Zone methods and the fermion sign problem

Ali Alavi, Alex J. W. Thom

Dean Lee

We present a new approach to the problem of alternating signs for fermionic many body Monte Carlo simulations. We demonstrate that the exchange of identical fermions is typically short-ranged even when the underlying physics is dominated by long distance correlations. We show that the exchange process has a maximum characteristic range of sqrt[2*(1-f)*beta*h] lattice sites, where beta is the inverse temperature, h is the hopping parameter, and f is the filling fraction. We introduce the notion of permutation zones, special regions of the lattice where identical fermions may interchange and outside of which they may not. Using successively larger permutation zones, one can extrapolate to obtain thermodynamic observables in regimes where direct simulation is impossible.

Yunuo Xiong, Hongwei Xiong

The fermion sign problem remains the primary obstacle in simulating the thermodynamic properties of various fermionic systems. In this work, we present a sign-blocking method to mitigate the numerical instability inherent in the sign problem. In the sign-blocking method, the Monte Carlo importance sampling remains identical to traditional methods; instead, the sign-blocking method is applied during the post-processing of signed samples. Given the significant progress in simulating the 2D Fermi-Hubbard model over the past decade, a wealth of energy benchmarks is available for comparison. Consequently, we use the 2D Fermi-Hubbard model as a benchmark to validate the sign-blocking method. Surprisingly, our results align exceptionally well with existing state-of-the-art benchmarks, even in regimes previously considered challenging. The physical mechanism of the sign-blocking method lies in uncovering the correlation between energy and sign factors through data blocking, thereby successfully inferring the fermionic system's energy. Our findings suggest that the sign-blocking method holds promise for complex quantum systems, particularly when combined with appropriate simulation techniques such as auxiliary-field formalisms that trace out the fermionic degrees of freedom.

Yunuo Xiong, Tommaso Morresi, Hongwei Xiong

For strongly quantum-degenerate systems at finite temperatures, the fermion sign problem remains the major obstacle to first-principles simulations. In this work, we apply the recently proposed pseudo-fermion method - designed to overcome the sign problem - to strongly quantum-degenerate uniform electron gases. We find that the pseudo-fermion method can efficiently and highly accurately infer the energy of the uniform electron gas while being free from the fermion sign problem. For example, in the strongly quantum-degenerate regime where RPIMC fails (33 spin-polarized electrons at the density parameter $r_s = 0.5$), the relative deviation between the pseudo-fermion method and the exact CPIMC result is only 0.6%. In particular, the pseudo-fermion method bridges the gap where neither CPIMC nor RPIMC can accurately simulate the regime $1 \le r_s \le 2$ at the reduced temperature $θ= 0.0625$. This work demonstrates that the pseudo-fermion method opens a new pathway for studying strongly quantum-degenerate systems in a sign-problem-free manner.