Tensor network implementation of bulk entanglement spectrum

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.

Figure 1

(a) A segment of a matrix product state partitioned into two spatial subspaces, $A$ and $B$. The open-ended vertical links represent physical degrees of freedom, and the horizontal links represent “virtual” degrees of freedom which are summed as in Eq. (2). (b) The reduced density matrix obtained by tracing out $B$. The MPS transfer matrix is shown by the box.

Figure 2

The tensor network representation of the BES procedure. The reduced density matrices ${\rho }_A$ corresponding to different extensive partitions are shown. Tuning the extensive partition of a topologically nontrivial ground state $\Psi$ realizes a topological phase transition that occurs at the partition where $A$ and $B$ are symmetric (middle). If too little is traced out (top), ${\rho }_A\approx |\Psi \rangle \langle \Psi |$ is nontrivial. If too much is traced out (bottom), ${\rho }_A\approx {\otimes }_{i\in A}{\rho }_i$ is trivial.

Figure 3

The partition function corresponding to the entanglement Hamiltonian from an extensive partition, tracing out blocks of $N_1$ sites and leaving $N_2$ sites in between. Boxed in blue and green are the building blocks of the partition function, namely, the transfer matrices propagating two virtual degrees of freedom from one layer to the next.

Figure 4

The partition function from the entanglement Hamiltonian at the symmetric partition is identified as a classical partition function. (top) The transfer matrix is interpreted as a vertex interaction between Ising variables (large dots). (bottom) The resulting classical model is defined on a new lattice.