We investigate the e↵ect of a spherical cuto↵ in the single particle momentum on the lattice calculations of unitary fermions, and use this to explore the ground state configuration. The unitarity limit, corresponding to an infinite s-wave scattering length and zero interaction range, is achieved by tuning the lattice coupling of four-fermi interactions to reproduce the scattering phase shift of σ = π/2 in the continuum. We particularly compute the ground state energies of three unitary fermions at zero and non-zero net momenta. We show that in the continuum limit the spherical cuto↵ does not a↵ect the results, while it violates the unitarity for a boosted pair of fermions. Our findings suggest that the pair of fermions in the ground state of three unitary fermions has zero momentum, while the total net momentum of the system is carried by the unpaired fermion.We investigate the e↵ect of a spherical cuto↵ in the single particle momentum on the lattice calculations of unitary fermions, and use this to explore the ground state configuration. The unitarity limit, corresponding to an infinite s-wave scattering length and zero interaction range, is achieved by tuning the lattice coupling of four-fermi interactions to reproduce the scattering phase shift of σ = π/2 in the continuum. We particularly compute the ground state energies of three unitary fermions at zero and non-zero net momenta. We show that in the continuum limit the spherical cuto↵ does not a↵ect the results, while it violates the unitarity for a boosted pair of fermions. Our findings suggest that the pair of fermions in the ground state of three unitary fermions has zero momentum, while the total net momentum of the system is carried by the unpaired fermion.

Aug 5, 20260 citations

Ground state energy at unitarity

Dean Lee

We consider two-component fermions on the lattice in the unitarity limit. This is an idealized limit of attractive fermions where the range of the interaction is zero and the scattering length is infinite. Using Euclidean time projection, we compute the ground state energy using four computationally different but physically identical auxiliary-field methods. The best performance is obtained using a bounded continuous auxiliary field and a nonlocal updating algorithm called the hybrid Monte Carlo. With this method, we calculate results for 10 and 14 fermions at lattice volumes ${4}^{3},{5}^{3},{6}^{3},{7}^{3},{8}^{3}$ and extrapolate to the continuum limit. For 10 fermions in a periodic cube, the ground state energy is $0.292(12)$ times the ground state energy for noninteracting fermions. For 14 fermions, the ratio is $0.329(5)$ .We consider two-component fermions on the lattice in the unitarity limit. This is an idealized limit of attractive fermions where the range of the interaction is zero and the scattering length is infinite. Using Euclidean time projection, we compute the ground state energy using four computationally different but physically identical auxiliary-field methods. The best performance is obtained using a bounded continuous auxiliary field and a nonlocal updating algorithm called the hybrid Monte Carlo. With this method, we calculate results for 10 and 14 fermions at lattice volumes ${4}^{3},{5}^{3},{6}^{3},{7}^{3},{8}^{3}$ and extrapolate to the continuum limit. For 10 fermions in a periodic cube, the ground state energy is $0.292(12)$ times the ground state energy for noninteracting fermions. For 14 fermions, the ratio is $0.329(5)$.

arXiv:0803.1280Physical Review CAug 13, 200867 citations

Precision benchmark calculations for four particles at unitarity

Shahin Bour, Xin Li, Dean Lee +2 more

The unitarity limit describes interacting particles where the range of the interaction is zero and the scattering length is infinite. We present precision benchmark calculations for two-component fermions at unitarity using three different ab initio methods: Hamiltonian lattice formalism using iterated eigenvector methods, Euclidean lattice formalism with auxiliary-field projection Monte Carlo methods, and continuum diffusion Monte Carlo methods with fixed and released nodes. We have calculated the ground-state energy of the unpolarized four-particle system in a periodic cube as a dimensionless fraction of the ground-state energy for the noninteracting system. We obtain values of $0.211(2)$ and $0.210(2)$ using two different Hamiltonian lattice representations, $0.206(9)$ using Euclidean lattice formalism, and an upper bound of $0.212(2)$ from fixed-node diffusion Monte Carlo methods. Released-node calculations starting from the fixed-node result yield a decrease of less than $0.002$ over a propagation of $0.4{E}_{F}^{\ensuremath{-}1}$ in Euclidean time, where ${E}_{F}$ is the Fermi energy. We find good agreement among all three ab initio methods.The unitarity limit describes interacting particles where the range of the interaction is zero and the scattering length is infinite. We present precision benchmark calculations for two-component fermions at unitarity using three different ab initio methods: Hamiltonian lattice formalism using iterated eigenvector methods, Euclidean lattice formalism with auxiliary-field projection Monte Carlo methods, and continuum diffusion Monte Carlo methods with fixed and released nodes. We have calculated the ground-state energy of the unpolarized four-particle system in a periodic cube as a dimensionless fraction of the ground-state energy for the noninteracting system. We obtain values of $0.211(2)$ and $0.210(2)$ using two different Hamiltonian lattice representations, $0.206(9)$ using Euclidean lattice formalism, and an upper bound of $0.212(2)$ from fixed-node diffusion Monte Carlo methods. Released-node calculations starting from the fixed-node result yield a decrease of less than $0.002$ over a propagation of $0.4{E}_{F}^{\ensuremath{-}1}$ in Euclidean time, where ${E}_{F}$ is the Fermi energy. We find good agreement among all three ab initio methods.

arXiv:1104.2102Physical Review AJun 15, 201122 citations

Ground-state energy of spin- 12 fermions in the unitary limit

Dean Lee

We present lattice results for the ground-state energy of a spin- $\frac{1}{2}$ fermion system in the unitary limit, where the effective range of the interaction is zero and the scattering length is infinite. We compute the ground-state energy for a system of 6, 10, 14, 18, and 22 particles, with equal numbers of up and down spins in a periodic cube. We estimate that in the limit of large number of particles, the ground-state energy is 0.25(3) times the ground-state energy of the free Fermi system.We present lattice results for the ground-state energy of a spin-$\frac{1}{2}$ fermion system in the unitary limit, where the effective range of the interaction is zero and the scattering length is infinite. We compute the ground-state energy for a system of 6, 10, 14, 18, and 22 particles, with equal numbers of up and down spins in a periodic cube. We estimate that in the limit of large number of particles, the ground-state energy is 0.25(3) times the ground-state energy of the free Fermi system.

arXiv:cond-mat/0511332Physical Review BMar 17, 200666 citations

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

Rongzheng He, Ning Li, Bing-Nan Lu +1 more

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.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.

arXiv:1910.01257Physical Review AJun 11, 202011 citations

Spin 1/2 Fermions in the Unitary Limit.III

H. S. Køhler

In scattering theory, the unitary limit is defined by an infinite scattering-length and a zero effective range, corresponding to a phase-shift π/2, independent of energy. This condition is satisfied by a rank-1 separable potential V(k,k')=-v(k)v(k') with v^{2}(k)=(4π)^{2}(Λ^{2}-k^{2})^{-1/2}, Λbeing the cut-off in momentum space.Previous calculations using a Pauli-corrected ladder summation to calculate the energy of a zero temperature many body system of spin 1/2 fermions with this interaction gave ξ=0.24 (in units of kinetic energy) independent of density and with Λ---->infinity. This value of ξis appreciably smaller than the experimental and that obtained from other calculations, most notably from Monte Carlo, which in principle would be the most reliable. Our previous work did however also show a strong dependence on effective range r_0 (with r_0=0 at unitarity). With an increase to r_0=1.0 the energy varied from ξ~0.38 at k_f=0.6 1/fm to ~0.45 at k_f=1.8 1/fm which is somewhat closer to the Monte-Carlo results. These previous calculations are here extended by including the effect of the previously neglected mean-field propagation, the dispersion correction. This is repulsive and found to increase drastically with decreasing effective range. It is large enough to suggest a revised value of ξ~0.4 ~0.5 independent of r_0. Off-shell effects are also investigated by introducing a rank-2 (phase-shift equivalent) separable potential. Effects of 10% or more in energy could be demonstrated for r_0>0. It is pointed out that a computational cut-off in momentum-space brings in another scale in the in otherwise scale-less unitary problem.In scattering theory, the unitary limit is defined by an infinite scattering-length and a zero effective range, corresponding to a phase-shift π/2, independent of energy. This condition is satisfied by a rank-1 separable potential V(k,k')=-v(k)v(k') with v^{2}(k)=(4π)^{2}(Λ^{2}-k^{2})^{-1/2}, Λbeing the cut-off in momentum space.Previous calculations using a Pauli-corrected ladder summation to calculate the energy of a zero temperature many body system of spin 1/2 fermions with this interaction gave ξ=0.24 (in units of kinetic energy) independent of density and with Λ---->infinity. This value of ξis appreciably smaller than the experimental and that obtained from other calculations, most notably from Monte Carlo, which in principle would be the most reliable. Our previous work did however also show a strong dependence on effective range r_0 (with r_0=0 at unitarity). With an increase to r_0=1.0 the energy varied from ξ~0.38 at k_f=0.6 1/fm to ~0.45 at k_f=1.8 1/fm which is somewhat closer to the Monte-Carlo results. These previous calculations are here extended by including the effect of the previously neglected mean-field propagation, the dispersion correction. This is repulsive and found to increase drastically with decreasing effective range. It is large enough to suggest a revised value of ξ~0.4 ~0.5 independent of r_0. Off-shell effects are also investigated by introducing a rank-2 (phase-shift equivalent) separable potential. Effects of 10% or more in energy could be demonstrated for r_0>0. It is pointed out that a computational cut-off in momentum-space brings in another scale in the in otherwise scale-less unitary problem.

arXiv:1008.3884Aug 23, 20100 citations

{4}^{3},{5}^{3},{6}^{3},{7}^{3},{8}^{3}

and extrapolate to the continuum limit. For 10 fermions in a periodic cube, the ground state energy is

0.292(12)

times the ground state energy for noninteracting fermions. For 14 fermions, the ratio is

0.329(5)

0.211(2)

and

0.210(2)

using two different Hamiltonian lattice representations,

0.206(9)

using Euclidean lattice formalism, and an upper bound of

0.212(2)

from fixed-node diffusion Monte Carlo methods. Released-node calculations starting from the fixed-node result yield a decrease of less than

0.002

over a propagation of $0.4{E}_{F}^{\ensuremath{-}1}$ in Euclidean time, where

{E}_{F}

is the Fermi energy. We find good agreement among all three ab initio methods.

We present lattice results for the ground-state energy of a spin-

\frac{1}{2}

fermion system in the unitary limit, where the effective range of the interaction is zero and the scattering length is infinite. We compute the ground-state energy for a system of 6, 10, 14, 18, and 22 particles, with equal numbers of up and down spins in a periodic cube. We estimate that in the limit of large number of particles, the ground-state energy is 0.25(3) times the ground-state energy of the free Fermi system.

{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.