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Progress of Theoretical Physics

Quantum Group and q-Virasoro Symmetries in Fermion Systems

Haru-Tada Sato

Published Jan 1, 1995• 3 citations
Publisher

Abstract

We discuss a generalization of the quantum group Uq(sl(2)) to the q-Virasoro algebra in two-dimensional electrons system under uniform magnetic field. It is shown that the integral representations of both algebras are reduced to those in a (1 + 1)-dimensional fermion. As an application of the quantum group symmetry, we discuss a model of quantum group current on the analogy of the Hall current.

DOI
10.1143/ptp.93.195

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We discuss a generalization of the quantum group Uq(sl(2)) to the q-Virasoro algebra in two-dimensional electrons system under uniform magnetic field. It is shown that the integral representations of both algebras are reduced to those in a (1 + 1)-dimensional fermion. As an application of the quantum group symmetry, we discuss a model of quantum group current on the analogy of the Hall current.
Quantum Hall effect wavefunctions corresponding to the filling factors 1/2p+1, 2/2p+1,..., 2p/2p+1, 1, are shown to form a basis of irreducible cyclic representation of the quantum algebra U_q(sl(2)) at q^{2p+1}=1. Thus, the wavefunctions Ψ_{P/Q} possessing filling factors P/Q<1 where Q is odd and P, Q are relatively prime integers are classified in terms of U_q(sl(2)). Adopted as dynamical symmetry this leads to non--existence of a ``universal microscopic theory" of the quantum Hall effect, defined as the eigenvalue problem of a differential operator O: O Ψ_ν=\ell_νΨ_ν. for ν=1,1/3,2/3,..., in the complex plane.
The magnetic translation algebra plays an important role in the quantum Hall effect. Murthy and Shankar, arXiv:1207.2133, have shown how to realize this algebra using fermionic bilinears defined on a two-dimensional square lattice. We show that, in any dimension ddd, it is always possible to close the magnetic translation algebra using fermionic bilinears, whether in the continuum or on the lattice. We also show that these generators are complete in even, but not odd, dimensions, in the sense that any fermionic Hamiltonian in even dimensions that conserves particle number can be represented in terms of the generators of this algebra, whether or not time-reversal symmetry is broken. As an example, we reproduce the fff-sum rule of interacting electrons at vanishing magnetic field using this representation. We also show that interactions can significantly change the bare bandwidth of lattice Hamiltonians when represented in terms of the generators of the magnetic translation algebra.
I discuss many-body models for correlated fermions in two space dimensions which can be solved exactly using group theory. The simplest example is a model of a quantum Hall system: two-dimensional (2D) fermions in a constant magnetic field and a particular non-local four-point interaction. It is exactly solvable due to a dynamical symmetry corresponding to the Lie algebra gl ∞ ⊕ gl ∞ . There is an algorithm to construct all energy eigenvalues and eigenfunctions of this model. The latter are, in general, many-body states with spatial correlations. The model also has a non-trivial zero temperature phase diagram. I point out that this QH model can be obtained from a more realistic one using a truncation procedure generalizing a similar one leading to mean field theory. Applying this truncation procedure to other 2D fermion models I obtain various simplified models of increasing complexity which generalize mean field theory by taking into account non-trivial correlations but nevertheless are treatable by exact methods.