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arXiv:1409.1374 · Physical Review E

Reexamination of the nonperturbative renormalization-group approach to the Kosterlitz-Thouless transition

Paweł Jakubczyk, N. Dupuis, Bertrand Delamotte

Published Dec 1, 2014• 56 citations
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Abstract

We reexamine the two-dimensional linear $\mathrm{O}(2)$ model (${\ensuremath{\varphi}}^{4}$ theory) in the framework of the nonperturbative renormalization-group. From the flow equations obtained in the derivative expansion to second order and with optimization of the infrared regulator, we find a transition between a high-temperature (disordered) phase and a low-temperature phase displaying a line of fixed points and algebraic order. We obtain a picture in agreement with the standard theory of the Kosterlitz-Thouless (KT) transition and reproduce the universal features of the transition. In particular, we find the anomalous dimension $\ensuremath{\eta}({T}_{\mathrm{KT}})\ensuremath{\simeq}0.24$ and the stiffness jump ${\ensuremath{\rho}}_{s}({T}_{\mathrm{KT}}^{\ensuremath{-}})\ensuremath{\simeq}0.64$ at the transition temperature ${T}_{\mathrm{KT}}$, in very good agreement with the exact results $\ensuremath{\eta}({T}_{\mathrm{KT}})=1/4$ and ${\ensuremath{\rho}}_{s}({T}_{\mathrm{KT}}^{\ensuremath{-}})=2/\ensuremath{\pi}$, as well as an essential singularity of the correlation length in the high-temperature phase as $T\ensuremath{\rightarrow}{T}_{\mathrm{KT}}$.

DOI
10.1103/physreve.90.062105

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We study the quasi-two-dimensional quantum O(2) model, a quantum generalization of the Lawrence-Doniach model, within the nonperturbative renormalization-group approach and propose a generic phase diagram for layered three-dimensional systems with an O(2)-symmetric order parameter. Below the transition temperature we identify a wide region of the phase diagram where the renormalization-group flow is quasi-two-dimensional for length scales smaller than a Josephson length lJ{l}_{J}lJ​, leading to signatures of Kosterlitz-Thouless physics in the temperature dependence of physical observables. In particular the order parameter varies as a power law of the interplane coupling with an exponent which depends on the anomalous dimension (itself related to the stiffness) of the strictly two-dimensional low-temperature Kosterlitz-Thouless phase.
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A new set of renormalization equations for the two-dimensional Coulomb gas is derived and solved numerically. These equations give a new picture of the Kosterlitz-Thouless transition. A new temperature T∗{T}^{\mathrm{*}}T∗ is found. Above T∗{T}^{\mathrm{*}}T∗ the value of the dielectric constant at the transition is ${\ensuremath{\epsilon}}_{c}$=1/(4Tc{T}_{c}Tc​) as before whereas below T∗{T}^{\mathrm{*}}T∗ the new result ${\ensuremath{\epsilon}}_{c}1/(41/(41/(4{T}_{c}$) is obtained. The singular behavior at T∗{T}^{\mathrm{*}}T∗ is explored. The results are translated into a renormalization-group flow diagram. An interesting consequence is that the universal jump prediction for the Kosterlitz-Thouless transition turns into a nonuniversal jump prediction for critical temperatures below T∗{T}^{\mathrm{*}}T∗.
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