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arXiv:cond-mat/9510122 · Physical review. B, Condensed matter

Integer quantum Hall effect for hard-core bosons and a failure of bosonic Chern-Simons mean-field theories for electrons at a half-filled Landau level

Olle Heinonen, M. D. Johnson

Published Jan 15, 1996• 0 citations
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Abstract

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.

DOI
10.1103/physrevb.53.1517

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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 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.
The integer quantum Hall state occurs when Landau levels are fully occupied by fermions, while the fractional quantum Hall state usually emerges when the Landau level is partially filled by strongly correlated fermions or bosons. Here, we report two fractional Chern insulating states of hard-core bosons in a multiband lattice model hosting topological flat bands with a high Chern number. The previously proposed $\ensuremath{\nu}=1/3$ fractional Chern insulating state inherited from the high Chern number C=2C=2C=2 of the lowest topological flat band is revisited by the infinite density matrix renormalization group algorithm. In particular, we numerically identify a bosonic 1/21/21/2-Laughlin-like fractional Chern insulating state at the integer fillings. We show two lower topological flat bands jointly generate an effective C=1C=1C=1 Chern band with half filling. Furthermore, we find a strictly particle-hole-like symmetry between the $\ensuremath{\nu}$ and $3\ensuremath{-}\ensuremath{\nu}$ filling in our model. These findings extend our understanding of quantum Hall states and offer a route to realize fractional states in the system with multibands and high Chern numbers.
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.
We study two-dimensional bosons and fermions in a magnetic field with vanishing range interactions. This model was previously proposed for fermions to illustrate the exactness and uniqueness of Laughlin's wave function as a ground state within the lowest Landau level. We show that the restriction to the lowest Landau level is not a valid approximation of these models for arbitrarily high (but finite) cyclotron frequency, since it predicts fractional quantum Hall effect (FQHE) at an incorrect filling factor. In particular, in contrast to the lowest-Landau-level theory, hard-core bosons are shown not to exhibit any FQHE. However, the set of filling factors for the FQHE are identical in the two cases, thus reaffirming the universality of the FQHE.
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.
It is argued that a short-range interaction induces a quasi-Landau-level structure in the filling-factor dependence of the ground-state energy of electrons in the lowest Landau level. The discussion is based on exact numerical results obtained from small-system calculations, and the adiabatic principle of Greiter and Wilczek. It is shown that by attaching a single negative flux quantum to each electron adiabatically, the noninteracting electron ground states occupying higher Landau levels are smoothly transformed into the ground states of a hard-core boson gas in the lowest Landau level. This implies a close correspondence between the noninteracting fermions and the hard-core bosons in the lowest Landau level. Numerical results suggest the possibility of a smooth transition from hard-core bosons to fermions with a short-range interaction, by continuously varying the interaction potential.