Third-order perturbative lattice and complex Langevin analyses of the finite-temperature equation of state of nonrelativistic fermions in one dimension

Abstract

We analyze the pressure and density equations of state of unpolarized non-relativistic fermions at finite temperature in one spatial dimension. For attractively interacting regimes, we perform a third-order lattice perturbation theory calculation, assess its convergence properties by comparing with hybrid Monte Carlo results (there is no sign problem in this regime), and demonstrate agreement with real Langevin calculations. For repulsive interactions, we present lattice perturbation theory results as well as complex Langevin calculations, with a modified action to prevent uncontrolled excursions in the complex plane. Although perturbation theory is a common tool, our implementation of it is unconventional; we use a Hubbard-Stratonovich transformation to decouple the system and automate the application of Wick's theorem, thus generating the diagrammatic expansion, including symmetry factors, at any desired order. We also present an efficient technique to tackle nested Matsubara frequency sums without relying on contour integration, which is independent of dimension and applies to both relativistic and non-relativistic systems, as well as all energy-independent interactions. We find exceptional agreement between perturbative and non-perturbative results at weak couplings, and furnish predictions based on complex Langevin at strong couplings. We additionally present perturbative calculations of up to the fifth-order virial coefficient for repulsive and attractive couplings. Both the lattice perturbation theory and complex Langevin formalisms can easily be extended to a variety of situations including polarized systems, bosons, and higher dimension.

Virial coefficients of one-dimensional and two-dimensional Fermi gases by stochastic methods and a semiclassical lattice approximation

“…We thank A. C. Loheac for discussions and for the perturbation theory data of Ref. [36], W. J. Porter for discussions in the early stages of the project, and J. Levinsen and C. Ordóñez for comments on the manuscript. We also thank the authors of Ref.…”

Thermodynamics of spin-orbit-coupled bosons in two dimensions from the complex Langevin method

“…This prevents the use of traditional Monte Carlo methods, since they use e −S as a probability weight. One alternative in this scenario is the complex Langevin technique, which has been used to study theories with sign problems such as those with repulsive interactions [24] as well as polarized [25,26] and massimbalanced fermions [27] (see Ref. [28] for a review), finite density QCD with staggered quarks [29][30][31][32], random matrix models [33,34], rotating bosons [35,36], superstringinspired matrix models [37], among others.…”

Equation of state of non-relativistic matter from automated perturbation theory and complex Langevin

“…2 (right panel) (which appears in our work of Ref. [6]) is the first time such a calculation was carried out. From the density, by integration over βµ, we obtain the pressure, which is shown in Fig.…”

“…Details of the lattice perturbation theory and CL formalisms, as well as further comments on the results for the unpolarized system, can be found in Ref. [6]. Additionally, we provide a first estimate for the finite-temperature particle density of the polarized Fermi gas at unitarity in Sec.…”

“…The functions f i (σ) correspond to a sum of i-th order Feynman diagrams with associated symmetry factors (see Ref. [6] for details). The resulting perturbative expansion for the pressure P up to N3LO normalized by the non-interacting counterpart P 0 in terms of Feynman diagrams is displayed in Fig.…”