Entanglement entropy of periodic sublattices

We study the entanglement entropy (EE) of Gaussian systems on a lattice with periodic boundary conditions, both in the vacuum and at nonzero temperatures. By restricting the reduced subsystem to periodic sublattices, we can compute the entanglement spectrum and EE exactly. We illustrate this for a free (1+1)-dimensional massive scalar field at a fixed temperature. Consistent with previous literature, we demonstrate that for a sufficiently large periodic sublattice the EE grows extensively, even in the vacuum. Furthermore, the analytic expression for the EE allows us to probe its behavior both in the massless limit and in the continuum limit at any temperature.

Figure 1

Example of a one-dimensional periodic lattice with $N=12$, with subsystem $A$ consisting of the filled lattice points. On the left $N_A=6$ and $p=2$, while on the right $N_A=4$ and $p=3$.

Figure 2

Continuum limit where $p,N\to \infty$ while keeping $N/p=N_A$ fixed. Here, $N_A=4$.

Figure 3

Two coupled harmonic oscillators, corresponding to $N=2,N_A=1$.

Figure 4

Vacuum EE per lattice site for $p=2$, as a function of $m\epsilon$, for values of $N_A=5$, 10, and 500. Convergence to extensivity is achieved vary rapidly for sufficiently large $m\epsilon$. The EE is decreasing along the flow $m\epsilon$.

Figure 5

Vacuum EE per lattice site for $p=10$, as a function of $m\epsilon$, for values of $N_A=1$, 2, and 100 (chosen so that the value for total $N$ is the same as that in Fig. 4).

Figure 6

Left: The degenerate eigenvalue ${\lambda }_0$ in the continuum limit $p,N\to \infty$. For $m\epsilon \ll 1$ the divergence is slower (logarithmic) than any finite $p$, where it diverges as ${\left(m\epsilon \right)}^{-1/2}$. Right: The corresponding EE for a point on the circle.

Figure 7

Mutual information between two opposite points in the circular lattice with $N$ lattice sites with $m\epsilon ={10}^{-6}$ and $T=1/\beta =0$. In the limit $N\to \infty$ the mutual information goes to zero, albeit rather slowly.