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arXiv:1109.5327 · Physical Review A

Zero-density-limit extrapolation of the superfluid transition temperature in a unitary atomic Fermi gas on a lattice

Qijin Chen

Published Aug 9, 2012• 8 citations
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Abstract

The superfluid transition temperature ${T}_{c}$ of a unitary Fermi gas on a three-dimensional isotropic lattice with an attractive on-site interaction is investigated as a function of density $n$, from half filling down to $5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}7}$ per unit cell, using a pairing fluctuation theory. We show that except at very low densities (${n}^{1/3}<0.2$), where ${T}_{c}/{E}_{F}$ is linear in ${n}^{1/3}$, ${T}_{c}/{E}_{F}$ exhibits a significant higher-order nonlinear dependence on ${n}^{1/3}$. This calls for extra caution against possible significant error in the zero-density-limit value of ${T}_{c}/{E}_{F}$ from typical quantum Monte Carlo (QMC) simulations, obtained by linearly extrapolating data points at intermediate and high densities. Our result, ${T}_{c}/{E}_{F}=0.256$, at the $n=0$ limit, is close to, and should be compared with, the maximum ${T}_{c}/{E}_{F}$ near the Bose-Einstein condensation regime obtained from QMC calculations.

DOI
10.1103/physreva.86.023610

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The superfluid transition temperature Tc{T}_{c}Tc​ of a unitary Fermi gas on a three-dimensional isotropic lattice with an attractive on-site interaction is investigated as a function of density nnn, from half filling down to 5\ifmmode×\else\texttimes\fi10\ensuremath−75\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}7}5\ifmmode×\else\texttimes\fi10\ensuremath−7 per unit cell, using a pairing fluctuation theory. We show that except at very low densities (n1/3<0.2{n}^{1/3}<0.2n1/3<0.2), where Tc/EF{T}_{c}/{E}_{F}Tc​/EF​ is linear in n1/3{n}^{1/3}n1/3, Tc/EF{T}_{c}/{E}_{F}Tc​/EF​ exhibits a significant higher-order nonlinear dependence on n1/3{n}^{1/3}n1/3. This calls for extra caution against possible significant error in the zero-density-limit value of Tc/EF{T}_{c}/{E}_{F}Tc​/EF​ from typical quantum Monte Carlo (QMC) simulations, obtained by linearly extrapolating data points at intermediate and high densities. Our result, Tc/EF=0.256{T}_{c}/{E}_{F}=0.256Tc​/EF​=0.256, at the n=0n=0n=0 limit, is close to, and should be compared with, the maximum Tc/EF{T}_{c}/{E}_{F}Tc​/EF​ near the Bose-Einstein condensation regime obtained from QMC calculations.
The unitary Fermi gas (UFG) is a strongly correlated system of two-species (spin-1/2) fermions with a short-range attractive interaction modeled by a contact interaction and has attracted much interest across different disciplines. The UFG is considered a paradigm for strongly correlated superfluids and has been investigated extensively, with generally good agreement found between theory and experiment. However, the extent of a pseudogap regime above the critical temperature TcT_cTc​ for superfluidity is still debated both theoretically and experimentally. Here we study thermodynamic properties of the UFG across the superfluid phase transition using finite-temperature lattice auxiliary-field quantum Monte Carlo (AFMC) methods in the canonical ensemble of fixed particle numbers. We extrapolate our lattice AFMC results to the continuous time and continuum limits, thus removing the systematic error associated with the finite filling factor of previous AFMC studies. We determine the critical temperature to be Tc=0.16(1) TFT_c=0.16(1)\, T_{F}Tc​=0.16(1)TF​. For the largest particle number studied N=114N=114N=114, the energy-staggering pairing gap is suppressed above a pairing scale temperature of T∗≈0.2 TFT^{*}\approx 0.2\,T_FT∗≈0.2TF​. The spin susceptibility displays moderate suppression above TcT_cTc​ with a spin gap temperature of Ts≈0.2 TFT_s\approx 0.2 \,T_FTs​≈0.2TF​. We also calculate a free energy-staggering pairing gap, which shows substantially reduced statistical errors when compared with the energy-staggering gap, allowing for a clear signature of pairing correlations in the finite-size system. All results indicate that the pseudogap regime is narrow, with pseudogap signatures emerging at temperatures below T∗≈0.2 TFT^{*}\approx 0.2 \, T_FT∗≈0.2TF​. The reduced statistical errors of the free energy gap enable an extrapolation at low temperatures, allowing an estimate of the zero-temperature pairing gap ΔE=0.576(24) εFΔ_E = 0.576(24) \, ε_FΔE​=0.576(24)εF​.
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.
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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 calculate the ground-state properties of an unpolarized two-component Fermi gas with the aid of the diffusion quantum Monte Carlo (DMC) methods. Using an extrapolation to the zero effective range of the attractive two-particle interaction, we find E/EfreeE/{E}_{\mathrm{free}}E/Efree​ in the unitary limit to be 0.212(2), 0.407(2), 0.409(3), and 0.398(3) for 4, 14, 38, and 66 atoms, respectively. Our calculations indicate that the dependence of the total energy on the effective range of the interaction Reff{R}_{\mathrm{eff}}Reff​ is sizable and the extrapolation to Reff=0{R}_{\mathrm{eff}}=0Reff​=0 is therefore important for reaching the true unitary limit. To test the quality of nodal surfaces and to estimate the impact of the fixed-node approximation, we perform released-node DMC calculations for 4 and 14 atoms. Analysis of the released-node and the fixed-node results suggests that the main sources of the fixed-node errors are long-range correlations, which are difficult to sample in the released-node approaches due to the fast growth of the bosonic noise. Besides energies, we evaluate the two-body density matrix and the condensate fraction. We find that the condensate fraction for the 66-atom system converges to 0.56(1) after the extrapolation to the zero interaction range.