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arXiv:cond-mat/0312108 · Physical Review B

Truncated-determinant diagrammatic Monte Carlo for fermions with contact interaction

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

Published Nov 4, 2004• 16 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.70.193101

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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.
We present a simple trick that allows us to consider the sum of all connected Feynman diagrams at fixed position of interaction vertices for general fermionic models, such that the thermodynamic limit can be taken analytically. With our approach one can achieve superior performance compared to conventional diagrammatic Monte Carlo algorithm, while rendering the algorithmic part dramatically simpler. By considering the sum of all connected diagrams at once, we allow for massive cancellations between different diagrams, greatly reducing the sign problem. In the end, the computational effort increases only exponentially with the order of the expansion, which should be contrasted with the factorial growth of the standard diagrammatic technique. We illustrate the efficiency of the technique for the two-dimensional Fermi-Hubbard model.
We introduce a Monte Carlo determinantal algorithm that allows to efficiently sample dynamical quantities, such as the self-energy, of fermionic quantum systems in their thermodynamic limit. The method expresses the sum of a factorial number of one-particle-irreducible diagrams as a recursive sum of determinants with exponential complexity. By comparing results of this algorithm for the lattice Hubbard model with those obtained from diagrammatic Monte Carlo and the recently introduced connected determinant approach, we show that we can reach higher perturbation orders and greater accuracy for the same computational effort.
As an intrinsically-unbiased approach, quantum Monte Carlo (QMC) is of vital importance in understanding correlated phases of matter. Unfortunately, it often suffers notorious sign problem when simulating interacting fermion models. Here, we show for the first time that there exist interacting fermion models whose sign problem becomes less severe for larger system sizes and eventually disappears in the thermodynamic limit, which we dub as "asymptotic sign free". We demonstrate asymptotically-free sign in determinant QMC for various interacting models. Moreover, based on renormalization-group-like ideas we propose a heuristic understanding of the feature of asymptotic sign free. We believe that asymptotic sign free behavior could shed new lights to deepening our understanding of sign problem. More importantly, it can provide a promising way to decipher intriguing physics in correlated models which were conventionally thought not accessible by QMC.
Reliable simulations of correlated quantum systems, including high-temperature superconductors and frustrated magnets, are increasingly desired nowadays to further our understanding of essential features in such systems. Quantum Monte Carlo (QMC) is a unique numerically exact and intrinsically unbiased method to simulate interacting quantum many-body systems. More importantly, when QMC simulations are free from the notorious fermion sign problem, they can reliably simulate interacting quantum models with large system size and low temperature to reveal low-energy physics such as spontaneously broken symmetries and universal quantum critical behaviors. Here, we concisely review recent progress made in developing new sign-problem-free QMC algorithms, including those employing Majorana representation and those utilizing hot-spot physics. We also discuss applications of these novel sign-problem-free QMC algorithms in simulations of various interesting quantum many-body models. Finally, we discuss possible future directions of designing sign-problem-free QMC methods.
Sign problem in quantum Monte Carlo (QMC) simulation appears to be an extremely hard yet interesting problem. In this article, we present a pedagogical overview on the origin of the sign problem in various quantum Monte Carlo simulation techniques, ranging from the world-line and stochastic series expansion Monte Carlo for boson and spin systems to the determinant and momentum-space quantum Monte Carlo for interacting fermions. We point out the basis dependency of the sign problem and summarize the progresses to cure, ease and even make use of the sign problem over the years, such as symmetry analysis of the underlying Hamiltonian, basis optimization in writting down the partition functions and many others. Moreover, we state that although traditional lore saying that in case of sign problem, the average sign in QMC simulation approaches zero exponentially fast with the space-time volume of the configurational space, there are recent breakthroughs showing this is not always the case and based on the properties of the partition function for finite size systems, one could distinguish when the average sign has the usual exponential scaling and when it is bestowed with an algebraic scaling at the low temperature limit. Fermionic QMC simulations with such algebraic sign problems have been successfully carried out for extended Hubbard-type and quantum Moiré lattice models.
We introduce and compare three different Monte Carlo determinantal algorithms that allow one to compute dynamical quantities, such as the self-energy, of fermionic systems in their thermodynamic limit. We show that the most efficient approach expresses the sum of a factorial number of one-particle-irreducible diagrams as a recursive sum of determinants with exponential complexity. By comparing results for the two-dimensional Hubbard model with those obtained from state-of-the-art diagrammatic Monte Carlo, we show that we can reach higher perturbation orders and greater accuracy for the same computational effort.