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arXiv:1902.04588 · Physical review. B./Physical review. B

Dynamic properties of superconductors: Anderson-Bogoliubov mode and Berry phase in the BCS and BEC regimes

Dmitry Mozyrsky, Andrey V. Chubukov

Published May 17, 2019• 6 citations
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Abstract

We analyze the evolution of the dynamics of a neutral $s$-wave superconductor between the BCS and BEC regimes. We consider 2d case when a BCS-BEC crossover occurs already at weak coupling as a function of the ratio of the two scales---the Fermi energy ${E}_{F}$ and the bound state energy for two fermions in a vacuum, ${E}_{0}$. The BCS and BEC limits correspond to ${E}_{F}\ensuremath{\gg}{E}_{0}$ and ${E}_{F}\ensuremath{\ll}{E}_{0}$, respectively. The chemical potential $\ensuremath{\mu}={E}_{F}\ensuremath{-}{E}_{0}$ changes the sign between the two regimes. We use the effective action approach, derive the leading terms in the expansion of the effective action in the spatial and time derivative of the slowly varying superconducting order parameter $\mathrm{\ensuremath{\Delta}}(r,\ensuremath{\tau})$, and express the action in terms of a derivatives of the phase $\ensuremath{\phi}(r,\ensuremath{\tau})$ of $\mathrm{\ensuremath{\Delta}}(r,\ensuremath{\tau})=\mathrm{\ensuremath{\Delta}}{e}^{i\ensuremath{\phi}(r,\ensuremath{\tau})}$. The action contains ${(\ensuremath{\nabla}\ensuremath{\phi})}^{2}$ and ${\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\phi}}}^{2}$ terms, which determine the dispersion of collective phase fluctuations, and $i\ensuremath{\pi}A\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\phi}}$ term. For continuous $\ensuremath{\phi}(r,\ensuremath{\tau})$, the latter reduces to the contribution from the boundary and does not affect the dynamics. We show that this long-wavelength action does not change through the BCS-BEC crossover. We apply our approach to a moving vortex, for which $\ensuremath{\phi}$ is singular at the center of the vortex core, and $i\ensuremath{\pi}{A}_{\mathrm{vort}}\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\phi}}$ term affects vortex dynamics. We find that this term has two contributions. One comes from the states away from the vortex core and has ${A}_{\mathrm{vort},1}=n/2$, where $n$ is the fermion density. The other comes from electronic states inside the vortex core and has ${A}_{\mathrm{vort},2}=\ensuremath{-}{n}_{0}/2$, where ${n}_{0}$ is the fermion density at the vortex core. This last term comes from the continuous part of the electronic spectrum and has no contribution from discrete levels inside the core; it also does not change if we add impurities. We interpret this term as the contribution to vortex dynamics in the continuum limit, when the spacing between energy levels $\ensuremath{\omega}$ is set to zero, while fermionic lifetime $\ensuremath{\tau}$ can be arbitrary. The total ${A}_{\mathrm{vort}}=(n\ensuremath{-}{n}_{0})/2$ determines the transversal force acting on the vortex core, $\ensuremath{\pi}{A}_{\mathrm{vort}}\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\mathbf{R}}\ifmmode\times\else\texttimes\fi{}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$, where $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\mathbf{R}}$ is the velocity of the vortex core and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$ a unit vector perpendicular to the 2d sample. The difference $(n\ensuremath{-}{n}_{0})/2$ changes through the BEC-BCS crossover as ${n}_{0}$ nearly compensates $n$ in the BCS regime, but vanishes in the BEC regime.

DOI
10.1103/physrevb.99.174510

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Physical review. B, Condensed matter•Jun 1, 1996•120 citations
We analyze the evolution of the dynamics of a neutral sss-wave superconductor between the BCS and BEC regimes. We consider 2d case when a BCS-BEC crossover occurs already at weak coupling as a function of the ratio of the two scales---the Fermi energy EF{E}_{F}EF​ and the bound state energy for two fermions in a vacuum, E0{E}_{0}E0​. The BCS and BEC limits correspond to EF\ensuremath≫E0{E}_{F}\ensuremath{\gg}{E}_{0}EF​\ensuremath≫E0​ and EF\ensuremath≪E0{E}_{F}\ensuremath{\ll}{E}_{0}EF​\ensuremath≪E0​, respectively. The chemical potential \ensuremathμ=EF\ensuremath−E0\ensuremath{\mu}={E}_{F}\ensuremath{-}{E}_{0}\ensuremathμ=EF​\ensuremath−E0​ changes the sign between the two regimes. We use the effective action approach, derive the leading terms in the expansion of the effective action in the spatial and time derivative of the slowly varying superconducting order parameter \ensuremathΔ(r,\ensuremathτ)\mathrm{\ensuremath{\Delta}}(r,\ensuremath{\tau})\ensuremathΔ(r,\ensuremathτ), and express the action in terms of a derivatives of the phase \ensuremathϕ(r,\ensuremathτ)\ensuremath{\phi}(r,\ensuremath{\tau})\ensuremathϕ(r,\ensuremathτ) of \ensuremathΔ(r,\ensuremathτ)=\ensuremathΔei\ensuremathϕ(r,\ensuremathτ)\mathrm{\ensuremath{\Delta}}(r,\ensuremath{\tau})=\mathrm{\ensuremath{\Delta}}{e}^{i\ensuremath{\phi}(r,\ensuremath{\tau})}\ensuremathΔ(r,\ensuremathτ)=\ensuremathΔei\ensuremathϕ(r,\ensuremathτ). The action contains (\ensuremath∇\ensuremathϕ)2{(\ensuremath{\nabla}\ensuremath{\phi})}^{2}(\ensuremath∇\ensuremathϕ)2 and \ensuremathϕ\ifmmode˙\else˙\fi2{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\phi}}}^{2}\ensuremathϕ\ifmmode˙\else˙\fi​2 terms, which determine the dispersion of collective phase fluctuations, and i\ensuremathπA\ensuremathϕ\ifmmode˙\else˙\fii\ensuremath{\pi}A\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\phi}}i\ensuremathπA\ensuremathϕ\ifmmode˙\else˙\fi​ term. For continuous \ensuremathϕ(r,\ensuremathτ)\ensuremath{\phi}(r,\ensuremath{\tau})\ensuremathϕ(r,\ensuremathτ), the latter reduces to the contribution from the boundary and does not affect the dynamics. We show that this long-wavelength action does not change through the BCS-BEC crossover. We apply our approach to a moving vortex, for which \ensuremathϕ\ensuremath{\phi}\ensuremathϕ is singular at the center of the vortex core, and i\ensuremathπAvort\ensuremathϕ\ifmmode˙\else˙\fii\ensuremath{\pi}{A}_{\mathrm{vort}}\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\phi}}i\ensuremathπAvort​\ensuremathϕ\ifmmode˙\else˙\fi​ term affects vortex dynamics. We find that this term has two contributions. One comes from the states away from the vortex core and has Avort,1=n/2{A}_{\mathrm{vort},1}=n/2Avort,1​=n/2, where nnn is the fermion density. The other comes from electronic states inside the vortex core and has Avort,2=\ensuremath−n0/2{A}_{\mathrm{vort},2}=\ensuremath{-}{n}_{0}/2Avort,2​=\ensuremath−n0​/2, where n0{n}_{0}n0​ is the fermion density at the vortex core. This last term comes from the continuous part of the electronic spectrum and has no contribution from discrete levels inside the core; it also does not change if we add impurities. We interpret this term as the contribution to vortex dynamics in the continuum limit, when the spacing between energy levels \ensuremathω\ensuremath{\omega}\ensuremathω is set to zero, while fermionic lifetime \ensuremathτ\ensuremath{\tau}\ensuremathτ can be arbitrary. The total Avort=(n\ensuremath−n0)/2{A}_{\mathrm{vort}}=(n\ensuremath{-}{n}_{0})/2Avort​=(n\ensuremath−n0​)/2 determines the transversal force acting on the vortex core, \ensuremathπAvortR\ifmmode˙\else˙\fi\ifmmode×\else\texttimes\fiz\ifmmode^\elseˆ\fi\ensuremath{\pi}{A}_{\mathrm{vort}}\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\mathbf{R}}\ifmmode\times\else\texttimes\fi{}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}\ensuremathπAvort​R\ifmmode˙\else˙\fi​\ifmmode×\else\texttimes\fiz\ifmmode^\elseˆ\fi, where R\ifmmode˙\else˙\fi\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\mathbf{R}}R\ifmmode˙\else˙\fi​ is the velocity of the vortex core and z\ifmmode^\elseˆ\fi\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}z\ifmmode^\elseˆ\fi a unit vector perpendicular to the 2d sample. The difference (n\ensuremath−n0)/2(n\ensuremath{-}{n}_{0})/2(n\ensuremath−n0​)/2 changes through the BEC-BCS crossover as n0{n}_{0}n0​ nearly compensates nnn in the BCS regime, but vanishes in the BEC regime.
We use variational approach to analyze the evolution of the dynamics of a neutral s-wave superconductor between BCS and BEC regimes. We consider 2D case, when BCS-BEC crossover occurs already at weak coupling and is governed by the ratio of the two scales—the Fermi energy EF and the bound state energy for two fermions in a vacuum, E0. BCS and BEC regimes correspond to EF ≫ E0 and EF ≪ E0, respectively. We compute the spectrum of low-energy bosonic excitations and show that the velocity of phase fluctuations remains v F / 2 through BCS-BEC crossover. We also discuss the topological A ϕ ̇ term in the effective action.
A standard perturbative expansion around the mean-field solution is used to derive the low-energy effective action for superconductors at T=0.T=0.T=0. Taking into account the density fluctuations at the outset we get the effective action where the density $\ensuremath{\rho}$ is the conjugated momentum to the phase $\ensuremath{\theta}$ of the order parameter. In the hydrodynamic regime, the dynamics of the superconductor is described by a time-dependent nonlinear Schr\"odinger (TDNLS) equation for the field $\ensuremath{\Psi}(x)=\sqrt{\ensuremath{\rho}/2}{e}^{i\ensuremath{\theta}}.$ The evolution of the density fluctuations in the crossover from weak-coupling (BCS) to strong-coupling (Bose condensation of localized pairs) superconductivity is discussed for the attractive Hubbard model. In the bosonic limit, the TDNLS equation reduces to the Gross-Pitaevskii equation for the order parameter, as in the standard description of superfluidity. The conditions under which a phase-only action can be derived in the presence of a long-range interaction to describe the physics of the superconductivity of ``bad metals'' are discussed.
We derive an expression for the superfluid density of a uniform two-component Fermi gas through the BCS-BEC crossover in terms of the thermodynamic potential in the presence of an imposed superfluid flow. Treating the pairing fluctuations in a Gaussian approximation following the approach of Nozi\`eres and Schmitt-Rink, we use this definition of ${\ensuremath{\rho}}_{s}$ to obtain an explicit result which is valid at finite temperatures and over the full BCS-BEC crossover. It is crucial that the BCS gap $\ensuremath{\Delta}$, the chemical potential $\ensuremath{\mu}$, and ${\ensuremath{\rho}}_{s}$ all include the effect of fluctuations at the same level in a self-consistent manner. We show that the normal fluid density ${\ensuremath{\rho}}_{n}\ensuremath{\equiv}n\ensuremath{-}{\ensuremath{\rho}}_{s}$ naturally separates into a sum of contributions from Fermi BCS quasiparticles $({\ensuremath{\rho}}_{n}^{F})$ and Bose collective modes $({\ensuremath{\rho}}_{n}^{B})$. The expression for ${\ensuremath{\rho}}_{n}^{F}$ is just Landau's formula for a BCS Fermi superfluid but now calculated over the BCS-BEC crossover. The expression for the Bose contribution ${\ensuremath{\rho}}_{n}^{B}$ is more complicated and only reduces to Landau's formula for a Bose superfluid in the extreme BEC limit, where all the fermions have formed stable Bose pairs and the Bogoliubov excitations of the associated molecular Bose condensate are undamped. In a companion paper, we present numerical calculations of ${\ensuremath{\rho}}_{s}$ using an expression equivalent to the one derived in this paper, over the BCS-BEC crossover, including unitarity, and at finite temperatures.
We review efforts to unify both the Bardeen, Cooper and Schrieffer (BCS) and Bose-Einstein condensation (BEC) pictures of superconductivity. We have finally achieved this in terms of a "\textit{complete} boson-fermion (BF) model" (CBFM) that reduces in special cases to all the main continuum (as opposed to "spin") statistical theories of superconductivity. Our BF model is "complete" in the sense that not only two-electron (2e) but also two-hole (2h) Cooper pairs (CPs) are allowed in arbitrary proportions. In contrast, BCS-Bogoliubov theory--which can also be considered as the theory of a mixture of kinematically independent electrons, 2e- and 2h-CPs--allows only equal, 50%-50%, mixtures of the two kinds of CPs. This is obvious from the perfect symmetry about μμμ, the electron chemical potential, of the well-known Bogoliubov v2(ε)v^{2}(ε)v2(ε) and u2(ε)u^{2}(ε)u2(ε) coefficients, where εεε is the electron energy. The CBFM is then applied to see: a) whether the BCS model interaction for the electron-phonon dynamical mechanism is sufficient to predict the unusually high values of TcT_{c}Tc​ (in units of the Fermi temperature) of ≃0.01−0.1\simeq 0.01-0.1≃0.01−0.1 exhibited by the so-called ``exotic'' superconductors \cite{Brandow} in both 2D and 3D--relative to the low values of ≲10−3\lesssim 10^{-3}≲10−3 more or less correctly predicted by BCS theory for conventional, elemental superconductors; and b) whether it can at least suggest, if not explain, why "hole superconductors" have higher TcT_{c}Tc​'s.
We analyze the interplay between superconductivity and the formation of bound pairs of fermions (BCS-BEC crossover) in a 2D model of interacting fermions with small Fermi energy EF{E}_{F}EF​ and weak attractive interaction, which extends to energies well above EF{E}_{F}EF​. The 2D case is special because a two-particle bound state forms at arbitrary weak interaction, and already at weak coupling, one has to distinguish between the bound-state formation and superconductivity. We briefly review the situation in the one-band model and then consider two different two-band models: one with one hole band and one electron band and another with two hole or two electron bands. In each case, we obtain the bound-state energy 2E02{E}_{0}2E0​ for two fermions in a vacuum and solve the set of coupled equations for the pairing gaps and the chemical potentials to obtain the onset temperature of the pairing Tins{T}_{\mathrm{ins}}Tins​ and the quasiparticle dispersion at T=0T=0T=0. We then compute the superfluid stiffness ${\ensuremath{\rho}}_{s}(T=0)$ and obtain the actual Tc{T}_{c}Tc​. For definiteness, we set EF{E}_{F}EF​ in one band to be near zero and consider different ratios of E0{E}_{0}E0​ and EF{E}_{F}EF​ in the other band. We show that at ${E}_{F}\ensuremath{\gg}{E}_{0}$, the behavior of both two-band models is BCS-like in the sense that ${T}_{c}\ensuremath{\approx}{T}_{\mathrm{ins}}\ensuremath{\ll}{E}_{F}$ and $\mathrm{\ensuremath{\Delta}}\ensuremath{\sim}{T}_{c}$. At ${E}_{F}\ensuremath{\ll}{E}_{0}$, the two models behave differently: in the model with two hole/two electron bands, ${T}_{\mathrm{ins}}\ensuremath{\sim}{E}_{0}/ln\frac{{E}_{0}}{{E}_{F}}, \mathrm{\ensuremath{\Delta}}\ensuremath{\sim}{({E}_{0}{E}_{F})}^{1/2}$, and ${T}_{c}\ensuremath{\sim}{E}_{F}$, like in the one-band model. In between Tins{T}_{\mathrm{ins}}Tins​ and Tc{T}_{c}Tc​, the system displays a preformed pair behavior. In the model with one hole and one electron bands, Tc{T}_{c}Tc​ remains of order Tins{T}_{\mathrm{ins}}Tins​, and both remain finite at EF=0{E}_{F}=0EF​=0 and of the order of E0{E}_{0}E0​. The preformed pair behavior still does exist in this model because Tc{T}_{c}Tc​ is numerically smaller than Tins{T}_{\mathrm{ins}}Tins​. For both models, we reexpress Tins{T}_{\mathrm{ins}}Tins​ in terms of the fully renormalized two-particle scattering amplitude by extending to the two-band case (the method pioneered by Gorkov and Melik-Barkhudarov back in 1961). We apply our results for the model with a hole and an electron band to Fe pnictides and Fe chalcogenides in which a superconducting gap has been detected on the bands that do not cross the Fermi level, and to FeSe, in which the superconducting gap is comparable to the Fermi energy. We apply the results for the model with two electron bands to Nb-doped SrTiO3{\mathrm{SrTiO}}_{3}SrTiO3​ and argue that our theory explains the rapid increase of Tc{T}_{c}Tc​ when both bands start crossing the Fermi level.
We consider a fermionic system at zero temperature interacting through an effective nonretarded potential of the type introduced by Nozi\`eres and Schmitt-Rink, and calculate the phase coherence length ${\ensuremath{\xi}}_{\mathrm{phase}}$ (associated with the spatial fluctuations of the superconducting order parameter) by exploiting a functional-integral formulation for the correlation functions and the associated loop expansion. This formulation is especially suited to follow the evolution of the fermionic system from a BCS-type superconductor for weak coupling to a Bose-condensed system for strong coupling, since in the latter limit a direct mapping of the original fermionic system onto an effective system of bosons with a residual boson-boson interaction can be established. Explicit calculations are performed at the one-loop order. The phase coherence length ${\ensuremath{\xi}}_{\mathrm{phase}}$ is compared with the coherence length ${\ensuremath{\xi}}_{\mathrm{pair}}$ for two-electron correlation, which is relevant to distinguish the weak- (kF{\mathit{k}}_{\mathit{F}}kF​${\ensuremath{\xi}}_{\mathrm{pair}}$\ensuremath{\gg}1) from the strong- (kF{\mathit{k}}_{\mathit{F}}kF​${\ensuremath{\xi}}_{\mathrm{pair}}$\ensuremath{\ll}1) coupling limits (kF{\mathit{k}}_{\mathit{F}}kF​ being the Fermi wave vector) as well as to follow the crossover in between. It is shown that ${\ensuremath{\xi}}_{\mathrm{phase}}$ coincides with ${\ensuremath{\xi}}_{\mathrm{pair}}$ down to kF{\mathit{k}}_{\mathit{F}}kF​${\ensuremath{\xi}}_{\mathrm{pair}}$\ensuremath{\simeq}10, ${\ensuremath{\xi}}_{\mathrm{pair}}$ in turn coinciding with the Pippard coherence length. In the strong-coupling limit we find instead that ${\ensuremath{\xi}}_{\mathrm{phase}}$\ensuremath{\gg}${\ensuremath{\xi}}_{\mathrm{pair}}$, with ${\ensuremath{\xi}}_{\mathrm{pair}}$ coinciding with the radius of the bound-electron pair. From the mapping onto an effective system of bosons in the strong-coupling limit we further relate ${\ensuremath{\xi}}_{\mathrm{pair}}$ with the ``range'' of the residual boson-boson interaction, which is physically the only significant length associated with the dynamics of the bosonic system. \textcopyright{} 1996 The American Physical Society.