Anomalies in the entanglement properties of the square-lattice Heisenberg model

We compute the bipartite entanglement properties of the spin-half square-lattice Heisenberg model by a variety of numerical techniques that include valence-bond quantum Monte Carlo (QMC), stochastic series expansion QMC, high-temperature series expansions, and zero-temperature coupling constant expansions around the Ising limit. We find that the area law is always satisfied, but in addition to the entanglement entropy per unit boundary length, there are other terms that depend logarithmically on the subregion size, arising from broken symmetry in the bulk and from the existence of corners at the boundary. We find that the numerical results are anomalous in several ways. First, the bulk term arising from broken symmetry deviates from an exact calculation that can be done for a mean-field Néel state. Second, the corner logs do not agree with the known results for noninteracting Boson modes. And, third, even the finite-temperature mutual information shows an anomalous behavior as $T$ goes to zero, suggesting that the $T\to 0$ and $L\to \infty$ limits do not commute. These calculations show that entanglement entropy demonstrates a very rich behavior in $d>1$, which deserves further attention.

Figure 1

A possible simulation cell diagram for a six-site system, including loops, operators, the initial valence bond states, and the compatible initial spin states. Up (down) spins are shown in gray (black). (Off-) diagonal operators are shown in (blue) black, giving a total of $M=3$ operators in this example. The dashed line denotes $|V_L\rangle$ (propagated from the trial state on left) and $|V_R\rangle$ (similarly from the right).

Figure 2

One possible simulation cell configuration for the loop ratio algorithm on a six-site system where $\alpha =2$ and region $A^{\prime }$ contains the first two sites of the system. Spins between the usually noninteracting copies are connected through loops via the swap operator (red). The green and orange links are used to show the connections between the sheets. The loop on the left side of the swap operator on the top sheet is connected to the right side on the bottom sheet and so forth.

Figure 3

(Left) The Renyi entropy for a $20\times 20$ PBC system versus the width ($x$) of region $A$ for both the square and strip geometries. The boundary length does not change with the region width for strip geometry (since its height $y$ traverses the periodic lattice), thus the entropy approaches a constant value. For square geometry the boundary length is $4x$. In both geometries the entropy scales with the length of the boundary, with that scaling becoming better as the width of region $A$ approaches half the system size. At right, the “strip” (top) and “square” (bottom) geometries on a $6\times 6$ lattice.

Figure 4

Percentage error in $S_2$ versus the number of operators per site ($m$) for different $L\times L$ lattices. For $L=4,6,8$ the lattices have open boundaries and region $A$ is half the system using the strip geometry. The exact values were found using density matrix renormalization group (DMRG) simulations. The $L=20$ lattice has periodic boundaries and $A$ is a $2\times 2$ square. The “exact” value is taken from the $m=10$ simulation. Each data set was fit to an exponential function.

Figure 5

$S_2/\ell$ vs. $\ell$, where $\ell$ is the boundary length of region $A$, for regions with square and strip geometry embedded in $L\times L$ systems with periodic boundary conditions. The data sets outlined in black correspond to region $A$ with width $x=L/2$ for both geometries. The other data sets use smaller region $A$ of the same geometries. Fits to Eq. (9) are included for all data, and the coefficients found are listed in Table .

Figure 6

The finite-temperature Renyi mutual information for $L\times L$ periodic systems with strip geometry (note $\ell =2L$) for region $A$ with SSE QMC results for $L=8,12,16,24,32$, and [4/6], [5/5], [6/5], and [6/4] Pade approximants for the HTE in the variable $w=\tanh \beta$. The horizontal lines on the left-hand side correspond to the zero temperature results (from VB QMC) for each of the finite-temperature SSE data sets.

Figure 7

The inverse crossover temperature of Renyi mutual information for the system sizes shown in Fig. 6, where the crossover temperature is taken as the temperature at the highest point of the $T>0$ mutual information peak. The $x$ axis is logarithmic, showing that the crossover temperature scales as $\sim$$\log \left(L\right)$.