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arXiv:cond-mat/9804196 · Physical review. B, Condensed matter

Critical properties of the topological Ginzburg-Landau model

C. de Calan, Adolfo P. C. Malbouisson, Flavio S. Nogueira, Nami Fux Svaiter

Published Jan 1, 1999• 17 citations
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Abstract

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 $\mathrm{XY}$ value, ${\ensuremath{\nu}}_{\mathrm{XY}}=0.67.$ This shows that the Chern-Simons term does not affect considerably the $\mathrm{XY}$ scaling of superconductors. We discuss briefly the possible phenomenological applications of this model.

DOI
10.1103/physrevb.59.554

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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 \ensuremathν\ensuremath{\nu}\ensuremathν exponent does not vary very much from the XY\mathrm{XY}XY value, \ensuremathνXY=0.67.{\ensuremath{\nu}}_{\mathrm{XY}}=0.67.\ensuremathνXY​=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 study the critical exponents of the superconducting phase transition in the context of renormalization group theory starting from a dual formulation of the Ginzburg-Landau theory. The dual formulation describes a loop gas of Abrikosov flux tubes which proliferate when the critical temperature is approached from below. In contrast to the Ginzburg-Landau theory, it has a spontaneously broken global symmetry and possesses an infrared stable fixed point. The exponents coincide with those of a superfluid with reversed temperature axis.
Abstract In this work, we define and calculate critical exponents associated with higher order thermodynamic phase transitions. Such phase transitions can be classified into two classes: with or without a local order parameter. For phase transitions involving a local order parameter, we write down the Landau theory and calculate critical exponents using the saddle point approximation. Further, we investigate fluctuations about the saddle point and demarcate when such fluctuations dominate over saddle point calculations by introducing the generalized Ginzburg criteria. We use Wilsonian renormalization group (RG) to derive scaling forms for observables near criticality and obtain scaling relations between the critical exponents. Afterwards, we find out fixed points of the RG flow using the one-loop beta function and calculate critical exponents about the fixed points for third and fourth order phase transitions.
The perturbative renormalization of the Ginzburg-Landau model is reconsidered based on the Feynman diagram technique. We derive renormalization group (RG) flow equations, exactly calculating all vertices appearing in the perturbative renormalization of the phi^4 model up to the epsilon^3 order of the epsilon-expansion. In this case, the phi^2, phi^4, phi^6, and phi^8 vertices appear. All these vertices are relevant. We have tested the expected basic properties of the RG flow, such as the semigroup property. Under repeated RG transformation R_s, appropriately represented RG flow on the critical surface converges to certain s-independent fixed point. The Fourier-transformed two-point correlation function G(k) has been considered. Although the epsilon-expansion of X(k)=1/G(k) is well defined on the critical surface, we have revealed an inconsistency of the perturbative method with the exact rescaling of X(k), represented as an expansion in powers of k at k --> 0. We have discussed also some aspects of the perturbative renormalization of the two-point correlation function slightly above the critical point. Apart from the epsilon-expansion, we have tested and briefly discussed also a modified approach, where the phi^4 coupling constant u is the expansion parameter at a fixed spatial dimensionality d.
Topology plays a cardinal role in explaining phases and quantum phase transitions beyond the Landau-Ginzburg-Wilson paradigm. In this study, we formulate a set of models of Dirac fermions in 2+1 dimensions with SU(NNN)×\times×SU(2)×\times×U(1) symmetry that have the potential to host critical points described by field theories with topological terms. For N=2N=2N=2 it shows a rich phase diagram containing semimetallic, quantum spin Hall insulating, Kekulé valence bond solid and s-wave superconducting phases and features multiple Landau-Ginzburg-Wilson phase transitions driven by interaction strength. At N=1N=1N=1 a deconfined quantum critical point is observed. At N=2N=2N=2 one expects the critical theory to correspond to a level 2 Wess-Zumino-Witten theory in 2+1 dimensions. Here the numerical results however show a strong first order transition. Another transition can be governed by a topological θθθ-term which is rendered irrelevant for even values of NNN thus leading to Landau-Ginzburg-Wilson behaviour.
Different perturbation theory treatments of the Ginzburg-Landau phase transition model are discussed. This includes a criticism of the perturbative renormalization group (RG) approach and a proposal of a novel method providing critical exponents consistent with the known exact solutions in two dimensions. The usual perturbation theory is reorganized by appropriate grouping of Feynman diagrams of φ4 model with O(n) symmetry. As a result, equations for calculation of the two-point correlation function are obtained which allow to predict possible exact values of critical exponents in two and three dimensions by proving relevant scaling properties of the asymptotic solution at (and near) the criticality. The new values of critical exponents are discussed and compared to the results of numerical simulations and experiments.
We study the two-dimensional N-component Landau-Ginzburg Hamiltonian with cubic anisotropy. We compute and analyze the fixed-dimension perturbative expansion of the renormalization-group functions to four loops. The relations of these models with N-color Ashkin-Teller models, discrete cubic models, the planar model with fourth-order anisotropy, and the structural phase transition in adsorbed monolayers are discussed. Our results for N=2N=2N=2 (XY(XY(XY model with cubic anisotropy) are compatible with the existence of a line of fixed points joining the Ising and the O(2)O(2)O(2) fixed points. Along this line the exponent $\ensuremath{\eta}$ has the constant value 1/4,1/4,1/4, while the exponent $\ensuremath{\nu}$ runs in a continuous and monotonic way from 1 to $\ensuremath{\infty}$ [from Ising to O(2)].O(2)].O(2)]. In the four-loop approximation, for N> 3N>~3N> 3 we find a cubic fixed point in the region u,v> 0.u,v>~0.u,v> 0.