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Ein Gitter-Pairing-Field Zugang zu Ultrakalten Fermigasen

Ehmann, Florian

Published Jan 1, 2023• 0 citations
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Abstract

Two-component Fermi gases model the behavior of many systems in different fields of physics, and one of their interesting features is that they condense into superfluids at low temperatures. Ultracold atoms experiments represent one realization of such Fermi gases, and their great flexibility sparked active research into their phase structure. In the literature, there are studies of the phase structure using functional methods, mean-field approximations, and other approaches. In the present work, we aim to perform ab-initio Monte-Carlo simulations of the system to probe their phase structure for inhomogeneous phases in the presence of spin imbalance. Such simulations generally require a bosonization of the theoretical description of the system that rewrites the theory in terms of an auxiliary bosonic field. For this auxiliary field, many possible choices achieve this, and previous studies often use a field that corresponds to a density of fermions. In the present work, we develop a novel approach to this problem by bosonizing the system in terms of the so-called pairing field which corresponds to the superfluid order parameter. Even in the absence of mass- or spin-imbalance, this approach exhibits a sign problem that presents a problem to many Monte-Carlo methods. To circumvent this sign problem, we base our simulation on the Complex Langevin method. The central question of this thesis is if a simulation of a pairing-field-based formalism using the Complex-Langevin method is suitable to study the phase structure of two-component Fermi gases in the presence of spin imbalance. To this end, we develop a lattice theory based on the pairing field by discretizing the continuous Hamiltonian and performing a rigorous derivation of the path integral from there. We pay special attention to the derivation of lattice derivative operators and rescale the theory to be dimensionless. Beyond that, we develop an efficient notation for lattice theories that makes their handling arguably easier than that of continuum theories. With the obtained theory, we derive the Langevin equation for the system together with the expressions we need to sample observables and develop a numerical simulation on that basis. We use the simulation to study 0+1-dimensional systems as a proof-of-concept. Specifically, we calculate density equations of state and two-point functions for a range of dimensionless couplings and compare them to exact solutions. Additionally, we compare our obtained results for the two-point functions to general discussions of the analytic properties of correlation functions to gain additional insight into the qualitative and numerical behavior of our simulation. We found that our results for both density equations of state and two-point functions are in excellent agreement with the exact solutions. Going forward, we plan to study systems in d>0 spatial dimensions, beginning with simulations of 1+1-dimensional systems. In d=3 dimensions the pairing field's correspondence to the superfluid order parameter may allow us to efficiently study the spontaneous breaking of the U(1) symmetry of the system. On that basis, we can probe the phase diagram of the system for inhomogeneous phases at finite spin imbalance in future studies.

DOI
10.26083/tuprints-00024103

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Two-component Fermi gases model the behavior of many systems in different fields of physics, and one of their interesting features is that they condense into superfluids at low temperatures. Ultracold atoms experiments represent one realization of such Fermi gases, and their great flexibility sparked active research into their phase structure. In the literature, there are studies of the phase structure using functional methods, mean-field approximations, and other approaches. In the present work, we aim to perform ab-initio Monte-Carlo simulations of the system to probe their phase structure for inhomogeneous phases in the presence of spin imbalance. Such simulations generally require a bosonization of the theoretical description of the system that rewrites the theory in terms of an auxiliary bosonic field. For this auxiliary field, many possible choices achieve this, and previous studies often use a field that corresponds to a density of fermions. In the present work, we develop a novel approach to this problem by bosonizing the system in terms of the so-called pairing field which corresponds to the superfluid order parameter. Even in the absence of mass- or spin-imbalance, this approach exhibits a sign problem that presents a problem to many Monte-Carlo methods. To circumvent this sign problem, we base our simulation on the Complex Langevin method. The central question of this thesis is if a simulation of a pairing-field-based formalism using the Complex-Langevin method is suitable to study the phase structure of two-component Fermi gases in the presence of spin imbalance. To this end, we develop a lattice theory based on the pairing field by discretizing the continuous Hamiltonian and performing a rigorous derivation of the path integral from there. We pay special attention to the derivation of lattice derivative operators and rescale the theory to be dimensionless. Beyond that, we develop an efficient notation for lattice theories that makes their handling arguably easier than that of continuum theories. With the obtained theory, we derive the Langevin equation for the system together with the expressions we need to sample observables and develop a numerical simulation on that basis. We use the simulation to study 0+1-dimensional systems as a proof-of-concept. Specifically, we calculate density equations of state and two-point functions for a range of dimensionless couplings and compare them to exact solutions. Additionally, we compare our obtained results for the two-point functions to general discussions of the analytic properties of correlation functions to gain additional insight into the qualitative and numerical behavior of our simulation. We found that our results for both density equations of state and two-point functions are in excellent agreement with the exact solutions. Going forward, we plan to study systems in d>0 spatial dimensions, beginning with simulations of 1+1-dimensional systems. In d=3 dimensions the pairing field's correspondence to the superfluid order parameter may allow us to efficiently study the spontaneous breaking of the U(1) symmetry of the system. On that basis, we can probe the phase diagram of the system for inhomogeneous phases at finite spin imbalance in future studies.
Two-component Fermi gases model the behavior of many systems in different fields of physics, and one of their interesting features is that they condense into superfluids at low temperatures. Ultracold atoms experiments represent one realization of such Fermi gases, and their great flexibility sparked active research into their phase structure. In the literature, there are studies of the phase structure using functional methods, mean-field approximations, and other approaches. In the present work, we aim to perform ab-initio Monte-Carlo simulations of the system to probe their phase structure for inhomogeneous phases in the presence of spin imbalance. Such simulations generally require a bosonization of the theoretical description of the system that rewrites the theory in terms of an auxiliary bosonic field. For this auxiliary field, many possible choices achieve this, and previous studies often use a field that corresponds to a density of fermions. In the present work, we develop a novel approach to this problem by bosonizing the system in terms of the so-called pairing field which corresponds to the superfluid order parameter. Even in the absence of mass- or spin-imbalance, this approach exhibits a sign problem that presents a problem to many Monte-Carlo methods. To circumvent this sign problem, we base our simulation on the Complex Langevin method. The central question of this thesis is if a simulation of a pairing-field-based formalism using the Complex-Langevin method is suitable to study the phase structure of two-component Fermi gases in the presence of spin imbalance. To this end, we develop a lattice theory based on the pairing field by discretizing the continuous Hamiltonian and performing a rigorous derivation of the path integral from there. We pay special attention to the derivation of lattice derivative operators and rescale the theory to be dimensionless. Beyond that, we develop an efficient notation for lattice theories that makes their handling arguably easier than that of continuum theories. With the obtained theory, we derive the Langevin equation for the system together with the expressions we need to sample observables and develop a numerical simulation on that basis. We use the simulation to study 0+1-dimensional systems as a proof-of-concept. Specifically, we calculate density equations of state and two-point functions for a range of dimensionless couplings and compare them to exact solutions. Additionally, we compare our obtained results for the two-point functions to general discussions of the analytic properties of correlation functions to gain additional insight into the qualitative and numerical behavior of our simulation. We found that our results for both density equations of state and two-point functions are in excellent agreement with the exact solutions. Going forward, we plan to study systems in d>0 spatial dimensions, beginning with simulations of 1+1-dimensional systems. In d=3 dimensions the pairing field's correspondence to the superfluid order parameter may allow us to efficiently study the spontaneous breaking of the U(1) symmetry of the system. On that basis, we can probe the phase diagram of the system for inhomogeneous phases at finite spin imbalance in future studies.
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.
We develop a pairing-field formalism for ab initio studies of non-relativistic two-component fermions on a (d+1)-dimensional spacetime lattice. More specifically, we focus on theories where the interaction between the two components can be described by the exchange of a corresponding pairing field. The introduction of a pairing field may indeed be convenient for studies of, e.g., the finite-temperature phase structure and critical behavior of, e.g., ultracold atomic Fermi gases. Moreover, such a formalism allows to directly compute the momentum and frequency dependence of the pair propagator, from which the pair-correlation function can be extracted. For a first illustration of the application of our formalism, we compute the density equation of state and the superfluid order parameter for a gas of unpolarized fermions in (0+1) dimensions by employing the complex Langevin approach to surmount the sign problem.
We develop a pairing-field formalism for ab initio studies of non-relativistic two-component fermions on a (d+1)-dimensional spacetime lattice. More specifically, we focus on theories where the interaction between the two components can be described by the exchange of a corresponding pairing field. The introduction of a pairing field may indeed be convenient for studies of, e.g., the finite-temperature phase structure and critical behavior of, e.g., ultracold atomic Fermi gases. Moreover, such a formalism allows to directly compute the momentum and frequency dependence of the pair propagator, from which the pair-correlation function can be extracted. For a first illustration of the application of our formalism, we compute the density equation of state and the superfluid order parameter for a gas of unpolarized fermions in (0+1) dimensions by employing the complex Langevin approach to surmount the sign problem.
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.