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arXiv:1805.08219 · Annual Review of Condensed Matter Physics

Sign-Problem-Free Fermionic Quantum Monte Carlo: Developments and Applications

Zi-Xiang Li, Hong Yao

Published Dec 19, 2018• 126 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1146/annurev-conmatphys-033117-054307

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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.
Understanding strongly correlated electrons is an important long-term goal, not only for uncovering fundamental physics behind, but also for their emergence of lots of novel states which have potential applications in quantum control and quantum computations. Meanwhile, the strongly correlated electrons are usually extremely hard problems, and it is generally impossible to understand them unbiasedly. Quantum Monte Carlo is a typical unbiased numeric method, which does not depend on any perturbation, and it can help us to exactly understand the strongly correlated electrons, so that it is widely used in high energy and condensed matter physics. However, quantum Monte Carlo usually suffers from the notorious sign problem. In this paper, we introduce general ideas to design sign problem free models and discuss the sign bound theory we proposed recently. In the sign bound theory, we build a direct connection between the average sign and the ground state properties of the system. We find usually the average sign has the conventional exponential decay with system size increasing, leading to exponential complexity; but for some cases it can have algebraic decay, so that quantum Monte Carlo simulation still has polynomial complexity. By designing sign problem free or algebraic sign behaved strongly correlated electron models, we can approach to several long outstanding problems, such as the itinerant quantum criticality, the competition between unconventional superconductivity and magnetism, as well as the recently found correlated phases and phase transitions in moiré quantum matter.
Metallic quantum critical phenomena are believed to play a key role in many strongly correlated materials, including high-temperature superconductors. Theoretically, the problem of quantum criticality in the presence of a Fermi surface has proven to be highly challenging. However, it has recently been realized that many models used to describe such systems are amenable to numerically exact solution by quantum Monte Carlo (QMC) techniques, without suffering from the fermion sign problem. In this review, we examine the status of the understanding of metallic quantum criticality and the recent progress made by QMC simulations. We focus on the cases of spin-density wave and Ising nematic criticality. We describe the results obtained so far and their implications for superconductivity, non-Fermi liquid behavior, and transport near metallic quantum critical points. Some of the outstanding puzzles and future directions are highlighted.
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.