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arXiv:2309.12811

Fractional quantum Hall states with variational Projected Entangled-Pair States: a study of the bosonic Harper-Hofstadter model

Erik Lennart Weerda, Matteo Rizzi

Published Sep 22, 2023• 1 citation
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Abstract

An important class of model Hamiltonians for investigation of topological phases of matter consists of mobile, interacting particles on a lattice subject to a semiclassical gauge field, as exemplified by the bosonic Harper-Hofstadter model. A unique method for investigations of two-dimensional quantum systems are the infinite projected-entangled pair states, as they avoid spurious finite-size effects that can alter the phase structure. However, due to no-go theorems in related cases, this was often conjectured to be impossible in the past. In this Letter, we show that upon variational optimization, the infinite projected-entangled pair states can be used to this end by identifying fractional Hall states in the bosonic Harper-Hofstadter model. The obtained states are characterized by showing exponential decay of bulk correlations, as dictated by a bulk gap, as well as chiral edge modes via the entanglement spectrum.

DOI
10.34734/fzj-2025-02116

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An important class of model Hamiltonians for investigation of topological phases of matter consists of mobile, interacting particles on a lattice subject to a semiclassical gauge field, as exemplified by the bosonic Harper-Hofstadter model. A unique method for investigations of two-dimensional quantum systems are the infinite projected-entangled pair states, as they avoid spurious finite-size effects that can alter the phase structure. However, due to no-go theorems in related cases, this was often conjectured to be impossible in the past. In this Letter, we show that upon variational optimization, the infinite projected-entangled pair states can be used to this end by identifying fractional Hall states in the bosonic Harper-Hofstadter model. The obtained states are characterized by showing exponential decay of bulk correlations, as dictated by a bulk gap, as well as chiral edge modes via the entanglement spectrum.
An important class of model Hamiltonians for investigation of topological phases of matter consists of mobile, interacting particles on a lattice subject to a semiclassical gauge field, as exemplified by the bosonic Harper-Hofstadter model. A unique method for investigations of two-dimensional quantum systems are the infinite projected-entangled pair states, as they avoid spurious finite-size effects that can alter the phase structure. However, due to no-go theorems in related cases, this was often conjectured to be impossible in the past. In this Letter, we show that upon variational optimization, the infinite projected-entangled pair states can be used to this end by identifying fractional Hall states in the bosonic Harper-Hofstadter model. The obtained states are characterized by showing exponential decay of bulk correlations, as dictated by a bulk gap, as well as chiral edge modes via the entanglement spectrum. Published by the American Physical Society 2024
Infinite projected entangled-pair states (iPEPS) provide a powerful variational framework for two-dimensional quantum matter and have been widely used to capture bosonic topological order, including chiral spin liquids. Here we extend this approach to \emph{fermionic} topological order by variationally optimizing U(1)U(1)U(1)-symmetric fermionic iPEPS for a fractional Chern insulator (FCI), with bond dimensions up to D=9D=9D=9. We find evidence for a critical bond dimension, above which the ansatz faithfully represents the FCI phase. The FCI state is characterized using bulk observables, including the equal-time single-particle Green's function and the pair-correlation function, as well as the momentum-resolved edge entanglement spectrum. To enable entanglement-spectrum calculations for large iPEPS unit cells, we introduce a compression scheme and show that the low-lying part of the spectrum is already well converged at relatively small cutoff dimensions.
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.
Infinite projected entangled-pair states (iPEPSs) provide a powerful variational framework for two-dimensional quantum matter and have been widely used to capture bosonic topological order, including chiral spin liquids. Here, we extend this approach to fermionic topological order by variationally optimizing U(1)-symmetric fermionic iPEPSs for a fractional Chern insulator (FCI), with bond dimensions up to D=9. We find evidence for a critical bond dimension, above which the ansatz faithfully represents the FCI phase. The FCI state is characterized using bulk observables, including the equal-time single-particle Green's function and the pair-correlation function, as well as the momentum-resolved edge entanglement spectrum. To enable entanglement-spectrum calculations for large iPEPS unit cells, we introduce a compression scheme and show that the low-lying part of the spectrum is already well converged at relatively small cutoff dimensions.
A simple spin-12\frac{1}{2}21​ frustrated antiferromagnetic Heisenberg model (AFHM) on the square lattice---including chiral plaquette cyclic terms---was argued [A. E. B. Nielsen, G. Sierra, and J. I. Cirac, Nat. Commun. 4, 2864 (2013)] to host a bosonic Kalmeyer-Laughlin (KL) fractional quantum Hall ground state [V. Kalmeyer and R. B. Laughlin, Phys. Rev. Lett. 59, 2095 (1987)]. Here, we construct generic families of chiral projected entangled pair states (chiral PEPS) with low bond dimension (D=3,4,5D=3,4,5D=3,4,5) which, upon optimization, provide better variational energies than the KL Ansatz. The optimal D=3D=3D=3 PEPS exhibits chiral edge modes described by the Wess-Zumino-Witten SU(2)1\text{SU}{(2)}_{1}SU(2)1​ model, as expected for the KL spin liquid. However, we find evidence that, in contrast to the KL state, the PEPS spin liquids have power-law dimer-dimer correlations and exhibit a gossamer long-range tail in the spin-spin correlations. We conjecture that these features are genuine to local chiral AFHM on bipartite lattices.
Protected chiral edge modes are a well-known signature of topologically ordered phases, such as the fractional quantum Hall states (FQHS). Using the framework of projected entangled pair states (PEPS) on the square lattice, the authors construct a family of chiral spin-12\frac{1}{2}21​ quantum spin liquids with Z2{\mathbb{Z}}_{2}Z2​ gauge symmetry and analyze in full detail the properties of the edge modes. Surprisingly, the results show that the latter can be well described by a chiral conformal field theory of free bosons (SU(2)1{}_{1}1​), as is the case for the $\ensuremath{\nu}$=12\frac{1}{2}21​ (bosonic) gapped Laughlin state, despite the fact that the numerical data here suggest a critical bulk. The authors thus propose that our family of PEPS physically describes a boundary between a chiral topological phase and a trivial phase and might be closely connected to an (unknown) analogous FQHS.