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

References in corpus (23)

• Topological Insulators
• Topological insulators and superconductors
• Non-Abelian Anyons and Topological Quantum Computation
• $Z_2$ Topological Order and the Quantum Spin Hall Effect
• Topological entanglement entropy
• Detecting topological order in a ground state wave function
• Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States
• Entanglement spectrum of a topological phase in one dimension
• Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order
• Tensor renormalization group approach to 2D classical lattice models
• General Relationship Between the Entanglement Spectrum and the Edge State Spectrum of Topological Quantum States
• Entanglement spectrum of topological insulators and superconductors
• Quasi-particle Statistics and Braiding from Ground State Entanglement
• Bipartite entanglement and entropic boundary law in lattice spin systems
• Ground state entanglement and geometric entropy in the Kitaev's model
• Entanglement entropy and entanglement spectrum of the Kitaev model
• Bulk-Edge Correspondence in the Entanglement Spectra
• A geometric proof of the equality between entanglement and edge spectra
• Boundary-locality and perturbative structure of entanglement spectra in gapped systems
• Entanglement spectra of complex paired superfluids
• Bulk Entanglement Spectrum Reveals Quantum Criticality within a Topological State
• Critical theories of phase transition between symmetry protected topological states and their relation to the gapless boundary theories
• New gapped quantum phases for S=1 spin chain with D2h symmetry

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• Relationship between Symmetry Protected Topological Phases and Boundary Conformal Field Theories via the Entanglement Spectrum
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• Entanglement Spectrum of a Random Partition: Connection with the Localization Transition
• Matrix product states for topological phases with parafermions
• Momentum-space entanglement after a quench in one-dimensional disordered fermionic systems
• Disentangled topological numbers by a purification of entangled mixed states for non-interacting fermion systems
• SU(3) quantum critical model emerging from a spin-1 topological phase
• Meson content of entanglement spectra after integrable and nonintegrable quantum quenches
• Topologically distinct critical theories emerging from the bulk entanglement spectrum of integer quantum Hall states on a lattice
• Bulk Entanglement Spectrum in Gapped Spin Ladders
• Quantum antiferromagnetic Heisenberg half-odd integer spin model as the entanglement Hamiltonian of the integer spin Affleck-Kennedy-Lieb-Tasaki states
• Entanglement of a alternating bipartition in spin chains: Relation with classical integrable models
• Manifestations of topological band crossings in bulk entanglement spectrum: An analytical study for integer quantum Hall states
• Refined Characterization of Lattice Chern Insulators by Bulk Entanglement Spectrum
• Emergent infinite-randomness fixed points from the extensive random bipartitions of the spin-1 Affleck-Kennedy-Lieb-Tasaki topological state