DMRG study of FQHE systems in the open cylinder geometry

TL;DR

This paper applies the DMRG algorithm to study the fractional quantum Hall effect in a cylinder geometry, revealing ground states and phase behaviors, including bubble and Wigner crystal states, at various filling factors.

Contribution

It demonstrates the effectiveness of finite-size DMRG in analyzing FQHE systems in cylinder geometry and compares methods for Coulomb interaction regularization.

Findings

Ground state is a compressible two-electron bubble phase.

Wigner crystal state observed at small filling factors.

Comparison of Coulomb regularization methods enhances understanding.

Abstract

The study of the fractional quantum Hall liquid state of two-dimensional electrons requires a non-perturbative treatment of interactions. It is possible to perform exact diagonalizations of the Hamiltonian provided one considers only a small number of electrons in an appropriate geometry. Many insights have been obtained in the past from considering electrons moving on a sphere or on a torus. In the Landau gauge it is also natural to impose periodic boundary conditions in only one direction, the cylinder geometry. The interacting problem now looks formally like a one-dimensional problem that can be attacked by the standard DMRG algorithm. We have studied the efficiency of this algorithm to study the ground state properties of the electron liquid at lowest Landau level filling factor $\nu=1/3$ when the interactions are truncated to the two most important repulsive hard-core components. Use of finite-size DMRG allows us to conclude that the ground state is a compressible two-electron bubble phase in agreement with previous Hartree-Fock calculations. We discuss the treatment of Coulomb interactions in the cylinder geometry. To regularize the long-distance behavior of the Coulomb potential, we compare two methods: using a Yukawa potential or forbidding arbitrary long distances by defining the interelectron distance as the chord distance through the cylinder. This allows us to observe the Wigner crystal state for small filling factor.