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arXiv:cond-mat/9407014 · Physical review. B, Condensed matter

Fractional quantum Hall effect on a lattice

Fakher F. Assaad, S. Biskamp

Published Jan 15, 1995• 9 citations
arXivHTMLPDF

Abstract

Starting from the Hofstadter butterfly, we define lattice versions of Landau levels as well as a continuum limit that ensures they scale to continuum Landau levels. By including a next-neighbor repulsive interaction and projecting onto the lowest-lattice Landau level, we show that incompressible ground states exist on the lattice at odd-filling fractions. Already for values of ${\mathit{l}}_{0}$/a\ensuremath{\sim}2 where ${\mathit{l}}_{0}$ (a) is the magnetic length (lattice constant), the continuum \ensuremath{\nu}=1/3 Laughlin state is reproduced with perfect accuracy. Hence, a is argued to be an irrelevant length scale.

DOI
10.1103/physrevb.51.1605

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Starting from the Hofstadter butterfly, we define lattice versions of Landau levels as well as a continuum limit that ensures they scale to continuum Landau levels. By including a next-neighbor repulsive interaction and projecting onto the lowest-lattice Landau level, we show that incompressible ground states exist on the lattice at odd-filling fractions. Already for values of l0{\mathit{l}}_{0}l0​/a\ensuremath{\sim}2 where l0{\mathit{l}}_{0}l0​ (a) is the magnetic length (lattice constant), the continuum \ensuremath{\nu}=1/3 Laughlin state is reproduced with perfect accuracy. Hence, a is argued to be an irrelevant length scale.
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.
We present numerical evidence in support of the view that the physical origin of the fractional quantum Hall effect at \ensuremath{\nu}=n/(2n+1) is associated with the fact that this is the largest filling factor at which an n-component system of electrons can avoid having any pair of electrons occur in a state of relative angular momentum 1. Our calculations, for n=2, are based on a model in which the Hilbert space is truncated to the two lowest Landau levels and the Landau-level separation, \ensuremath{\Elzxh}${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$, is a parameter of the model. For \ensuremath{\Elzxh}${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$=0 and short-range repulsive interactions, a zero-energy incompressible ground state occurs at \ensuremath{\nu}=2/5. Our numerical results suggest that, in agreement with arguments advanced by J. K. Jain, this incompressibility survives to the strong-field limit \ensuremath{\Elzxh}${\mathrm{\ensuremath{\omega}}}_{\mathit{c}}$=\ensuremath{\infty}.
A simple one-dimensional model is proposed, in which N spinless repulsively interacting fermions occupy M > N degenerate states. It is argued that the energy spectrum and the wavefunctions of this system strongly resemble the spectrum and wavefunctions of 2D electrons in the lowest Landau level (the problem of the Fractional Quantum Hall Effect). In particular, Laughlin-type wavefunctions describe ground states at filling factors v = N/M = 1/ q , q odd.. Within this model the complimentary wavefunction for v =1−1/ q is found explicitly, and extremely simple ground state wavefunctions for arbitrary odd-denominator filling factors are proposed.
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.
We study lattice models of charged particles in uniform magnetic fields. We show how longer range hopping can be engineered to produce a massively degenerate manifold of single-particle ground states with wave functions identical to those making up the lowest Landau level of continuum electrons in a magnetic field. We find that in the presence of local interactions, and at the appropriate filling factors, Laughlin's fractional quantum Hall wave function is an exact many-body ground state of our lattice model. The hopping matrix elements in our model fall off as a Gaussian, and when the flux per plaquette is small compared to the fundamental flux quantum one only needs to include nearest and next-nearest neighbor hoppings. We suggest how to realize this model using atoms in optical lattices, and describe observable consequences of the resulting fractional quantum Hall physics.
The authors discuss the existence of fractional quantised Hall states in other than the lowest (n=0) Landau level. They present the results of numerical studies of systems with up to nine particles in the n=1 Landau level. They find that a fractional quantised Hall state is likely for filling fractions of the n=1 Landau level, nu 1 , with nu 1 -1 =5. At nu 1 -1 =3 they find that the Laughlin trial wavefunctions is not a good candidate ground-state wavefunction. If there is a gap at nu 1 -1 it is predicted to be small. At nu 1 -1 = 7 / 2 they predict the existence of a fractional quantised Hall state despite the small or non-existent gap at nu 1 -1 =3. They also show that introducing truncated pseudopotential interactions is not always a physical procedure outside of the lowest Landau level.