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Collective branch in the continuum of BCS superconductors and superfluid Fermi gases

Hadrien Kurkjian, Serghei N. Klimin, Jacques Tempere, Yvan Castin

Published May 1, 2018• 0 citations
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Abstract

We demonstrate the existence of a collective excitation branch in the pair-breaking continuum of superfluid Fermi gases and BCS superconductors, as suggested by Littlewood and Varma in 1982. We analytically continue the RPA equation on the collective mode energy through its branch cut associated with the continuum, and obtain the full complex dispersion relation, including in the strong coupling regime. For $\Delta$/$\mu$ > 1.210 (very close to unitarity in a superfluid Fermi gas), where $\Delta$ is the order parameter and $\mu$ the chemical potential, the real part of the branch is wholly within the band gap [0, 2$\Delta$].In the long wavelength limit, the branch varies quadratically with the wave number, with a complex effective mass that we compute analytically. This contradicts the result of Littlewood and Varma that prevailed so far.

Related papers

Pair-Breaking Collective Branch in BCS Superconductors and Superfluid Fermi Gases

Hadrien Kurkjian, Serghei N. Klimin, Jacques Tempere +1 more

We demonstrate the existence of a collective excitation branch in the pair-breaking continuum of superfluid Fermi gases and BCS superconductors. At zero temperature, we analytically continue the equation on the collective mode energy in Anderson's Random Phase Approximation or Gaussian fluctuations through its branch cut associated with the continuum, and obtain the full complex dispersion relation, including in the strong coupling regime. The branch exists as long as the chemical potential μ is positive and the wave number below sqrt[2mμ]/ℏ (with m the fermion mass). In the long wavelength limit, the branch varies quadratically with the wave number, with a complex effective mass that we compute analytically for an arbitrary interaction strength.

arXiv:1805.02462•Physical Review Letters•Mar 8, 2019•43 citations

Collective excitation branch in the continuum of pair-condensed Fermi gases : analytical study and scaling laws

Yvan Castin, Hadrien Kurkjian

The pair-condensed unpolarized spin-$1/2$ Fermi gases have a collective excitation branch in their pair-breaking continuum (V.A. Andrianov, V.N. Popov, 1976). We study it at zero temperature, with the eigenenergy equation deduced from the linearized time-dependent BCS theory and extended analytically to the lower half complex plane through its branch cut, calculating both the dispersion relation and the spectral weights (quasiparticle residues) of the branch. In the case of BCS superconductors, so called because the effect of the ion lattice is replaced by a short-range electron-electron interaction, we also include the Coulomb interaction and we restrict ourselves to the weak coupling limit $Δ/μ\to 0^+$ ($Δ$ is the order parameter, $μ$ the chemical potential) and to wavenumbers $q=O(1/ξ)$ where $ξ$ is the size of a pair; when the complex energy $z_q$ is expressed in units of $Δ$ and $q$ in units of $1/ξ$, the branch follows a universal law insensitive to the Coulomb interaction. In the case of cold atoms in the BEC-BCS crossover, only a contact interaction remains, but the coupling strength $Δ/μ$ can take arbitrary values, and we study the branch at any wave number. At weak coupling, we predict three scales, that already mentioned $q\approx 1/ξ$, that $q\approx(Δ/μ)^{-1/3}/ξ$ where the real part of the dispersion relation has a minimum and that $q\approx (μ/Δ)/ξ\approx k_{\rm F}$ ($k_{\rm F}$ is the Fermi wave number) where the branch reaches the edge of its existence domain. Near the point where the chemical potential vanishes on the BCS side, $μ/Δ\to 0^+$, where $ξ\approx k_{\rm F}$, we find two scales $q\approx(μ/Δ)^{1/2}/ξ$ and $q\approx 1/ξ$. In all cases, the branch has a limit $2Δ$ and a quadratic start at $q=0$. These results were obtained for $μ>0$, where the eigenenergy equation admits at least two branch points $ε_a(q)$ and $ε_b(q)$ on the positive real axis, and for an analytic continuation through the interval $[ε_a(q),ε_b(q)] $. We find new continuum branches by performing the analytic continuation through $[ε_b(q),+\infty[$ or even, for $q$ low enough, where there is a third real positive branch point $ε_c(q)$, through $[ε_b(q),ε_c(q)]$ and $[ε_c(q),+\infty[$. On the BEC side $μ<0$ not previously studied, where there is only one real positive branch point $ ε_a(q)$, we also find new collective excitation branches under the branch cut $[ε_a (q),+\infty[$. For $μ>0$, some of these new branches have a low-wavenumber exotic hypoacoustic $z_q\approx q^{3/2}$ or hyperacoustic $z_q\approx q^{4/5}$ behavior. For $μ<0$, we find a hyperacoustic branch and a nonhypoacoustic branch, with a limit $2Δ$ and a purely real quadratic start at $q=0$ for $Δ/|μ|<0.222$.

arXiv:1907.12238•Jul 29, 2019•2 citations

Collective excitations in the critical regime of superfluid Fermi gases

Serghei N. Klimin, Jacques Tempere, Hadrien Kurkjian

Studying the collective pairing phenomena in a two-component Fermi gas, we predict the appearance near the transition temperature $T_c$ of a well-resolved collective mode of quadratic dispersion. The mode is visible both above and below $T_c$ in the system's response to a driving pairing field. When approaching $T_c$ from below, the phononic and pair-breaking branches, characteristic of the zero temperature behavior, reduce to a very low energy-momentum region when the pair correlation length reaches its critical divergent behavior $\xi_{\rm pair}\propto|T_c-T|^{-1/2}$; elsewhere, they are replaced by the quadratically-dispersed pairing resonance, which thus acts as a precursor of the phase transition. In the strong-coupling and Bose-Einstein Condensate regime, this mode is a weakly-damped propagating mode associated to a Lorentzian resonance. Conversely, in the BCS limit it is a relaxation mode of pure imaginary eigenenergy. At large momenta, the resonance disappears when it is reabsorbed by the lower-edge of the pairing continuum. At intermediate temperatures between 0 and $T_c$, we unify the newly found collective phenomena near $T_c$ with the phononic and pair-breaking branches predicted from previous studies, and we exhaustively classify the roots of the analytically continued dispersion equation, and show that they provided a very good summary of the pair spectral functions.

•Mar 9, 2021•0 citations

Collective excitations of superfluid Fermi gases near the transition temperature

Serghei N. Klimin, Jacques Tempere, Hadrien Kurkjian

Studying the collective pairing phenomena in a two-component Fermi gas, we predict the appearance near the transition temperature ${T}_{c}$ of a well-resolved collective mode of quadratic dispersion. The mode is visible both above and below ${T}_{c}$ in the system's response to a driving pairing field. When approaching ${T}_{c}$ from below, the phononic and pair-breaking branches, characteristic of the zero temperature behavior, reduce to a relatively low energy-momentum region. Elsewhere they are replaced by the quadratically dispersed pairing resonance, which thus acts as a precursor of the phase transition. In the strong-coupling and Bose-Einstein condensate regime, this mode is a weakly damped propagating mode associated with a Lorentzian resonance. Conversely, in the BCS limit it is a relaxation mode of pure imaginary eigenenergy. At large momenta, the resonance disappears when it is reabsorbed by the lower edge of the pairing continuum. At intermediate temperatures between 0 and ${T}_{c}$ we unify the newly found collective phenomena near ${T}_{c}$ with the phononic and pair-breaking branches predicted from previous studies, and we exhaustively classify the roots of the analytically continued dispersion equation, and show that they provided a very good summary of the pair spectral functions.

arXiv:2103.05733•Physical Review A•Apr 30, 2021•13 citations

Leggett collective excitations in a two-band Fermi superfluid at finite temperatures

S N Klimin, Hadrien Kurkjian, J Tempere

Abstract The Leggett collective excitations for a two-band Fermi gas with s-wave pairing and Josephson interband coupling in the BCS-BEC crossover at finite temperatures are investigated within the Gaussian pair fluctuation approach. Eigenfrequencies and damping factors for Leggett modes are determined in a nonperturbative way, using the analytic continuation of the fluctuation propagator through a branch cut in the complex frequency plane, as in Kurkjian et al (2019 Phys. Rev. Lett. 122 093403). The treatment is performed beyond the low-energy expansion, which is necessary when the collective excitation energy reaches the pair-breaking continuum edge. The results are applied in particular to cold atomic gases at the orbital Feshbach resonance and in a regime far from BEC, which can be relevant for future experiments.

arXiv:1908.11795•New Journal of Physics•Nov 1, 2019•10 citations

Bragg spectroscopy and pair-breaking-continuum mode in a superfluid Fermi gas

Yvan Castin

The superfluid, pair condensed spin-1/2 Fermi gases are supposed to exhibit at nonzero wave vector a still unobserved collective excitation mode in their pair-breaking continuum. Using BCS theory at zero temperature and in the long wavelength limit, we predict that this mode is quantitatively observable (in frequency, width and spectral weight) in the response of a cold atom gas to a laser Bragg excitation, if one measures the perturbation induced on the order parameter modulus rather than on the density.

arXiv:1911.10950•Nov 25, 2019•0 citations
We demonstrate the existence of a collective excitation branch in the pair-breaking continuum of superfluid Fermi gases and BCS superconductors, as suggested by Littlewood and Varma in 1982. We analytically continue the RPA equation on the collective mode energy through its branch cut associated with the continuum, and obtain the full complex dispersion relation, including in the strong coupling regime. For Δ\DeltaΔ/μ\muμ > 1.210 (very close to unitarity in a superfluid Fermi gas), where Δ\DeltaΔ is the order parameter and μ\muμ the chemical potential, the real part of the branch is wholly within the band gap [0, 2Δ\DeltaΔ].In the long wavelength limit, the branch varies quadratically with the wave number, with a complex effective mass that we compute analytically. This contradicts the result of Littlewood and Varma that prevailed so far.
We demonstrate the existence of a collective excitation branch in the pair-breaking continuum of superfluid Fermi gases and BCS superconductors. At zero temperature, we analytically continue the equation on the collective mode energy in Anderson's Random Phase Approximation or Gaussian fluctuations through its branch cut associated with the continuum, and obtain the full complex dispersion relation, including in the strong coupling regime. The branch exists as long as the chemical potential μ is positive and the wave number below sqrt[2mμ]/ℏ (with m the fermion mass). In the long wavelength limit, the branch varies quadratically with the wave number, with a complex effective mass that we compute analytically for an arbitrary interaction strength.
The pair-condensed unpolarized spin-1/21/21/2 Fermi gases have a collective excitation branch in their pair-breaking continuum (V.A. Andrianov, V.N. Popov, 1976). We study it at zero temperature, with the eigenenergy equation deduced from the linearized time-dependent BCS theory and extended analytically to the lower half complex plane through its branch cut, calculating both the dispersion relation and the spectral weights (quasiparticle residues) of the branch. In the case of BCS superconductors, so called because the effect of the ion lattice is replaced by a short-range electron-electron interaction, we also include the Coulomb interaction and we restrict ourselves to the weak coupling limit Δ/μ→0+Δ/μ\to 0^+Δ/μ→0+ (ΔΔΔ is the order parameter, μμμ the chemical potential) and to wavenumbers q=O(1/ξ)q=O(1/ξ)q=O(1/ξ) where ξξξ is the size of a pair; when the complex energy zqz_qzq​ is expressed in units of ΔΔΔ and qqq in units of 1/ξ1/ξ1/ξ, the branch follows a universal law insensitive to the Coulomb interaction. In the case of cold atoms in the BEC-BCS crossover, only a contact interaction remains, but the coupling strength Δ/μΔ/μΔ/μ can take arbitrary values, and we study the branch at any wave number. At weak coupling, we predict three scales, that already mentioned q≈1/ξq\approx 1/ξq≈1/ξ, that q≈(Δ/μ)−1/3/ξq\approx(Δ/μ)^{-1/3}/ξq≈(Δ/μ)−1/3/ξ where the real part of the dispersion relation has a minimum and that q≈(μ/Δ)/ξ≈kFq\approx (μ/Δ)/ξ\approx k_{\rm F}q≈(μ/Δ)/ξ≈kF​ (kFk_{\rm F}kF​ is the Fermi wave number) where the branch reaches the edge of its existence domain. Near the point where the chemical potential vanishes on the BCS side, μ/Δ→0+μ/Δ\to 0^+μ/Δ→0+, where ξ≈kFξ\approx k_{\rm F}ξ≈kF​, we find two scales q≈(μ/Δ)1/2/ξq\approx(μ/Δ)^{1/2}/ξq≈(μ/Δ)1/2/ξ and q≈1/ξq\approx 1/ξq≈1/ξ. In all cases, the branch has a limit 2Δ2Δ2Δ and a quadratic start at q=0q=0q=0. These results were obtained for μ>0μ>0μ>0, where the eigenenergy equation admits at least two branch points εa(q)ε_a(q)εa​(q) and εb(q)ε_b(q)εb​(q) on the positive real axis, and for an analytic continuation through the interval $[ε_a(q),ε_b(q)] $. We find new continuum branches by performing the analytic continuation through [εb(q),+∞[[ε_b(q),+\infty[[εb​(q),+∞[ or even, for qqq low enough, where there is a third real positive branch point εc(q)ε_c(q)εc​(q), through [εb(q),εc(q)][ε_b(q),ε_c(q)][εb​(q),εc​(q)] and [εc(q),+∞[[ε_c(q),+\infty[[εc​(q),+∞[. On the BEC side μ<0μ<0μ<0 not previously studied, where there is only one real positive branch point $ ε_a(q)$, we also find new collective excitation branches under the branch cut [εa(q),+∞[[ε_a (q),+\infty[[εa​(q),+∞[. For μ>0μ>0μ>0, some of these new branches have a low-wavenumber exotic hypoacoustic zq≈q3/2z_q\approx q^{3/2}zq​≈q3/2 or hyperacoustic zq≈q4/5z_q\approx q^{4/5}zq​≈q4/5 behavior. For μ<0μ<0μ<0, we find a hyperacoustic branch and a nonhypoacoustic branch, with a limit 2Δ2Δ2Δ and a purely real quadratic start at q=0q=0q=0 for Δ/∣μ∣<0.222Δ/|μ|<0.222Δ/∣μ∣<0.222.
Studying the collective pairing phenomena in a two-component Fermi gas, we predict the appearance near the transition temperature TcT_cTc​ of a well-resolved collective mode of quadratic dispersion. The mode is visible both above and below TcT_cTc​ in the system's response to a driving pairing field. When approaching TcT_cTc​ from below, the phononic and pair-breaking branches, characteristic of the zero temperature behavior, reduce to a very low energy-momentum region when the pair correlation length reaches its critical divergent behavior ξpair∝∣Tc−T∣−1/2\xi_{\rm pair}\propto|T_c-T|^{-1/2}ξpair​∝∣Tc​−T∣−1/2; elsewhere, they are replaced by the quadratically-dispersed pairing resonance, which thus acts as a precursor of the phase transition. In the strong-coupling and Bose-Einstein Condensate regime, this mode is a weakly-damped propagating mode associated to a Lorentzian resonance. Conversely, in the BCS limit it is a relaxation mode of pure imaginary eigenenergy. At large momenta, the resonance disappears when it is reabsorbed by the lower-edge of the pairing continuum. At intermediate temperatures between 0 and TcT_cTc​, we unify the newly found collective phenomena near TcT_cTc​ with the phononic and pair-breaking branches predicted from previous studies, and we exhaustively classify the roots of the analytically continued dispersion equation, and show that they provided a very good summary of the pair spectral functions.
Studying the collective pairing phenomena in a two-component Fermi gas, we predict the appearance near the transition temperature Tc{T}_{c}Tc​ of a well-resolved collective mode of quadratic dispersion. The mode is visible both above and below Tc{T}_{c}Tc​ in the system's response to a driving pairing field. When approaching Tc{T}_{c}Tc​ from below, the phononic and pair-breaking branches, characteristic of the zero temperature behavior, reduce to a relatively low energy-momentum region. Elsewhere they are replaced by the quadratically dispersed pairing resonance, which thus acts as a precursor of the phase transition. In the strong-coupling and Bose-Einstein condensate regime, this mode is a weakly damped propagating mode associated with a Lorentzian resonance. Conversely, in the BCS limit it is a relaxation mode of pure imaginary eigenenergy. At large momenta, the resonance disappears when it is reabsorbed by the lower edge of the pairing continuum. At intermediate temperatures between 0 and Tc{T}_{c}Tc​ we unify the newly found collective phenomena near Tc{T}_{c}Tc​ with the phononic and pair-breaking branches predicted from previous studies, and we exhaustively classify the roots of the analytically continued dispersion equation, and show that they provided a very good summary of the pair spectral functions.
Abstract The Leggett collective excitations for a two-band Fermi gas with s-wave pairing and Josephson interband coupling in the BCS-BEC crossover at finite temperatures are investigated within the Gaussian pair fluctuation approach. Eigenfrequencies and damping factors for Leggett modes are determined in a nonperturbative way, using the analytic continuation of the fluctuation propagator through a branch cut in the complex frequency plane, as in Kurkjian et al (2019 Phys. Rev. Lett. 122 093403). The treatment is performed beyond the low-energy expansion, which is necessary when the collective excitation energy reaches the pair-breaking continuum edge. The results are applied in particular to cold atomic gases at the orbital Feshbach resonance and in a regime far from BEC, which can be relevant for future experiments.
The superfluid, pair condensed spin-1/2 Fermi gases are supposed to exhibit at nonzero wave vector a still unobserved collective excitation mode in their pair-breaking continuum. Using BCS theory at zero temperature and in the long wavelength limit, we predict that this mode is quantitatively observable (in frequency, width and spectral weight) in the response of a cold atom gas to a laser Bragg excitation, if one measures the perturbation induced on the order parameter modulus rather than on the density.