S-Duality Constraints on 1D Patterns Associated with Fractional Quantum Hall States

Abstract

Using the modular invariance of the torus, constraints on the 1D patterns are derived that are associated with various fractional quantum Hall ground states, e.g. through the thin torus limit. In the simplest case, these constraints enforce the well known odd-denominator rule, which is seen to be a necessary property of all 1D patterns associated to quantum Hall states with minimum torus degeneracy. However, the same constraints also have implications for the non-Abelian states possible within this framework. In simple cases, including the ν = 1 Moore-Read state and the ν = 3/2 level 3 Read-Rezayi state, the filling factor and the torus degeneracy uniquely specify the possible patterns, and thus all physical properties that are encoded in them. It is also shown that some states, such as the "strong p-wave pairing state", cannot in principle be described through patterns.Introduction. The study of fractional quantum Hall (FQH) liquids has been among the most intriguing problems in condensed matter physics during the past few decades, in both theory and experiment. On the theoretical side, the construction of variational many-body wave functions has traditionally played a pivotal role[1]. In principle, the possible variational constructions are limitless. A systematic classification of FQH phases therefore requires additional constraints, such as simplicity in a composite fermion picture [2]. Another program to implement such constraints is to require that the trial wave functions can be obtained as conformal blocks in certain conformal field theories [3]. The problem is then relegated to identifying all conformal field theories leading to permissible wave functions. On the other hand, it has recently become appreciated that a large class of trial wave functions can be characterized by simple sequences of integers, either through the thin torus limit and adiabatic continuity [4][5][6][7][8][9], or through Jack polynomials [10]. The patterns of integers associated with viable quantum Hall states are in turn subject to a number of consistency requirements, such as rotational invariance of the associated Jack polynomials, or constraints on the associated "patterns of zeros" studied in Ref. 11. A complete set of consistency requirements is desirable in order to understand the possible quantum numbers of all quantum Hall phases that are accessible within this framework. In this paper, it will be shown that the one-dimensional (1D) patterns associated with the ground state sectors of a quantum Hall phase are highly constrained by modular invariance on the torus. In the simplest case, the implication of these constraints is the well-known "odddenominator-rule", which is found to be required within this framework for all quantum Hall states that have the "minimum torus degeneracy". Such states are necessarily Abelian, and include states in the Haldane-Halperin hierarchy. Furthermore, in some other cases of interest, it is found that the filling factor and the torus degeneracy already completely determine the associated se...

Abelian and Non-Abelian Statistics in the Coherent State Representation

“…We can view the orbitals ϕ n as forming a 1D periodic "lattice" along the x direction, with each orbital representing a lattice site. Note that we have ϕ n+L (z) = ϕ n (z), and in this sense the "orbital lattice" satisfies ordinary periodic boundary conditions in n. A "thin torus limit" [21][22][23][24][25][26][27][28][29][30][31][32][33]50] can be defined as κ 1. In this limit, the orbitals in the basis (2) are well separated and have negligible overlap.…”

“…B. Outline of the method Our method of inferring the statistics of a quantum Hall state can be broken down into a few elementary steps, which can in principle be applied to any quantum Hall state that can be assigned well-defined ground-state patterns through a thin torus limit [21][22][23][24][25][26][29][30][31][32][33]. Here we give a brief summary of the individual steps and the underlying ideas.…”

“…These patterns may be identified through various interrelated approaches. We we focus here on the approach based on the thin torus limit and adiabatic continuity [21][22][23][24][25][26][27][28][29][30][31][32][33]. This is not done for mere convenience, as it turns out that adiabatic continuity will play an essential role in the following.…”

Absence of a gap in the Gaffnian state

“…It has certainly some desirable properties like good overlap with the true ground state for Coulomb interactions. Several studies have been devoted to its properties [12][13][14][15][16][17][18][19]. However, there is a potential problem which has been raised originally by Read [20][21][22].…”

Thin torus perturbative analysis of elementary excitations in the Gaffnian and Haldane-Rezayi quantum Hall states

“…12 The TT limit is yet another way to achieve a two-dimensional -one-dimensional (2D-1D) correspondence in the context of quantum Hall systems. [13][14][15][16][17][18] In Ref. 12, the very knowledge of the TT limit of the Haldane-Rezayi (HR) wave functions was used to argue that charge-neutral gapless excitations must exist in the TT limit, and the latter have been characterized as certain extended equal-amplitude superpositions of defects (see below).…”