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Journal of Applied Physics

Generalized flux states and the quantum Hall effect (abstract)

Gergely T. Zimányi, Franco Nori, Elihu Abrahams

Published Apr 15, 1991• 0 citations
Publisher

Abstract

We investigate certain generalized flux phases arising in a mean-field approach to the t-J model. First we establish that the energy of noninteracting electrons moving in a uniform magnetic field has an absolute minimum as a function of the flux at exactly one flux quantum per particle. Using this result we show that if the hard-core nature of the hole bosons is taken into account, then the generalized flux states in the mean-field approximation constitute a solution where both the spinons and the holons experience an average flux of one flux quantum per particle. We elaborate the relation to the integer and fractional quantum Hall effect. Based on this analogy one expects that only states with certain fractional fillings (ν=1/odd integer) are stable against the fluctuations of the (gauge) field. We speculate that at arbitrary fillings the system minimizes its energy by phase separating into these stable states. Finally we work out the ground state energy for nonuniform (staggered) flux phases and establish that the often claimed equivalence of frustration and doping does not hold for these states.

DOI
10.1063/1.348073

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Physical review. B, Condensed matter•Mar 1, 1991•11 citations
We investigate certain generalized flux phases arising in a mean-field approach to the t-J model. First we establish that the energy of noninteracting electrons moving in a uniform magnetic field has an absolute minimum as a function of the flux at exactly one flux quantum per particle. Using this result we show that if the hard-core nature of the hole bosons is taken into account, then the generalized flux states in the mean-field approximation constitute a solution where both the spinons and the holons experience an average flux of one flux quantum per particle. We elaborate the relation to the integer and fractional quantum Hall effect. Based on this analogy one expects that only states with certain fractional fillings (ν=1/odd integer) are stable against the fluctuations of the (gauge) field. We speculate that at arbitrary fillings the system minimizes its energy by phase separating into these stable states. Finally we work out the ground state energy for nonuniform (staggered) flux phases and establish that the often claimed equivalence of frustration and doping does not hold for these states.
We investigate certain generalized flux phases arising in a mean-field approach to the t-J model. First, we establish that the energy of noninteracting electrons moving in a uniform magnetic field has an absolute minimum as a function of the flux at exactly one flux quantum per particle. Using this result, we show that if the hard-core nature of the hole bosons is taken into account, then the slave-boson mean-field approximation for the t-J Hamiltonian allows for a solution where both the spinons and the holons experience an average flux of one flux quantum per particle. This enables them to achieve the lowest possible energy within the manifold of spatially uniform flux states. In the case of the continuum model, this is possible only for certain fractional fillings and we speculate that the system may react to this frustration effect by phase separation.
Through a hard-core slave-boson mean-field approach to the t-J model, which describes the dynamics of holes moving in a quantum antiferromagnet, we investigate its generalized flux phases. We first study the motion of an electron gas for arbitrary fillings and fields and establish the linear proportionality between the filling factor and the optimal flux with high accuracy. Using this result, we show that if the hard-core nature of the hole bosons is taken into account, then the slave-boson mean-field approximation for the t-J Hamiltonian allows for a solution where both the spinons and the holons experience an average flux of one flux quantum per particle. This enables them to achieve the lowest possible energy within the manifold of spatially-uniform flux states. In the case of the continuum model, this is possible only for certain fractional fillings and we suggest that the system may react to this frustration effect by phase separation.
We study the ground-state phase diagram of the strongly interacting Harper–Hofstadter–Mott model at quarter flux on a quasi-one-dimensional lattice consisting of a single magnetic flux quantum in y -direction. In addition to superfluid phases with various density patterns, the ground-state phase diagram features quasi-one-dimensional analogs of fractional quantum Hall phases at fillings ν = 1/2 and 3/2, where the latter is only found thanks to the hopping anisotropy and the quasi-one-dimensional geometry. At integer fillings—where in the full two-dimensional system the ground-state is expected to be gapless—we observe gapped non-degenerate ground-states: at ν = 1 it shows an odd 'fermionic' Hall conductance, while the Hall response at ν = 2 consists of the transverse transport of a single particle–hole pair, resulting in a net zero Hall conductance. The results are obtained by exact diagonalization and in the reciprocal mean-field approximation.
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.
We study wave functions for generalized flux phases by the variational Monte Carlo method. Flux tubes are attached to the holes, which produce additional phase factors besides those due to the motion of electrons. The phase factors cancel each other when the electrons and holes interchange position at some special electron densities and, as a result, there are no orbital currents in these states. We investigate especially the one-third filled case and find a slightly lower kinetic energy than the original commensurate flux phase. We also calculate the chirality to confirm that time-reversal symmetry is also broken in these states. The analogy to the quantum-Hall-effect wave functions leads us to study modified forms of off-diagonal long-range order in the pair-correlation function. The existence of this long-range order is shown explicitly by variational Monte Carlo calculations and renormalized mean-field approximations. The flux phases do not have the usual off-diagonal long-range order but do have a modified form similar to that in the quantum Hall effect. The relationships to the slave-boson mean-field theories and anyon systems are also discussed.