Bulk Entanglement Spectrum Reveals Quantum Criticality within a Topological State

A quantum phase transition is usually achieved by tuning physical parameters in a Hamiltonian at zero temperature. Here, we show that the ground state of a topological phase itself encodes critical properties of its transition to a trivial phase. To extract this information, we introduce an extensive partition of the system into two subsystems both of which extend throughout the bulk in all directions. The resulting bulk entanglement spectrum has a low-lying part that resembles the excitation spectrum of a bulk Hamiltonian, which allows us to probe a topological phase transition from a single wave function by tuning either the geometry of the partition or the entanglement temperature. As an example, this remarkable correspondence between the topological phase transition and the entanglement criticality is rigorously established for integer quantum Hall states.

Figure 1

Extensive partition of a topological state yields a topological phase transition in the bulk entanglement Hamiltonian $H_A$ of a subsystem. The horizontal and vertical sequences represent two ways of realizing the transition: the horizontal sequence denotes geometrically tuning the partition towards quantum criticality at the symmetric point (center); the vertical sequence comes from changing the entanglement temperature ($T=1\to T^{\prime }$) at an earlier stage of an asymmetric partition. Dotted arrows always indicate the tracing out procedure. The schematic bulk entanglement spectrum and topological invariant for $A$ are shown below every stage of partition. $C_A=1$ denotes the topological order of the original topological ground state, and $C_A=0$ denotes topologically trivial order.

Figure 2

(a) At the symmetric partition of the integer quantum Hall state, the low-lying states in the bulk entanglement spectrum percolate. (b) The corresponding single particle bulk entanglement spectrum for the symmetric partition of the integer quantum Hall state. The model Hamiltonian used here is $H(k)=(\cos k_x+\cos k_y-\mu ){\sigma }_z+\sin k_x{\sigma }_x+\sin k_y{\sigma }_y$. Each block of the partition used is $5\times 5$ sites. Note the massless Dirac dispersion at $\mathrm{\Gamma }$. (c) An asymmetric partition in which the low-lying states do not percolate. The Hilbert space is the same as that of (a), but the lattice constant is different. The resulting BES is gapped (d).