# Low-Lying Collective Excitations of Superconductors and Charged Superfluids

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## Abstract

**:**

## 1. Introduction

## 2. Gaussian Pair-and-Density Fluctuation Method

## 3. Spectra of Collective Excitations

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

GPDF | Gaussian Pair-and-Density Fluctuation |

RPA | Random Phase Approximation |

## References

- Ohashi, Y.; Takada, S. On the plasma oscillation in superconductivity. J. Phys. Soc. Jpn.
**1998**, 67, 551–559. [Google Scholar] [CrossRef] - Shimano, R.; Tsuji, N. Higgs Mode in Superconductors. Annu. Rev. Condens. Matter Phys.
**2020**, 11, 103–124. [Google Scholar] [CrossRef] - Carlson, R.V.; Goldman, A.M. Propagating Order-Parameter Collective Modes in Superconducting Films. Phys. Rev. Lett.
**1975**, 34, 11–15. [Google Scholar] [CrossRef] - Hoinka, S.; Dyke, P.; Lingham, M.G.; Kinnunen, J.J.; Bruun, G.M.; Vale, C.J. Goldstone mode and pair-breaking excitations in atomic Fermi superfluids. Nat. Phys.
**2017**, 13, 943–946. [Google Scholar] [CrossRef] - Carusotto, I.; Castin, Y. Atom Interferometric Detection of the Pairing Order Parameter in a Fermi Gas. Phys. Rev. Lett.
**2005**, 94, 223202. [Google Scholar] [CrossRef] [PubMed] - Matsunaga, R.; Hamada, Y.I.; Makise, K.; Uzawa, Y.; Terai, H.; Wang, Z.; Shimano, R. Higgs Amplitude Mode in the BCS Superconductors Nb
_{1-x}Ti_{x}N Induced by Terahertz Pulse Excitation. Phys. Rev. Lett.**2013**, 111, 057002. [Google Scholar] [CrossRef] [PubMed] - Ohashi, Y.; Takada, S. Goldstone mode in charged superconductivity: Theoretical studies of the Carlson-Goldman mode and effects of the Landau damping in the superconducting state. J. Phys. Soc. Jpn.
**1997**, 66, 2437–2458. [Google Scholar] [CrossRef] - Anderson, P.W. Random-Phase Approximation in the Theory of Superconductivity. Phys. Rev.
**1958**, 112, 1900–1916. [Google Scholar] [CrossRef] - Repplinger, T.; Klimin, S.; Gélédan, M.; Tempere, J.; Kurkjian, H. Plasmons in three-dimensional superconductors. Phys. Rev. B
**2023**, 107, 014504. [Google Scholar] [CrossRef] - Klimin, S.N.; Tempere, J.; Repplinger, T.; Kurkjian, H. Collective excitations of a charged Fermi superfluid in the BCS-BEC crossover. arXiv
**2023**, arXiv:2208.09757. [Google Scholar] - Sá de Melo, C.A.R.; Randeria, M.; Engelbrecht, J.R. Crossover from BCS to Bose superconductivity: Transition temperature and time-dependent Ginzburg-Landau theory. Phys. Rev. Lett.
**1993**, 71, 3202–3205. [Google Scholar] [CrossRef] - Castin, Y.; Kurkjian, H. Collective excitation branch in the continuum of pair-condensed Fermi gases: Analytical study and scaling laws. Comptes Rendus Phys.
**2020**, 21, 253–310. [Google Scholar] [CrossRef] - Nozières, P. Le Problème à N Corps: Propriétés Générales des gaz de Fermions; Dunod: Paris, France, 1963. [Google Scholar]
- Reijnders, K.J.A.; Tudorovskiy, T.; Katsnelson, M.I. Semiclassical theory for plasmons in spatially inhomogeneous media. Ann. Phys.
**2022**, 446, 169116. [Google Scholar] [CrossRef] - Kurkjian, H.; Klimin, S.N.; Tempere, J.; Castin, Y. Pair-Breaking Collective Branch in BCS Superconductors and Superfluid Fermi Gases. Phys. Rev. Lett.
**2019**, 122, 093403. [Google Scholar] [CrossRef] [PubMed] - Klimin, S.N.; Tempere, J.; Kurkjian, H. Collective excitations of superfluid Fermi gases near the transition temperature. Phys. Rev. A
**2021**, 103, 043336. [Google Scholar] [CrossRef] - Kurkjian, H.; Castin, Y.; Sinatra, A. Three-Phonon and Four-Phonon Interaction Processes in a Pair-Condensed Fermi Gas. Ann. Phys.
**2017**, 529, 1600352. [Google Scholar] [CrossRef] - Abrikosov, A.A.; Gorkov, L.P.; Dzyaloshinski, I.E. Methods of Quantum Field Theory in Statistical Physics; Dover Publications: London, UK, 1975; ISBN 978-0199232727. [Google Scholar]
- Klimin, S.N.; Tempere, J.; Kurkjian, H. Phononic collective excitations in superfluid Fermi gases at nonzero temperatures. Phys. Rev. A
**2019**, 100, 063634. [Google Scholar] [CrossRef]

**Figure 1.**(Adapted from Ref. [16].) Angular-point frequencies of the GPDF matrix elements for $1/{k}_{F}{a}_{s}=0.5$ and $T=0.9{T}_{c}$. The areas between curves determine intervals for the analytic continuation. The momentum is multiplied by the coherence length $\xi \equiv {v}_{F}/{T}_{c}$, where ${v}_{F}$ is the Fermi velocity (${v}_{F}=2$ in the present units).

**Figure 2.**Contour plots of the spectral weight functions at $T=0.9{T}_{c}$ (

**a**,

**b**) and at $T=0.99{T}_{c}$ (

**c**,

**d**) for the phase (

**a**,

**c**) and modulus (

**b**,

**d**) response of a charged Fermi superfluid with the inverse s-wave scattering length $1/{k}_{F}{a}_{s}=-0.5$. The momentum is measured in units of $1/\xi $, where $\xi \equiv {v}_{F}/{T}_{c}$ with the Fermi velocity ${v}_{F}=\hslash {k}_{F}/m$. The clipping area above the plot range for the spectral weights is shown by red color.

**Figure 3.**Spectral weight functions of the phase (

**a**,

**c**) and amplitude (

**b**,

**d**) pair field response at $T/{T}_{c}=0.9$ (

**a**,

**b**) and $T/{T}_{c}=0.99$ (

**c**,

**d**) for several values of the field momentum q.

**Figure 4.**Heavy solid, dashed and dotted curves: eigenfrequency (

**a**,

**c**) and damping factor (

**b**,

**d**) of low-lying collective excitations for the inverse scattering length $1/{k}_{F}{a}_{s}=-0.5$ (

**a**,

**b**) and $1/{k}_{F}{a}_{s}=0$ (

**c**,

**d**) at the relative temperature $T/{T}_{c}=0.9$. Thin solid, dashed and dotted curves show angular-point frequencies, which indicate bounds of energy intervals for different scattering processes.

**Figure 5.**Eigenfrequency (

**a**,

**c**) and damping factor (

**b**,

**d**) of low-lying collective excitations for the inverse scattering length $1/{k}_{F}{a}_{s}=-0.5$ (

**a**,

**b**) and $1/{k}_{F}{a}_{s}=0$ (

**c**,

**d**) at the relative temperature $T/{T}_{c}=0.99$. The notations are the same as in Figure 4.

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**MDPI and ACS Style**

Klimin, S.; Tempere, J.; Kurkjian, H.
Low-Lying Collective Excitations of Superconductors and Charged Superfluids. *Condens. Matter* **2023**, *8*, 42.
https://doi.org/10.3390/condmat8020042

**AMA Style**

Klimin S, Tempere J, Kurkjian H.
Low-Lying Collective Excitations of Superconductors and Charged Superfluids. *Condensed Matter*. 2023; 8(2):42.
https://doi.org/10.3390/condmat8020042

**Chicago/Turabian Style**

Klimin, Serghei, Jacques Tempere, and Hadrien Kurkjian.
2023. "Low-Lying Collective Excitations of Superconductors and Charged Superfluids" *Condensed Matter* 8, no. 2: 42.
https://doi.org/10.3390/condmat8020042