High-efficiency quantum Monte Carlo algorithm for extracting entanglement entropy in interacting fermion systems

The entanglement entropy probing novel phases and phase transitions numerically via quantum Monte Carlo has made great achievements in large-scale interacting spin/boson systems. In contrast, the numerical exploration in interacting fermion systems is rare, even though fermion systems attract more attentions in condensed matter. The fundamental restrictions is that the computational cost of fermion quantum Monte Carlo ($\sim \beta N^3$) is much higher than that of spin/boson ($\sim \beta N$). Here, $N$ is the total number of sites and $\beta$ is the inverse temperature or projection length. To tackle this problem, we propose a fermionic quantum Monte Carlo algorithm based on the incremental technique along physical parameters, which greatly improves the efficiency of extracting entanglement entropy. We benchmark the developed algorithm by calculating the scaling behavior of the entanglement entropy in a two-dimensional square lattice Hubbard model. The obtained phase diagram including Fermi surface and Goldstone modes validates the correctness of the algorithm. Remarkably, our method shows the high-efficiency with respect to the existing algorithms, while keeping the high computation precision. We proceed to apply this algorithm to explore the scaling behavior of the entanglement entropy and particularly its derivative at Gross-Neveu criticality. Our results elucidate that such critical behavior can be quantified by the correlation length exponent.