Demonstrating the wormhole mechanism of the entanglement spectrum via a perturbed boundary

The Li-Haldane conjecture is one of the most famous conjectures in physics and opens a new research area in the quantum entanglement and topological phase. Although a lot of theoretical and numerical works have confirmed the conjecture in topological states with bulk-boundary correspondence, the cases with gapped boundary and the systems in high dimension are widely unknown. What is the valid scope of the Li-Haldane conjecture? Via the newly developed quantum Monte Carlo scheme, we are now able to extract the large-scale entanglement spectrum (ES) and study its relation with the edge energy spectrum generally. Taking the two-dimensional Affleck-Kennedy-Lieb-Tasaki model with a tunable boundary on the square-octagon lattice as an example, we find several counterexamples which cannot be explained by the Li-Haldane conjecture; e.g., the low-lying entanglement spectrum does not always show similar behaviors as the energy spectrum on the virtual boundary, and sometimes the ES resembles the energy spectrum of the edge even if it is gapped. Finally, we demonstrate that the newly proposed “wormhole mechanism” on the path integral of a reduced density matrix is the formation principle of the general ES. We find that the Li-Haldane conjecture is a particular case in some limit of the wormhole picture while all the examples of the conjecture we have studied can totally be explained within the wormhole mechanism framework. Our results provide important evidence for demonstrating that the wormhole mechanism is the fundamental principle to explain the ES.

Figure 1

(a) A geometrical structure of the partition function ${\mathcal{Z}}_A^{\left(n\right)}$. The subsystem $A$ is entangled with the environment $\overline{A}$. Each replica connects with each other in $A$, while the environment $\overline{A}$ for each replica is independent. ${\beta }_A=n$ represents the imaginary-time length in $A$ and $\beta$ is $1/T$ for the total system. (b) A schematic picture of worldlines crossing a replica. The subsystem $A$ is coupled with the environment $\overline{A}$ via $J_y$. These worldlines which cross the bulk will decay to zero as $\beta \to \infty$ in principle. Meanwhile, the ones which cross the imaginary-time edge of $\overline{A}$ will arrive at the next replica without much cost.

Figure 2

Schematic figures of the model Eq. (2) on the square-octagon lattice under periodic boundary condition (PBC) along the $x$ direction and a tunable boundary in the $y$ direction. (a) Open boundary condition (OBC) in the $y$ direction and $J_s$ perturbations at the lower edge. (b, c) A tunable boundary coupling $J_y$ along the $y$ direction and $J_s$ perturbations at the entanglement boundary which is marked by blue dashed lines. $A$ and $\overline{A}$ denote the subsystem and environment respectively. The $J_y$ coupling can change boundary conditions along the $y$ direction. In particular, the system is under PBC or OBC when $J_y=1$ or 0 respectively.

Figure 3

Dynamical spectra for the $J_1-J_2$ Heisenberg model on the square-octagon lattice under a modified boundary as in Fig. 2 with $L=32$ and $\beta =64$. Upper row: Dynamical spectra on the bottom boundary. Lower row: Dynamical spectra on the top boundary. For better presentation, we set the color bars to be logarithmic scale above 8.0.

Figure 4

Entanglement spectra for the $J_1-J_2$ Heisenberg model on the square-octagon lattice under a modified entanglement boundary as in Figs. 2 and 2 with $L=32$, ${\beta }_A=64$, and $\beta =50$. Upper row: Entanglement spectra of the bottom edge as shown in Fig. 2. Lower row: Entanglement spectra of the bottom edge as shown in Fig. 2. For better presentation, we set the color bars to be logarithmic scale above 8.0.

Figure 5

The entanglement spectra of Fig. 2 with $J_s=0.3$, $L=32$, $\beta =50$, and ${\beta }_A=64$, where $J_y$ is (a)–(f) 0.0, 0.1, 0.2, 0.4, 0.6, and 1.0. For better presentation, we set the color bars to be logarithmic scale above 8.0.

Figure 6

The original entanglement spectra of Fig. 2 with $J_s=0.0$, $L=32$, $\beta =50$, and ${\beta }_A=64$, where $J_y$ is (a)–(f) 0.0, 0.1, 0.2, 0.4, 0.6, and 1.0. For better presentation, we set the color bars to be logarithmic scale above 8.0.

Figure 7

The square-octagon lattice with the PBC in $x$ direction. (a) The lattice with OBC in $y$ direction with $J_y=1$. (b) The lattice with PBC in $y$ direction. The blue dashed line cuts it into two entangled parts, $A$ and $\overline{A}$. And $J_s$ is added to the lower boundary of subsystem $A$ and upper boundary of $\overline{A}$ as green dashed lines show. (c) The lattice with PBC in $y$ direction, where $J_s$ is added to the upper boundary and lower boundary of environment $\overline{A}$ as green dashed lines show.

Figure 8

The original dynamical spectra ($\beta =64$) and entanglement spectra ($\beta =50$ and ${\beta }_A=64$) of Fig. 7 with $J_y=1$ and $L=32$. (a) Dynamical spectrum with $J_s=0.3$. (b) Entanglement spectrum of the lower boundary of subsystem $A$ at Fig. 7 with $J_s=0.3$. (c) Entanglement spectrum of the lower boundary of subsystem $A$ at Fig. 7 with $J_s=0.3$. (d) Entanglement spectrum of the upper boundary of subsystem $A$ at Fig. 7 with $J_s=0.3$. For better presentation, we set the color bars to be logarithmic scale above 8.0.

Figure 9

Different lattice sizes $L$ of boundary equal-time correlation functions $C_k\left(r\right)$ at $J_s=0.3$, $J_y=0.0$, ${\beta }_A=2L$, and $\beta =50$. (a) The original boundary equal-time correlation $|C_1\left(r\right)|$ and $|C_2\left(r\right)|$ with $L=16$, 24, 32, and 40. (b) Rescaling of $|C_1\left(r\right)|$ and $|C_2\left(r\right)|$ for different sizes $L$. The exponent $\eta =1$ is the anomalous dimension. (c) $C_k\left(r\right)$ is marked at different entangled edges on the square-octagon lattice ($k=1,2,3$, and 4).

Figure 10

The finite-size scaling of the entanglement spectra gap for Fig. 2 with $J_s=0.3$, $\beta =50$, and ${\beta }_A=2L$. (a) Imaginary-time correlation function $G\left(\tau \right)$ at $q=\pi$ with $J_y=0.0$. (b) Imaginary-time correlation function $G\left(\tau \right)$ at $q=\pi$ with $J_y=0.1$. (c) Different sizes of entanglement spectra gap with $J_y=0.0$ and 0.1.

Figure 11

Imaginary-time correlation function $G\left(\tau \right)$ of the real Hamiltonian (a) and entanglement Hamiltonian (b) at $q=\pi$ with the perturbation $J_s=0.3$ on the boundary of subsystem $A$, as Figs. 2 and 2 shown.