Tensor Network Implementation of Bulk Entanglement Spectrum

Abstract

Many topologically nontrivial states of matter possess gapless degrees of freedom on the boundary, and when these boundary states delocalize into the bulk, a phase transition occurs and the system becomes topologically trivial. We show that tensor networks provide a natural framework for analyzing such topological phase transitions in terms of the boundary degrees of freedom which mediate it. To do so, we make use of a correspondence between a topologically nontrivial ground state and its phase transition to a trivial phase established in [1]. This involved computing the bulk entanglement spectrum (BES) of the ground state, upon tracing out an extensive subsystem. This work implements BES via tensor network representations of ground states. In this framework, the universality class of the quantum critical entanglement Hamiltonian in d spatial dimensions is either derived analytically or mapped to a classical statistical model in d + 1 dimensions, which can be studied using Monte Carlo or tensor renormalization group methods. As an example, we analytically derive the universality classes of topological phase transitions from the spin-1 chain Haldane phase and demonstrate that the AKLT wavefunction (and its generalizations) remarkably contains critical six-vertex (and in general eight-vertex) models within it.

5 pages, 4 figures, 2 pages of Supplementary Material

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• Non-Abelian Anyons and Topological Quantum Computation
• Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States
• Tensor renormalization group approach to 2D classical lattice models
• Bipartite entanglement and entropic boundary law in lattice spin systems
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Cited by in corpus (4)

• Entanglement Chern number for an extensive partition of a topological ground state
• Entanglement Spectrum of a Random Partition: Connection with the Localization Transition
• Disentangled topological numbers by a purification of entangled mixed states for non-interacting fermion systems
• Topologically distinct critical theories emerging from the bulk entanglement spectrum of integer quantum Hall states on a lattice