Entanglement Entropy from Nonequilibrium Work

The Rényi entanglement entropy in quantum many-body systems can be viewed as the difference in free energy between partition functions with different trace topologies. We introduce an external field $\lambda$ that controls the partition function topology, allowing us to define a notion of nonequilibrium work as $\lambda$ is varied smoothly. Nonequilibrium fluctuation theorems of the work provide us with statistically exact estimates of the Rényi entanglement entropy. This framework also naturally leads to the idea of using quench functions with spatially smooth profiles, providing us a way to average over lattice scale features of the entanglement entropy while preserving long distance universal information. We use these ideas to extract universal information from quantum Monte Carlo simulations of SU($N$) spin models in one and two dimensions. The vast gain in efficiency of this method allows us to access unprecedented system sizes up to $192\times 96$ spins for the square lattice Heisenberg antiferromagnet.

Figure 1

(a) The quench protocol outlined in the general method section. The external field $\lambda$ is taken to be spatially constant in the $A$ subsystem (here three sites of a six site chain) and varying in time from 0 to 1. (b) The quench function in Eq. (7) shown at five different time steps. Here the site at $x=2$ is smoothly brought into the entangling region as the subsystem partition is moved from left to right. (c) The same quench function at one instant in time shown for several values of $\delta$. When $\delta$ is small each site is quenched individually, computing exactly the Rényi entanglement entropy of a block subsystem. For larger $\delta$ the subsystem boundary is spread over several sites, generalizing the Rényi entanglement entropy in such a way that lattice scale features are suppressed while universal information is preserved.

Figure 2

Here we compute the Rényi entanglement entropy using the quench function in Eq. (7) for an $L=120$ chain for SU(2), SU(3) and SU(4) with $\delta =1.1$. The colored curves with shaded error bar are the QMC data and black lines are numerical fits. We find perfect agreement with the universal form in Eq. (9) with central charge $c=N-1$.

Figure 3

In the main plot we show the data collected using our two dimensional version of the quench function in Eq. (7) on the staggered SU(2) model for $L=6,8,10,\dots ,20$, $J_2/J_1=1$, and $\delta =0.7$. We use the data from the center cut to fit to the universal scaling form in Eq. (11) and in the inset we plot the center cut data with the area law piece subtracted on semilog axes along with the best fit (black line). This procedure is repeated for SU(3) and SU(4) (also shown in the inset) with $J_2/J_1=2$ and $J_2/J_1=3.5$, respectively. We find excellent agreement with the number of Goldstone modes $N_g=2(N-1)$.

Figure 4

Half-system second Rényi entanglement entropy for the spin-$\frac{1}{2}$ square lattice Heisenberg model. By using $T=0$ projector QMC simulations in the valence bond basis we can much more efficiently reach the ground state. This allows for precise calculations on unprecedented system sizes (up to size $192\times 96$ shown here). Without adding a next-nearest neighbor coupling, we observe very slow convergence to $N_g=2$ when fitting to the scaling form (11).