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FRACTIONAL QUANTUM HALL STATES ON A SQUARE LATTICE

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Published Sep 10, 2026• 0 citations
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Abstract

We calculate the ground state energies and wavefunctions for small systems of fermions on a lattice using singly and doubly periodic boundary conditions. We find minima in the ground state energy when the magnetic flux per plaquette corresponds to filling fractions with odd denominators. We interpret these minima in terms of fractional quantum Hall states. The pair correlation function at nearest and next-nearest neighbours is calculated as a function of the repulsive interaction strength between particles.

DOI
10.82444/warw.33555325

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We calculate the ground state energies and wavefunctions for small systems of fermions on a lattice using singly and doubly periodic boundary conditions. We find minima in the ground state energy when the magnetic flux per plaquette corresponds to filling fractions with odd denominators. We interpret these minima in terms of fractional quantum Hall states. The pair correlation function at nearest and next-nearest neighbours is calculated as a function of the repulsive interaction strength between particles.
The authors calculate the ground state energies and wavefunctions for small systems of fermions on a lattice using singly and doubly periodic boundary conditions. They find minima in the ground state energy when the magnetic flux per plaquette corresponds to filling fractions with odd denominators. They interpret these minima in terms of fractional quantum Hall states. The pair correlation function at nearest and next-nearest neighbours is calculated as a function of the repulsive interaction strength between particles.
A simple one-dimensional model is proposed, in which N spinless repulsively interacting fermions occupy M > N degenerate states. It is argued that the energy spectrum and the wavefunctions of this system strongly resemble the spectrum and wavefunctions of 2D electrons in the lowest Landau level (the problem of the Fractional Quantum Hall Effect). In particular, Laughlin-type wavefunctions describe ground states at filling factors v = N/M = 1/ q , q odd.. Within this model the complimentary wavefunction for v =1−1/ q is found explicitly, and extremely simple ground state wavefunctions for arbitrary odd-denominator filling factors are proposed.
We study lattice models of charged particles in uniform magnetic fields. We show how longer range hopping can be engineered to produce a massively degenerate manifold of single-particle ground states with wave functions identical to those making up the lowest Landau level of continuum electrons in a magnetic field. We find that in the presence of local interactions, and at the appropriate filling factors, Laughlin's fractional quantum Hall wave function is an exact many-body ground state of our lattice model. The hopping matrix elements in our model fall off as a Gaussian, and when the flux per plaquette is small compared to the fundamental flux quantum one only needs to include nearest and next-nearest neighbor hoppings. We suggest how to realize this model using atoms in optical lattices, and describe observable consequences of the resulting fractional quantum Hall physics.
A two-dimensional system of electrons with repulsive two-particle interaction is considered in a homogeneous magnetic field. Assuming the limit of high magnetic field the motion of the electrons is restricted to the lowest Landau level. We present quantum-mechanical wave functions for the many-particle system of N spin-polarized fermions on a parallelogram with periodic boundary conditions and with ${\mathit{N}}_{\mathrm{\ensuremath{\Phi}}}$ quanta of magnetic flux flowing through its area. These wave functions are generalizations of the Laughlin wave function to more general rational values of the occupation fraction \ensuremath{\nu}=N/${\mathit{N}}_{\mathrm{\ensuremath{\Phi}}}$. We divide the N electrons into n groups and assume Laughlin-Jastrow-type correlations for electrons within the same group, while electrons in different groups are correlated with a second Jastrow function. The corresponding wave functions exhibit a special structure that depends on \ensuremath{\nu} but is not affected by the interaction between the particles. Furthermore, we consider excited states with quasielectrons and quasiholes. Our theory leads to a unified view of the composite fermion theory of Jain and the hierarchy theory of Haldane, Laughlin, and Halperin. We present a qualitative argument why the fractions of Jain are observed preferably in the experiments.
Finite-size two-dimensional systems are investigated by numerical diagonalization of the Hamiltonian. Both fermion systems and boson systems are investigated. The energy of the liquidlike ground state of the fermion system shows a downward cusp at 13\frac{1}{3}31​ filling of the lowest Landau level. This behavior is consistent with the observed fractional quantum Hall effect. The short-range behavior of the pair distribution function of the ground state is analyzed and good agreement with Laughlin's trial wave function is obtained.