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arXiv:1710.09729 · Physical review. B./Physical review. B

Lattice Monte Carlo for quantum Hall states on a torus

Jie Wang, Scott Geraedts, Edward H. Rezayi, F. D. M. Haldane

Published Mar 15, 2019• 39 citations
arXivHTMLPDF

Abstract

The Monte Carlo method is very useful for studying various model states proposed for the fractional quantum Hall systems. In this paper, we introduce a lattice Monte Carlo method based on an exact lattice formalism for quantum Hall problems defined on a torus. This ``lattice representation'' is applied to study the many body Berry phase of the composite-Fermi liquid phase in a half-filled Landau level. The Monte Carlo result agrees with the exact numerical result found on small systems sizes and is consistent with the prediction from the Dirac fermion low-energy effective theory. Many other aspects of other model states, including the structure factor, Coulomb energy and particle-hole symmetry, are studied and discussed.

DOI
10.1103/physrevb.99.125123

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The Monte Carlo method is very useful for studying various model states proposed for the fractional quantum Hall systems. In this paper, we introduce a lattice Monte Carlo method based on an exact lattice formalism for quantum Hall problems defined on a torus. This ``lattice representation'' is applied to study the many body Berry phase of the composite-Fermi liquid phase in a half-filled Landau level. The Monte Carlo result agrees with the exact numerical result found on small systems sizes and is consistent with the prediction from the Dirac fermion low-energy effective theory. Many other aspects of other model states, including the structure factor, Coulomb energy and particle-hole symmetry, are studied and discussed.
We consider spin-polarized electrons in a single Landau level on a torus. The quantum Hall problem is mapped onto a one-dimensional lattice model with lattice constant $2\ensuremath{\pi}/{L}_{1}$, where L1{L}_{1}L1​ is a circumference of the torus (in units of the magnetic length). In the Tao-Thouless limit ${L}_{1}\ensuremath{\rightarrow}0$, the interacting many-electron problem is exactly diagonalized at any rational filling factor $\ensuremath{\nu}=p/q\ensuremath{\le}1$. For odd qqq, the ground state has the same qualitative properties as a bulk $({L}_{1}\ensuremath{\rightarrow}\ensuremath{\infty})$ quantum Hall hierarchy state and the lowest-energy quasiparticle excitations have the same fractional charges as in the bulk. These states are the ${L}_{1}\ensuremath{\rightarrow}0$ limits of the Laughlin and Jain wave functions for filling fractions where these exist. We argue that the exact solutions generically, for odd qqq, are continuously connected to the two-dimensional bulk quantum Hall hierarchy states---i.e., that there is no phase transition as ${L}_{1}\ensuremath{\rightarrow}\ensuremath{\infty}$ for filling factors where such states can be observed. For even-denominator fractions, a phase transition occurs as L1{L}_{1}L1​ increases. For $\ensuremath{\nu}=1/2$ this leads to the system being mapped onto a Luttinger liquid of neutral particles at small but finite L1{L}_{1}L1​; this then develops continuously into the composite fermion wave function that is believed to describe the bulk $\ensuremath{\nu}=1/2$ system. The analysis generalizes to non-Abelian quantum Hall states.
Composite-Boson Ansatz for Fractional Quantum Hall Manifolds of Lattice Bosons at Generic Fillings ν<1/2ν<1/2ν<1/2
Hofstadter systems provide a lattice route to fractional quantum Hall physics, but their low-energy manifolds often lack simple wave-function descriptions. Here we present a composite-boson ansatz for a broad family of finite-size lattice-split Laughlin-quasihole manifolds in the low-flux Hofstadter--Bose--Hubbard model on a torus at generic fillings ν<1/2ν<1/2ν<1/2. Attaching two vortices to each boson yields reduced-flux orbitals from which we construct a many-body trial basis. Its translation-resolved rank reproduces the expected low-energy-manifold dimension and provides a composite-boson interpretation of the established quasihole counting previously inferred from generalized exclusion rules and thin-torus arguments. For smaller systems, diagonalization within the lowest-band-projected ansatz span closely reproduces the exact projected spectrum, while variational Monte Carlo extends the rank and energetic analysis to larger systems for which explicit subspace construction becomes costly. To probe the fractional quantum Hall character of these manifolds, we calculate their many-body Chern-numbers and show that localized added-flux excitations exhibit quasihole-like behavior, with fractional density depletion and an Aharonov--Bohm-free braiding phase that both track the effective filling. Together, these results establish a microscopic composite-boson framework for organizing a broad family of sub-half-filled fractional quantum Hall manifolds of lattice bosons.
We study the two-dimensional electron gas in a magnetic field at filling fraction ν=12ν=\frac{1}{2}ν=21​. At this filling the system is in a gapless state which can be interpreted as a Fermi liquid of composite fermions. We construct trial wave functions for the system on a torus, based on this idea, and numerically compare these to exact wave functions for small systems found by exact diagonalization. We find that the trial wave functions give an excellent description of the ground state of the system, as well as its charged excitations, in all momentum sectors. We analyze the dispersion of the composite fermions and the Berry phase associated with dragging a single fermion around the Fermi surface and comment on the implications of our results for the current debate on whether composite fermions are Dirac fermions.
We consider the lowest Landau level on a torus as a function of its circumference L 1 . When L 1 → 0, the ground state at general rational filling fraction is a crystal with a gap—a Tao–Thouless state. For filling fractions ν = p /(2 pm + 1), these states are the limits of Laughlin's or Jain's wavefunctions describing the gapped quantum Hall states when L 1 → ∞ . For the half-filled Landau level, there is a transition to a Fermi sea of non-interacting neutral fermions (dipoles), or rather to a Luttinger liquid modification thereof, at L 1 ∼ 5 magnetic lengths. Using exact diagonalization we identify this state as a version of the Rezayi–Read state, and find that it develops continuously into the state that is believed to describe the observed metallic phase as L 1 → ∞ . Furthermore, the effective Landau level structure that emerges within the lowest Landau level is found to be a consequence of the magnetic symmetries.