exponents vary continuously along the fixed line, the longitudinal conductivity is constant due to a general conformal sum rule for Kac-Moody algebras (derived here), and is computed exactly. We also obtain a closed expression for the extended zero-energy wave function for every realization of disorder and compute its exact multifractal spectrum f(\ensuremath{\alpha}) and the exponents of all participation ratios. One point on the fixed line corresponds to a recently proposed model by Gade and Wegner. (iii) The model in the presence of a random on-site potential scales to a strong disorder regime, which is argued to be described by a symplectic nonlinear-sigma-model fixed point. (iv) We find a plausible global phase diagram in which all forms of disorder are simultaneously considered. In this generic case, the presence of random-phase factors in the edge-state description indicates that the transition is described by a Chalker-Coddington model, with a so far analytically inaccessible fixed point.
In this work we study a system of interacting fermions on a triangular lattice in the presence of an external magnetic field. We neglect spin and fix a density of one third, with one unit of magnetic flux per particle. The infinite density matrix renormalization group algorithm is used to compute the ground state of this generalized Fermi-Hubbard model. Increasing the strength of the nearest-neighbor repulsion, we find a first order transition between an Integer Quantum Hall phase and a crystalline, generalized Wigner crystal state. The first-order nature of the phase transition is consistent with a Ginzburg-Landau argument. We expect our results to be relevant for moiré heterostructures of two-dimensional materials.
Field-theoretical methods have been shown to be useful in constructing simple effective theories for two-dimensional (2D) systems. These effective theories are usually studied by perturbing around a mean-field approximation, so the question as to whether such an approximation is meaningful arises immediately. We here study 2D interacting electrons in a half-filled Landau level mapped onto interacting hard-core bosons in a magnetic field. We argue that an interacting hard-core boson system in a uniform external field such that there is one flux quantum per particle (unit filling) exhibits an integer quantum Hall effect. As a consequence, the mean-field approximation for mapping electrons at half-filling to a boson system at integer filling fails. \textcopyright{} 1996 The American Physical Society.
lattice model exhibiting the integer quantum Hall effect. We consider an interacting version of the model, and prove that for short-range interactions, smaller than the bandwidth, the Hall conductivity is quantized, for all the values of the parameters outside two critical curves, across which the model undergoes a 'topological' phase transition: the Hall coefficient remains integer and constant as long as we continuously deform the parameters without crossing the curves; when this happens, the Hall coefficient jumps abruptly to a different integer. Previous works were limited to the perturbative regime, in which the interaction is much smaller than the bare gap, so they were restricted to regions far from the critical lines. The non-renormalization of the Hall conductivity arises as a consequence of lattice conservation laws and of the regularity properties of the current-current correlations. Our method provides a full construction of the critical curves, which are modified ('dressed') by the electron-electron interaction. The shift of the transition curves manifests itself via apparent infrared divergences in the naive perturbative series, which we resolve via renormalization group methods.
We study interacting two-component hard-core bosons on a square lattice for which, in the presence of staggered magnetic flux, its ground state is a bosonic integer quantum Hall (BIQH) state. By using a coupled-wire bosonization approach, we show analytically that this model exhibits a BIQH state at total charge half filling associated with a symmetry-protected topological phase under U(1)\text{U}(1)U(1) charge conservation. These theoretical expectations are verified, using the infinite density matrix renormalization group method, by providing numerical evidence for (i) a quantized Hall conductance ${\ensuremath{\sigma}}_{xy}=\ifmmode\pm\else\textpm\fi{}2$, and (ii) two counterpropagating gapless edge modes. Our model is a bosonic cousin of the fermionic Haldane model and serves as an additional case of analogy between bosonic and fermionic quantum Hall states.
In this work we study a system of interacting fermions on a triangular lattice in the presence of an external magnetic field. We assume electrons are spin polarized and fix a density of one third, with one unit of magnetic flux per particle. The infinite density matrix renormalization group algorithm is used to compute the ground state of this generalized Fermi-Hubbard model. Increasing the strength of the nearest-neighbor repulsion, we find a first-order transition between an integer quantum Hall phase and a crystalline, generalized Wigner crystal state. The first-order nature of the phase transition is consistent with a Ginzburg-Landau argument. We expect our results to be relevant for moir & eacute; heterostructures of two-dimensional materials.