Entanglement entropy of the two-dimensional Heisenberg antiferromagnet

We compute the von Neumann and generalized Rényi entanglement entropies in the ground-state of the spin-$1/2$ antiferromagnetic Heisenberg model on the square lattice using the modified spin-wave theory for finite lattices. The addition of a staggered magnetic field to regularize the zero modes associated with symmetry breaking is shown to be essential for obtaining well-behaved values for the entanglement entropy. The von Neumann and Rényi entropies obey an area law with additive logarithmic corrections, and are in good quantitative agreement with numerical results from valence bond quantum Monte Carlo and density matrix renormalization group calculations. We also compute the spin fluctuations and observe a multiplicative logarithmic correction to the area law in excellent agreement with quantum Monte Carlo calculations.

Figure 1

Ground-state spin fluctuations in the spin-$1/2$ Heisenberg antiferromagnet on $L\times L$ square lattices with PBCs. For each $L$, the subsystem is half of the total system, i.e., an $L/2\times L$ region as shown in the inset. Dashed lines show fluctuations computed in the spin-wave approximation for different $L$ at fixed values of the staggered magnetic field. Solid lines show fits to the form $\mathcal{F}(L)=aL\ln L+\mathit{bL}+c$, with $a\simeq 0.023,b\simeq 0.065$ for the spin-wave results and $a\simeq 0.023,b\simeq 0.066$ for the QMC results.

Figure 2

Spin-wave results for the ground-state Rényi entropy ${\mathcal{S}}_2$ in the spin-$1/2$ Heisenberg antiferromagnet on $L\times L$ square lattices with PBCs. For each $L$, the subsystems are $\ell \times \ell$ squares as shown in the inset (cf. Ref. 6).

Figure 3

Ground-state von Neumann entanglement entropy $\mathcal{S}$ and Rényi entropies ${\mathcal{S}}_2$, ${\mathcal{S}}_3$, ${\mathcal{S}}_4$ in the spin-$1/2$ Heisenberg antiferromagnet on $L\times L$ square lattices with PBCs. The setup is the same as Fig. 1, i.e., subsystems of size $L/2\times L$. Dashed lines show the von Neumann entropy computed in the spin-wave approximation for different $L$ at fixed values of the staggered magnetic field. Solid lines show fits to the form $\mathcal{S}(L)=aL+b\ln L+c$ (cf. Fig. 4).

Figure 4

Extrapolated values for $a$, $b$, $c$ in fits of the entanglement entropies of Fig. 3 to the form $\mathcal{S}(L)=aL+b\ln L+c$, indicated by dashed lines. Inset shows the scaling of $a$ with the order of the Rényi entropy $\alpha$. Solid line is a fit to $a_{\alpha }=a_{\infty }{e}^{w/\alpha }$ with $a_{\infty }\simeq 0.095$, $w\simeq 1.40$.

Figure 5

Ground-state spin fluctuations $\mathcal{F}$ (scaled by 4), von Neumann entanglement entropy $\mathcal{S}$, and Rényi entropy ${\mathcal{S}}_2$ in the spin-$1/2$ antiferromagnetic Heisenberg model on $4N\times N$ ladders with OBCs. As shown in the inset, for each $N$, the subsystem is the left half, i.e., a $2N\times N$ region. Solid (spin-wave) and dashed (DMRG) lines are guides for the eyes only. QMC results for $\mathcal{F}$ are shown as evidence for the convergence of the entropies, especially for $N=$ 7, 8 (cf. Ref. 5).