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arXiv:0903.5524 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Comparison of renormalization group schemes for sine-Gordon-type models

I. Nándori, S. Nagy, Kornél Sailer, Andrea Trombettoni

Published Jul 17, 2009• 30 citations

Abstract

The scheme dependence of the renormalization group (RG) flow has been investigated in the local potential approximation for two-dimensional periodic, sine-Gordon type field-theoretic models discussing the applicability of various functional RG methods in detail. It was shown that scheme-independent determination of such physical parameters is possible as the critical frequency (temperature) at which Kosterlitz-Thouless-Berezinskii type phase transition takes place in the sine-Gordon and the layered sine-Gordon models, and the critical ratio characterizing the Ising-type phase transition of the massive sine-Gordon model. For the latter case, the Maxwell construction represents a strong constraint on the RG flow, which results in a scheme-independent infrared value for the critical ratio. For the massive sine-Gordon model also the shrinking of the domain of the phase with spontaneously broken periodicity is shown to take place due to the quantum fluctuations.

DOI: 10.1103/physrevd.80.025008

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The scheme dependence of the renormalization group (RG) flow has been investigated in the local potential approximation for two-dimensional periodic, sine-Gordon type field-theoretic models discussing the applicability of various functional RG methods in detail. It was shown that scheme-independent determination of such physical parameters is possible as the critical frequency (temperature) at which Kosterlitz-Thouless-Berezinskii type phase transition takes place in the sine-Gordon and the layered sine-Gordon models, and the critical ratio characterizing the Ising-type phase transition of the massive sine-Gordon model. For the latter case, the Maxwell construction represents a strong constraint on the RG flow, which results in a scheme-independent infrared value for the critical ratio. For the massive sine-Gordon model also the shrinking of the domain of the phase with spontaneously broken periodicity is shown to take place due to the quantum fluctuations.

The renormalization group (RG) flow for the two-dimensional sine-Gordon model is determined by means of Polchinski's RG equation at next-to-leading order in the derivative expansion. In this work we have two different goals, (i) to consider the renormalization scheme-dependence of Polchinski's method by matching Polchinski's equation with the Wegner-Houghton equation and with the real space RG equations for the two-dimensional dilute Coulomb-gas, (ii) to go beyond the local potential approximation in the gradient expansion in order to clarify the supposed role of the field-dependent wave-function renormalization. The well-known Coleman fixed point of the sine-Gordon model is recovered after linearization, whereas the flow exhibits strong dependence on the choice of the renormalization scheme when non-linear terms are kept. The RG flow is compared to those obtained in the Wegner-Houghton approach and in the dilute gas approximation for the two-dimensional Coulomb-gas.

The functional renormalization group treatment is presented for the two-dimensional sine-Gordon model including a bilocal term in the potential, which contributes to the flow at the tree level. It is shown that the flow of the bilocal term can substitute the evolution of the wave function renormalization constant, since it can recover the Kosterlitz–Thouless type phase transition. The flows can also reveal the connection between the sine-Gordon and the noninteracting Thirring models at a special value of the wave number parameter.

Abstract. The phase structure of the layered sine-Gordon (LSG) model is investigated in terms of symmetry considerations by means of differential renormalization group (RG) method using the local potential approximation. The RG analysis of the general N-layer model provides us with the possibility to consider the dependence of the vortex dynamics on the number of layers. The Lagrangians are distinguished according to the number of zero eigenvalues of their mass matrices where the number is found to be decisive with respect to the phase structure of the N-layer models, with neighbouring layers being coupled by terms quadratic in the field variables. It is shown that the LSG model with N layers undergoes a Kosterlitz–Thouless type phase transition at the critical value of the parameter β2 c = 8Nπ. In the limit of infinitely many layers the LSG model can be considered as the discretized version of the three-dimensional sine-Gordon model which has been shown to have a single phase within the local potential approximation. The infinite critical value of the parameter β2 c for the LSG model in the continuum limit (N → ∞) is consistent with the latter observation. PACS numbers: 11.10.Hi,11.10.Gh,11.10.Kk

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.

We introduce and study the properties of a periodic model interpolating between the sine-- and the sinh--Gordon theories in

1+1

dimensions. This model shows the peculiarities, due to the preservation of the functional form of their potential across RG flows, of the two limiting cases: the sine-Gordon, not having conventional order/magnetization at finite temperature, but exhibiting Berezinskii-Kosterlitz-Thouless (BKT) transition; and the sinh-Gordon, not having a phase transition, but being integrable. The considered interpolation, which we term as {\em sn-Gordon} model, is performed with potentials written in terms of Jacobi functions. The critical properties of the sn-Gordon theory are discussed by a renormalization-group approach. The critical points, except the sinh-Gordon one, are found to be of BKT type. Explicit expressions for the critical coupling as a function of the elliptic modulus are given.