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arXiv:1910.01257 · Physical Review A

Superfluid condensate fraction and pairing wave function of the unitary Fermi gas

Rongzheng He, Ning Li, Bing-Nan Lu, Dean Lee

Published Jun 11, 2020• 11 citations
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Abstract

The unitary Fermi gas is a many-body system of two-component fermions with zero-range interactions tuned to infinite scattering length. Despite much activity and interest in unitary Fermi gas and its universal properties, there have been great difficulties in performing accurate calculations of the superfluid condensate fraction and pairing wave function. In this paper, we present auxiliary-field lattice Monte Carlo simulations using a lattice interaction which accelerates the approach to the continuum limit, thereby allowing for robust calculations of these difficult observables. As a benchmark test, we compute the ground-state energy of 33 spin-up and 33 spin-down particles. As a fraction of the free Fermi gas energy ${E}_{\text{FG}}$, we find ${E}_{0}/{E}_{\text{FG}}=0.369(2),0.372(2),$ using two different definitions of the finite-system energy ratio, in agreement with the latest theoretical and experimental results. We then determine the condensate fraction by measuring off-diagonal long-range order in the two-body density matrix. We find that the fraction of condensed pairs is $\ensuremath{\alpha}=0.43(2)$. We also extract the pairing wave function and find the pair correlation length to be ${\ensuremath{\zeta}}_{p}{k}_{F}=1.8(3)\ensuremath{\hbar}$, where ${k}_{F}$ is the Fermi momentum. Provided that the simulations can be performed without severe sign oscillations, the methods we present here can be applied to superfluid neutron matter as well as more exotic $P$-wave and $D$-wave superfluids.

DOI
10.1103/physreva.101.063615

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The unitary Fermi gas is a many-body system of two-component fermions with zero-range interactions tuned to infinite scattering length. Despite much activity and interest in unitary Fermi gas and its universal properties, there have been great difficulties in performing accurate calculations of the superfluid condensate fraction and pairing wave function. In this paper, we present auxiliary-field lattice Monte Carlo simulations using a lattice interaction which accelerates the approach to the continuum limit, thereby allowing for robust calculations of these difficult observables. As a benchmark test, we compute the ground-state energy of 33 spin-up and 33 spin-down particles. As a fraction of the free Fermi gas energy EFG{E}_{\text{FG}}EFG​, we find E0/EFG=0.369(2),0.372(2),{E}_{0}/{E}_{\text{FG}}=0.369(2),0.372(2),E0​/EFG​=0.369(2),0.372(2), using two different definitions of the finite-system energy ratio, in agreement with the latest theoretical and experimental results. We then determine the condensate fraction by measuring off-diagonal long-range order in the two-body density matrix. We find that the fraction of condensed pairs is \ensuremathα=0.43(2)\ensuremath{\alpha}=0.43(2)\ensuremathα=0.43(2). We also extract the pairing wave function and find the pair correlation length to be \ensuremathζpkF=1.8(3)\ensuremathℏ{\ensuremath{\zeta}}_{p}{k}_{F}=1.8(3)\ensuremath{\hbar}\ensuremathζp​kF​=1.8(3)\ensuremathℏ, where kF{k}_{F}kF​ is the Fermi momentum. Provided that the simulations can be performed without severe sign oscillations, the methods we present here can be applied to superfluid neutron matter as well as more exotic PPP-wave and DDD-wave superfluids.
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.
In the two-component Fermi gas with a contact interaction, a pseudogap regime can exist at temperatures between the superfluid critical temperature T_{c} and a temperature T^{*}>T_{c}. This regime is characterized by pairing correlations without superfluidity. However, in the unitary limit of infinite scattering length, the existence of this regime is still debated. To help address this, we have applied finite-temperature auxiliary-field quantum Monte Carlo (AFMC) methods to study the thermodynamics of the superfluid phase transition and signatures of the pseudogap in the spin-balanced homogeneous unitary Fermi gas. We present results at finite filling factor ν≃0.06 for the condensate fraction, an energy-staggering pairing gap, the spin susceptibility, and the heat capacity, and compare them to experimental data when available. In contrast to previous AFMC simulations, 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. The canonical ensemble AFMC framework enables the calculation of a model-independent gap, providing direct information on pairing correlations without the need for numerical analytic continuation. We use finite-size scaling to estimate T_{c} at the corresponding filling factor. We find that the energy-staggering pairing gap vanishes above T_{c}, showing no pseudogap effects, and that the spin susceptibility shows a substantially reduced signature of a spin gap compared to previously reported AFMC simulations.
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​.
Superfluidity in the cold atomic two-species Fermi gas system in the unitary limit of infinite scattering length remains incompletely understood. In particular, a pseudogap phase has been proposed to exist above the superfluid critical temperature. Here we apply the auxiliary-field quantum Monte Carlo method to perform ab initio calculations of the temperature dependence of three quantities---the energy-staggering pairing gap, the condensate fraction, and the heat capacity---in a trapped finite-size cold atom system in the unitary limit of infinite scattering length. As the calculations of the energy-staggering pairing gap require the use of the canonical ensemble, we employ an efficient algorithm for the stabilization of particle-number projection that is essential for reaching convergence in the size of the model space. We observe clear signatures of the superfluid phase transition in all three quantities, including a signature of the recently measured $\ensuremath{\lambda}$ peak in the heat capacity. We also find that the energy-staggering gap does not lead the condensate fraction as the temperature decreases.
We solve the zero-temperature unitary Fermi gas problem by incorporating a BCS importance function into the auxiliary-field quantum Monte Carlo method. We demonstrate that this method does not suffer from a sign problem and that it increases the efficiency of standard techniques by many orders of magnitude for strongly paired fermions. We calculate the ground-state energies exactly for unpolarized systems with up to 66 particles on lattices of up to 273{27}^{3}273 sites, obtaining an accurate result for the universal parameter $\ensuremath{\xi}$. We also obtain results for interactions with different effective ranges and find that the energy is consistent with a universal linear dependence on the product of the Fermi momentum and the effective range. This method will have many applications in superfluid cold atom systems and in both electronic and nuclear structures where pairing is important.
We investigate the zero-temperature properties of a diluted homogeneous Bose gas made of NNN particles interacting via a two-body square-well potential by performing Monte Carlo simulations. We tune the interaction strength to achieve arbitrary positive values of the scattering length and compute by Monte Carlo quadrature the energy per particle E/NE/NE/N and the condensate fraction N0/N{N}_{0}/NN0​/N of this system by using a Jastrow ansatz for the many-body wave function, which avoids the formation of the self-bound ground state and describes instead a (metastable) gaseous state with uniform density. In the unitarity limit, where the scattering length diverges while the range of the interatomic potential is much smaller than the average distance between atoms, we find a finite energy per particle [$E/N=0.70\phantom{\rule{4pt}{0ex}}{\ensuremath{\hbar}}^{2}{(6{\ensuremath{\pi}}^{2}n)}^{2/3}/2m$, with nnn the number density] and a quite large condensate fraction (N0/N=0.83{N}_{0}/N=0.83N0​/N=0.83).