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Physical review. B, Condensed matter

New renormalization equations for the Kosterlitz-Thouless transition

Petter Minnhagen

Published Sep 1, 1985• 59 citations
Publisher

Abstract

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 ${T}^{\mathrm{*}}$ is found. Above ${T}^{\mathrm{*}}$ the value of the dielectric constant at the transition is ${\ensuremath{\epsilon}}_{c}$=1/(4${T}_{c}$) as before whereas below ${T}^{\mathrm{*}}$ the new result ${\ensuremath{\epsilon}}_{c}$1/(4${T}_{c}$) is obtained. The singular behavior at ${T}^{\mathrm{*}}$ 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 ${T}^{\mathrm{*}}$.

DOI
10.1103/physrevb.32.3088

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A new and more direct derivation is given of an extended set of renormalization equations for the two-dimensional Coulomb gas. Our equations are equivalent to those proposed by Minnhagen, but are cast in the form of ordinary renormalization equations. It turns out that the difference from the usual renormalization equations is due to the fact that the boundary conditions to be used are of a mixed type: two of them appear as initial conditions; the remaining ones as conditions at the end of a trajectory. The striking new features found by Minnhagen, such as the nonuniversal jump in the inverse dielectric constant and the presence of a first-order transition at higher densities, are a direct result of the possibility (due to the mixed boundary conditions) of having crossing trajectories. These crossing trajectories were also found by Minnhagen et al., but only in the high-temperature region as multiple solutions of their integral equations. Finally, we show that the rather ad hoc criterion adopted by Minnhagen and co-workers for selecting the equilibrium solution indeed corresponds to the one with the smaller free energy, at least in the cases considered by these authors.

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arXiv:2207.13748•Jul 27, 2022•2 citations
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 \ensuremathϵc{\ensuremath{\epsilon}}_{c}\ensuremathϵc​=1/(4Tc{T}_{c}Tc​) as before whereas below T∗{T}^{\mathrm{*}}T∗ the new result \ensuremathϵc{\ensuremath{\epsilon}}_{c}\ensuremathϵc​1/(4Tc{T}_{c}Tc​) 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∗.
A new and more direct derivation is given of an extended set of renormalization equations for the two-dimensional Coulomb gas. Our equations are equivalent to those proposed by Minnhagen, but are cast in the form of ordinary renormalization equations. It turns out that the difference from the usual renormalization equations is due to the fact that the boundary conditions to be used are of a mixed type: two of them appear as initial conditions; the remaining ones as conditions at the end of a trajectory. The striking new features found by Minnhagen, such as the nonuniversal jump in the inverse dielectric constant and the presence of a first-order transition at higher densities, are a direct result of the possibility (due to the mixed boundary conditions) of having crossing trajectories. These crossing trajectories were also found by Minnhagen et al., but only in the high-temperature region as multiple solutions of their integral equations. Finally, we show that the rather ad hoc criterion adopted by Minnhagen and co-workers for selecting the equilibrium solution indeed corresponds to the one with the smaller free energy, at least in the cases considered by these authors.
We reexamine the two-dimensional linear O(2)\mathrm{O}(2)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 TKT{T}_{\mathrm{KT}}TKT​, 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 $T\ensuremath{\rightarrow}{T}_{\mathrm{KT}}$.
The width of the critical region for the Kosterlitz-Thouless transition is estimated and found to be extremely narrow. The estimate is based on a numerical solution of a set of renormalization equations for the two-dimensional Coulomb gas. It is concluded that the Kosterlitz-Thouless critical behavior cannot be observed in the resistance data for quasi-two-dimensional superconductors. The implications for the quasi-two-dimensionality of high-Tc{\mathit{T}}_{\mathit{c}}Tc​ superconductors are discussed.
We investigate the general features of the renormalization-group flow at the Berezinskii-Kosterlitz-Thouless (BKT) transition, providing a thorough quantitative description of the asymptotc critical behavior, including the multiplicative and subleading logarithmic corrections. For this purpose, we consider the RG flow of the sine-Gordon model around the renormalizable point which describes the BKT transition. We reduce the corresponding $\ensuremath{\beta}$ functions to a universal canonical form, valid to all perturbative orders. Then we determine the asymptotic solutions of the RG equations in various critical regimes: the infinite-volume critical behavior in the disordered phase, the finite-size scaling limit for homogeneous systems of finite size, and the trap-size scaling limit occurring in two-dimensional bosonic particle systems trapped by an external space-dependent potential.
Fundamental properties of the Kosterlitz-Thouless-Berezinskii (KTB) transition which occurs in systems in the universality class of the two-dimensional X-Y model are reviewed here with an emphasis on the real-space renormalization-group analysis used to derive the recursion relations. A derivation of the unique temperature dependence of the correlation length of this system will be presented and the signatures of the KTB transition in electronic transport measurements will be explained. Extensions of this model and technique to layered superconductors will also be presented. The size of the three-dimensional (3D) critical region, the nature of the 3D cross-over, and the effect of a finite electric current will be discussed.
This is a set of notes recalling some of the most important results on the XY model from the ground up. They are meant for a junior researcher wanting to get accustomed to the Kosterlitz-Thouless phase transition in the context of the 2D classical XY model. The connection to the 2D Coulomb gas is presented in detail, as well as the renormalization group flow obtained from this dual representation. A numerical Monte-Carlo approach to the classical XY model is presented. Finally, two physical setting that realize the celebrated Kosterlitz-Thouless phase transition are presented: superfluid and liquid crystal thin films.