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arXiv:0707.3637 · Physical Review Letters

Model Fractional Quantum Hall States and Jack Polynomials

Bogdan Andrei Bernevig, F. D. M. Haldane

Published Jun 19, 2008• 335 citations
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Abstract

We describe an occupation-number-like picture of fractional quantum Hall states in terms of polynomial wave functions characterized by a dominant occupation-number configuration. The bosonic variants of single-component Abelian and non-Abelian fractional quantum Hall states are modeled by Jack symmetric polynomials (Jacks), characterized by dominant occupation-number configurations satisfying a generalized Pauli principle. In a series of well-known quantum Hall states, including the Laughlin, Read-Moore, and Read-Rezayi, the Jack polynomials naturally implement a "squeezing rule" that constrains allowed configurations to be restricted to those obtained by squeezing the dominant configuration. The Jacks presented in this Letter describe new trial uniform states, but it is yet to be determined to which actual experimental fractional quantum Hall effect states they apply.

DOI
10.1103/physrevlett.100.246802

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We describe an occupation-number-like picture of fractional quantum Hall states in terms of polynomial wave functions characterized by a dominant occupation-number configuration. The bosonic variants of single-component Abelian and non-Abelian fractional quantum Hall states are modeled by Jack symmetric polynomials (Jacks), characterized by dominant occupation-number configurations satisfying a generalized Pauli principle. In a series of well-known quantum Hall states, including the Laughlin, Read-Moore, and Read-Rezayi, the Jack polynomials naturally implement a "squeezing rule" that constrains allowed configurations to be restricted to those obtained by squeezing the dominant configuration. The Jacks presented in this Letter describe new trial uniform states, but it is yet to be determined to which actual experimental fractional quantum Hall effect states they apply.
Many bosonic (fermionic) fractional quantum Hall states, such as Laughlin, Moore-Read, and Read-Rezayi wave functions, belong to a special class of orthogonal polynomials: the Jack polynomials (times a Vandermonde determinant). This fundamental observation allows one to point out two different recurrence relations for the coefficients of the permanent (Slater) decomposition of the bosonic (fermionic) states. Here we provide an explicit Fock space representation for these wave functions by introducing a two-body squeezing operator which represents them as a Jastrow operator applied to reference states, which are, in general, simple periodic one-dimensional patterns. Remarkably, this operator representation is the same for bosons and fermions, and the different nature of the two recurrence relations is an outcome of particle statistics.
We compute the physical properties of non-Abelian fractional quantum Hall (FQH) states described by Jack polynomials at general filling nu = k/r. For r = 2, these states are the Zk Read-Rezayi parafermions, whereas for r > 2 they represent new FQH states. The r = k+1 states, multiplied by a Vandermonde determinant, are a non-Abelian alternative construction of states at fermionic filling 2/5,3/7,4/9,.... We obtain the thermal Hall coefficient, the quantum dimensions, the electron scaling exponent, and the non-Abelian quasihole propagator. The properties of the r > 2 Jack polynomials indicate they are correlators of fields of nonunitary conformal field theories (CFT), but the CFT-FQH connection fails when invoked to compute physical properties such as the quasihole propagator. The quasihole wave function, written as a coherent state representation of Jack polynomials, has an identical structure for all non-Abelian states.
We deduce a new set of symmetries and relations between the coefficients of the expansion of Abelian and non-Abelian fractional quantum Hall (FQH) states in free (bosonic or fermionic) many-body states. Our rules allow us to build an approximation of a FQH model state with an overlap increasing with growing system size (that may sometimes reach unity) while using a fraction of the original Hilbert space. We prove these symmetries by deriving a previously unknown recursion formula for all the coefficients of the Slater expansion of the Laughlin, Read-Rezayi, and many other states (all Jacks multiplied by Vandermonde determinants), which completely removes the current need for diagonalization procedures for these model Hamiltonians.
We develop a general algebraic scheme to decompose fractional quantum Hall (FQH) wave functions based on the operator contraction multiplication. By introducing fermionic and bosonic operators and establishing three fundamental contraction rules, we achieve an exact decomposition of Laughlin states. This approach naturally extends to multi-component systems by factorizing coupled Jastrow factors via resultants and elementary symmetric polynomials, enabling the first complete decomposition of Halperin states. For Halperin (2,2,12,2,12,2,1) state, we explicitly derive its basic expansion, identify root configurations, and reveal intra- and inter-color squeezing operators, thereby uncovering the underlying generalized Pauli principle. Using this method, we compute orbital entanglement spectra for up to 161616 particles with decomposition dimensions exceeding 101110^{11}1011, obtaining edge excitation sequences that precisely match chiral Luttinger liquid theory. Our framework breaks through the longstanding limitations of Jack polynomials, provides a unified decomposition for both single- and multi-component FQH states, and opens a new avenue for exploring wave functions for more complex FQH states.
Model wavefunctions are essential for studying fractional quantum Hall (FQH) phases. They reveal the intricate connections between model FQH states, conformal field theory, Jack polynomials, and pseudopotential Hamiltonians. However, most model wavefunctions studied so far are in the continuum. In this work, we construct model wavefunctions and the corresponding exact parent Hamiltonians for the entire ZkZ_kZk​ Read--Rezayi series on a lattice. Our model wavefunctions obey a generalized clustering rule implemented on the lattice, which we find is encoded in the Macdonald polynomials providing a unified algebraic description of our lattice-specific model states. This ensures that the model states are exact zero-energy ground states of local density interactions and exhibit an infinitely large gap in the particle entanglement spectrum. Numerical studies of the Laughlin and Moore--Read series find the expected zero-mode and one-flux quasihole counting and positive energy gaps at the system sizes studied, and the available finite-size scaling suggests nonzero thermodynamic gaps. Tuning the model interaction range can drive a phase transition out of the FQH phase. Our work not only advances the theoretical understanding of the underlying mathematical structure of lattice model FQH states but also is potentially useful for designing exotic many-body phases in engineered quantum platforms.
Fractional quantum Hall eect is a remarkable behaviour of correlated electrons, observed exclusively in two dimensions, at low temperatures, and in strong magnetic elds.The most prominent fractional quantum Hall state occurs at Landau level lling factor ν = 1/3 and it is described by the famous Laughlin wave function, which (apart from the trivial Gaussian factor) is an example of Jack polynomial.Fermionic Jack polynomials also describe another pair of candidate fractional quantum Hall states: MooreRead and ReadRezayi states, believed to form at the ν = 1/2 and 3/5 llings of the second Landau level, respectively.Bosonic Jacks on the other hand are candidates for certain correlated states of cold atoms.We examine here a continuous family of fermionic Jack polynomials whose special case is the Laughlin state as approximate wave functions for the 1/3 fractional quantum Hall eect.