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arXiv:1710.10176 · Journal of Physics Conference Series

Advances in non-relativistic matter via complex Langevin approaches

Joaquín E. Drut

Published Jun 1, 2018• 3 citations
arXivHTMLPDF

Abstract

The recent progress in understanding the mathematics of complex stochastic quantization, as well as its application to quantum chromodynamics in situations that have a complex phase problem (e.g. finite quark density, real time), has opened up an intriguing possibility for non-relativistic many-body physics which has so far remained largely unexplored. In this brief contribution, I review a few specific examples of advances in the characterization of the thermodynamics of non-relativistic matter in a variety of one-dimensional cases affected by the sign problem: repulsive interactions, finite polarization, finite mass imbalance, and projection to finite systems to obtain virial coefficients.

DOI
10.1088/1742-6596/1041/1/012005

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The recent progress in understanding the mathematics of complex stochastic quantization, as well as its application to quantum chromodynamics in situations that have a complex phase problem (e.g. finite quark density, real time), has opened up an intriguing possibility for non-relativistic many-body physics which has so far remained largely unexplored. In this brief contribution, I review a few specific examples of advances in the characterization of the thermodynamics of non-relativistic matter in a variety of one-dimensional cases affected by the sign problem: repulsive interactions, finite polarization, finite mass imbalance, and projection to finite systems to obtain virial coefficients.
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.
From ultracold atoms to quantum chromodynamics, reliable ab initio studies of strongly interacting fermions require numerical methods, typically in some form of quantum Monte Carlo calculation. Unfortunately, (non)relativistic systems at finite density (spin polarization) generally have a sign problem, such that those ab initio calculations are impractical. It is well-known, however, that in the relativistic case imaginary chemical potentials solve this problem, assuming the data can be analytically continued to the real axis. Is this feasible for nonrelativistic systems? Are the interesting features of the phase diagram accessible in this manner? By introducing complex chemical potentials, for real total particle number and imaginary polarization, the sign problem is avoided in the nonrelativistic case. To give a first answer to the above questions, we perform a mean-field study of the finite-temperature phase diagram of spin-1/2 fermions with imaginary polarization.
From ultracold atoms to quantum chromodynamics, reliable ab initio studies of strongly interacting fermions require numerical methods, typically in some form of quantum Monte Carlo calculation. Unfortunately, (non)relativistic systems at finite density (spin polarization) generally have a sign problem, such that those ab initio calculations are impractical. It is well-known, however, that in the relativistic case imaginary chemical potentials solve this problem, assuming the data can be analytically continued to the real axis. Is this feasible for nonrelativistic systems? Are the interesting features of the phase diagram accessible in this manner? By introducing complex chemical potentials, for real total particle number and imaginary polarization, the sign problem is avoided in the nonrelativistic case. To give a first answer to the above questions, we perform a mean-field study of the finite-temperature phase diagram of spin-1/2 fermions with imaginary polarization.
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.
Strongly coupled quantum matter displays a rich phenomenology including phase transitions and often unexpected collective behavior. Remarkable advances in experiments with ultracold Fermi gases allow us to gain deep insight into these intriguing systems. Their theoretical description, however, is often challenging as exact analytic solutions are available only in a few special cases, and approximate techniques such as mean-field or perturbation theory are of limited use. Numerical treatment with Monte Carlo (MC) methods has led to profound success in this regard. Unfortunately, for many systems - and especially for asymmetric quantum gases - the infamous sign problem slows progress due to an exponentially scaling of the computational effort with inceasing system size. In this thesis, we set out to explore the rich physics of two-component Fermi gases in the presence of finite spin polarization and/or mass imbalance. To surmount an arising sign problem, we learn from methodological advances made in the field of quantum chromodynamics and further develop these lattice approaches in the context of nonrelativistic Fermi gases. An extensive overview of the numerical methods is presented, including several toy problems to detail the capabilities and shortcomings of the developed approaches. With these tools in hand, we perform extensive benchmarks of the hybrid Monte Carlo method with imaginary asymmetries (iHMC) and the complex Langevin (CL) method, which is based on a complex version of stochastic quantization. Both approaches are shown to yield excellent results for the ground-state energy equation of state of mass-imbalanced Fermi gases in one spatial dimension. Due to its great versatility, the CL method is subsequently employed to study pairing in one-dimensional Fermi gases, for which suitable two-body correlations are computed, revealing unexpected pairing patterns for spin- and mass-imbalanced systems. Another major system of interest in this thesis is the paradigmatic unitary Fermi gas which is investigated at finite temperature and spin polarization. A precise determination of the density equation of state in the normal phase enables us to explore a broad range of thermodynamic properties. We infer valuable information on the finite-temperature phase diagram, such as a flat phase boundary of the normal-to-superfluid transition near the balanced limit and indications for the absence of an extensive pseudogap phase above this transition. The presented results provide experimentally testable ab initio predictions for a range of previously inaccessible thermodynamic quantities.