Monte Carlo simulations of fermion systems: The determinant method

Evgueni Bourovski, Nikolay V. Prokof'ev, Boris V. Svistunov

For some models of interacting fermions the known solution to the notorious sign problem in Monte Carlo (MC) simulations is to work with macroscopic fermionic determinants; the price, however, is a macroscopic scaling of the numerical effort spent on elementary local updates. We find that the ratio of two macroscopic determinants can be found with any desired accuracy by considering truncated (local in space and time) matices. In this respect, MC for interacting fermionic systems becomes similar to that for the sign-problem-free bosonic systems with system-size independent update cost. We demonstrate the utility of the truncated-determinant method by simulating the attractive Hubbard model within the MC scheme based on partially summed Feynman diagrams. We conjecture that similar approach may be useful in other implementations of the sign-free determinant schemes.

Yasuhiro Hatsugai, Masatoshi Imada

Xiao Yan Xu, Yang Qi, Junwei Liu +2 more

The self-learning Monte Carlo method is a powerful general-purpose numerical method recently introduced to simulate many-body systems. In this work, we extend it to an interacting fermion quantum system in the framework of the widely used determinant quantum Monte Carlo. This method can generally reduce the computational complexity and moreover can greatly suppress the autocorrelation time near a critical point. This enables us to simulate an interacting fermion system on a $100\ifmmode\times\else\texttimes\fi{}100$ lattice even at the critical point and obtain critical exponents with high precision.

I. Morgenstern, Diethelm W�rtz

Alexander P. Lyubartsev, P. N. Vorontsov‐Velyaminov

A rigorous expression for the partition function of N identical fermions with spin 1/2 is obtained in a form suitable for a path-integral Monte Carlo (PIMC) simulation of a many-electron system at a finite temperature. A sum over permutations is reduced to a sum over classes with readily determinable factors. A distribution over projections and values of the total spin is also obtained, enabling PIMC calculations of spin-dependent quantities. An MC algorithm is proposed which accounts simultaneously for all types of trajectory linkages. A test calculation, four electrons in a spherical cavity with a radius of 1 nm at a temperature of 300--2000 K, is performed.

Francesco Fucito, Enzo Marinari, Giorgio Parisi A +1 more