The sign problem is a notorious obstacle for classically simulating quantum theories with fermions. We propose an effective field theory method for analyzing the sign problem. At high temperatures, a $d$+1 dimensional field theory reduces to a bosonic $d$-dimensional theory; the phase of the Pfaffian in the higher dimensional theory is encoded in an operator in the lower dimensional theory. We apply this framework to the D0-brane/BFSS matrix quantum mechanics, where the phase becomes an operator in a bosonic multi-matrix integral. Our results show that the continuum theory has a sign problem that persists in the large-$N$ 't Hooft regime. However, detecting the sign problem involves going to 10-loop order in the high-temperature expansion. This delayed onset follows from the fact that the Pfaffian phase transforms as an $O(9)$ pseudoscalar. Furthermore, the relevant diagrams give a numerically small prefactor. Consequently, ignoring the sign problem leads to a relatively small fractional error in thermodynamic quantities for temperatures $T \gtrsim λ^{1/3}$. However, at stronger coupling in the 't Hooft regime, the sign problem may become more severe. Finally, we initiate the application of this framework to higher-dimensional maximally supersymmetric Yang-Mills theories.
The sign problem is a notorious obstacle for classically simulating quantum theories with fermions. We propose an effective field theory method for analyzing the sign problem. At high temperatures, a d+1 dimensional field theory reduces to a bosonic d-dimensional theory; the phase of the Pfaffian in the higher dimensional theory is encoded in an operator in the lower dimensional theory. We apply this framework to the D0-brane/BFSS matrix quantum mechanics, where the phase becomes an operator in a bosonic multi-matrix integral. Our results show that the continuum theory has a sign problem that persists in the large-N 't Hooft regime. However, detecting the sign problem involves going to 10-loop order in the high-temperature expansion. This delayed onset follows from the fact that the Pfaffian phase transforms as an O(9) pseudoscalar. Furthermore, the relevant diagrams give a numerically small prefactor. Consequently, ignoring the sign problem leads to a relatively small fractional error in thermodynamic quantities for temperatures T≳λ1/3. However, at stronger coupling in the 't Hooft regime, the sign problem may become more severe. Finally, we initiate the application of this framework to higher-dimensional maximally supersymmetric Yang-Mills theories.
Monte Carlo calculations in the framework of lattice field theory provide non-perturbative access to the equilibrium physics of quantum fields. When applied to certain fermionic systems, or to the calculation of out-of-equilibrium physics, these methods encounter the so-called sign problem, and computational resource requirements become impractically large. These difficulties prevent the calculation from first principles of the equation of state of quantum chromodynamics, as well as the computation of transport coefficients in quantum field theories, among other things. This thesis details two methods for mitigating or avoiding the sign problem. First, via the complexification of the field variables and the application of Cauchy's integral theorem, the difficulty of the sign problem can be changed. This requires searching for a suitable contour of integration. Several methods of finding such a contour are discussed, as well as the procedure for integrating on it. Two notable examples are highlighted: in one case, a contour exists which entirely removes the sign problem, and in another, there is provably no contour available to improve the sign problem by more than a (parametrically) small amount. As an alternative, physical simulations can be performed with the aid of a quantum computer. The formal elements underlying a quantum computation - that is, a Hilbert space, unitary operators acting on it, and Hermitian observables to be measured - can be matched to those of a quantum field theory. In this way an error-corrected quantum computer may be made to serve as a well controlled laboratory. Precise algorithms for this task are presented, specifically in the context of quantum chromodynamics.
Monte Carlo calculations in the framework of lattice field theory provide non- perturbative access to the equilibrium physics of quantum fields. When applied to certain fermionic systems, or to the calculation of out-of-equilibrium physics, these methods encounter the so-called sign problem, and computational resource require- ments become impractically large. These difficulties prevent the calculation from first principles of the equation of state of quantum chromodynamics, as well as the computation of transport coefficients in quantum field theories, among other things. This thesis details two methods for mitigating or avoiding the sign problem. First, via the complexification of the field variables and the application of Cauchy’s integral theorem, the difficulty of the sign problem can be changed. This requires searching for a suitable contour of integration. Several methods of finding such a contour are discussed, as well as the procedure for integrating on it. Two notable examples are highlighted: in one case, a contour exists which entirely removes the sign problem, and in another, there is provably no contour available to improve the sign problem by more than a (parametrically) small amount. As an alternative, physical simulations can be performed with the aid of a quantum computer. The formal elements underlying a quantum computation — that is, a Hilbert space, unitary operators acting on it, and Hermitian observables to be measured — can be matched to those of a quantum field theory. In this way an error-corrected quantum computer may be made to serve as a well controlled laboratory. Precise algorithms for this task are presented, specifically in the context of quantum chromodynamics.
The phase diagram of QCD is a central research topic in modern particle physics. Its calculation, however, is notoriously difficult especially when not only nonzero temperatures T but also non-vanishing baryon densities are considered – as is relevant for heavy‑ion collisions or inside large neutron stars. At the theoretical level, nonzero baryon densities are achieved by introducing a nonzero chemical potential μ, which creates a particle-antiparticle asymmetry. It turns out that for nonzero chemical potential, the fermion determinant is no longer real but complex, and therefore standard approaches based on importance sampling break down. In the literature, this is commonly referred to as the sign problem. In order to circumvent this sign problem, a variety of methods has been proposed. Examples include a Taylor expansion in μ, reweighting, complex Langevin, thimbles or path optimization. Nevertheless, none of the proposed remedies can successfully reach regimes in which μ/T>1, and therefore even today the full QCD phase diagram in the μ−T plane cannot be calculated. There is a promising further approach, called tensor-network approach, that differs crucially from all of the aforementioned approaches. In this approach, the partition function is written as a contraction of a large number of local tensors. Of special relevance is the higher order tensor renormalization group (HOTRG) approach, which can be applied for arbitrary number of dimensions, and in particular its generalization, the Grassmann HOTRG (GHOTRG) approach, which is applicable to theories including fermions. Recently, GHOTRG was tailored to strong-coupling QCD using one flavor of staggered quarks in two dimensions. It was observed that, for a non‑zero chemical potential, no numerical instabilities arise. In this thesis, we go beyond both the strong-coupling limit and the single-flavor case. We express the partition function of lattice QCD with staggered fermions in the strong-coupling expansion in terms of a tensor network. This is done for an arbitrary number of dimensions, flavors, and orders of the strong-coupling expansion. Further, the GHOTRG approach is extended such that every coefficient Zn of the partition function Z=∑nZnβn can be calculated separately. Finally, numerical results in two dimensions are studied.
In the functional-integral technique an auxiliary field, coupled to appropriate operators such as spins, linearizes the interaction term in a quantum many-body system. The partition function is then averaged over this time-dependent stochastic field. Quantum Monte Carlo methods evaluate this integral numerically, but suffer from the sign (or phase) problem: the integrand may not be positive definite (or not real). It is shown that, in certain cases that include the many-band Hubbard model and the Heisenberg model, the sign problem is inevitable on fundamental grounds. Here, Monte Carlo simulations generate a distribution of incompatible operators---a Wigner function---from which expectation values and correlation functions are to be calculated; in general no positive-definite distribution of this form exists. The distribution of time-averaged auxiliary fields is the convolution of this operator distribution with a Gaussian of variance proportional to temperature, and is interpreted as a Boltzmann distribution exp(-\ensuremath{\beta}Veff) in classical configuration space. At high temperatures and large degeneracies this classical effective Hamiltonian Veff tends to the static approximation as a classical limit. In the low-temperature limit the field distribution becomes a Wigner function, the sign problem occurs, and Veff is complex. Interpretations of the distributions, and a criterion for their positivity, are discussed. The theory is illustrated by an exact evaluation of the Wigner function for spin s and the effective classical Hamiltonian for the spin-1/2 van der Waals model. The field distribution can be negative here, more noticeably if the number of spins is odd.
Beyond the analysis of the Lee-Yang (LY) zero of ξ at 0 K presented by our previous work [He et. al. Phys. Rev. E 113, 24115 (2026)], it is important but intricate to understand how these zeros evolve with temperature (T). Here, we use an analytically solvable noninteracting one-dimensional particle-on-a-ring model to address this. We determine the trajectories of these zeros and analyze how their evolution with T reshapes the analytic structure of the partition function. In particular, the zero originating from ξ=−1 at T=0 remains close to −1 at low T, where it governs the sign factor and strongly constrains continuation along the real ξ axis. This explains why both direct extrapolation and implicit schemes such as contour-based fitting can fail in the low-T regime, even at high fitting order, while becoming reasonable again once the relevant zeros move away at higher Ts. Furthermore, based on the polynomial structure of the partition function, we propose a new fitting strategy for low-T fermionic properties. The key is to first obtain reliable high-T fermionic properties by continuing sign-problem-free data in ξ∈[0,1] to ξ=−1, and then extend this information toward lower T through T-fitting of the ξ-independent remainder ϕ(β)=ZF. These results provide a solvable benchmark for diagnosing the validity of analytic continuation and suggest a possible route toward treating more realistic interacting fermionic systems.
The ab initio thermodynamic simulation of correlated Fermi systems is of central importance for many applications, such as warm dense matter, electrons in quantum dots, and ultracold atoms. Unfortunately, path integral Monte Carlo (PIMC) simulations of fermions are severely restricted by the notorious fermion sign problem (FSP). In this paper, we present a hands-on discussion of the FSP and investigate in detail its manifestation with respect to temperature, system size, interaction-strength and -type, and the dimensionality of the system. Moreover, we analyze the probability distribution of fermionic expectation values, which can be non-Gaussian and fat-tailed when the FSP is severe. As a practical application, we consider electrons and dipolar atoms in a harmonic confinement, and the uniform electron gas in the warm dense matter regime. In addition, we provide extensive PIMC data, which can be used as a reference for the development of new methods and as a benchmark for approximations.