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Physical review. B, Condensed matter

Analysis of a new set of renormalization equations for the Kosterlitz-Thouless transition

Joseph Marie Thijssen, H. J. F. Knops

Published Nov 1, 1988• 28 citations

Abstract

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.

DOI: 10.1103/physrevb.38.9080

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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.

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{*}}

.

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-

{\mathit{T}}_{\mathit{c}}

superconductors are discussed.

The screening length xi and the singular part of the free energy of the two-dimensional Coulomb gas are calculated for small fugacity, y, in the vicinity of the transition temperature, T= 1 / 4 , from Kosterlitz's renormalisation-group equations. The results are expressed in terms of a single 'scaling function'. As special cases the authors recover known behaviour in (i) the critical region and (ii) the low-density, high-temperature limit where xi approximately A(T)y - nu with nu =2T/(4T-1). They also obtain the new result that, as T to 1 / 4 , A(T) has the singular form A(T) approximately (T- 1 / 4 ) v .

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.

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 $T\ensuremath{\rightarrow}{T}_{\mathrm{KT}}$.

The existence of a discontinuity in the inverse dielectric constant of the two-dimensional (2D) Coulomb gas is demonstrated on purely numerical grounds. This is done by expanding the free energy in an applied twist and performing a finite-size scaling analysis of the coefficients of higher-order terms. The phase transition, driven by unbinding of dipoles, corresponds to the Kosterlitz-Thouless transition in the 2D

XY

model. The method developed is also used for investigating the possibility of a Kosterlitz-Thouless phase transition in a three-dimensional system of point charges interacting with a logarithmic pair-potential, a system related to effective theories of low-dimensional strongly correlated systems. We also contrast the finite-size scaling of the fluctuations of the dipole moments of the two-dimensional Coulomb gas and the three-dimensional logarithmic system to those of the three-dimensional Coulomb gas.