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arXiv:1108.5638 · Physical Review B

Flow equation approach to the linear response theory of superconductors

M. Zapalska, T. Domański

Published Nov 23, 2011• 7 citations
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Abstract

We apply the flow equation method for studying the current-current response function of electron systems with the pairing instability. To illustrate the specific scheme in which the flow equation procedure determines the two-particle Green's functions, we reproduce the standard response kernel of the BCS superconductor. We next generalize this nonperturbative treatment considering the pairing field fluctuations. Our study indicates that the residual diamagnetic behavior detected above the transition temperature in the cuprate superconductors can originate from the noncondensed preformed pairs.

DOI
10.1103/physrevb.84.174520

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We apply the flow equation method for studying the current-current response function of electron systems with the pairing instability. To illustrate the specific scheme in which the flow equation procedure determines the two-particle Green's functions, we reproduce the standard response kernel of the BCS superconductor. We next generalize this nonperturbative treatment considering the pairing field fluctuations. Our study indicates that the residual diamagnetic behavior detected above the transition temperature in the cuprate superconductors can originate from the noncondensed preformed pairs.
We address the linear response of the electron system with the pairing instabilities using nonperturbative framework of the continuous unitary transformation technique. For the case of the Bose-Einstein condensed pairs we have analytically derived the BCS result, which in the static and long wavelength limit accounts for the Meissner effect. We next apply our treatment to the mixture of the non-condensed (preformed) pairs interacting with the mobile electrons through the charge-exchange Andreev scattering. We determine the leading contributions to the current-current response function, expressing the vertices by the set of the corresponding flow equations. Our study indicates that the residual diamagnetic behavior detected by the torque magnetometry well above the transition temperature [L. Li et al, Phys. Rev. B 81, 054510 (2010)] can originate from the noncondensed preformed pairs.
We apply the flow equation method for studying the fermion systems where pairing interactions can either trigger the BCS instability with the symmetry breaking manifested by the off-diagonal order parameter or lead to the gaped single particle spectrum without any symmetry breaking. We construct the continuous Bogoliubov transformation in a scheme resembling the renormalization group procedure. We further extend this continuous transformation to a case where fermion pairs interact with the boson field. Due to temporal quantum fluctuations the single particle excitation spectrum develops a gap which is centered around the renormalized boson energy. When bosons undergo the Bose Einstein condensation this structure evolves into the BCS spectrum.
The equations for the Green's functions of the theory of superconductivity are developed in an iterative scheme in which the two-particle function is rewritten as the functional derivative of the single-particle function with respect to external-source terms. When a source coupled to electron pairs in addition to an external potential coupled to the charge density is included in the Hamiltonian, the possibility for a gap in the single-particle excitations is found. The first order in the iteration scheme for the solution to the Green's function equations is independent of the way in which the two-particle function is generated, as the final result must be. Equations for the vertex functions are obtained and used to find the linear response of the current to an applied electromagnetic field. An exact solution to the vertex equation of interest is used to show that no current will be induced by a static longitudinal vector potential. In addition, all the functions calculated are found to have translationally invariant solutions. Thus, the original invariances of the Hamiltonian are restored when the source terms are set equal to zero. Corrections to the self-energy are estimated by using another of the solutions to the vertex equations.
This chapter reviews the calculation of a number of system properties within the pairing approximation. Since the BCS theory was originally proposed, attempts to justify the pairing correlations basic to the theory have proceeded along two lines. The first approach has been to apply the BCS theory to a wide variety of phenomena in superconductors and check the theoretical predictions of the pairing approximation against experiment. The second approach has been to treat by various approximate methods the residual interactions neglected within the pairing scheme, hoping to show that these residual interactions introduce no major change in the predicted properties of the system. The interaction Hamiltonian, being a one-body operator, can at most change the state of one electron. While the majority of the electron–phonon interaction has been accounted for in forming the wave functions for the normal and superconducting states, there remains the part corresponding to resonant phonon absorption and emission processes.
The normal state of strongly coupled superconductors is characterized by the presence of ``preformed'' Cooper pairs well above the superconducting critical temperature. In this regime, the electrons are paired, but they lack the phase coherence necessary for superconductivity. The existence of preformed pairs implies the existence of a characteristic energy scale associated with a pseudogap. Preformed pairs are often invoked to interpret systems where some signatures of pairing are present without actual superconductivity, but an unambiguous theoretical characterization of a preformed-pair system is still lacking. To fill this gap, we consider the response to an external pairing field of an attractive Hubbard model, which hosts one of the cleanest realizations of a preformed pair phase, and a repulsive model where sss-wave superconductivity cannot be realized. Using dynamical mean-field theory to study this response, we identify the characteristic features which distinguish the reaction of a preformed pair state from a normal metal without any precursor of pairing. The theoretical detection of preformed pairs is associated with the behavior of the second derivative of the order parameter with respect to the external field, as confirmed by analytic calculations in limiting cases. Our findings provide a solid test bed for the interpretation of state-of-the-art calculations for the normal state of the doped Hubbard model in terms of ddd-wave preformed pairs and, in perspective, of nonequilibrium experiments in high-temperature superconductors.