Entanglement spectra of interacting fermions in quantum Monte Carlo simulations

In a recent article, Grover [Phys. Rev. Lett. 111, 130402 (2013)] introduced a simple method to compute Renyi entanglement entropies in the realm of the auxiliary field quantum Monte Carlo algorithm. Here, we develop this approach further and provide a stabilization scheme to compute higher-order Renyi entropies and an extension to access the entanglement spectrum. The method is tested on systems of correlated topological insulators.

Figure 1

Honeycomb lattice and the regions A and B. In the figure, region A has a width of $W_{\mathrm{A}}=4$.

Figure 2

Renyi entanglement entropies as a function of the regularization parameter $\Lambda$ (a) and order $n$ (b) for a $6\times 6$ Kane-Mele Hubbard model at $U/t=2$, $\lambda /t=0.2$, and $t^{\prime }/t=1$ and the region A defined in Fig. 1. Error bars are smaller than the symbol size. For these simulations, we have used a projection parameter $\Theta t=40$. This guarantees convergence within our error bars. For each replica, we have carried out $500\times {10}^3$ sweeps. Each sweep consists of sequentially updating each Ising spin in the space-imaginary time lattice.

Figure 3

Entanglement spectrum of the dimerized Kane-Mele model at $\lambda /t=0.2$ and $U/t=0$. In (a) and (b) the dimerization runs along the ${\delta }_1$ direction (see Fig. 1). In (c) the dimerization runs along the ${\delta }_2$ direction. The spectrum has been obtained by inverting Eq. (26).

Figure 4

Replica time displaced single-particle Green function (left) and the corresponding single-particle spectral function (right) for the Kane-Mele Hubbard model at $U/t=2$, $\lambda /t=0.2$, and different values of the dimerization $t^{\prime }/t$ across the transition from the QSH insulator to the topologically trivial band insulator. In computing the single-particle spectral function, we have considered the $12\times 12$ lattice with $W_{\mathrm{A}}=8$. For the largest system sizes, $L=12$, we have used $40\times {10}^3$ sweeps per replica and set the imaginary time propagation parameter to $\Theta t=40$. The spectral functions are obtained with the stochastic maximum-entropy method [38, 39].