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arXiv:0803.1349 · Physical Review B

Dynamical mean-field approximation for a two-component Fermi gas

Nir Barnea

Published Jan 17, 2008• 4 citations
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Abstract

The properties of nonrelativistic two-component Fermi gas with attractive two-body contact interaction are studied within dynamical mean-field theory. To this end, the problem is approximated by the lattice Hubbard Hamiltonian. We have found that the low-density gas, corresponding to the limit of empty lattice, can be appropriately described in this theory. Moreover, we have also found that the main properties of the BCS-BEC transition are qualitatively captured as well.

DOI
10.1103/physrevb.77.020501

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The properties of nonrelativistic two-component Fermi gas with attractive two-body contact interaction are studied within dynamical mean-field theory. To this end, the problem is approximated by the lattice Hubbard Hamiltonian. We have found that the low-density gas, corresponding to the limit of empty lattice, can be appropriately described in this theory. Moreover, we have also found that the main properties of the BCS-BEC transition are qualitatively captured as well.
We study the properties of a dilute low-energy two-component Fermi gas. To this end, we approximate the Fermi gas by the Hubbard lattice Hamiltonian and solve it using the dynamical mean-field approximation. At unitarity, this approximation yields the value $\ensuremath{\xi}\ensuremath{\approx}0.44$ for the ratio between the ground-state energy per particle of the interacting and free systems $(E∕N=\ensuremath{\xi}{E}_{FG})$, in remarkable agreement with full quantum Monte Carlo simulations. Investigating the evolution of the system with temperature, we have found that below the critical temperature Tc{T}_{c}Tc​, the system is a superfluid and the energy gap is decreasing monotonously. For temperatures above Tc{T}_{c}Tc​, the system is an insulator and the corresponding energy gap is monotonously increasing.
We investigate the approach to the universal regime of the dilute unitary Fermi gas as the density is reduced to zero in a lattice model. To this end we study the chemical potential, superfluid order parameter and internal energy of the attractive Hubbard model in three different lattices with densities of states (DOSs) which share the same low-energy behavior of fermions in three-dimensional free space: a cubic lattice, a ``Bethe lattice'' with a semicircular DOS, and a ``lattice gas'' with parabolic dispersion and a sharp energy cutoff that ensures the normalization of the DOS. The model is solved using dynamical mean-field theory, that treats directly the thermodynamic limit and arbitrarily low densities, eliminating finite-size effects. At densities on the order of one fermion per site the lattice and its specific form dominate the results. The evolution to the low-density limit is smooth and it does not allow to define an unambiguous low-density regime. Such finite-density effects are significantly reduced using the lattice gas, and they are maximal for the three-dimensional cubic lattice. Even though dynamical mean-field theory is bound to reduce to the more standard static mean field in the limit of zero density due to the local nature of the self-energy and of the vertex functions, it compares well with accurate Monte Carlo simulations down to the lowest densities accessible to the latter.
We investigate the superfluid phase transition in a gas of Fermi atoms loaded on a three-dimensional optical lattice. When the lattice potential is strong, this system can be well described by an attractive Hubbard model. In this model, we calculate the superfluid phase transition temperature Tc{T}_{\mathrm{c}}Tc​, including both superfluid and (spin and charge) density fluctuations within the self-consistent ttt-matrix theory and fluctuation exchange approximation, respectively. Since we treat these fluctuations in a consistent manner, our theory satisfies the required particle-hole symmetry over the entire BCS--Bose-Einstein-condensation (BEC) crossover region. We show that charge density fluctuations compete against superfluid fluctuations near the half-filling, leading to the suppression of Tc{T}_{\mathrm{c}}Tc​. As a result, the maximum Tc{T}_{\mathrm{c}}Tc​ is obtained away from the half-filling. Since the strong density fluctuations originate from the nesting property of the Fermi surface at the half-filling (which is absent in a uniform gas with no lattice potential), our results would be useful in considering lattice effects on strong-coupling superfluidity.
We investigate the superfluid phase transition in a gas of Fermi atoms loaded on a three-dimensional optical lattice. When the lattice potential is strong, this system can be well described by an attractive Hubbard model. In this model, we calculate the superfluid phase transition temperature Tc, including both superfluid and (spin and charge) density fluctuations within the self-consistent t-matrix theory and fluctuation exchange approximation, respectively. Since we treat these fluctuations in a consistent manner, our theory satisfies the required particle-hole symmetry over the entire BCS-BEC crossover region. We show that charge density fluctuations compete against superfluid fluctuations near the half-filling, leading to the suppression of Tc. As a result, the maximum Tc is obtained away the half-filling. Since the strong density fluctuations originate from the nesting property of the Fermi surface at the half filling (which is absent in a uniform gas with no lattice potential), our results would be useful in considering lattice effects on strong-coupling superfluidity.
We consider a mixture of two-component Fermi gases at low temperature. The density profile of this degenerate Fermi gas is calculated under the semiclassical approximation. The results show that the fermion-fermion interactions make a large correction to the density profile at low temperature. The phase separation of such a mixture is also discussed for both attractive and repulsive interatomic interactions, and the numerical calculations demonstrate the existance of a stable temperature region T c1 < T < T c2 for the mixture. In addition, we give the critical temperature of the BCS-type transition in this system beyond the semiclassical approximation.
The ground-state properties of a two-component Fermi gas with attractive short-range interactions are calculated using the fixed-node diffusion Monte Carlo method. The interaction strength is varied over a wide range by tuning the value a of the s-wave scattering length of the two-body potential. We calculate the energy per particle, the one- and two-body density matrix as a function of the interaction strength. Results for the momentum distribution of the atoms, as obtained from the Fourier transform of the one-body density matrix, are reported as a function of the interaction strength. Off-diagonal long-range order in the system is investigated through the asymptotic behavior of the two-body density matrix. The condensate fraction of pairs is calculated in the unitary limit and on both sides of the BCS-BEC crossover.