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Charged Fixed Point Found in Superconductor Below Tc

Hagen M. Kleinert, Flavio S. Nogueira

Published Apr 30, 2001• 3 citations
OpenAlex

Abstract

We present a perturbative approach which yields, for the first time, an infrared-stable fixed point in the Ginzburg-Landau (GL) model at the oneloop level. The calculations are done in d = 3 dimensions and below Tc, where the renormalization group functions can be expressed directly as functions of the Ginzburg parameter κ which is the ratio between the two fundamental scales of the problem, the penetration depth λ and the correlation length ξ. We find a charged fixed point for κ> 1 / √ 2, that is, in the type II regime, where ∆κ ≡ κ − 1 / √ 2 is shown to be a natural expansion parameter. This parameter controls a momentum space instability in the two-point correlation function of the order field. This instability appears at a nonzero wave-vector p0 whose magnitude scales like ∼ ∆κ ¯ β, with a critical exponent ¯ β = 1/2 in the one-loop approximation, a behavior known from magnetic systems with a Lifshitz point in the phase diagram. This momentum space instability is argued to be the origin of the negative η-exponent of the order field.

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We present a perturbative approach which yields, for the first time, an infrared-stable fixed point in the Ginzburg-Landau (GL) model at the oneloop level. The calculations are done in d = 3 dimensions and below Tc, where the renormalization group functions can be expressed directly as functions of the Ginzburg parameter κ which is the ratio between the two fundamental scales of the problem, the penetration depth λ and the correlation length ξ. We find a charged fixed point for κ> 1 / √ 2, that is, in the type II regime, where ∆κ ≡ κ − 1 / √ 2 is shown to be a natural expansion parameter. This parameter controls a momentum space instability in the two-point correlation function of the order field. This instability appears at a nonzero wave-vector p0 whose magnitude scales like ∼ ∆κ ¯ β, with a critical exponent ¯ β = 1/2 in the one-loop approximation, a behavior known from magnetic systems with a Lifshitz point in the phase diagram. This momentum space instability is argued to be the origin of the negative η-exponent of the order field.
We study the nature of the multicritical point in the three-dimensional $\mathrm{O}(3)\ensuremath{\bigoplus}\mathrm{O}(2)$ symmetric Landau-Ginzburg-Wilson theory, which describes the competition of two order parameters that are O(3) and O(2) symmetric, respectively. This study is relevant for SO(5) theory of high-Tc{T}_{c}Tc​ superconductors, which predicts the existence of a multicritical point in the temperature-doping phase diagram, where the antiferromagnetic and superconducting transition lines meet. We investigate whether $\mathrm{O}(3)\ensuremath{\bigoplus}\mathrm{O}(2)$ symmetry gets effectively enlarged to O(5) approaching the multicritical point. For this purpose, we study the stability of the O(5) fixed point. By means of a Monte Carlo simulation, we show that the O(5) fixed point is unstable with respect to the spin-4 quartic perturbation with the crossover exponent ${\ensuremath{\phi}}_{4,4}=0.180(15)$, in substantial agreement with recent field-theoretical results. This estimate is much larger than the one-loop ϵϵϵ-expansion estimate ${\ensuremath{\phi}}_{4,4}=1∕26$, which has often been used in the literature to discuss the multicritical behavior within SO(5) theory. Therefore, no symmetry enlargement is generically expected at the multicritical transition. We also perform a five-loop field-theoretical analysis of the renormalization-group flow. It shows that bicritical systems are not in the attraction domain of the stable decoupled fixed point. Thus, in these systems---high-Tc{T}_{c}Tc​ cuprates should belong to this class---the multicritical point corresponds to a first-order transition.
A renormalization-group scheme is developed for the three-dimensional O(2N)O(2N)O(2N)-symmetric Ginzburg-Landau-Wilson model, which is consistent with the use of a 1/N1/N1/N expansion as a systematic method of approximation. It is motivated by an application to the critical properties of superconductors, reported in a separate paper. Within this scheme, the infrared stable fixed point controlling critical behavior appears at z=0,z=0,z=0, where $z={\ensuremath{\lambda}}^{\ensuremath{-}1}$ is the inverse of the quartic coupling constant, and an efficient renormalization procedure consists of the minimal subtraction of ultraviolet divergences at z=0.z=0.z=0. This scheme is implemented at next-to-leading order, and the standard results for critical exponents calculated by other means are recovered. An apparently novel result of this nonperturbative method of approximation is that corrections to scaling (or confluent singularities) do not, as in perturbative analyses, appear as simple power series in the variable ${y=zt}^{\ensuremath{\omega}\ensuremath{\nu}}.$ At least in three dimensions, the power series are modified by powers of lny.\mathrm{ln}y.lny.
We consider a Ginzburg-Landau model for superconductivity with a Chern-Simons term added. The flow diagram contains two charged fixed points corresponding to the tricritical and infrared stable fixed points. The topological coupling controls the fixed-point structure and eventually the region of first-order transitions disappears. We compute the critical exponents as a function of the topological coupling. We obtain that the value of the $\ensuremath{\nu}$ exponent does not vary very much from the XY\mathrm{XY}XY value, ${\ensuremath{\nu}}_{\mathrm{XY}}=0.67.$ This shows that the Chern-Simons term does not affect considerably the XY\mathrm{XY}XY scaling of superconductors. We discuss briefly the possible phenomenological applications of this model.
We consider the scaling behavior in the critical domain of superconductors at zero external magnetic field. The first part of the paper is concerned with the Ginzburg-Landau model in the zero-magnetic-field Meissner phase. We discuss the scaling behavior of the superfluid density and we give an alternative proof of Josephson's relation for a charged superfluid. This proof is obtained as a consequence of an exact renormalization group equation for the photon mass. We obtain Josephson's relation directly in the form ${\ensuremath{\rho}}_{s}\ensuremath{\sim}{t}^{\ensuremath{\nu}};$ that is, we do not need to assume that the hyperscaling relation holds. Next, we give an interpretation of a recent experiment performed in thin films of ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}.$ We argue that the measured mean-field-like behavior of the penetration depth exponent ${\ensuremath{\nu}}^{\ensuremath{'}}$ is possibly associated with a nontrivial critical behavior and we predict the exponents $\ensuremath{\nu}=1$ and $\ensuremath{\alpha}=\ensuremath{-}1$ for the correlation length and specific heat, respectively. In the second part of the paper we discuss the scaling behavior in the continuum dual Ginzburg-Landau model. After reviewing lattice duality in the Ginzburg-Landau model, we discuss the continuum dual version by considering a family of scalings characterized by a parameter $\ensuremath{\zeta}$ introduced such that ${m}_{h,0}^{2}\ensuremath{\sim}{t}^{\ensuremath{\zeta}},$ where mh,0{m}_{h,0}mh,0​ is the bare mass of the magnetic induction field. We discuss the difficulties in identifying the renormalized magnetic induction mass with the photon mass. We show that the only way to have a critical regime with ${\ensuremath{\nu}}^{\ensuremath{'}}=\ensuremath{\nu}\ensuremath{\approx}2/3$ is having $\ensuremath{\zeta}\ensuremath{\approx}4/3,$ that is, with mh,0{m}_{h,0}mh,0​ having the scaling behavior of the renormalized photon mass.
We present in detail a functional renormalization group (FRG) study of a Landau-Ginzburg model of type-II superconductors (generalized to N2\frac{N}{2}2N​ complex fields) in an external magnetic field, both for a pure system and also in the presence of quenched random impurities. If the coupling functions are restricted to the space of functions with nonzero support only at reciprocal lattice vectors corresponding to the Abrikosov lattice, we find a stable FRG fixed point in the presence of disorder for 1<N<41<N<41<N<4, identical to that of the disordered O(N)O(N)O(N) model in $d\ensuremath{-}2$ dimensions. This implies a continuous transition from the vortex crystal to vortex liquid in the presence of disorder, but only for d>4d>4d>4. The nonzero-temperature transition will disappear in physical dimensions. The pure system has a stable fixed point only for N>4N>4N>4. Therefore the physical case (N=2N=2N=2) is likely to have a first-order transition in the absence of quenched disorder. We give a full discussion of both the motivation of the model and the details of the FRG calculation. We also place our results in context with regard to the current experimental scene concerning the high-Tc{T}_{c}Tc​ compounds. In particular, we discuss the relevance of our results to the recently discovered critical end point in the phase diagram of Bi-Sr-Ca-Cu-O. The main results of this analysis were previously reported in the form of a Letter [M.A. Moore and T.J. Newman, Phys. Rev. Lett. 75, 533 (1995)].
The critical dynamics of superconductors in the charged regime is reconsidered within field-theory. For the dynamics the Ginzburg-Landau model with complex order parameter coupled to the gauge field suggested earlier [Lannert et al. Phys. Rev. Lett. 92, 097004 (2004)] is used. Assuming relaxational dynamics for both quantities the renormalization group functions within one loop approximation are recalculated for different choices of the gauge. A gauge independent result for the divergence of the measurable electric conductivity is obtained only at the weak scaling fixed point unstable in one loop order where the time scales of the order parameter and the gauge field are different.