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arXiv:0909.3693 · Physical Review B

Density matrix renormalization group for bosonic quantum Hall effect

Dmitry L. Kovrizhin

Published Mar 30, 2010• 19 citations
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Abstract

We developed a density matrix renormalization group technique to study quantum Hall fractions of fast rotating bosons. In this paper, we present a discussion of the method together with the results which we obtain in three distinct cases of the narrow-channel, cylinder, and spherical geometries. In the narrow-channel case, which is relevant to anisotropic confining traps in the limit of extremely fast rotation, we find a series of zero-temperature phase transitions in the strongly interacting regime as a function of the interaction strength between bosons. We compute energies and density profiles for different filling fractions on a cylinder and compare the convergence rates of the method in the cylinder and a sphere geometries.

DOI
10.1103/physrevb.81.125130

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We developed a density matrix renormalization group technique to study quantum Hall fractions of fast rotating bosons. In this paper, we present a discussion of the method together with the results which we obtain in three distinct cases of the narrow-channel, cylinder, and spherical geometries. In the narrow-channel case, which is relevant to anisotropic confining traps in the limit of extremely fast rotation, we find a series of zero-temperature phase transitions in the strongly interacting regime as a function of the interaction strength between bosons. We compute energies and density profiles for different filling fractions on a cylinder and compare the convergence rates of the method in the cylinder and a sphere geometries.
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In this paper, the density-matrix renormalization group method is employed to investigate the fractional quantum Hall effect at filling fractions $\ensuremath{\nu}=1/3$ and 5/25/25/2. We first present benchmark results at both filling fractions for large system sizes to show the accuracy as well as the capability of the numerical algorithm. Furthermore, we show that by keeping a large number of basis states, one can also obtain an accurate entanglement spectrum at $\ensuremath{\nu}=5/2$ for large systems with electron numbers up to Ne=34{N}_{e}=34Ne​=34, much larger than systems previously studied. Based on a finite-size scaling analysis, we demonstrate that the entanglement gap defined by Li and Haldane [Phys. Rev. Lett. 101, 010504 (2008)] is finite in the thermodynamic limit, which characterizes the topological order of the fractional quantum Hall effect state.
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We consider hardcore bosons in two coupled chains of one dimensional lattices at half filling with repulsive intrachain interaction and interchain attraction. This can be mapped onto a coupled chain of spin-1/2 XXZXXZXXZ model with interchain ferromagnetic coupling. We investigate various phases of hardcore bosons (and related spin model) at zero temperature by a density matrix renormalization group method. Apart from the usual superfluid and density wave phases, pairing of interchain bosons leads to the formation of phases like pair superfluid and density wave of strongly bound pairs. We discuss the possible experimental realization of such correlated phases in the context of cold dipolar gas.