Tensor network implementation of bulk entanglement spectrum

2014/07/03 by Timothy H. Hsieh, Liang Fu, Xiao-Liang Qi
Mathematics · Physics and Astronomy · #Ground state #Hamiltonian (control theory) #Mathematics #Phase transition #Physics #Quantum #Quantum and electron transport phenomena #Quantum entanglement #Quantum many-body systems #Quantum mechanics #Renormalization group #Topological Materials and Phenomena #Topology (electrical circuits) #Universality (dynamical systems) #cond-mat.str-el #hep-th

paper · pdf · doi:10.1103/physrevb.90.085137

published as Phys. Rev. B 90, 085137 (2014) · 5 pages, 4 figures, 2 pages of Supplementary Material

arxiv created 2014/07/03 · openalex publication_date 2014/08/25 · arxiv updated 2014/09/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

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 T. Hsieh and L. Fu (arXiv:1305.1949). 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 Affleck-Kennedy-Lieb-Tasaki (AKLT) wave function (and its generalizations) remarkably contains critical six-vertex (and, in general, eight-vertex) models within it.

Citations

• Bulk Entanglement Spectrum Reveals Quantum Criticality within a Topological State
• Critical theory of quantum spin chains
• TOPOLOGICAL ORDERS IN RIGID STATES
• Detecting Topological Order in a Ground State Wave Function
• Topological insulators and superconductors
• Ground-state degeneracy of the fractional quantum Hall states in the presence of a random potential and on high-genus Riemann surfaces
• Quantized Hall Conductance in a Two-Dimensional Periodic Potential
• Non-abelian bosonization in two dimensions
• Entanglement spectrum of a topological phase in one dimension
• Ground state entanglement and geometric entropy in the Kitaev model
• Bipartite entanglement and entropic boundary law in lattice spin systems
• The birth of topological insulators
• Topological Insulators
• Entanglement Entropy and Entanglement Spectrum of the Kitaev Model
• Non-Abelian anyons and topological quantum computation
• Continuum dynamics of the 1-D Heisenberg antiferromagnet: Identification with the O(3) nonlinear sigma model