arXiv:cond-mat/0511332 · Physical Review B

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

Dean Lee

Published Mar 17, 2006• 66 citations

Abstract

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.

DOI: 10.1103/physrevb.73.115112

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arXiv:cond-mat/0511332 · Physical Review B

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

Dean Lee

Published Mar 17, 2006• 66 citations

arXiv HTML PDF

Abstract

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.

DOI: 10.1103/physrevb.73.115112

Related papers

Constrained Path Monte Carlo method for spin $ 1/2 $ fermions at unitarity limit

Faisal Etminan, Mohammad Mehdi Firoozabadi

We present calculations for spin $ 1/2 $ fermions at unitarity 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 20 particles, with equal numbers of up and down spins in a periodic cube in the full ground-state constrained-path Monte Carlo (CPMC) method using the extended, attractive Hubbard model. Our results in a careful extrapolation to the thermodynamic limit may suggest that the ratio of the ground-state energy to that of a free Fermi gas is $ ξ= 0.43(4) $, which can compare with recent experimental results and consistent with the fixed node Green's function Monte Carlo and novel lattice approaches results. 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.We present calculations for spin $ 1/2 $ fermions at unitarity 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 20 particles, with equal numbers of up and down spins in a periodic cube in the full ground-state constrained-path Monte Carlo (CPMC) method using the extended, attractive Hubbard model. Our results in a careful extrapolation to the thermodynamic limit may suggest that the ratio of the ground-state energy to that of a free Fermi gas is $ ξ= 0.43(4) $, which can compare with recent experimental results and consistent with the fixed node Green's function Monte Carlo and novel lattice approaches results. 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.

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Ground state energy at unitarity

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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 $0.329(5)$.

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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.

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Jong-Wan Lee, Hwancheol Jeong

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.

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Spin 1/2 Fermions in the Unitary Limit.I

H. S. Køhler

This report concerns the energy of a zero-temperature many-body system of spin 1/2 fermions interacting via a two-body potential with a free space infinite scattering length and zero effective range; the Unitary limit. Given the corresponding phase-shift $δ(k)=π/2$ a one-term separable potential is obtained by inverse scattering assuming a momentum cut-off $Λ$ such that $δ(k)=0$ for $k>Λ$ . The \it effective \rm interaction in the many-body system is calculated in a pp-ladder approximation with Pauli-blocking but neglecting mean-field (dispersion) corrections; effective mass $m^{*}=1$ . Using only the zero relative momentum component of this interaction the total energy is $ξ=4/9$ (in units of the fermigas), a result reported by several previous authors. Integrating the momentum dependent interaction over the Fermi sea this energy is revised to $ξ=0.24.$ This result is independent of density and of the cut-off $Λ$ if $Λ> \sim 3k_{f}$ . With $m^{*}\neq 1$ there is however a strong dependence on this cut-off. Including hh-ladders estimates give $ξ=0.4\leftrightarrow 0.6$ , but a reliable result would in this case require a Green's function calculation.This report concerns the energy of a zero-temperature many-body system of spin 1/2 fermions interacting via a two-body potential with a free space infinite scattering length and zero effective range; the Unitary limit. Given the corresponding phase-shift $δ(k)=π/2$ a one-term separable potential is obtained by inverse scattering assuming a momentum cut-off $Λ$ such that $δ(k)=0$ for $k>Λ$. The \it effective \rm interaction in the many-body system is calculated in a pp-ladder approximation with Pauli-blocking but neglecting mean-field (dispersion) corrections; effective mass $m^{*}=1$. Using only the zero relative momentum component of this interaction the total energy is $ξ=4/9$ (in units of the fermigas), a result reported by several previous authors. Integrating the momentum dependent interaction over the Fermi sea this energy is revised to $ξ=0.24.$ This result is independent of density and of the cut-off $Λ$ if $Λ> \sim 3k_{f}$. With $m^{*}\neq 1$ there is however a strong dependence on this cut-off. Including hh-ladders estimates give $ξ=0.4\leftrightarrow 0.6$, but a reliable result would in this case require a Green's function calculation.

arXiv:0705.0944May 7, 20070 citations

Superfluidity and excitations at unitarity

Dean Lee

We present lattice results for spin- $1∕2$ fermions at unitarity, where the effective range of the interaction is zero and the scattering length is infinite. We measure the spatial coherence of difermion pairs for a system of 6, 10, 14, 18, 22, 26 particles with equal numbers of up and down spins in a periodic cube. Using Euclidean time projection, we analyze ground-state properties and transient behavior due to low-energy excitations. At asymptotically large values of $t$ we see long-range order consistent with spontaneously broken U(1) fermion-number symmetry and a superfluid ground state. At intermediate times we see exponential decay in the $t$ -dependent signal due to an unknown low-energy excitation. We probe this low-energy excitation further by calculating two-particle correlation functions. We find that the excitation has the properties of a chain of particles extending across the periodic lattice.We present lattice results for spin-$1∕2$ fermions at unitarity, where the effective range of the interaction is zero and the scattering length is infinite. We measure the spatial coherence of difermion pairs for a system of 6, 10, 14, 18, 22, 26 particles with equal numbers of up and down spins in a periodic cube. Using Euclidean time projection, we analyze ground-state properties and transient behavior due to low-energy excitations. At asymptotically large values of $t$ we see long-range order consistent with spontaneously broken U(1) fermion-number symmetry and a superfluid ground state. At intermediate times we see exponential decay in the $t$-dependent signal due to an unknown low-energy excitation. We probe this low-energy excitation further by calculating two-particle correlation functions. We find that the excitation has the properties of a chain of particles extending across the periodic lattice.

arXiv:cond-mat/0606706Physical Review BApr 3, 200718 citations

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

\frac{1}{2}

{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)

δ(k)=π/2

a one-term separable potential is obtained by inverse scattering assuming a momentum cut-off

Λ

such that

δ(k)=0

for

k>Λ

. The \it effective \rm interaction in the many-body system is calculated in a pp-ladder approximation with Pauli-blocking but neglecting mean-field (dispersion) corrections; effective mass

m^{*}=1

. Using only the zero relative momentum component of this interaction the total energy is

ξ=4/9

(in units of the fermigas), a result reported by several previous authors. Integrating the momentum dependent interaction over the Fermi sea this energy is revised to

ξ=0.24.

This result is independent of density and of the cut-off

Λ

Λ> \sim 3k_{f}

. With

m^{*}\neq 1

there is however a strong dependence on this cut-off. Including hh-ladders estimates give

ξ=0.4\leftrightarrow 0.6

, but a reliable result would in this case require a Green's function calculation.

We present lattice results for spin-

1∕2

fermions at unitarity, where the effective range of the interaction is zero and the scattering length is infinite. We measure the spatial coherence of difermion pairs for a system of 6, 10, 14, 18, 22, 26 particles with equal numbers of up and down spins in a periodic cube. Using Euclidean time projection, we analyze ground-state properties and transient behavior due to low-energy excitations. At asymptotically large values of

t

we see long-range order consistent with spontaneously broken U(1) fermion-number symmetry and a superfluid ground state. At intermediate times we see exponential decay in the

t

-dependent signal due to an unknown low-energy excitation. We probe this low-energy excitation further by calculating two-particle correlation functions. We find that the excitation has the properties of a chain of particles extending across the periodic lattice.