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Particle Projection Using a Complex Langevin Method

C. R. Shill, J.E. Drut

Published Nov 9, 2022• 0 citations
Publisher

Abstract

Using complex stochastic quantization, we implement a particle-number projection technique on the partition function of spin-1/2 fermions at finite temperature on the lattice. We discuss the method, its application towards obtaining the thermal properties of finite Fermi systems in three spatial dimensions, and results for the first five virial coefficients of one-dimensional, attractively interacting fermions.

DOI
10.17615/0nb6-wy56

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Using complex stochastic quantization, we implement a particle-number projection technique on the partition function of spin-1/2 fermions at finite temperature on the lattice. We discuss the method, its application towards obtaining the thermal properties of finite Fermi systems in three spatial dimensions, and results for the first five virial coefficients of one-dimensional, attractively interacting fermions.
We map out the interaction effects on the first six virial coefficients of one-dimensional Fermi gases with zero-range attractive and repulsive interactions and the first four virial coefficients of the two-dimensional analog with attractive interactions. To that end, we use two nonperturbative stochastic methods: projection by complex stochastic quantization, which allows us to determine high-order coefficients at weak coupling and estimate the radius of convergence of the virial expansion, and a path-integral representation of the virial coefficients. To complement our numerical calculations, we present leading-order results in a semiclassical lattice approximation, which we find to be surprisingly close to the expected answers.
We analyze the pressure and density equations of state of unpolarized nonrelativistic fermions at finite temperature in one spatial dimension with contact interactions. 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 nonrelativistic systems, as well as all energy-independent interactions. We find exceptional agreement between perturbative and nonperturbative 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.
The theoretical treatment of Fermi systems consisting of particles with unequal masses is challenging. Even in one spatial dimension analytic solutions are limited to special configurations and numerical progress with Monte Carlo simulations is hindered by the sign-problem. To circumvent this issue, we exploit the Complex Langevin approach and study one-dimensional mass-imbalanced two-component Fermi gases with attractive and repulsive interactions. We find perfect agreement with results obtained by other methods in a range of parameter space. Promisingly, our approach is not limited to the specific model presented here and can easily be extended to finite spin polarization and, most notably, can also be applied in higher dimensions.
The theoretical treatment of Fermi systems consisting of particles with unequal masses is challenging. Even in one spatial dimension analytic solutions are limited to special configurations and numerical progress with Monte Carlo simulations is hindered by the sign-problem. To circumvent this issue, we exploit the Complex Langevin approach and study one-dimensional mass-imbalanced two-component Fermi gases with attractive and repulsive interactions. We find perfect agreement with results obtained by other methods in a range of parameter space. Promisingly, our approach is not limited to the specific model presented here and can easily be extended to finite spin polarization and, most notably, can also be applied in higher dimensions.
The theoretical treatment of Fermi systems consisting of particles with unequal masses is challenging. Even in one spatial dimension analytic solutions are limited to special configurations and numerical progress with Monte Carlo simulations is hindered by the sign-problem. To circumvent this issue, we exploit the Complex Langevin approach and study one-dimensional mass-imbalanced two-component Fermi gases with attractive and repulsive interactions. We find perfect agreement with results obtained by other methods in a range of parameter space. Promisingly, our approach is not limited to the specific model presented here and can easily be extended to finite spin polarization and, most notably, can also be applied in higher dimensions.