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arXiv:1812.09008 · International Journal of Modern Physics A

Renormalization of the bilocal sine-Gordon model

I. Steib, S. Nagy

Published Jul 29, 2019• 2 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1142/s0217751x19501173

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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.
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 two-dimensional classical sine-Gordon model is investigated by using a two-parameter self-consistent renormalization method up to second order of renormalized vertices, where coupled self-consistent equations for renormalized mass m and renormalized temperature x are obtained. In high temperature region (x > 2), the self-consistent equations have a massless solution which stands for the existence of the Kosterlitz-Thouless phase, and lead to an equation for x which is the same as that given by José et al. and is consistent with the result of Wilson's renormalization group method. In low temperature region (x < 2), numerical calculation is carried out, and it is shown that the system is well described by the self-consistent harmonic approximation, which is a one-parameter (m) renormalization treatment, except near x = 2. Near x ≲ 2 our self-consistent equations have no solution, through this can be partly remedied by counting chain diagrams.
The renormalization group equations for the sine-Gordon system with a short wave-length cutoff in two dimensions are derived by means of a momentum-shell recursion method. In a special limit they agree with the Kosterlitz renormalization group equations which describe the critical properties of the phase transitions of the two-dimensional (2-D) neutral Coulomb gas with single charges ±1 and of the 2-D isotropic X-Y model in the absence of an external field. The results, therefore, give rise to a possibility that certain classes of phase transitions in low dimensions can be described in a unified way with the sine-Gordon Hamiltonian.
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.
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.