# BCS-BEC Crossover and Pairing Fluctuations in a Two Band Superfluid/Superconductor: A T Matrix Approach

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}[5] and iron based compounds [6]. From the theoretical viewpoint, a variety of non-trivial phenomena has been proposed (e.g., topological phase soliton [7,8], odd-frequency pairing [9], multiple Leggett mode [10], hidden criticality [11], stable Sarma phase [12], and Lifshitz transitions with resonance effects and amplification of the critical temperature [13,14]).

## 2. Hamiltonian

## 3. T Matrix Approximation for Two Band Systems

## 4. Results

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Bardeen, J.; Cooper, L.N.; Schrieffer, J.R. Theory of Superconductivity. Phys. Rev.
**1957**, 108, 1175. [Google Scholar] [CrossRef] [Green Version] - Suhl, H.; Matthias, B.T.; Walker, L.R. Bardeen–Cooper–Schrieffer Theory of Superconductivity in the Case of Overlapping Bands. Phys. Rev. Lett.
**1959**, 3, 552. [Google Scholar] [CrossRef] - Tanaka, Y. Multicomponent superconductivity based on multiband superconductors. Supercond. Sci. Technol.
**2015**, 28, 034002. [Google Scholar] [CrossRef] - Milošević, M.V.; Perali, A. Emergent phenomena in multicomponent superconductivity: An introduction to the focus issue. Supercond. Sci. Technol.
**2015**, 28, 060201. [Google Scholar] [CrossRef] [Green Version] - Nagamatsu, J.; Nakagawa, N.; Muranaka, T.; Zenitani, Y.; Akimitsu, J. Superconductivity at 39K in magnesium diboride. Nature
**2001**, 410, 63–64. [Google Scholar] [CrossRef] - Kamihara, Y.; Watanabe, T.; Hirano, M.; Hosono, H. Iron-Based Layered Superconductor La[O
_{1−x}F_{x}]FeAs (x = 0.05–0.12) with T_{c}= 26 K. J. Am. Chem. Soc.**2008**, 130, 3296–3297. [Google Scholar] [CrossRef] [PubMed] - Tanaka, Y. Soliton in Two-Band Superconductor. Phys. Rev. Lett.
**2001**, 88, 017002. [Google Scholar] [CrossRef] [PubMed] - Kuplevakhsky, S.V.; Omelyanchouk, A.N.; Yerin, Y.S. Soliton states in mesoscopic two band-superconducting cylinders. Low Temp. Phys.
**2011**, 37, 667. [Google Scholar] [CrossRef] [Green Version] - Black-Schaffer, A.M.; Balatsky, A.V. Odd-frequency superconducting pairing in multiband superconductors. Phys. Rev. B
**2013**, 88, 104514. [Google Scholar] [CrossRef] [Green Version] - Ota, Y.; Machida, M.; Koyama, T.; Aoki, H. Collective modes in multiband superfluids and superconductors: Multiple dynamic classes. Phys. Rev. B
**2011**, 83, 060507(R). [Google Scholar] [CrossRef] [Green Version] - Komendová, L.; Chen, Y.; Shanenko, A.A.; Milošević, M.V.; Peeters, F.M. Two-Band Superconductors: Hidden Criticality Deep in the Superconducting State. Phys. Rev. Lett.
**2012**, 108, 207002. [Google Scholar] [CrossRef] [PubMed] [Green Version] - He, L.; Zhuang, P. Stable Sarma state in two band Fermi systems. Phys. Rev. B
**2009**, 79, 024511. [Google Scholar] [CrossRef] [Green Version] - Valletta, A.; Bianconi, A.; Perali, A.; Saini, N.L. Electronic and superconducting properties of a superlattice of quantum stripes at the atomic limit. Z. Phys. B: Condens. Matter
**1997**, 104, 707–713. [Google Scholar] [CrossRef] - Mazziotti, M.V.; Valletta, A.; Campi, G.; Innocenti, D.; Perali, A.; Bianconi, A. Possible Fano resonance for high-T
_{c}multi-gap superconductivity in p-Terphenyl doped by K at the Lifshitz transition. Eur. Phys. Lett.**2017**, 118, 37003. [Google Scholar] [CrossRef] [Green Version] - Eagles, D.M. Possible pairing without superconductivity at low carrier concentrations in bulk and thin-film superconducting semiconductor. Phys. Rev.
**1969**, 186, 456. [Google Scholar] [CrossRef] - Leggett, A.J. Diatomic molecules and Cooper pairs. In Modern Trends in the Theory of Condensed Matter; Peralski, A., Przystawa, J.A., Eds.; Springer: Berlin, Germany, 1980. [Google Scholar]
- Regal, C.A.; Greiner, M.; Jin, D.S. Observation of Resonance Condensation of Fermionic Atom Pairs. Phys. Rev. Lett.
**2004**, 92, 040403. [Google Scholar] [CrossRef] [Green Version] - Zwierlein, M.W.; Stan, C.A.; Schunck, C.H.; Raupach, S.M.F.; Kerman, A.J.; Ketterle, W. Condensation of Pairs of Fermionic Atoms near a Feshbach Resonance. Phys. Rev. Lett.
**2004**, 92, 120403. [Google Scholar] [CrossRef] [Green Version] - Giorgini, S.; Pitaevskii, L.P.; Stringari, S. Theory of ultracold atomic Fermi gases. Rev. Mod. Phys.
**2008**, 80, 1215. [Google Scholar] [CrossRef] [Green Version] - Bloch, I.; Dalibard, J.; Zwerger, W. Many-body physics with ultracold gases. Rev. Mod. Phys.
**2008**, 80, 885. [Google Scholar] [CrossRef] [Green Version] - Calvanese Strinati, G.; Pieri, P.; Röpke, G.; Schuck, P.; Urban, M. The BCS-BEC crossover: From ultra-cold Fermi gases to nuclear systems. Phys. Rep.
**2018**, 738, 1. [Google Scholar] [CrossRef] [Green Version] - Kasahara, S.; Watashige, T.; Hanaguri, T.; Kohsaka, Y.; Yamashita, T.; Shimoyama, Y.; Mizukami, Y.; Endo, R.; Ikeda, H.; Aoyama, K.; et al. Field-induced superconducting phase of FeSe in the BCS-BEC cross-over. Proc. Natl. Acad. Sci. USA
**2014**, 111, 16309. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kasahara, S.; Yamashita, T.; Shi, A.; Kobayashi, R.; Shimoyama, Y.; Watashige, T.; Ishida, K.; Terashima, T.; Wolf, T.; Hardy, F.; et al. Giant superconducting fluctuations in the compensated semimetal FeSe at the BCS-BEC crossover. Nat. Commun.
**2016**, 7, 12843. [Google Scholar] [CrossRef] [PubMed] - Rinott, S.; Chashka, K.B.; Ribak, A.; Rienks Emile, D.L.; Taleb-Ibrahimi, A.; Le Fevre, P.; Bertran, F.; Randeria, M.; Kanigel, A. Tuning across the BCS-BEC crossover in the multiband superconductor Fe
_{1+y}Se_{x}Te_{1−x}: An angle-resolved photoemission study. Sci. Adv.**2017**, 3, e1602372. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Innocenti, D.; Poccia, N.; Ricci, A.; Valletta, A.; Caprara, S.; Perali, A.; Bianconi, A. Resonant and crossover phenomena in a multiband superconductor: Tuning the chemical potential near a band edge. Phys. Rev. B
**2010**, 82, 184528. [Google Scholar] [CrossRef] [Green Version] - Salasnich, L.; Shanenko, A.A.; Vagov, A.; Albino Aguiar, J.; Perali, A. Screening of pair fluctuations in superconductors with coupled shallow and deep bands: A route to higher-temperature superconductivity. Phys. Rev. B
**2019**, 100, 064510. [Google Scholar] [CrossRef] [Green Version] - Tajima, H.; Yerin, Y.; Perali, A.; Pieri, P. Enhanced critical temperature, pairing fluctuation effects, and BCS-BEC crossover in a two band Fermi gas. Phys. Rev. B
**2019**, 99, 180503(R). [Google Scholar] [CrossRef] [Green Version] - Hanaguri, T.; Kasahara, S.; Böker, J.; Eremin, I.; Shibauchi, T.; Matsuda, Y. Quantum Vortex Core and Missing Pseudogap in the Multiband BCS-BEC Crossover Superconductor FeSe. Phys. Rev. Lett.
**2019**, 122, 077001. [Google Scholar] [CrossRef] [Green Version] - Perali, A.; Pieri, P.; Strinati, G.C.; Castellani, C. Pseudogap and spectral function from superconducting fluctuations to the bosonic limit. Phys. Rev. B
**2002**, 66, 024510. [Google Scholar] [CrossRef] [Green Version] - Tsuchiya, S.; Watanabe, R.; Ohashi, Y. Single-particle properties and pseudogap effects in the BCS-BEC crossover regime of an ultracold Fermi gas above T
_{c}. Phys. Rev. A**2009**, 80, 033613. [Google Scholar] [CrossRef] [Green Version] - Palestini, F.; Perali, A.; Pieri, P.; Strinati, G.C. Dispersions, weights, and widths of the single-particle spectral function in the normal phase of a Fermi gas. Phys. Rev. B
**2012**, 85, 024517. [Google Scholar] [CrossRef] [Green Version] - Marsiglio, F.; Pieri, P.; Perali, A.; Palestini, F.; Strinati, G.C. Pairing effects in the normal phase of a two-dimensional Fermi gas. Phys. Rev. B
**2015**, 91, 054509. [Google Scholar] [CrossRef] [Green Version] - Takahashi, H.; Nabeshima, F.; Ogawa, R.; Ohmichi, E.; Ohta, H.; Maeda, A. Superconducting fluctuations in FeSe investigated by precise torque magnetometry. Phys. Rev. B
**2019**, 99, 060503(R). [Google Scholar] [CrossRef] [Green Version] - Gati, E.; Böhmer, A.E.; Bud’ko, S.L.; Canfield, P.C. Bulk superconductivity and role of fluctuations in the iron based superconductor FeSe at high pressures. Phys. Rev. Lett.
**2019**, 123, 167002. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lubashevsky, Y.; Lahoud, E.; Chashka, K.; Podolsky, D.; Kanigel, A. Shallow pockets and very strong coupling superconductivity in FeSe
_{x}Te_{1−x}. Nat. Phys.**2012**, 8, 309–312. [Google Scholar] [CrossRef] [Green Version] - Zhang, R.; Cheng, Y.; Zhai, H.; Zhang, P. Orbital Feshbach Resonance in Alkali-Earth Atoms. Phys. Rev. Lett.
**2015**, 115, 135301. [Google Scholar] [CrossRef] - Pagano, G.; Mancini, M.; Cappellini, G.; Livi, L.; Sias, C.; Catani, J.; Inguscio, M.; Fallani, L. Strongly Interacting Gas of Two-Electron Fermions at an Orbital Feshbach Reonance. Phys. Rev. Lett.
**2015**, 115, 265301. [Google Scholar] [CrossRef] [Green Version] - Höfer, M.; Riegger, L.; Scazza, F.; Hofrichter, C.; Fernandes, D.R.; Parish, M.M.; Levinsen, J.; Bloch, I.; Folling, S. Observation of an Orbital Interaction-Induced Feshbach Resonance in
^{173}Yb. Phys. Rev. Lett.**2015**, 115, 265302. [Google Scholar] [CrossRef] [Green Version] - He, L.; Hu, H.; Liu, X.-J. Two-band description of resonant supefluidity in atomic Fermi gases. Phys. Rev. A
**2015**, 91, 023622. [Google Scholar] [CrossRef] [Green Version] - He, L.; Wang, J.; Peng, S.-G.; Liu, X.-J.; Hu, H. Strongly correlated Fermi superfluid near an orbital Feshbach resonance: Stability, equation of state, and Leggett mode. Phys. Rev. A
**2016**, 94, 043624. [Google Scholar] [CrossRef] - Mondal, S.; Inotani, D.; Ohashi, Y. Single-particle Excitations and Strong-Coupling Effects in the BCS-BEC Crossover Regime of a Rare-Earth Fermi Gas with an Orbital Feshbach Resonance. J. Phys. Soc. Jpn.
**2018**, 87, 084302. [Google Scholar] [CrossRef] [Green Version] - Sedrakian, A.; Clark, J.W. Superfluidity in nuclear systems and neutron stars. Eur. Phys. Jour. A
**2019**, 55, 167. [Google Scholar] [CrossRef] [Green Version] - Ohashi, Y.; Tajima, H.; van Wyk, P. BCS-BEC crossover in cold atomic and in nuclear systems. Progr. Part. Nucl. Phys.
**2019**. [Google Scholar] [CrossRef] - Nozières, P.; Schmitt-Rink, S. Bose condensation in an attractive fermion gas: From weak to strong coupling superconductivity. J. Low Temp. Phys.
**1985**, 59, 195–211. [Google Scholar] [CrossRef] - Serene, J.W. Stability of two-dimensional Fermi liquids against pair fluctuations with large total momentum. Phys. Rev. B
**1989**, 40, 10873. [Google Scholar] [CrossRef] [PubMed] - Wolf, S.; Vagov, A.; Shanenko, A.A.; Axt, V.M.; Perali, A.; Albino Aguiar, J. BCS-BEC crossover induced by a shallow band: Pushing standard superconductivity types apart. Phys. Rev. B
**2017**, 95, 094521. [Google Scholar] [CrossRef] [Green Version] - Iskin, M.; Sá de Melo, C.A.R. Two-band superfluidity from the BCS to the BEC limit. Phys. Rev. B
**2006**, 74, 144517. [Google Scholar] [CrossRef] [Green Version] - Yerin, Y.; Tajima, H.; Pieri, P.; Perali, A. Coexistence of giant Cooper pairs with a bosonic condensate and anomalous behavior of energy gaps in the BCS-BEC crossover of a two band superfluid Fermi gas. Phys. Rev. B
**2019**, 100, 104528. [Google Scholar] [CrossRef] [Green Version] - 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. [Google Scholar] [CrossRef] - Pekker, D.; Babadi, M.; Sensarma, R.; Zinner, N.; Pollet, L.; Zwierlein, M.W.; Demler, E. Competition between Pairing and Ferromagnetic Instabilities in Ultracold Fermi Gases near Feshbach Resonances. Phys. Rev. Lett.
**2011**, 106, 05402. [Google Scholar] [CrossRef] [Green Version] - Tan, S. Energetics of a strongly correlated Fermi gas. Ann. Phys.
**2008**, 323, 2952. [Google Scholar] [CrossRef] [Green Version] - Tan, S. Large momentum part of a strongly correlated Fermi gas. Ann. Phys.
**2008**, 323, 2971. [Google Scholar] [CrossRef] [Green Version] - Tan, S. Generalized virial theorem and pressure relation for a strongly correlated Fermi gas. Ann. Phys.
**2008**, 323, 2987. [Google Scholar] [CrossRef] [Green Version] - Palestini, F.; Perali, A.; Pieri, P.; Strinati, G.C. Temperature and coupling dependence of the universal contact intensity for an ultracold Fermi gas. Phys. Rev. A
**2010**, 82, 021605(R). [Google Scholar] [CrossRef] [Green Version] - Pini, M.; Pieri, P.; Strinati, G.C. Fermi gas throughout the BCS-BEC crossover: Comparative study of t matrix approaches with various degrees of self-consistency. Phys. Rev. B
**2019**, 99, 094502. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Feynman diagrams of (

**a**) the self-energy ${\Sigma}_{i}$ and (

**b**) many body T matrix ${T}_{\mathrm{MB}}$ where single lines represent bare Green’s functions ${G}_{i}^{0}$. The black box indicates the matrix of coupling constants $\widehat{U}$.

**Figure 2.**Momentum distribution functions ${\overline{n}}_{i}\left(k\right)$ at the superfluid critical temperature ${T}_{\mathrm{c}}$ identified by Equation (15). The left column corresponds to ${\lambda}_{12}=1$ and (

**a**) ${\left({k}_{\mathrm{F},2}{a}_{22}\right)}^{-1}=-1$, (

**b**) ${\left({k}_{\mathrm{F},2}{a}_{22}\right)}^{-1}=0$, and (

**c**) ${\left({k}_{\mathrm{F},2}{a}_{22}\right)}^{-1}=1$, while the right column corresponds to ${\lambda}_{12}=2$ and (

**d**) ${\left({k}_{\mathrm{F},2}{a}_{22}\right)}^{-1}=-1$, (

**e**) ${\left({k}_{\mathrm{F},2}{a}_{22}\right)}^{-1}=0$, and (

**f**) ${\left({k}_{\mathrm{F},2}{a}_{22}\right)}^{-1}=1$. The coupling in the deep band is fixed at ${\left({k}_{\mathrm{F},1}{a}_{11}\right)}^{-1}=-4$. The dotted curves correspond to the free Fermi distribution functions $f\left({\xi}_{\mathbf{k},i}\right)$ calculated with the same chemical potential and temperature.

**Figure 3.**Superfluid critical temperature ${T}_{\mathrm{c}}$ in the Bardeen–Cooper–Schrieffer–Bose–Einstein condensation (BCS-BEC) crossover regime of the shallow band calculated for different values of the pair-exchange coupling ${\lambda}_{12}$ as a function of the coupling ${\left({k}_{\mathrm{F},2}{a}_{22}\right)}^{-1}$ in the shallow band. The black dotted curve represents the numerical result of the single band counterpart. The dashed line ${T}_{\mathrm{BEC}}^{2\mathrm{b}}=0.218{(n/{n}_{2}^{0})}^{2/3}{T}_{\mathrm{F},2}=0.633{T}_{\mathrm{F},2}$ is the molecular BEC temperature when all particles in both bands form tightly bound molecules. The coupling in the deep band is fixed at ${\left({k}_{\mathrm{F},1}{a}_{11}\right)}^{-1}=-4$.

**Figure 4.**Chemical potential ${\mu}_{2}\equiv \mu -{E}_{0}$ referred to the bottom of the shallow band, at $T={T}_{\mathrm{c}}$ for different values of the pair-exchange coupling ${\lambda}_{12}$ as a function of the coupling ${\left({k}_{\mathrm{F},2}{a}_{22}\right)}^{-1}$ in the shallow band. The horizontal line $-{E}_{0}$ corresponds to the bottom of the lower band. The dotted curves represent half of the two body binding energy $-{E}_{\mathrm{b}}/2$ in our two band configuration. The coupling in the deep band is fixed at ${\left({k}_{\mathrm{F},1}{a}_{11}\right)}^{-1}=-4$.

**Figure 5.**Occupation number density ${n}_{1}$ (decreasing functions) and ${n}_{2}$ (increasing functions) at $T={T}_{\mathrm{c}}$ for different values of the pair-exchange coupling ${\lambda}_{12}$ as a function of the coupling ${\left({k}_{\mathrm{F},2}{a}_{22}\right)}^{-1}$ in the shallow band. The horizontal dashed lines represent ${n}_{1,0}=0.798n$ (upper line) and ${n}_{2,0}=0.202n$ (lower line). The coupling in the deep band is fixed at ${\left({k}_{\mathrm{F},1}{a}_{11}\right)}^{-1}=-4$.

**Figure 6.**Momentum distribution functions ${\overline{n}}_{i}\left(k\right)$ multiplied by ${(k/{k}_{\mathrm{F},\mathrm{t}})}^{4}$ at ${\left({k}_{\mathrm{F},2}{a}_{22}\right)}^{-1}=0$ and $T={T}_{\mathrm{c}}$. Solid (dashed) lines show the results of a deep (shallow) band. The coupling in the deep band is fixed at ${\left({k}_{\mathrm{F},1}{a}_{11}\right)}^{-1}=-4$.

**Figure 7.**(

**a**) Critical temperatures within the T matrix approach (${T}_{\mathrm{c}}$, full lines) and Nozières–Schmitt–Rink (NSR) approximation (${T}_{\mathrm{c}}^{\mathrm{NSR}}$, dotted lines) for different values of the coupling ${\lambda}_{12}$ as a function of the coupling ${\left({k}_{\mathrm{F},2}{a}_{22}\right)}^{-1}$ in the shallow band. (

**b**) Relative difference between the two critical temperatures $({T}_{\mathrm{c}}-{T}_{\mathrm{c}}^{\mathrm{NSR}})/{T}_{\mathrm{c}}$ along the BCS-BEC crossover in the shallow band. The dashed curve is the asymptotic behavior for the single band counterpart given by Equation (20) [55]. The coupling in the deep band is fixed at ${\left({k}_{\mathrm{F},1}{a}_{11}\right)}^{-1}=-4$.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Tajima, H.; Perali, A.; Pieri, P.
BCS-BEC Crossover and Pairing Fluctuations in a Two Band Superfluid/Superconductor: A *T* Matrix Approach. *Condens. Matter* **2020**, *5*, 10.
https://doi.org/10.3390/condmat5010010

**AMA Style**

Tajima H, Perali A, Pieri P.
BCS-BEC Crossover and Pairing Fluctuations in a Two Band Superfluid/Superconductor: A *T* Matrix Approach. *Condensed Matter*. 2020; 5(1):10.
https://doi.org/10.3390/condmat5010010

**Chicago/Turabian Style**

Tajima, Hiroyuki, Andrea Perali, and Pierbiagio Pieri.
2020. "BCS-BEC Crossover and Pairing Fluctuations in a Two Band Superfluid/Superconductor: A *T* Matrix Approach" *Condensed Matter* 5, no. 1: 10.
https://doi.org/10.3390/condmat5010010