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arXiv:1008.3348 · Physical Review A

Thermodynamics of balanced and slightly spin-imbalanced Fermi gases at unitarity

Olga Goulko, Matthew Wingate A

Published Nov 19, 2010• 71 citations
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Abstract

In this article we present a Monte Carlo calculation of the critical temperature and other thermodynamic quantities for the unitary Fermi gas with a population imbalance (unequal number of fermions in the two spin components). We describe an improved worm-type algorithm that is less prone to autocorrelations than the previously available methods and show how this algorithm can be applied to simulate the unitary Fermi gas in presence of a small imbalance. Our data indicate that the critical temperature remains almost constant for small imbalances $h=\ensuremath{\Delta}\ensuremath{\mu}/{\ensuremath{\varepsilon}}_{F}\ensuremath{\lesssim}0.2$. We obtain the continuum result ${T}_{c}=0.171(5){\ensuremath{\varepsilon}}_{F}$ in units of Fermi energy and derive a lower bound on the deviation of the critical temperature from the balanced limit, ${T}_{c}(h)\ensuremath{-}{T}_{c}(0)>\ensuremath{-}0.5{\ensuremath{\varepsilon}}_{F}{h}^{2}$. Using an additional assumption a tighter lower bound can be obtained. We also calculate the energy per particle and the chemical potential in the balanced and imbalanced cases.

DOI
10.1103/physreva.82.053621

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In this article we present a Monte Carlo calculation of the critical temperature and other thermodynamic quantities for the unitary Fermi gas with a population imbalance (unequal number of fermions in the two spin components). We describe an improved worm-type algorithm that is less prone to autocorrelations than the previously available methods and show how this algorithm can be applied to simulate the unitary Fermi gas in presence of a small imbalance. Our data indicate that the critical temperature remains almost constant for small imbalances h=\ensuremathΔ\ensuremathμ/\ensuremathεF\ensuremath≲0.2h=\ensuremath{\Delta}\ensuremath{\mu}/{\ensuremath{\varepsilon}}_{F}\ensuremath{\lesssim}0.2h=\ensuremathΔ\ensuremathμ/\ensuremathεF​\ensuremath≲0.2. We obtain the continuum result Tc=0.171(5)\ensuremathεF{T}_{c}=0.171(5){\ensuremath{\varepsilon}}_{F}Tc​=0.171(5)\ensuremathεF​ in units of Fermi energy and derive a lower bound on the deviation of the critical temperature from the balanced limit, Tc(h)\ensuremath−Tc(0)>\ensuremath−0.5\ensuremathεFh2{T}_{c}(h)\ensuremath{-}{T}_{c}(0)>\ensuremath{-}0.5{\ensuremath{\varepsilon}}_{F}{h}^{2}Tc​(h)\ensuremath−Tc​(0)>\ensuremath−0.5\ensuremathεF​h2. Using an additional assumption a tighter lower bound can be obtained. We also calculate the energy per particle and the chemical potential in the balanced and imbalanced cases.
The Fermi gas at unitarity is a particularly interesting system of cold atoms, being dilute and strongly interacting at the same time. It can be studied non-perturbatively with Monte Carlo methods, like the recently developed worm algorithm. We discuss our implementation and tests of this algorithm and suggest a modification that increases its efficiency by reducing autocorrelations. We then show how the worm algorithm can be applied to calculate the critical temperature of an imbalanced Fermi gas (unequal number of fermions in the two spin components). We finally present some results obtained with the modified algorithm, in the balanced as well as in the imbalanced case.
Lattice field theory is a useful tool for studying strongly interacting theories in condensed matter physics.A prominent example is the unitary Fermi gas: a two-component system of fermions interacting with divergent scattering length.With Monte Carlo methods this system can be studied from first principles.In the presence of an imbalance (unequal number of particles in the two components) a sign problem arises, which makes conventional algorithms inapplicable.We will show how to apply reweighting techniques to generalise the recently developed worm algorithm to the imbalanced case, and present results for the critical temperature, the energy per particle, the chemical potential and the contact density for equal, as well as unequal number of fermions in the two spin components.
We simulate the dilute attractive Fermi–Hubbard model in the unitarity regime using a diagrammatic determinant Monte Carlo (MC) algorithm with worm-type updates. We obtain the dependence of the critical temperature on the filling factor ν and, by extrapolating to ν → 0, determine the universal critical temperature of the continuum unitary Fermi gas in units of Fermi energy: T c /ε F = 0.152(7). We also determine the thermodynamic functions and show how the MC results can be used for accurate thermometry of a trapped unitary gas.
We simulate the dilute attractive Fermi-Hubbard model in the unitarity regime using a diagrammatic determinant Monte Carlo algorithm with worm-type updates. We obtain the dependence of the critical temperature on the filling factor ννν and, by extrapolating to ν→0ν\to 0ν→0, determine the universal critical temperature of the continuum unitary Fermi gas in units of Fermi energy: Tc/εF=0.152(7)T_c/ε_F=0.152(7)Tc​/εF​=0.152(7). We also determine the thermodynamic functions and show how the Monte Carlo results can be used for accurate thermometry of a trapped unitary gas.
Fermionic matter is ubiquitous in nature, from the electrons in metals and semiconductors or the neutrons in the inner crust of neutron stars, to gases of fermionic atoms, like 40K or 6Li that can be created and studied under laboratory conditions. It is especially interesting to study these systems at very low temperatures, where we enter the world of quantum mechanical phenomena. Due to the Fermi-Dirac statistics, a dilute system of spin-polarised fermions exhibits no interactions and can be viewed as an ideal Fermi gas. However, interactions play a crucial role for fermions of several spin species. This thesis addresses several questions concerning interacting Fermi gases, in particular the transition between the normal and the superfluid phase and dynamical properties at higher temperatures. First we will look at the unitary Fermi gas: a two-component system of fermions interacting with divergent scattering length. This system is particularly interesting as it exhibits universal behaviour. Due to the strong interactions perturbation theory is inapplicable and no exact theoretical description is available. I will describe the Determinant Diagrammatic Monte Carlo algorithm with which the unitary Fermi gas can be studied from first principles. This algorithm fails in the presence of a spin imbalance (unequal number of particles in the two components) due to a sign problem. I will show how to apply reweighting techniques to generalise the algorithm to the imbalanced case, and present results for the critical temperature and other thermodynamic observables at the critical point, namely the chemical potential, the energy per particle and the contact density. These are the first numerical results for the imbalanced unitary Fermi gas at finite temperature. I will also show how temperatures beyond the critical point can be accessed and present results for the equation of state and the temperature dependence of the contact density. At sufficiently high temperatures a semiclassical description captures all relevant physical features of the system. The dynamics of an interacting Fermi gas can then be studied via a numerical simulation of the Boltzmann equation. I will describe such a numerical setup and apply it to study the collision of two spin-polarised fermionic clouds. When the two components are separated in an elongated harmonic trap and then released, they collide and for sufficiently strong interactions can bounce off each other several times. I will discuss the different types of the qualitative behaviour, show how they can be interpreted in terms of the equilibrium properties of the system, and explain how they relate to the coupling between different excitation modes. I will also demonstrate how transport coefficients, for instance the spin drag, can be extracted from the numerical data.
In the two-component Fermi gas with contact interactions a pseudogap regime, in which pairing correlations are present without superfluidity, can exist at temperatures between the superfluid critical temperature TcT_cTc​ and a temperature T∗>TcT^* > T_cT∗>Tc​. However, the existence of a pseudogap in the unitary limit of infinite scattering length is debated. To help address this issue, we have used finite-temperature auxiliary-field quantum Monte Carlo (AFMC) methods to study the thermodynamics of the spin-balanced homogeneous unitary Fermi gas on a lattice. We present results for the thermal energy, heat capacity, condensate fraction, a model-independent pairing gap, and spin susceptibility, and compare them to experimental data when available. Our model space consists of the complete first Brillouin zone of the lattice, and our calculations are performed in the canonical ensemble of fixed particle number. We find that the energy-staggering pairing gap vanishes above TcT_cTc​ and that T∗T^*T∗ at unitarity, as determined from the spin susceptibility, is lower than previously reported in AFMC simulations.