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arXiv:2302.11262 · Physical Review Letters

Robust Fermi-Liquid Instabilities in Sign Problem-Free Models

Ori Grossman, Erez Berg

Published Aug 1, 2023• 13 citations

Abstract

Determinant quantum Monte Carlo (DQMC) is a powerful numerical technique to study many-body fermionic systems. In recent years, several classes of sign-free (SF) models have been discovered, where the notorious sign problem can be circumvented. However, it is not clear what the inherent physical characteristics and limitations of SF models are. In particular, which zero-temperature quantum phases of matter are accessible within such models, and which are fundamentally inaccessible? Here, we show that a model belonging to any of the known SF classes within DQMC cannot have a stable Fermi-liquid ground state in spatial dimension d≥2, unless the antiunitary symmetry that prevents the sign problem is spontaneously broken (for which there are currently no known examples in SF models). For SF models belonging to one of the symmetry classes (where the absence of the sign problem follows from a combination of nonunitary symmetries of the fermionic action), any putative Fermi liquid fixed point generically includes an attractive Cooper-like interaction that destabilizes it. In the recently discovered lower-symmetry classes of SF models, the Fermi surface (FS) is generically unstable even at the level of the quadratic action. Our results suggest a fundamental link between Fermi liquids and the fermion sign problem. Interestingly, our results do not rule out a non-Fermi-liquid ground state with a FS in a sign-free model.

DOI: 10.1103/physrevlett.131.056501

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Determinant quantum Monte Carlo (DQMC) is a powerful numerical technique to study many-body fermionic systems. In recent years, several classes of sign-free (SF) models have been discovered, where the notorious sign problem can be circumvented. However, it is not clear what the inherent physical characteristics and limitations of SF models are. In particular, which zero-temperature quantum phases of matter are accessible within such models, and which are fundamentally inaccessible? Here, we show that a model belonging to any of the known SF classes within DQMC cannot have a stable Fermi-liquid ground state in spatial dimension d≥2, unless the antiunitary symmetry that prevents the sign problem is spontaneously broken (for which there are currently no known examples in SF models). For SF models belonging to one of the symmetry classes (where the absence of the sign problem follows from a combination of nonunitary symmetries of the fermionic action), any putative Fermi liquid fixed point generically includes an attractive Cooper-like interaction that destabilizes it. In the recently discovered lower-symmetry classes of SF models, the Fermi surface (FS) is generically unstable even at the level of the quadratic action. Our results suggest a fundamental link between Fermi liquids and the fermion sign problem. Interestingly, our results do not rule out a non-Fermi-liquid ground state with a FS in a sign-free model.

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.

Quantum Monte Carlo (QMC) simulations involving fermions have a notorious sign problem. Some well-known exceptions to the auxiliary field QMC algorithm rely on the factorizibility of the fermion determinant. Recently, a fermionic QMC algorithm [C. Wu, J. Hu, and S. Zhang, Phys. Rev. Lett. 91, 186402 (2003)] has been found in which the fermion determinant may not necessarily be factorizable, but can instead be expressed as a product of complex conjugate pairs of eigenvalues, thus eliminating the sign problem for a much wider class of models. In this paper, we present the general conditions for the applicability of this algorithm and point out that it is deeply related to the time-reversal symmetry of the fermion matrix. We apply this method to various models of strongly correlated systems at all doping levels and lattice geometries, and show that many phases can be simulated without the sign problem.

The infamous sign problem leads to an exponential complexity in Monte Carlo simulations of generic many-body quantum systems.Nevertheless, many phases of matter are known to admit a sign-problem-free representative, allowing efficient simulations on classical computers.Motivated by long-standing open problems in many-body physics, as well as fundamental questions in quantum complexity, the possibility of intrinsic sign problems, where a phase of matter admits no sign-problem-free representative, was recently raised but remains largely unexplored.Here we establish the existence of an intrinsic sign problem in a broad class of gapped, chiral, topological phases of matter.Within this class, we exclude the possibility of stoquastic Hamiltonians for bosons (or "qudits") and of sign-problem-free determinantal Monte Carlo algorithms for fermions.The intrinsically sign-problematic class of phases we identify is defined in terms of topological invariants with clear observable signatures: the chiral central charge and the topological spins of anyons.We obtain analogous results for phases that are spontaneously chiral, and present evidence for an extension of our results that applies to both chiral and nonchiral topological matter.

Non-Fermi liquids are an important topic in condensed matter physics, as their characteristics challenge the framework of traditional Fermi liquid theory and reveal the complex behavior of electrons in strongly interacting systems. Both the experimentally observed smeared region and the theoretically predicted marginal Fermi liquid suggest that spatial disorder seems to be an important driver of these phenomena. By performing large-scale determinant quantum Monte Carlo (DQMC) simulations in the ferromagnetic spin-fermion model at finite

N

, beyond the large-

N

used in previous theoretical work, we investigated the role of spatial disorder in the critical Fermi surface (FS) of this model. We proposed a corrected theory of our system, which is based on a modified Eliashberg theory and a universal theory of strange metals. This theory agrees well with the data obtained from DQMC, particularly in capturing the

ω\ln ω

type self-energy characteristic of marginal Fermi liquid behavior, and observing the linear-in-temperature resistivity. Our findings offer strong and unbiased validation of the universal theory of strange metals, broaden the applicability of the modified Eliashberg theory, and provide insights for numerically searching for marginal Fermi liquid and linear-in-temperature resistivity.

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.

A powerful way to guarantee the absence of a sign problem in determinantal quantum Monte Carlo simulations is imposing a particular type of antiunitary symmetries. It is shown that these same symmetries give rise to constraints on correlation functions, which can be used to identify local operators whose correlations upper bound those of a large class of physically relevant operators, including for example superconducting order parameters. Our results also help understand why it is difficult to realize generic finite-momentum orders in sign problem free models.