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arXiv:hep-th/0008114 · Physical review. B, Condensed matter

Nonperturbative renormalization flow and essential scaling for the Kosterlitz-Thouless transition

Gero von Gersdorff, CHRISTOF WETTERICH

Published Jul 12, 2001• 136 citations
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Abstract

The Kosterlitz-Thouless phase transition is described by the nonperturbative renormalization flow of the two-dimensional ${\ensuremath{\varphi}}^{4}$ model. The observation of essential scaling demonstrates that the flow equation incorporates nonperturbative effects that have previously found an alternative description in terms of vortices. The duality between the linear and nonlinear $\ensuremath{\sigma}$ model gives a unified description of the long-distance behavior for $O(N)$ models in arbitrary dimension d. We compute critical exponents in first order in the derivative expansion.

DOI
10.1103/physrevb.64.054513

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The Kosterlitz-Thouless phase transition is described by the nonperturbative renormalization flow of the two-dimensional \ensuremathφ4{\ensuremath{\varphi}}^{4}\ensuremathφ4 model. The observation of essential scaling demonstrates that the flow equation incorporates nonperturbative effects that have previously found an alternative description in terms of vortices. The duality between the linear and nonlinear \ensuremathσ\ensuremath{\sigma}\ensuremathσ model gives a unified description of the long-distance behavior for O(N)O(N)O(N) models in arbitrary dimension d. We compute critical exponents in first order in the derivative expansion.
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 $T\ensuremath{\rightarrow}{T}_{\mathrm{KT}}$.
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.
The two-dimensional qqq-state clock model for $q\ensuremath{\ge}5$ undergoes two Berezinskii-Kosterlitz-Thouless (BKT) phase transitions as temperature decreases. Here we report an extensive worm-type simulation of the square-lattice clock model for q=5q=5q=5--9 in a pair of flow representations, from high- and low-temperature expansions, respectively. By finite-size scaling analysis of susceptibilitylike quantities, we determine the critical points with a precision improving over the existing results. Due to the dual flow representations, each point in the critical region is observed to simultaneously exhibit a pair of anomalous dimensions, which are ${\ensuremath{\eta}}_{1}=1/4$ and ${\ensuremath{\eta}}_{2}=4/{q}^{2}$ at the two BKT transitions. Further, the approximate self-dual points ${\ensuremath{\beta}}_{\mathrm{sd}}(L)$, defined by the stringent condition that the susceptibilitylike quantities in both flow representations are identical, are found to be nearly independent of system size LLL and behave as ${\ensuremath{\beta}}_{\mathrm{sd}}\ensuremath{\simeq}q/2\ensuremath{\pi}$ asymptotically at the large-qqq limit. The exponent $\ensuremath{\eta}$ at ${\ensuremath{\beta}}_{\mathrm{sd}}$ is consistent with 1/q1/q1/q within statistical error as long as $q\ensuremath{\ge}5$. Based on this, we further conjecture that $\ensuremath{\eta}({\ensuremath{\beta}}_{\mathrm{sd}})=1/q$ holds exactly and is universal for systems in the qqq-state clock universality class. Our work provides a vivid demonstration of rich phenomena associated with the duality and self-duality of the clock model in two dimensions.
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∗.