Surmounting the sign problem in nonrelativistic calculations: A case study with mass-imbalanced fermions

Abstract

The calculation of the ground state and thermodynamics of mass-imbalanced Fermi systems is a challenging many-body problem. Even in one spatial dimension, analytic solutions are limited to special configurations and numerical progress with standard Monte Carlo approaches is hindered by the sign problem. The focus of the present work is on the further development of methods to study imbalanced systems in a fully nonperturbative fashion. We report our calculations of the ground-state energy of mass-imbalanced fermions using two different approaches which are also very popular in the context of the theory of the strong interaction (quantum chromodynamics, QCD): (a) the hybrid Monte Carlo algorithm with imaginary mass imbalance, followed by an analytic continuation to the real axis; and (b) the complex Langevin algorithm. We cover a range of on-site interaction strengths that includes strongly attractive as well as strongly repulsive cases which we verify with nonperturbative renormalization group methods and perturbation theory. Our findings indicate that, for strong repulsive couplings, the energy starts to flatten out, implying interesting consequences for short-range and high-frequency correlation functions. Overall, our results clearly indicate that the Complex Langevin approach is very versatile and works very well for imbalanced Fermi gases with both attractive and repulsive interactions. arXiv:1708.03149v2 [cond-mat.quant-gas]

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

“…[14] and also Refs. [11,15] where this has also been observed in the case of imaginary mass imbalances. Apart from that, an understanding of the emergence of the discrepancy between CL and perturbation theory for repulsive interactions and βµ > 0 with increasing imbalances requires a more detailed analysis, also with respect to the applicability of CL in this regime [9,10].…”

“…However, for a moderately-sized window about ξ 0.1, the calculation becomes well-controlled and converges quickly to the expected value, see also Ref. [11] for a detailed study of the role of this parameter for mass-balanced Fermi gases. The distribution of values of the determinant also indicates that zeros in the determinants are not seen, at least for the couplings studied.…”

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.…”

$Ab~initio$ description of excited states of 1D uniform matter with the Hohenberg–Kohn-theorem-inspired functional-renormalization-group method

“…Therefore, our result suggests that the FRG-DFT is a promising way for the analysis of not only ground states but also excited states of the quantum many-body systems. The FRG-DFT is expected to be adapted to various systems because our formalism can be naturally extended to higher-dimensional systems, and systems with internal degrees of freedom [24,25], superfluidity, and finite temperature.…”

“…The FRG-DFT method has also been applied to a (1+1)-dimensional many-body model simulating one-dimensional nucleons with a fixed particlenumber formalism [23], although the bound-state energy was underestimated by approximately 30% in comparison with the exact solution in two-particle system and over 17% in comparison with the results obtained using the Monte Carlo method [27] when the number of particles is no more than eight. In addition, the study of (1+1)-dimensional systems composed of finite number of spin-1/2 fermions interacting via a contact interaction has been reported [24,25].…”