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arXiv:1705.06515 · Physical review. B./Physical review. B

Fractional quantum Hall effect in the interacting Hofstadter model via tensor networks

Matthias Gerster, Matteo Rizzi, Pietro Silvi, Marcello Dalmonte, Simone Montangero

Published Nov 13, 2017• 80 citations
arXivHTMLPDF

Abstract

We show via tensor network methods that the Harper-Hofstadter Hamiltonian for hard-core bosons on a square geometry supports a topological phase realizing the $\ensuremath{\nu}=\frac{1}{2}$ fractional quantum Hall (FQH) effect on the lattice. We address the robustness of the ground-state degeneracy and of the energy gap, measure the many-body Chern number, and characterize the system using Green functions, showing that they decay algebraically at the edges of open geometries, indicating the presence of gapless edge modes. Moreover, we estimate the topological entanglement entropy by taking a combination of lattice bipartitions that reproduces the topological structure of the original proposals by Kitaev and Preskill [Phys. Rev. Lett. 96, 110404 (2006)] and Levin and Wen [Phys. Rev. Lett. 96, 110405 (2006)]. The numerical results show that the topological contribution is compatible with the expected value $\ensuremath{\gamma}=\frac{1}{2}$. Our results provide extensive evidence that FQH states are within reach of state-of-the-art cold-atom experiments.

DOI
10.1103/physrevb.96.195123

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We show via tensor network methods that the Harper-Hofstadter Hamiltonian for hard-core bosons on a square geometry supports a topological phase realizing the \ensuremathν=12\ensuremath{\nu}=\frac{1}{2}\ensuremathν=21​ fractional quantum Hall (FQH) effect on the lattice. We address the robustness of the ground-state degeneracy and of the energy gap, measure the many-body Chern number, and characterize the system using Green functions, showing that they decay algebraically at the edges of open geometries, indicating the presence of gapless edge modes. Moreover, we estimate the topological entanglement entropy by taking a combination of lattice bipartitions that reproduces the topological structure of the original proposals by Kitaev and Preskill [Phys. Rev. Lett. 96, 110404 (2006)] and Levin and Wen [Phys. Rev. Lett. 96, 110405 (2006)]. The numerical results show that the topological contribution is compatible with the expected value \ensuremathγ=12\ensuremath{\gamma}=\frac{1}{2}\ensuremathγ=21​. Our results provide extensive evidence that FQH states are within reach of state-of-the-art cold-atom experiments.
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The fractional quantum Hall (FQH) effect illustrates the range of novel phenomena which can arise in a topologically ordered state in the presence of strong interactions. The possibility of realizing FQH-like phases in models with strong lattice effects has attracted intense interest as a more experimentally accessible venue for FQH phenomena which calls for more theoretical attention. Here we investigate the physical relevance of previously derived geometric conditions which quantify deviations from the Landau level physics of the FQHE. We conduct extensive numerical many-body simulations on several lattice models, obtaining new theoretical results in the process, and find remarkable correlation between these conditions and the many-body gap. These results indicate which physical factors are most relevant for the stability of FQH-like phases, a paradigm we refer to as the geometric stability hypothesis, and provide easily implementable guidelines for obtaining robust FQH-like phases in numerical or real-world experiments.
Abstract We demonstrate the emergence of a topological ordered phase for non-Hermitian systems. Specifically, we elucidate that systems with non-Hermitian two-body interactions show a fractional quantum Hall (FQH) state. The non-Hermitian Hamiltonian is considered to be relevant to cold atoms with dissipation. We conclude the emergence of the non-Hermitian FQH state by the presence of the topological degeneracy and by the many-body Chern number for the ground state multiplet showing Ctot = 1. The robust topological degeneracy against non-Hermiticity arises from the manybody translational symmetry. Furthermore, we discover that the FQH state emerges without any repulsive interactions, which is attributed to a phenomenon reminiscent of the continuous quantum Zeno effect.
We analyze a recently proposed method to create fractional quantum Hall (FQH) states of atoms confined in optical lattices [A. S\o{}rensen et al., Phys. Rev. Lett. 94, 086803 (2005)]. Extending the previous work, we investigate conditions under which the FQH effect can be achieved for bosons on a lattice with an effective magnetic field and finite on-site interaction. Furthermore, we characterize the ground state in such systems by calculating Chern numbers which can provide direct signatures of topological order and explore regimes where the characterization in terms of wave-function overlap fails. We also discuss various issues which are relevant for the practical realization of such FQH states with ultracold atoms in an optical lattice, including the presence of a long-range dipole interaction which can improve the energy gap and stabilize the ground state. We also investigate a detection technique based on Bragg spectroscopy to probe these systems in an experimental realization.