Mathematical Structure of Bosonic and Fermionic Jack States and Their Application in Fractional Quantum Hall Effect

Abstract

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.

Emergence of Jack ground states from two-body pseudopotentials in fractional quantum Hall systems

“…The Jack polynomial [12][13][14][15]21,22,[28][29][30][31][32][33][34] , called simply a "Jack" and denoted by J α λ , is a symmetric polynomial indexed by the partition λ and the real number α. The partition is a sequence λ = (λ 1 , λ 2 , .…”

Quantum Hall State ν = 1/3 and Antilexicographic Order of Partitions

“…Algebraic combinatorics brought to our attention that certain trial FQH wave functions are related to well known symmetric polynomials like Jack polynomials J α λ (parameter α is a real number, λ is a partition) [14][15][16][17][18][19][20][21]. Among those FQH states are the Laughlin series ν = 1/r (when r is even one gets bosonic states and for odd r fermionic states), the Moore-Read state ν = 1/2 in the LL1, "parafermion" sequence ν = k/(k + 2) and "Gaffnian" wave functions (ν = 2/3 for bosons, ν = 2/5 for fermions) [22,23].…”

Ground States of Quantum Hall Three-Body ``Short-Range'' Repulsion and Mean Field Approximation: Correlation Functions and Overlaps

“…Surprisingly, even though Pf and APf are described entirely in terms of three-body interaction, they seem to capture many features of ground states of two-body Coulomb interaction Hamiltonians in half-filled first excited Landau level (LL1). Remarkably, the Moore-Read state can also be characterized as a Jack polynomial which makes it fall into the category of the Jack states and allows for application of tools known from the symmetric functions theory [14][15][16][17][18][19][20][21][22]. This is especially useful when one generates coefficients of Pf wave function * corresponding author for large systems, as the Jack states can be computed with relatively fast recursion formula [17,18,23,24].…”