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arXiv:cond-mat/0502372

Critical temperature for fermion pairing using lattice field theory

Matthew Wingate A

Published Feb 15, 2005• 5 citations
arXivHTMLPDF

Abstract

Dilute gases of 2-component fermions are of great interest in atomic and nuclear physics. When interactions are strong enough so that a bound state is at threshold, universal behavior is expected. Lattice field theory provides a first principles approach to the study of strongly interacting systems such as this through Monte Carlo simulation. Results of exploratory simulations are presented here. In particular, the finite temperature phase transition between superfluid and normal states is studied. We present first results for the critical temperature T_c and describe the future work necessary to determine T_c as a function of interaction strength.

DOI
10.48550/arxiv.cond-mat/0502372

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Dilute gases of 2-component fermions are of great interest in atomic and nuclear physics. When interactions are strong enough so that a bound state is at threshold, universal behavior is expected. Lattice field theory provides a first principles approach to the study of strongly interacting systems such as this through Monte Carlo simulation. Results of exploratory simulations are presented here. In particular, the finite temperature phase transition between superfluid and normal states is studied. We present first results for the critical temperature T_c and describe the future work necessary to determine T_c as a function of interaction strength.
We study the system of a dilute gas of fermions in 3-dimensions, with attractive interactions tuned to the unitarity point, using the non-perturbative Restricted Path Integral Monte Carlo (R-PIMC) method. The pairing and superfluid properties of this system are calculated at finite temperature. The total energy at very low temperature from our results agrees closely with that of previous ground-state Quantum Monte Carlo calculations. We identify the temperature T � � 0.70ǫF below which pairing correlations develop, and estimate the critical temperature for the superfluid transition Tc � 0.25ǫF from a finite size scaling analysis of the superfluid density. PACS numbers: The recent experiments, starting with generating a degenerate gas of cold atoms[3], the use of Feshbach resonance to tune the effective interaction between the fermions[4, 5], the measurements of the gap to identify a pairing scale[6] and the measurements of vortices[7] to identify the superfluid state, have all given an impetus to the study of superfluidity in the BCS-BEC crossover regime. The nature of superconductivity or superfluidity in a many-particle system with an increasing pairing attraction was shown [1, 2] to interpolate smoothly between the BCS regime for weak attraction between the fermions, and the Bose-Einstein condensation (BEC) regime for strong coupling. In the weak coupling regime, the Cooper pair size is much larger than the interparticle spacing, and the simultaneous pairing and condensation of fermions is well described by the BCS theory. In the strong coupling regime, fermions form strongly bound bosonic molecules at a pairing temperature scale T ∗ that is considerably higher than the BEC temperature Tc. In a two species system of fermions with an attractive interaction between them, the unitary point is defined by the divergence of the zero energy s-channel scattering length, as, for two particles in free space. The unitary point is interesting since it describes a strongly correlated system with universal properties[8]. In general, there are two length scales that define the system: the interparticle spacing ∝ n −1/3 , and the scattering length as which contains information about the interaction potential. At the unitary point however, since as diverges, all the properties of the system are described by a single length scale k −1
We consider pairing in a three-component gas of degenerate fermions. In particular, we solve the finite-temperature mean-field theory of an interacting gas for a system where both interaction strengths and fermion masses can be unequal. At zero temperature, we find the possibility of a quantum phase transition between states associated with pairing between different pairs of fermions. On the other hand, finite-temperature behavior of the three-component system reveals some qualitative differences from the two-component gas: for a range of parameters it is possible to have two different critical temperatures. The lower one corresponds to a transition between different pairing channels, while the higher one corresponds to the usual superfluid-normal transition. We discuss how these phase transitions could be observed in ultracold gases of fermionic atoms.
We study the system of a dilute gas of fermions in three dimensions, with attractive interactions tuned to the unitarity point, using the nonperturbative restricted path-integral Monte Carlo method. The pairing and superfluid properties of this system are calculated at finite temperature. The total energy at very low temperature from our results agrees closely with that of previous ground-state quantum Monte Carlo calculations. We identify the temperature ${T}^{*}\ensuremath{\approx}0.70{ϵ}_{F}$, below which pairing correlations develop, and estimate the critical temperature for the superfluid transition ${T}_{c}\ensuremath{\approx}0.245{ϵ}_{F}$ from a finite size scaling analysis of the superfluid density.
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.
The strongly correlated regime of the crossover from Bardeen-Cooper-Schrieffer pairing to Bose-Einstein condensation can be realized by diluting a system of two-component fermions with a short-range attractive interaction. We investigate this system via a novel continuous-space-time diagrammatic determinant Monte Carlo method and determine the universal curve Tc/epsilonF for the transition temperature between the normal and the superfluid states as a function of the scattering length with the maximum on the Bose-Einstein condensation side. At unitarity, we confirm that Tc/epsilonF=0.152(7).