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arXiv:2406.13541 · Physical review. B./Physical review. B

Pairing amplification induced by nonadiabatic effects on the electron-phonon interaction throughout the BCS-BEC crossover

V.R. Velasco, Giovanni Midei, Massimo Capone, Andrea Perali

Published Feb 18, 2025• 2 citations
arXivHTMLPDF

Abstract

Nonadiabatic effects in the electron-phonon coupling are important whenever the ratio between the phononic and the electronic energy scales---the adiabatic ratio---is non-negligible. For superconducting systems, this gives rise to additional diagrams in the superconducting self-energy, the vertex, and cross-corrections. In this work, we explore these corrections in a two-dimensional single-band system through the crossover between the weak-coupling Bardeen-Cooper-Schrieffer (BCS) and strong-coupling Bose-Einstein regimes. By focusing on the pseudogap phase, we identify the parameter range in which the pairing amplitude is amplified by nonadiabatic effects and, due to controlled approximations, we map them throughout the BCS-BEC crossover. These effects become stronger as the system is driven deeply in the crossover regime, for phonon frequencies of the order of the hopping energy and for large enough electron-phonon coupling. Finally, we provide the phase space regions in which the effects of nonadiabaticity are more relevant for unconventional superconductors.

DOI
10.1103/physrevb.111.064512

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Nonadiabatic effects in the electron-phonon coupling are important whenever the ratio between the phononic and the electronic energy scales---the adiabatic ratio---is non-negligible. For superconducting systems, this gives rise to additional diagrams in the superconducting self-energy, the vertex, and cross-corrections. In this work, we explore these corrections in a two-dimensional single-band system through the crossover between the weak-coupling Bardeen-Cooper-Schrieffer (BCS) and strong-coupling Bose-Einstein regimes. By focusing on the pseudogap phase, we identify the parameter range in which the pairing amplitude is amplified by nonadiabatic effects and, due to controlled approximations, we map them throughout the BCS-BEC crossover. These effects become stronger as the system is driven deeply in the crossover regime, for phonon frequencies of the order of the hopping energy and for large enough electron-phonon coupling. Finally, we provide the phase space regions in which the effects of nonadiabaticity are more relevant for unconventional superconductors.
The condensation of paired fermions into superfluid states changes progressively depending on the coupling strength. At the midpoint of the crossover between Bardeen-Cooper-Schrieffer (BCS) weak-coupling and Bose-Einstein condensate (BEC) strong-coupling limits, paired fermions condensate most robustly, thereby leading to the emergence of a pseudogap due to enhanced pairing fluctuations. In the case of electrons in solids, excessively strong interactions often induce competing electronic orders instead of strong-coupling superconductivity, and experimental comprehension of the pseudogap remains incomplete. In this study, we provide experimental evidence demonstrating the opening of a pseudogap, marking the incipient stage of the BCS-BEC crossover in the organic system $\ensuremath{\kappa}\text{\ensuremath{-}}(\mathrm{BEDT}\text{\ensuremath{-}}{\mathrm{TTF})}_{2}X$. By controlling electron correlations, we investigate the thermodynamic properties of the BCS-BEC crossover and pseudogap phase. Since the superconductivity of $\ensuremath{\kappa}\text{\ensuremath{-}}{(\text{BEDT}\text{\ensuremath{-}}\mathrm{TTF})}_{2}X$ arises from a simple Fermi liquid that does not exhibit any other electronic orders, our study sheds light on the inherent nature of the BCS-BEC crossover.
We study the crossover from the weak coupling Bardeen-Cooper-Schrieffer (BCS) state to Bose-Einstein condensation (BEC) at strong coupling in a two-band superconductor with orbitals of opposite parity coexisting at a common Fermi surface in the metallic state. In such systems hybridization can play a role similar to spin-orbit interaction in fermionic spinor gases, enhancing interband pairing and opening the possibility for driving the BCS-BEC crossover. Building on a mean-field analysis, we investigate the BCS-BEC crossover induced by a variation of the hybridization strength. We show that the relevant scale for the crossover depends strongly on the ratio of effective masses, with the crossover favored in systems with one dispersive and one flat band. Including the effect of thermal pair fluctuations in a one-loop approximation, we calculate the dependence of the critical temperature on the microscopic parameters in the strong coupling regime.
The theory of unconventional superconductors continues to provide profound puzzles. The crossover between the weakly coupled Bardeen-Cooper-Schrieffer (BCS) state and the strong-pairing Bose-Einstein condensate (BEC) provides a useful perspective on how to address these questions. This paper describes a self-consistent framework for thinking about the crossover regime in between these two limits. The review discusses to what extent this BCS-BEC theory applies to a range of classes of superconducting materials including the cuprates, iron pnictides, twisted bilayer graphene, and interfacial superconductivity among others.
We study the superfluid critical temperature in a two-band attractive Fermi system with strong pairing fluctuations associated with both interband and intraband couplings. We focus specifically on a configuration where the intraband coupling is varied from weak to strong in a shallow band coupled to a weakly interacting deeper band. The whole crossover from the Bardeen-Cooper-Schrieffer (BCS) condensation of largely overlapping Cooper pairs to the Bose-Einstein condensation (BEC) of tightly bound molecules is covered by our analysis, which is based on the extension of the Nozi\`eres-Schmitt-Rink approach to a two-band system. In comparison with the single-band case, we find a strong enhancement of the critical temperature, a significant reduction of the preformed pair region where pseudogap effects are expected, and the entanglement of two kinds of composite bosons in the strong-coupling BEC regime.
The role of random onsite and hopping disorder in inducing a Bardeen–Cooper–Schrieffer superconductor to Bose–Einstein condensate (BCS–BEC) crossover is investigated using the Bogoliubov–de Gennes (BdG) method on a negative- U Hubbard model in two dimensions. In the strong disorder limit, the BdG state becomes highly inhomogeneous with the formation of superconducting ' droplets ' which are characterized by large pairing amplitudes. The different physical properties calculated for the inhomogeneous state differ significantly from the pure system at all values of electron densities, while we show that a crossover to a BEC state, if at all, is possible at low electron densities. The random onsite disorder supports such a crossover scenario, which is confirmed by observing the chemical potential to slip below the noninteracting band (Leggett criterion). The hopping disorder, however, renders a delocalized phase which allows for overlapping between the pairs, thus preventing the formation of a 'local pair' phase. Also the increase in bandwidth induced by hopping disorder always accommodates the chemical potential inside the band, and hence rules out a crossover phenomenon. However, another variant of the off-diagonal inhomogeneity, i.e. an anisotropic hopping, offers a pro-crossover scenario in the large anisotropy limit.
Recent experiments with ultracold fermionic atoms in optical lattices have provided a tuneable and clean realization of the attractive Hubbard model (AHM). In view of this, several physical properties may be thoroughly studied across the crossover between weak [Bardeen-Cooper-Schrieffer, (BCS)] and strong [Bose-Einstein condensation (BEC)] couplings. Here we report on extensive determinant Quantum Monte Carlo (DQMC) studies of the AHM on a square lattice from which several different quantities have been calculated and should be useful as a road map to experiments. We have obtained a detailed phase diagram for the critical superconducting temperature Tc{T}_{c}Tc​ in terms of the band filling $\ensuremath{\langle}n\ensuremath{\rangle}$ and interaction strength UUU from which we pinpoint a somewhat wide region $|U|/t\ensuremath{\approx}5\ifmmode\pm\else\textpm\fi{}1$ (ttt is the hopping amplitude) and $\ensuremath{\langle}n\ensuremath{\rangle}\ensuremath{\approx}0.79\ifmmode\pm\else\textpm\fi{}0.09$ leading to a maximum ${T}_{c}\ensuremath{\approx}0.16t$. Two additional temperature scales, namely, pairing Tp{T}_{p}Tp​ and degeneracy Td{T}_{d}Td​ have been highlighted: The former sets the scale for pair formation (believed to be closely related to the scale for the gap of spin excitations in cuprates), whereas the latter sets the scale for dominant quantum effects. Our DQMC data for the distribution of doubly occupied sites for the momentum distribution function, and for the quasiparticle weight show distinctive features on both sides of the BCS-BEC crossover, being also suggestive of an underlying crossover between Fermi- and non-Fermi liquid behaviors.