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arXiv:0712.0205 · Journal of Statistical Mechanics Theory and Experiment

The virial equation of state for unitary fermion thermodynamics with non-Gaussian correlations

Jisheng Chen, Jiarong Li, Yanping Wang, Xiang-jun Xia

Published Dec 15, 2008• 11 citations
arXivHTMLPDF

Abstract

We study the roles of the dynamical high order perturbation and statistically non-linear infrared fluctuation/correlation in the virial equation of state for the Fermi gas in the unitary limit. Incorporating the quantum level crossing rearrangement effects, the spontaneously generated entropy departing from the mean-field theory formalism leads to concise thermodynamical expressions. The dimensionless virial coefficients with complex non-local correlations are calculated up to the fourth order for the first time. The virial coefficients of unitary Fermi gas are found to be proportional to those of the ideal quantum gas with integer ratios through a general term formula. Counterintuitively, contrary to those of the ideal bosons ( ) or fermions ( ), the second virial coefficient a 2 of Fermi gas at unitarity is found to be equal to zero. With the vanishing leading order quantum correction, the BCS–BEC crossover thermodynamics manifests the famous pure classical Boyle’s law in the Boltzmann regime. The non-Gaussian correlation phenomena can be validated by studying the Joule–Thomson effect.

DOI
10.1088/1742-5468/2008/12/p12008

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We study the roles of the dynamical high order perturbation and statistically non-linear infrared fluctuation/correlation in the virial equation of state for the Fermi gas in the unitary limit. Incorporating the quantum level crossing rearrangement effects, the spontaneously generated entropy departing from the mean-field theory formalism leads to concise thermodynamical expressions. The dimensionless virial coefficients with complex non-local correlations are calculated up to the fourth order for the first time. The virial coefficients of unitary Fermi gas are found to be proportional to those of the ideal quantum gas with integer ratios through a general term formula. Counterintuitively, contrary to those of the ideal bosons ( ) or fermions ( ), the second virial coefficient a 2 of Fermi gas at unitarity is found to be equal to zero. With the vanishing leading order quantum correction, the BCS–BEC crossover thermodynamics manifests the famous pure classical Boyle’s law in the Boltzmann regime. The non-Gaussian correlation phenomena can be validated by studying the Joule–Thomson effect.
The Joule–Thomson effect reflects the interaction among constituent particles of macroscopic system. For classical ideal gas, the corresponding Joule–Thomson coefficient is vanishing while it is non-zero for ideal quantum gas due to the quantum degeneracy. In recent years, much attention is paid to the unitary Fermi gas with infinite two-body scattering length. According to universal analysis, the thermodynamical law of unitary Fermi gas is similar to that of non-interacting ideal gas, which can be explored by the virial theorem P = 2 E /3 V . Based on previous works, we further study the unitary Fermi gas properties. The effective chemical potential is introduced to characterize the nonlinear levels crossing effects in a strongly interacting medium. The changing behavior of the rescaled Joule–Thomson coefficient according to temperature manifests a quite different behavior from that for ideal Fermi gas.
An effective relativistic continuum massive Proca Lagrangian action is used to account for the Lorentz vector condensation effects on the equation of state of the strongly interacting fermions system. The interior quantum fluctuation effects are incorporated as an external field approximation indirectly through a fictive generalized Thomson Problem counterterm background. The general analytical formulas for the d -dimensional thermodynamics are given near the unitary limit region. In the non-relativistic limit for d = 3, the universal dimensionless coefficient ξ = 4/9 and energy gap Δ/ε f = 5/18 are reasonably consistent with the existing theoretical and experimental results. In the unitary limit for d = 2 and T = 0, the universal coefficient can even approach the extreme occasion ξ = 0 corresponding to the infinite effective fermion mass m * = ∞, which can be mapped to the strongly coupled two-dimensional electrons and is quite similar to the three-dimensional Bose–Einstein condensation of ideal boson gas. Instead, for d = 1, the universal coefficient ξ is negative, implying the non-existence of phase transition from superfluidity to normal state. The solutions manifest the quantum Ising universal class characteristic of the strongly coupled unitary fermions gas.
The dimensionless universal coefficient ξ defines the ratio of the unitary fermions energy density to that for the ideal non-interacting ones in the non-relativistic limit with T = 0. The classical Thomson problem is taken as a nonperturbative quantum many-body arm to address the ground state energy including the low energy nonlinear quantum fluctuation/correlation effects. With the relativistic Dirac continuum field theory formalism, the concise expression for the energy density functional of the strongly interacting limit fermions at both finite temperature and density is obtained. Analytically, the universal factor is calculated to be ξ = 4/9. The energy gap is Δ = (5/18) k f 2 /(2 m ).
We will present recent experimental work on the thermodynamics of strongly interacting Fermi gases. We have developed a general method to probe with high precision the Equation of State (EoS) of locally homogeneous ultracold gases. This allows stringent tests of recent many-body theories. First, we focus on the finite-temperature EoS of the unpolarized unitary gas. Precise thermometry is provided by adding to the Fermi gas of 6Li a trace of bosonic 7Li. We show that the low-temperature properties of the strongly interacting normal phase are well described by Fermi liquid theory and we localize the superfluid transition. Second, we address the zero-temperature EoS of the spin-polarized system in the BEC-BCS crossover. Surprisingly, despite strong interactions, the polarized phase behaves as a mixture of two ideal gases: a Fermi gas of majority atoms and a non-interacting gas of dressed quasi-particles, the Fermi polarons. We also obtain the equation of state of the superfluid state as a function of interaction strength, that we compare to Monte Carlo simulations and to the Lee-Huang-Yang corrections for low-density bosonic and fermionic superfluids.
We use the virial expansion to investigate the behavior of the two-component, attractive Fermi gas in the high-temperature limit, where the system smoothly evolves from weakly attractive fermions to weakly repulsive bosonic dimers as the short-range attraction is increased. We present a formalism for computing the virial coefficients that employs a diagrammatic approach to the grand potential and allows one to easily include an effective range R∗{R}^{*}R∗ in the interaction. In the limit where the thermal wavelength $\ensuremath{\lambda}\ensuremath{\ll}{R}^{*}$, the calculation of the virial coefficients is perturbative even at unitarity and the system resembles a weakly interacting Bose-Fermi mixture for all scattering lengths aaa. By interpolating from the perturbative limits $\ensuremath{\lambda}/|a|\ensuremath{\gg}1$ and ${R}^{*}/\ensuremath{\lambda}\ensuremath{\gg}1$, we estimate the value of the fourth virial coefficient at unitarity for R∗=0{R}^{*}=0R∗=0 and we find that it is close to the value obtained in recent experiments. We also derive the equations of state for the pressure, density, and entropy, as well as the spectral function at high temperatures.
Based on previous works, analytical calculational procedures for dealing with the strongly interacting fermion ground state are further developed through a medium-dependent potential in terms of the Bethe-Peierls contact interaction model. The methods are exact in the unitary limit regime and they lead to self-consistent equations analogous to the Hartree ones. The single particle energy spectrum rearrangement effects on the thermodynamics due to the Hugenholtz--van Hove theorem constraint are addressed. These effects lead to an additional instantaneous correlation potential contribution to the system physical chemical potential and pressure, i.e., equation of state, in order to enforce the classical thermodynamic consistency. The Dyson-Schwinger equations represent implicitly the various Bethe-Goldstone expansion ones. In a thermodynamically self-consistent way, the universal dimensionless factor is analytically calculated to be $\ensuremath{\xi}=\frac{4}{9}$, which defines the ratio of the unitary fermions energy density to that of the ideal noninteracting ones at T=0T=0T=0.