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arXiv:cond-mat/0104226 · Physical review. B, Condensed matter

Phase fluctuations in superconductors: From Galilean invariant to quantum XY models

L. Benfatto, A. Toschi, S. Caprara, Claudio Castellani

Published Sep 18, 2001• 20 citations
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Abstract

We analyze the corrections to the superfluid density due to phase fluctuations within both a continuum and a lattice model for s- and d-wave superconductors. We expand the phase-only action beyond the Gaussian level and compare our results with the quantum $\mathrm{XY}$ model both in the quantum and in the classical regime. We find dynamic anharmonic vertices, absent in the quantum $\mathrm{XY}$ model, which are responsible for the vanishing of the correction to the superfluid density at zero temperature in a continuum (Galilean invariant) model. Moreover, the phase-fluctuation effects are reduced with respect to the $\mathrm{XY}$ model by a factor at least of order ${1/(k}_{F}{\ensuremath{\xi}}_{0}{)}^{2}.$

DOI
10.1103/physrevb.64.140506

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We analyze the corrections to the superfluid density due to phase fluctuations within both a continuum and a lattice model for s- and d-wave superconductors. We expand the phase-only action beyond the Gaussian level and compare our results with the quantum XY\mathrm{XY}XY model both in the quantum and in the classical regime. We find dynamic anharmonic vertices, absent in the quantum XY\mathrm{XY}XY model, which are responsible for the vanishing of the correction to the superfluid density at zero temperature in a continuum (Galilean invariant) model. Moreover, the phase-fluctuation effects are reduced with respect to the XY\mathrm{XY}XY model by a factor at least of order 1/(kF\ensuremathξ0)2.{1/(k}_{F}{\ensuremath{\xi}}_{0}{)}^{2}.1/(kF​\ensuremathξ0​)2.
I study the effect of Coulomb interaction on superconducting order in a ddd-wave lattice superconductor at T=0T=0T=0 by considering the superconducting saddle point in the two-dimensional $t\text{\ensuremath{-}}J\text{\ensuremath{-}}U$ model with a repulsion UUU. The theory of low-energy superconducting phase fluctuations around this saddle point is derived in terms of the effective hard-core bosons (representing the density of spin-up electrons and the phase of the order parameter), interacting with the fluctuating density of spin-down electrons. Whereas the saddle-point value of the superconducting gap is found to continuously increase towards half filling, the phase stiffness at T=0T=0T=0 has a maximum, and then decreases with further underdoping. Right at half filling the phase stiffness vanishes for large UUU. This argues that the pseudogap phenomenon of the type observed in cuprates is in principle possible without a development of any competing order, purely as a result of growing correlations in the superconducting state. Implications for the finite temperature superconducting transition and the effects of static disorder are discussed qualitatively.
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We investigate the effect of white-noise disorder on the local phase stiffness and thermodynamic properties of a two-dimensional sss-wave superconductor. Starting from a local attractive model and using path-integral formalism, we derive an effective action by decoupling the superconducting order parameter into amplitude and phase components in a gauge-invariant manner. Perturbative techniques are applied to the phase fluctuation sector to derive an effective phase-only XY model for disordered superconducting systems. Solving the saddle-point Green's function using Bogoliubov-de Gennes theory, we calculate the distributions of nearest-neighbor couplings for various disorder strengths. A single-peak distribution is observed for low disorder strength, which becomes bimodal with one peak at negative couplings as the disorder strength increases. The local phase stiffness remains randomly distributed throughout the lattice and shows no correlation with pairing amplitudes. The temperature dependence of the superfluid stiffness (Js{J}_{s}Js​) is studied using Monte Carlo simulations. At strong disorder and low temperatures, Js{J}_{s}Js​ increases with increasing temperature, exhibiting anomalous behavior that may indicate the onset of a glassy transition. Additionally, calculations of the Edwards-Anderson order parameter in this disorder regime suggest the emergence of a phase-glass state at very low temperatures.
Introduction The quantum phase transitions considered so far in this book are mostly continuous, second order. They are associated with the divergence of a correlation length and of a characteristic time that renders the quantum critical point scale invariant. However, there are many important phase transitions at which the order parameter goes abruptly to zero and, more important, at which the characteristics of length and critical time do not diverge. In general there is also a latent heat , or work required, to transform one phase into another. In this and the next chapter we study these discontinuous quantum phase transitions. We proceed by considering initially some important cases of what is known as fluctuation induced quantum phase transitions to next treat a more general case of a first-order quantum phase transition. We will show through the study of these problems that the concepts of critical exponents and scaling remain useful, even though these systems do not exhibit scale invariance at their critical point. In Chapter 15 we present the scaling theory of first-order quantum phase transitions and use its results to treat some general problems. One of the most interesting cases of fluctuation-induced transitions occurs in superconductors due to the coupling of the charged carriers to the electromagnetic field. We consider initially a wider class of systems with macroscopic wave functions, the superfluids, of which superconductors are the charged version. We adopt a macroscopic approach to this problem and describe the superfluid by a complex order parameter φ , such that | φ | 2 is proportional to the density of condensed particles. The zero temperature phase diagram is expressed in terms of a control parameter, m 2 , such that, at m 2 = 0, the system presents a quantum phase transition from a superfluid state, for m 2 < 0, φ ≠ 0, to a normal state for m 2 > 0 where φ = 0. In the case of the superconductor with particles of charge q = 2 e , we will include the coupling of these particles to the electromagnetic field and study its effect in the phase diagram of these systems. We show that in the neutral superfluid, the zero temperature transition to the normal state is continuous and occurs at a quantum critical point.
In this work we consider the superfluid stiffness of a generically non-Galilean invariant interacting system and investigate under what conditions the stiffness may nonetheless approach the Galilean-invariant value n/mn/mn/m. Within Eliashberg theory we find that the renormalized stiffness is approximately given by n/mn/mn/m in the case when the l=0l=0l=0 and l=1l=1l=1 components of the effective Fermi-surface projected interaction are approximately equal over a range of frequencies. This holds, in particular, when the interaction is peaked at zero momentum transfer. We examine this result through three complementary lenses: the δ(ω)δ(ω)δ(ω) term in the conductivity, the phase dependence of the Luttinger-Ward free energy, and the coupling of the amplitude and phase sectors in the Hubbard-Stratonovich collective mode action. From these considerations we show that the value of the stiffness is determined by the strength of renormalization of the current vertex and that the latter can be interpreted as the shift of the self-consistent solution due to flow of the condensate, or alternatively as coupling of the phase mode to l=1l=1l=1 fluctuations of the order parameter. We highlight that even though the superfluid stiffness in some non-Galilean systems approaches the Galilean value, this is not enforced by symmetry, and in general the stiffness may be strongly suppressed from its BCS value. As a corollary we obtain the generic form of the phase action within Eliashberg theory and charge and spin Ward identities for a superconductor with frequency dependent gap function.