We construct systematic expansions around four and two spatial dimensions for a Fermi gas near the unitarity limit. Near four spatial dimensions such a Fermi gas can be understood as a weakly interacting system of fermionic and bosonic degrees of freedom. To the leading and next-to-leading orders in the expansion over $ϵ=4\ensuremath{-}d$, with $d$ being the dimensionality of space, we calculate the thermodynamic quantities and the fermion quasiparticle spectrum, both at unitarity and around the unitarity point. Then the phase structure of the polarized Fermi gas in the unitary regime is studied in the $ϵ$ expansion. At unitarity the unpolarized superfluid state and a fully polarized normal state are separated by a first order phase transition. However, on the BEC side of the unitarity point, in a certain range of the two-body binding energy, these two phases are separated in the phase diagram by more exotic phases, including gapless superfluid phases with one or two Fermi surfaces and a superfluid phase with a spatially varying condensate. We also show that the unitary Fermi gas near two spatial dimensions is weakly interacting, and calculate the thermodynamic quantities and the fermion quasiparticle spectrum in the expansion over $\overline{ϵ}=d\ensuremath{-}2$.We construct systematic expansions around four and two spatial dimensions for a Fermi gas near the unitarity limit. Near four spatial dimensions such a Fermi gas can be understood as a weakly interacting system of fermionic and bosonic degrees of freedom. To the leading and next-to-leading orders in the expansion over $ϵ=4\ensuremath{-}d$, with $d$ being the dimensionality of space, we calculate the thermodynamic quantities and the fermion quasiparticle spectrum, both at unitarity and around the unitarity point. Then the phase structure of the polarized Fermi gas in the unitary regime is studied in the $ϵ$ expansion. At unitarity the unpolarized superfluid state and a fully polarized normal state are separated by a first order phase transition. However, on the BEC side of the unitarity point, in a certain range of the two-body binding energy, these two phases are separated in the phase diagram by more exotic phases, including gapless superfluid phases with one or two Fermi surfaces and a superfluid phase with a spatially varying condensate. We also show that the unitary Fermi gas near two spatial dimensions is weakly interacting, and calculate the thermodynamic quantities and the fermion quasiparticle spectrum in the expansion over $\overline{ϵ}=d\ensuremath{-}2$.

arXiv:cond-mat/0607835Physical Review AJun 18, 200798 citations

ϵ Expansion for a Fermi Gas at Infinite Scattering Length

Yusuke Nishida, Dam Thanh Son

We show that there exists a systematic expansion around four spatial dimensions for Fermi gas in the unitarity regime. We perform the calculations to leading and next-to-leading orders in the expansion over $ϵ=4\ensuremath{-}d$, where $d$ is the dimensionality of space. We find the ratio of chemical potential and Fermi energy to be $\ensuremath{\mu}/{\ensuremath{\epsilon}}_{F}=\frac{1}{2}{ϵ}^{3/2}+\frac{1}{16}{ϵ}^{5/2}\mathrm{ln}ϵ\ensuremath{-}0.0246{ϵ}^{5/2}+\ensuremath{\cdots}$ and the ratio of the gap in the fermion quasiparticle spectrum and the chemical potential to be $\ensuremath{\Delta}/\ensuremath{\mu}=2{ϵ}^{\ensuremath{-}1}\ensuremath{-}0.691+\ensuremath{\cdots}$. The minimum of the fermion dispersion curve is located at $|\mathbit{p}|=(2m{\ensuremath{\epsilon}}_{0}{)}^{1/2}$, where ${\ensuremath{\epsilon}}_{0}/\ensuremath{\mu}=2+O(ϵ)$. Extrapolation to $d=3$ gives results consistent with Monte Carlo simulations.We show that there exists a systematic expansion around four spatial dimensions for Fermi gas in the unitarity regime. We perform the calculations to leading and next-to-leading orders in the expansion over $ϵ=4\ensuremath{-}d$, where $d$ is the dimensionality of space. We find the ratio of chemical potential and Fermi energy to be $\ensuremath{\mu}/{\ensuremath{\epsilon}}_{F}=\frac{1}{2}{ϵ}^{3/2}+\frac{1}{16}{ϵ}^{5/2}\mathrm{ln}ϵ\ensuremath{-}0.0246{ϵ}^{5/2}+\ensuremath{\cdots}$ and the ratio of the gap in the fermion quasiparticle spectrum and the chemical potential to be $\ensuremath{\Delta}/\ensuremath{\mu}=2{ϵ}^{\ensuremath{-}1}\ensuremath{-}0.691+\ensuremath{\cdots}$. The minimum of the fermion dispersion curve is located at $|\mathbit{p}|=(2m{\ensuremath{\epsilon}}_{0}{)}^{1/2}$, where ${\ensuremath{\epsilon}}_{0}/\ensuremath{\mu}=2+O(ϵ)$. Extrapolation to $d=3$ gives results consistent with Monte Carlo simulations.

arXiv:cond-mat/0604500Physical Review LettersAug 4, 2006182 citations

Unitary Fermi gas at finite temperature in the ϵ expansion

Yusuke Nishida

Thermodynamics of the unitary Fermi gas at finite temperature is investigated from the perspective of the expansion over $ϵ=4\ensuremath{-}d$ with $d$ being the dimensionality of space. We show that the thermodynamics is dominated by bosonic excitations in the low temperature region $T\ensuremath{\ll}{T}_{c}$. Analytic formulas for the thermodynamic functions as functions of the temperature are derived to the lowest order in $ϵ$ in this region. In the high temperature region where $T\ensuremath{\sim}{T}_{c}$, bosonic and fermionic quasiparticles are excited. We determine the critical temperature ${T}_{c}$ of the superfluid phase transition and the thermodynamic functions around ${T}_{c}$ to the leading and next-to-leading orders in $ϵ$ .Thermodynamics of the unitary Fermi gas at finite temperature is investigated from the perspective of the expansion over $ϵ=4\ensuremath{-}d$ with $d$ being the dimensionality of space. We show that the thermodynamics is dominated by bosonic excitations in the low temperature region $T\ensuremath{\ll}{T}_{c}$. Analytic formulas for the thermodynamic functions as functions of the temperature are derived to the lowest order in $ϵ$ in this region. In the high temperature region where $T\ensuremath{\sim}{T}_{c}$, bosonic and fermionic quasiparticles are excited. We determine the critical temperature ${T}_{c}$ of the superfluid phase transition and the thermodynamic functions around ${T}_{c}$ to the leading and next-to-leading orders in $ϵ$.

arXiv:cond-mat/0608321Physical Review AJun 18, 200742 citations

Ground-state energy of the unitary Fermi gas from the ϵ expansion

Yusuke Nishida

We update the ground-state energy ratio of unitary Fermi gas to noninteracting Fermi gas $(\ensuremath{\xi})$ from the $ϵ$ expansion by including the next-to-next-to-leading-order (NNLO) term near two spatial dimensions. Interpolations of the NNLO $ϵ$ expansions around four and two spatial dimensions with the use of Pad\'e approximants give $\ensuremath{\xi}\ensuremath{\approx}0.360\ifmmode\pm\else\textpm\fi{}0.020$ in three dimensions with the uncertainty due to different interpolation functions. This value is consistent with the previous interpolations of the NLO $ϵ$ expansions $\ensuremath{\xi}\ensuremath{\approx}0.377\ifmmode\pm\else\textpm\fi{}0.014$ in spite of the large NNLO corrections.We update the ground-state energy ratio of unitary Fermi gas to noninteracting Fermi gas $(\ensuremath{\xi})$ from the $ϵ$ expansion by including the next-to-next-to-leading-order (NNLO) term near two spatial dimensions. Interpolations of the NNLO $ϵ$ expansions around four and two spatial dimensions with the use of Pad\'e approximants give $\ensuremath{\xi}\ensuremath{\approx}0.360\ifmmode\pm\else\textpm\fi{}0.020$ in three dimensions with the uncertainty due to different interpolation functions. This value is consistent with the previous interpolations of the NLO $ϵ$ expansions $\ensuremath{\xi}\ensuremath{\approx}0.377\ifmmode\pm\else\textpm\fi{}0.014$ in spite of the large NNLO corrections.

arXiv:0808.3826Physical Review AJan 27, 200921 citations

Unitary p -wave Fermi gas in one dimension

Hiroyuki Tajima, Shoichiro Tsutsui, Takahiro M. Doi +1 more

We elucidate universal many-body properties of a one-dimensional, two-component ultracold Fermi gas near the $p$ -wave Feshbach resonance. The low-energy scattering in this system can be characterized by two parameters, that is, $p$ -wave scattering length and effective range. At the unitarity limit where the $p$ -wave scattering length diverges and the effective range is reduced to zero without conflicting with the causality bound, the system obeys universal thermodynamics as observed in a unitary Fermi gas with contact $s$ -wave interaction in three dimensions. It is in contrast to a Fermi gas with the $p$ -wave resonance in three dimensions in which the effective range is inevitably finite. We present the universal equation of state in this unitary $p$ -wave Fermi gas within the many-body $T$ -matrix approach as well as the virial expansion method. Moreover, we examine the single-particle spectral function in the high-density regime where the virial expansion is no longer valid. On the basis of the Hartree-like self-energy shift at the divergent scattering length, we conjecture that the equivalence of the Bertsch parameter across spatial dimensions holds even for a one-dimensional unitary $p$ -wave Fermi gas.We elucidate universal many-body properties of a one-dimensional, two-component ultracold Fermi gas near the $p$-wave Feshbach resonance. The low-energy scattering in this system can be characterized by two parameters, that is, $p$-wave scattering length and effective range. At the unitarity limit where the $p$-wave scattering length diverges and the effective range is reduced to zero without conflicting with the causality bound, the system obeys universal thermodynamics as observed in a unitary Fermi gas with contact $s$-wave interaction in three dimensions. It is in contrast to a Fermi gas with the $p$-wave resonance in three dimensions in which the effective range is inevitably finite. We present the universal equation of state in this unitary $p$-wave Fermi gas within the many-body $T$-matrix approach as well as the virial expansion method. Moreover, we examine the single-particle spectral function in the high-density regime where the virial expansion is no longer valid. On the basis of the Hartree-like self-energy shift at the divergent scattering length, we conjecture that the equivalence of the Bertsch parameter across spatial dimensions holds even for a one-dimensional unitary $p$-wave Fermi gas.

arXiv:2106.12909Physical Review AAug 23, 202113 citations

Universality of the unitary Fermi gas: a few-body perspective

Jesper Levinsen, Pietro Massignan, Shimpei Endo +1 more

Abstract We revisit the properties of the two-component Fermi gas with short-range interactions in three dimensions, in the limit where the s -wave scattering length diverges. Such a unitary Fermi gas possesses universal thermodynamic and dynamical observables that are independent of any interaction length scale. Focusing on trapped systems of N fermions, where N ≤ 10 , we investigate how well we can determine the zero-temperature behavior of the many-body system from published few-body data on the ground-state energy and the contact. For the unpolarized case, we find that the Bertsch parameters extracted from trapped few-body systems all lie within 15% of the established value. Furthermore, the few-body values for the contact are well within the range of values determined in the literature for the many-body system. In the limit of large spin polarization, we obtain a similar accuracy for the polaron energy, and we estimate the polaron’s effective mass from the dependence of its energy on N . We also compute an upper bound for the squared wave-function overlap between the unitary Fermi system and the non-interacting ground state, both for the trapped and uniform cases. This allows us to prove that the trapped unpolarized ground state at unitarity has zero overlap with its non-interacting counterpart in the many-body limit N → ∞ .Abstract We revisit the properties of the two-component Fermi gas with short-range interactions in three dimensions, in the limit where the s -wave scattering length diverges. Such a unitary Fermi gas possesses universal thermodynamic and dynamical observables that are independent of any interaction length scale. Focusing on trapped systems of N fermions, where N ≤ 10 , we investigate how well we can determine the zero-temperature behavior of the many-body system from published few-body data on the ground-state energy and the contact. For the unpolarized case, we find that the Bertsch parameters extracted from trapped few-body systems all lie within 15% of the established value. Furthermore, the few-body values for the contact are well within the range of values determined in the literature for the many-body system. In the limit of large spin polarization, we obtain a similar accuracy for the polaron energy, and we estimate the polaron’s effective mass from the dependence of its energy on N . We also compute an upper bound for the squared wave-function overlap between the unitary Fermi system and the non-interacting ground state, both for the trapped and uniform cases. This allows us to prove that the trapped unpolarized ground state at unitarity has zero overlap with its non-interacting counterpart in the many-body limit N → ∞ .

arXiv:1612.02131Journal of Physics B Atomic Molecular and Optical PhysicsJan 18, 201731 citations

d

being the dimensionality of space, we calculate the thermodynamic quantities and the fermion quasiparticle spectrum, both at unitarity and around the unitarity point. Then the phase structure of the polarized Fermi gas in the unitary regime is studied in the

ϵ

expansion. At unitarity the unpolarized superfluid state and a fully polarized normal state are separated by a first order phase transition. However, on the BEC side of the unitarity point, in a certain range of the two-body binding energy, these two phases are separated in the phase diagram by more exotic phases, including gapless superfluid phases with one or two Fermi surfaces and a superfluid phase with a spatially varying condensate. We also show that the unitary Fermi gas near two spatial dimensions is weakly interacting, and calculate the thermodynamic quantities and the fermion quasiparticle spectrum in the expansion over $\overline{ϵ}=d\ensuremath{-}2$.

d

is the dimensionality of space. We find the ratio of chemical potential and Fermi energy to be $\ensuremath{\mu}/{\ensuremath{\epsilon}}_{F}=\frac{1}{2}{ϵ}^{3/2}+\frac{1}{16}{ϵ}^{5/2}\mathrm{ln}ϵ\ensuremath{-}0.0246{ϵ}^{5/2}+\ensuremath{\cdots}$ and the ratio of the gap in the fermion quasiparticle spectrum and the chemical potential to be $\ensuremath{\Delta}/\ensuremath{\mu}=2{ϵ}^{\ensuremath{-}1}\ensuremath{-}0.691+\ensuremath{\cdots}$. The minimum of the fermion dispersion curve is located at $|\mathbit{p}|=(2m{\ensuremath{\epsilon}}_{0}{)}^{1/2}$, where ${\ensuremath{\epsilon}}_{0}/\ensuremath{\mu}=2+O(ϵ)$. Extrapolation to

d=3

gives results consistent with Monte Carlo simulations.

Thermodynamics of the unitary Fermi gas at finite temperature is investigated from the perspective of the expansion over $ϵ=4\ensuremath{-}d$ with

d

being the dimensionality of space. We show that the thermodynamics is dominated by bosonic excitations in the low temperature region $T\ensuremath{\ll}{T}_{c}$. Analytic formulas for the thermodynamic functions as functions of the temperature are derived to the lowest order in

ϵ

in this region. In the high temperature region where $T\ensuremath{\sim}{T}_{c}$, bosonic and fermionic quasiparticles are excited. We determine the critical temperature

{T}_{c}

of the superfluid phase transition and the thermodynamic functions around

{T}_{c}

to the leading and next-to-leading orders in

ϵ

We update the ground-state energy ratio of unitary Fermi gas to noninteracting Fermi gas $(\ensuremath{\xi})$ from the

ϵ

expansion by including the next-to-next-to-leading-order (NNLO) term near two spatial dimensions. Interpolations of the NNLO

ϵ

expansions around four and two spatial dimensions with the use of Pad\'e approximants give $\ensuremath{\xi}\ensuremath{\approx}0.360\ifmmode\pm\else\textpm\fi{}0.020$ in three dimensions with the uncertainty due to different interpolation functions. This value is consistent with the previous interpolations of the NLO

ϵ

expansions $\ensuremath{\xi}\ensuremath{\approx}0.377\ifmmode\pm\else\textpm\fi{}0.014$ in spite of the large NNLO corrections.

We elucidate universal many-body properties of a one-dimensional, two-component ultracold Fermi gas near the

p

-wave Feshbach resonance. The low-energy scattering in this system can be characterized by two parameters, that is,

p

-wave scattering length and effective range. At the unitarity limit where the

p

-wave scattering length diverges and the effective range is reduced to zero without conflicting with the causality bound, the system obeys universal thermodynamics as observed in a unitary Fermi gas with contact

s

-wave interaction in three dimensions. It is in contrast to a Fermi gas with the

p

-wave resonance in three dimensions in which the effective range is inevitably finite. We present the universal equation of state in this unitary

p

-wave Fermi gas within the many-body

T

-matrix approach as well as the virial expansion method. Moreover, we examine the single-particle spectral function in the high-density regime where the virial expansion is no longer valid. On the basis of the Hartree-like self-energy shift at the divergent scattering length, we conjecture that the equivalence of the Bertsch parameter across spatial dimensions holds even for a one-dimensional unitary

p

-wave Fermi gas.