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arXiv:1301.5996

Revival of Single-Particle Transport Theory for the Normal State of High-Tc Superconductors: II. Vertex Correction

Osamu Narikiyo

Published Jan 25, 2013• 2 citations
arXivHTMLPDF

Abstract

The vertex correction for the electric current is discussed on the basis of the Fermi-liquid theory. It does not alter the qualitative description of the electric transport by the relaxation-time approximation in the case of the normal state of high-Tc superconductors. The failure of the transport theory employing the fluctuation-exchange (FLEX) approximation is pointed out. It becomes manifest by considering the Ward identity for the fluctuation mode.

DOI
10.48550/arxiv.1301.5996

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The vertex correction for the electric current is discussed on the basis of the Fermi-liquid theory. It does not alter the qualitative description of the electric transport by the relaxation-time approximation in the case of the normal state of high-Tc superconductors. The failure of the transport theory employing the fluctuation-exchange (FLEX) approximation is pointed out. It becomes manifest by considering the Ward identity for the fluctuation mode.
On the basis of the self-consistent calculation scheme for the electron self-energy Σ with the use of the three-point vertex function Γ always satisfying the Ward identity, we find that the obtained quasi-particle dispersion in the normal state in gapped systems, such as semiconductors, insulators, and molecules, is well reproduced by that in the one-shot GW (or G0W0) approximation. In calculating the superconducting transition temperature Tc, we also find a similar situation; the result for Tc in the gauge-invariant self-consistent framework including the effect of Γ satisfying the Ward identity is different from that in the conventional Eliashberg theory (which amounts to the GW approximation for superconductivity) but is close to that in the G0W0 approximation. Those facts indicate that the G0W0 approximation actually takes proper account of both vertex and high-order self-energy corrections in a mutually cancelling manner and thus we can understand that the G0W0 approximation is better than the fully self-consistent GW one in obtaining some of physical quantities.
The Ward identities for the charge and heat currents are derived for particle-particle and particle-hole pairs. They are the exact constraints on the current-vertex functions imposed by conservation laws and should be satisfied by consistent theories. While the Ward identity for the charge current of electrons is well established, that for the heat current is not understood correctly. Thus the correct interpretation is presented. On this firm basis the Ward identities for pairs are discussed. As the application of the identity we criticize some inconsistent results in the studies of the superconducting fluctuation transport and the transport anomaly in the normal state of high- T c superconductors.
This thesis is organized in two independent parts, in which I study two different aspects of high Tc superconductivity. The first part begins with an introduction aimed to briefly introduce some relevant experimental and theoretical works performed in recent years, that have helped us to think about cuprates the way we do now. Afterwards, I introduce Landau Fermi liquid theory in a standard text book way. The question of validity of Fermi liquid theory in 2-d is then raised and investigated by searching for singularities in Landau's /-function. I show that the interaction function between two quasiparticles whose momenta approach each other near a curved point of the Fermi surface, contains a 1-d singularity not strong enough to change the Fermi liquid behavior. On the other hand, inflection points provide 2-d singularities that have to be taken seriously in Fermi liquid considerations. I then introduce nearly antiferromagnetic Fermi liquid theory (NAFL), which is a phenomenological theory proposed to describe high Tc systems. I mainly focus on the self-consistency of the theory in calculations. I criticize the theory on the basis of overlooking the vertex corrections in the strong coupling calculations of the transition temperature Tc. I calculate the first vertex correction for an optimally doped system and show that it is of the same order of magnitude as the bare vertex. Migdal's theorem is therefore not valid and Eliashberg formalism is not applicable to this situation. The same conclusion is obtained even after inclusion of the quasiparticle residue Z to the calculation. The sign of the vertex correction is then considered. I show that the positive sign of the vertex correction for the optimally doped system requires a phase transition of some sort as the doping is decreased. Part II of the thesis is devoted to the vortex lattice properties of high Tc superconductors. I establish a method to study vortex lattice properties of d-wave superconductors based on a generalization of the London model. The method has the advantage of simplicity as well as having very few free parameters (one at most) compared to other methods. The generalized London free energy is obtained from an s-d mixing Ginsburg-Landau free energy and also from the microscopic theory of Gorkov. The generalized London equation is found to be analytic at high temperatures. At very low temperatures however nonanalyticities arise as a result of the nodes on the superconducting gap. I then present the results of our calculations of some measurable quantities, such as the vortex lattice geometry and the effective penetration depth (as denned in μSR experiments). Comparison between our results and different experimental data is then performed. Especially our prediction for the magnetic field dependence of the effective penetration depth at T = 0, which is recently observed in //SR experiments with excellent agreement, is discussed.
The Fermi-liquid transport theory of electrical conductivity formulated by Eliashberg [Zh. Eksp. Teor. Fiz. 41, 1241 (1961)] is further developed, using the analytic continuation of the finite-temperature Ward-Takahashi relations obtained by the present author [Ann. Phys. (N.Y.) 173, 226 (1987)] to maintain the theoretical consistency of approximation schemes for solving the Bethe-Salpeter equation. In particular, the finite-temperature Ward-Takahashi relations corresponding to the local conservation of the electron number density and to the conservation of the electron momentum are considered. A general expression for the electrical conductivity in terms of the proper vertex part is also obtained. In the formula, cancellation of the quasiparticle renormalization factors at the Fermi surface is explicitly shown in a general way without assuming a particular model. In the case that the electrons are scattered by phonons, the formula gives the Gr\"uneisen-Bloch formula for the on-shell phonons. For the impurity resistivity, the theory of Edwards [Philos. Mag. 3, 1020 (1958)] is developed to finite temperature by introducing a hypothetical boson field that replaces the impurity averaging.
The effects of vertex corrections on the charge and spin responses of the Fermi liquid are considered. It is shown that the interaction in the particle - hole channel induces virtual pair excitations (Cooper pair like) in both the charge and the spin channels. It leads to the appearance of an anomalous Fermi-liquid term in both the charge and the spin responses of the Fermi liquid in addition to the normal response . Such a contribution of the anomalous term to the charge and spin responses is in agreement with the marginal Fermi-liquid hypothesis proposed by Varma et al to explain the universal anomalies in the normal state of cuprate high- superconductors.
It is widely believed that the t-J model contains some of the key physics of the high-temperature superconductors. It was proposed that the low-energy physics of the t-J model can be described by a gauge theory of fermions and bosons coupled to a gauge field. The normal state of these fermions was assumed to be a Fermi liquid. We show that perturbation theory around the Fermi-liquid state breaks down at low temperature in a wide class of gauge theories of high-temperature superconductors due to the interaction with an overdamped singular mode arising from the gauge field. This breakdown is present even if the coupling to the gauge field is arbitrary small.