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5 Acta Physica Polonica A

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Commutators of Jastrow Factors and Angular Momentum Operators

100%

Kuśmierz B. , Wójs A.

Acta Physica Polonica A

2017

tom 132

nr 2

405-407

We present detailed calculations of commutators of the Jastrow factor and certain differential operators useful in the fractional quantum Hall effect. In particular, we analyze action of the angular momentum operators projected from the Haldane sphere on an arbitrary composite fermions state. Examined L⁺ and L¯ momentum operators and following uniformity condition had proven to be useful in the search for candidates for quantum Hall ground states among many families of polynomials including the Jack polynomials.

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

100%

Kuśmierz B. , Wójs A.

Acta Physica Polonica A

2016

tom 130

nr 5

1183-1186

We focus on a certain aspect of trial wave function approach in the fractional quantum Hall effect. We analyze the role of partition orderings and discuss the possible numerical search for the partition determining the subspace of the Hilbert space containing a particular quantum Hall wave function. This research is inspired by analogical properties of certain polynomials which are the object of interest of the symmetric function theory, especially the Jack polynomials (related to the so-called "Jack states"). Presented method may be used in the search of candidate trial wave functions. We also justify (in certain cases) diagonalization of the Coulomb repulsion Hamiltonian restricted to certain subspaces. We focus on the states at filling factor ν=1/3 in the lowest and second Landau level.

Jack Polynomials and Fractional Quantum Hall Effect at ν = 1/3

100%

Kuśmierz B. , Wójs A.

Acta Physica Polonica A

2016

tom 130

nr 2

607-608

We investigate properties of strongly correlated, spinless electrons confined within given Landau level at filling factor ν = 1/3. Our analysis is based on the formalism of the Jack polynomials. Selected Jack polynomial wave functions are compared with ground states of the Coulomb interaction Hamiltonians, in different materials and the Landau levels, obtained by exact diagonalization. We show that certain Jack wave functions can be used as a description of fractional quantum Hall states.

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

80%

Kuśmierz B. , Wu Y. , Wójs A.

Acta Physica Polonica A

2014

tom 126

nr 5

1134-1136

Fractional quantum Hall effect is a remarkable behaviour of correlated electrons, observed exclusively in two dimensions, at low temperatures, and in strong magnetic fields. The most prominent fractional quantum Hall state occurs at Landau level filling 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: Moore-Read and Read-Rezayi states, believed to form at the ν = 1/2 and 3/5 fillings 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 effect.

Jack 3/5 State from Two-Body Interaction

80%

Kuśmierz B. , Wu Y. , Wójs A.

Acta Physica Polonica A

2016

tom 129

nr 1a

A-73-A-74

We investigate the Read-Rezayi parafermion state of correlated electrons at the fractional Landau level filling ν=3/5. It is a Jack polynomial generated by contact four-body repulsion. We show by exact diagonalization that it is also emerges from a suitable short-range two-body interaction. We find that it closely matches Coulomb ground state in the second Landau level of non-relativistic fermions, and thus possibly describes the ν=13/5 (and, by conjugation, ν=12/5) fractional quantum Hall effect in GaAs.