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Progress of Theoretical Physics

A Self-Consistent Treatment of the Kosterlitz-Thouless Transition of the Two-Dimensional Classical Sine-Gordon Model

T. Sakaguchi, Tsuguhiro Tamaribuchi, Satoshi Takada

Published Jul 1, 1982• 5 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.68.19

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•Jan 1, 2006•11 citations
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 quantum sine-Gordon model in two spatial dimensions is studied using the method of Wilson's renormalization group with the aid the Feynman path integral. The Kosterlitz-Thouless transition temperature is calculated and shown to be lower than that of the corresponding classical sine-Gordon model. A classical film system is shown to correspond to the quantum sine-Gordon model and its transition temperature is shown to increase with increasing thickness. Applicability of our treatment is examined.
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 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.
We investigate a gap-formation mechanism of the one-dimensional Kondo-necklace model at zero temperature. An analytic treatment using the bosonization method shows that the model undergoes the Kosterlitz-Thouless transition at the critical coupling Jc=0{J}_{c}=0Jc​=0, which is described by the quantum sine-Gordon model. This result is contrary to the previous investigations on the ground-state phase transition, where some finite critical coupling values have been predicted. To clarify the validity of our analytic calculation result, we estimate the singlet-triplet gap by employing White's density matrix renormalization group method; the numerical calculation is carried out for the systems up to 200 sites with open boundary condition. As a result, we find that the excitation gap observed in the large JJJ region becomes smaller quickly with the decrease of JJJ. However, the data show the existence of a small gap for J> 0.25J>~0.25J> 0.25, which seems to support Jc=0{J}_{c}=0Jc​=0.
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.
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