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arXiv:cond-mat/9901305 · Physical review. B, Condensed matter

Effective action for superconductors and BCS-Bose crossover

S. De Palo, Claudio Castellani, C. Di Castro, B. K. Chakraverty

Published Jul 1, 1999• 90 citations
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Abstract

A standard perturbative expansion around the mean-field solution is used to derive the low-energy effective action for superconductors at $T=0.$ 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.

DOI
10.1103/physrevb.60.564

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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 \ensuremathρ\ensuremath{\rho}\ensuremathρ is the conjugated momentum to the phase \ensuremathθ\ensuremath{\theta}\ensuremathθ 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 \ensuremathΨ(x)=\ensuremathρ/2ei\ensuremathθ.\ensuremath{\Psi}(x)=\sqrt{\ensuremath{\rho}/2}{e}^{i\ensuremath{\theta}}.\ensuremathΨ(x)=\ensuremathρ/2​ei\ensuremathθ. 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 study the two-dimensional attractive Hubbard model using the mapping onto the half-filled repulsive Hubbard model in a uniform magnetic field coupled to the fermion spins. The low-energy effective action for charge and pairing fluctuations is obtained in the hydrodynamic regime. We recover the action of a Bose superfluid where half the fermion density is identified as the conjugate variable of the phase of the superconducting order parameter. By integrating out charge fluctuations, we obtain a phase-only action. In the zero-temperature superconducting state, this action describes a collective phase mode smoothly evolving from the Anderson-Bogoliubov mode at weak coupling to the Bogoliubov mode of a Bose superfluid at strong coupling. At finite temperature, the phase-only action can be used to extract an effective XYXYXY model and thus obtain the Berezinskii-Kosterlitz-Thouless (BKT) phase transition temperature. We also identify a renormalized classical regime of superconducting fluctuations above the BKT phase transition, and a regime of incoherent pairs at higher temperature. Special care is devoted to the nearly half-filled case where the symmetry of the order parameter is enlarged to SO(3) due to strong $\mathbf{q}=(\ensuremath{\pi},\ensuremath{\pi})$ charge fluctuations. The low-energy effective action is then an SO(3) nonlinear $\ensuremath{\sigma}$ model with a (symmetry breaking) magnetic field proportional to the doping. In the strong-coupling limit, the attractive Hubbard model can be mapped onto the Heisenberg model in a magnetic field, which reduces to the quantum XYXYXY model (except for a weak magnetic field, i.e., in the low-density limit of the attractive model). In the low-density limit, the Heisenberg model allows one to recover the action of a Bose superfluid, including the ${(\mathbf{\ensuremath{\nabla}}\ensuremath{\rho})}^{2}$ term (with $\ensuremath{\rho}$ the density), and in turn the Gross-Pitaevskii equation.
We derive the low-energy effective action for three-dimensional superfluid Fermi systems in the strong-coupling limit, where superfluidity originates from Bose-Einstein condensation of composite bosons. Taking into account density and pairing fluctuations on the same footing, we show that the effective action involves only the fermion density ${\ensuremath{\rho}}_{\mathbf{r}}$ and its conjugate variable, the phase ${\ensuremath{\theta}}_{\mathbf{r}}$ of the pairing order parameter ${\ensuremath{\Delta}}_{\mathbf{r}}$. We recover the standard action of a Bose superfluid of density ${\ensuremath{\rho}}_{\mathbf{r}}∕2$, where the bosons have a mass mB=2m{m}_{B}=2mmB​=2m and interact via a repulsive contact potential with amplitude ${g}_{B}=4\ensuremath{\pi}{a}_{B}∕{m}_{B},{a}_{B}=2a$ (aaa the sss-wave scattering length associated to the fermion-fermion interaction in vacuum). For lattice models, the derivation of the effective action is based on the mapping of the attractive Hubbard model onto the Heisenberg model in a uniform magnetic field, and a coherent state path integral representation of the partition function. The effective description of the Fermi superfluid in the strong-coupling limit is a Bose-Hubbard model with an intersite hopping amplitude tB=J∕2{t}_{B}=J∕2tB​=J∕2 and an on-site repulsive interaction UB=2Jz{U}_{B}=2JzUB​=2Jz, where J=4t2∕UJ=4{t}^{2}∕UJ=4t2∕U (ttt and $\ensuremath{-}U$ are the intersite hopping amplitude and the on-site attraction in the (fermionic) Hubbard model, zzz the number of nearest-neighbor sites).
We consider a lattice version of the ``negative-U′′U''U′′ Hubbard model in three dimensions, to study the crossover from a superconducting phase transition (dominated by phase fluctuation) to a regime where the amplitude of the superconducting order parameter controls the critical temperature. Starting from a fermionic Hamiltonian, we give a microscopic derivation of the effective action in terms of the relevant physical fields: the modulus and phase of the superconducting order parameter. Furthermore, by employing to the resulting coarse--grained quantum XY\mathrm{XY}XY model the phase fluctuation algebra between number and phase operators (given by the Euclidean group E2),{E}_{2}),E2​), we map the ``phase-only'' action onto a solvable quantum spherical model. We establish the self--consistent theory (involving both fermionic and bosonic degrees of freedom), and calculate the superconducting phase coherence transition temperature Tc{T}_{c}Tc​ as a function of the coupling strength ∣U∣,|U|,∣U∣, exhibiting a maximum where the behavior crosses over from BCS to Bose-Einstein (BE) condensation. We examine a pseudogap in the normal state which emerges naturally as a precursor of superconductivity at the second characteristic temperature Tg{T}_{g}Tg​ due to a state with bound pairs but without long--range phase coherence. Furthermore, we demonstrate that the reduced gaps of the model $2{\ensuremath{\Delta}}_{0}{/k}_{B}{T}_{g}\ensuremath{\sim}4$ is almost independent on the pairing strength ∣U∣/t,|U|/t,∣U∣/t, whereas $2{\ensuremath{\Delta}}_{0}{/k}_{B}{T}_{c}$ increases with increasing ∣U∣|U|∣U∣---in striking analogy with the phenomenology of the high--temperature cuprate superconductors when interpreted in terms of the BCS-BC crossover scenario.
Abstract Starting from a model of free Fermions in two dimensions with an arbitrary strong effective interaction, we derive a Ginzburg‐Landau theory describing the crossover from BCS‐superconductivity to Bose‐condensation. We find a smooth crossover from the standard BCS‐limit to a Gross‐Pitaevski type equation for the order parameter in a Bose superfluid. The mean field transition temperature exhibits a maximum at a coupling strength, where the behaviour crosses over from BCS to Bose like with corresponding values of 2 Δ0/Tc ≈ 5 which are characteristic for high Tc superconductors.
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.
The (mean field based) BCS theory is considered one of the most successful theories in condensed matter physics. It is justified in ordinary metal superconductors the coherence length ξξξ is large, with two important features: the order parameter (OP) and excitation gap (EG) are identical, and the pair formation and their Bose condensation take place at the same temperature Tc. It fails to explain the underdoped cuprate superconductivity: EG is finite at Tc and thus distinct from OP. Since these superconductors belong to a large class of small ξξξ materials, this failure has the potential for widespread impact. Here we have extended BCS theory in a natural way to short ξξξ superconductors, based on a BCS--BEC crossover scenario, and arrived at a simple physical picture in which incoherent, finite momentum pairs become progressively more important as the pairing interaction becomes stronger, leading to the distinction between EG and OP. The superconductivity from the fermionic perspective and BEC from the bosonic perspective are just two sides of the same coin. Our theory is capable of making verifiable quantitative predictions. We obtain a cuprate phase diagram (with one free parameter) , in (semi-)quantitative agreement with experiment. The mutually compensating contributions from fermionic quasiparticles and bosonic pair excitations provides a natural explanation for the quasi-universal behavior of the in-plane superfluid density versus T. Our bosonic pair excitations also provide an intrinsic mechanism for the long mysterious linear T terms in the specific heat. Incoherent pair contributions lead to new low T power laws, consistent with existing experiments. Finally, we demonstrated that the onset of superconducting long range order leads to sharp features in the specific heat at Tc, consistent with experiment.