Quantum Monte Carlo calculation of entanglement Rényi entropies for generic quantum systems

We present a general scheme for the calculation of the Rényi entropy of a subsystem in quantum many-body models that can be efficiently simulated via quantum Monte Carlo. When the simulation is performed at very low temperature, the above approach delivers the entanglement Rényi entropy of the subsystem, and it allows us to explore the crossover to the thermal Rényi entropy as the temperature is increased. We implement this scheme explicitly within the stochastic series expansion as well as within path-integral Monte Carlo, and apply it to quantum spin and quantum rotor models. In the case of quantum spins, we show that relevant models in two dimensions with reduced symmetry ($XX$ model or hard-core bosons, transverse-field Ising model at the quantum critical point) exhibit an area law for the scaling of the entanglement entropy.

Figure 1

Transition from the ${\mathcal{Z}}^2$ to the ${\mathcal{Z}}_A^{\left(2\right)}$ sector by redefinition of the topology of the simulation box in the additional dimension. The blue/red regions and lines are associated with the action of the imaginary-time propagator $e^{-\beta \mathcal{H}}$. Solid lines link states which are connected by a propagation step.

Figure 2

(a) Entanglement entropy of the one-dimensional (1D) $XX$ chain; the solid line corresponds to exact diagonalization. (b) Entanglement entropy of the 1D O(2) quantum rotor model. Here $L=64$, $t=k_{B}T/J$, and $\epsilon =\Delta \tau U$; increments $\Delta l=1$ are used; the dashed lines correspond to fits to the CFT prediction $(1/4)\ln \left[C\left(l_A\right|L)\right]+s_1$.

Figure 3

Entanglement entropy of 2D spin models with various symmetries; the error bars are smaller than the symbol sizes. The dashed lines represent fits to the form $b+c\left(\ln l\right)/l+d/l$.

Figure 4

Rényi entropy for the 2D $XX$ model at increasing temperature. Error bars are smaller than the symbol size.

Figure 5

Evolution of the fit coefficients of $S_A^{\left(2\right)}$ when increasing the lower bound of the fit region $[l_{\min },4(L/2-1)]$.