Boundary-Locality and Perturbative Structure of Entanglement Spectra in Gapped Systems

The entanglement between two parts of a many-body system can be characterized in detail by the entanglement spectrum. Focusing on gapped phases of several one-dimensional systems, we show how this spectrum is dominated by contributions from the boundary between the parts. This contradicts the view of an “entanglement Hamiltonian” as a bulk entity. The boundary-local nature of the entanglement spectrum is clarified through its hierarchical level structure, through the combination of two single-boundary spectra to form a two-boundary spectrum, and finally through consideration of dominant eigenfunctions of the entanglement Hamiltonian. We show consequences of boundary-locality for perturbative calculations of the entanglement spectrum.

Figure 1

Single boundary ES of the $XXZ$ chain. (a) Long or infinite blocks. The ES levels appear at well-defined levels, which correspond to successive perturbative orders $n$. The accompanying integers denote degeneracies. On the right we show the total degeneracies at each order. The levels marked (i) and (ii) have dominant contributions shown in (b), where a filled-rectangle notation is introduced for domain walls. (c) Dominant contributions to low-lying levels. (d) Dominant contributions to low-lying $\delta S_A^z=0$ levels. (e) Finite-size block (12 spins in $A$); arbitrary-precision exact diagonalization data for $\Delta =10$. The effects of finite size are seen through breaking of degeneracies at order $\sim 8$ and higher. The integer-indicated degeneracies are now approximate.

Figure 2

Combining two single-boundary ES to form a two-boundary ES, for the $XXZ$ chain. Shading indicates the $A$ partition. (a and b) The single-boundary ES obtained by considering two different bipartitions. (c) Combination of single-boundary cases to form a two-boundary ES (large empty diamonds), compared with numerical diagonalization data at $\Delta =10$ (small filled diamonds), which match perfectly.

Figure 3

Single-boundary ES (DMRG data) for the (a) Heisenberg spin ladder and (b) Bose–Hubbard chain. For the ladder, we show numerical data for $J_{\mathrm{leg}}=0.1J_{\mathrm{rung}}$, in the rung singlet phase. The partitioning is shown in the top cartoon (region $A$ shaded as in Fig. 2). The integers next to the data points indicate approximate degeneracies. In the associated cartoons, we show with shading the rungs (degrees of freedom) which contribute to various levels of the ES. For the Bose–Hubbard case (data shown for $U=40$), the cartoons show dominant contributions to ES levels. Sites with occupancy $\ne 1$ are shaded; occupancies are indicated with integers on each site. Partition $A$ is again as in Fig. 2. In both panels the horizontal light dashed lines are guides to the eye indicating the perturbation order at which the levels appear.