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Many topologically nontrivial states of matter possess gapless degrees offreedom on the boundary, and when these boundary states delocalize into thebulk, a phase transition occurs and the system becomes topologically trivial.We show that tensor networks provide a natural framework for analyzing suchtopological phase transitions in terms of the boundary degrees of freedom whichmediate it. To do so, we make use of a correspondence between a topologicallynontrivial ground state and its phase transition to a trivial phase establishedin [1]. This involved computing the bulk entanglement spectrum (BES) of theground state, upon tracing out an extensive subsystem. This work implements BESvia tensor network representations of ground states. In this framework, theuniversality class of the quantum critical entanglement Hamiltonian in dspatial dimensions is either derived analytically or mapped to a classicalstatistical model in d + 1 dimensions, which can be studied using Monte Carloor tensor renormalization group methods. As an example, we analytically derivethe universality classes of topological phase transitions from the spin-1 chainHaldane phase and demonstrate that the AKLT wavefunction (and itsgeneralizations) remarkably contains critical six-vertex (and in generaleight-vertex) models within it.

Tensor Network Implementation of Bulk Entanglement Spectrum