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arXiv:cond-mat/0511023 · Physical Review Letters

Phase Fluctuations in Strongly Coupled d -Wave Superconductors

Matthias Mayr, Gonzalo Álvarez, Cengiz Şen, Elbio R. Dagotto

Published Jun 3, 2005• 59 citations
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Abstract

We present a numerically exact solution for the BCS Hamiltonian at any temperature, including the degrees of freedom associated with classical phase, as well as amplitude fluctuations via a Monte Carlo integration. This allows for an investigation over the whole range of couplings: from weak attraction, as in the well-known BCS limit, to the mainly unexplored strong-coupling regime of pronounced phase fluctuations. In the latter, two characteristic temperatures T(*) and T(c), associated with short- and long-range ordering, respectively, can be identified in a mean-field-motivated Hamiltonian. T(*) at the same time corresponds to the opening of a gap in the excitation spectrum. In addition to introducing a novel procedure to study strongly coupled d-wave superconductors, our results indicate that classical phase fluctuations are not sufficient to explain the pseudogap features of high-temperature superconductors.

DOI
10.1103/physrevlett.94.217001

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We present a numerically exact solution for the BCS Hamiltonian at any temperature, including the degrees of freedom associated with classical phase, as well as amplitude fluctuations via a Monte Carlo integration. This allows for an investigation over the whole range of couplings: from weak attraction, as in the well-known BCS limit, to the mainly unexplored strong-coupling regime of pronounced phase fluctuations. In the latter, two characteristic temperatures T() and T(c), associated with short- and long-range ordering, respectively, can be identified in a mean-field-motivated Hamiltonian. T() at the same time corresponds to the opening of a gap in the excitation spectrum. In addition to introducing a novel procedure to study strongly coupled d-wave superconductors, our results indicate that classical phase fluctuations are not sufficient to explain the pseudogap features of high-temperature superconductors.
One of the most prominent problems in high temperature superconductivity is the nature of the pseudogap phase in underdoped regimes; particularly important is the role of phase fluctuations. The Gutzwiller-projected BCS Hamiltonian is a useful model for high temperature superconductivity due to an exact mapping to the Heisenberg model at half filling and generally a very close connection to the t-J model at moderate doping. We develop the dynamical mean field theory for the d-wave BCS Hamiltonian with on-site repulsive interaction, UUU, physically imposing the partial Gutzwiller projection. For results, two pseudogap energy scales are identified: one associated with the bare pairing gap for the singlet formation and the other with the local phase coherence. The real superconducting gap determined from sharp coherence peaks in the density of states shows strong renormalization from the bare value due to UUU.
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 In weakly coupled BCS superconductors, only electrons within a tiny energy window around the Fermi energy, EF, form Cooper pairs. This may not be the case in strong coupling superconductors such as cuprates, FeSe, SrTiO3 or cold atom condensates where the pairing scale, EB, becomes comparable or even larger than EF. In cuprates, for example, a plausible candidate for the pseudogap state at low doping is a fluctuating pair density wave, but no microscopic model has yet been found which supports such a state. In this work, we write an analytically solvable model to examine pairing phases in the strongly coupled regime and in the presence of anisotropic interactions. Already for moderate coupling we find an unusual finite temperature phase, below an instability temperature Ti, where local pair correlations have non-zero center-of-mass momentum but lack long-range order. At low temperature, this fluctuating pair density wave can condense either to a uniform d-wave super- conductor or the widely postulated pair-density wave phase depending on the interaction strength. Our minimal model offers a unified microscopic framework to understand the emergence of both fluctuating and long range pair density waves in realistic systems.
The effect of random on-site disorder on s-wave (BCS) superconductors described by a two dimensional negative-U Hubbard model is studied using Bogoliubov-de Gennes (BdG) method. The mean field pairing amplitudes become spatially inhomogeneous at large values of disorder where the system breaks up into superconducting islands with large pairing amplitudes, separated by insulating strips. The amplitude fluctuations are correctly accounted for via BdG calculations, however it misses phase fluctuations which are inherent to low dimensions. The phase fluctuations affect superconducting properties strongly, and the effect is more pronounced in the limit of large disorder. We provide a close estimate of the actual transition temperature, Tcby incorporating phase fluctuations about the inhomogeneous BdG state. This is obtained by relating the jump in renormalized Ds(obtained from a self consistent Harmonic approximation on a phase-only Hamiltonian) at the Kosterlitz-Thouless (KT) transition and the KT transition temperature, TKTobtained from Ds(T) = (2/π)TKTas temperature tends to TKTfrom below. This yields opening of a large region sandwiched between [Formula: see text] (obtained by the vanishing of [Formula: see text]) and TKTwhere there is no phase coherence between the pairs, however amplitude correlations continue to exist, reminiscent of a pseudogap phase in high-Tcsuperconductors which is marked by short ranged preformed pairs without any definite phase relation between them. Further, the appearance of the superconducting islands from a homogeneous phase indicates the evolution of a system consisting of large and overlapping pairs to one that contains short and tightly bound pairs - a scenario termed as BCS-BEC crossover. We have investigated in details the crossover phenomenon as a function of disorder and confirmed it's existence at small values of electron concentration, however, absent at larger densities.
We present an effective Hamiltonian based real-space approach for studying the weak-coupling BCS to the strong-coupling Bose-Einstein condensate crossover in the two-dimensional attractive Hubbard model at finite temperatures. We introduce and justify an effective classical Hamiltonian to describe the thermal fluctuations of the relevant auxiliary fields. Our results for Tc{T}_{c}Tc​ and phase diagrams compare very well with those obtained from more sophisticated and CPU-intensive numerical methods. We demonstrate that the method works in the presence of disorder and can be a powerful tool for a real-space description of the effect of disorder on superconductivity. From a combined analysis of the superconducting order parameter, the distribution of auxiliary fields, and the quasiparticle density of states, we identify the regions of metallic, insulating, superconducting, and pseudogapped behavior. Our finding of the importance of phase fluctuations for the pseudogap behavior is consistent with the conclusions drawn from recent experiments on NbN superconductors. The method can be generalized to study superconductors with nontrivial order-parameter symmetries by identifying the relevant auxiliary variables.
The pseudogap state of cuprate high-temperature superconductors has been often viewed as either a yet unknown competing order or a precursor state to superconductivity. While awaiting the resolution of the pseudogap problem in cuprates, we demonstrate that local pairing fluctuations, vortex liquid dynamics, and other precursor phenomena can emerge quite generally whenever fermionic excitations remain gapped across the superconducting transition, regardless of the gap origin. Our choice of a tractable model is a lattice band insulator with short-range attractive interactions between fermions in the sss-wave channel. An effective crossover between Bardeen-Cooper-Schrieffer (BCS) and Bose-Einstein condensate (BEC) regimes can be identified in any band insulator above two dimensions, while in two dimensions only the BEC regime exists. The superconducting transition is ``unconventional'' (non-pair-breaking) in the BEC regime, identified by either the bosonic mean field or XY\mathit{XY}XY universality class. The insulator adjacent to the superconductor in the BEC regime is a bosonic Mott insulator of Cooper pairs, which may be susceptible to charge density wave ordering. We construct a function of the many-body excitation spectrum whose nonanalytic changes define a sharp distinction between band and Mott insulators. The corresponding ``second-order transition'' can be observed out of equilibrium by driving a Cooper pair laser in the Mott insulator. We explicitly show that the gap for charged bosonic excitations lies below the threshold for Cooper pair breakup in any BEC regime, despite quantum fluctuations. Our discussion ends with a view of possible consequences for cuprates, where antinodal pair dynamics has certain features in common with our simple sss-wave picture.