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Progress of Theoretical Physics

The Kosterlitz-Thouless Transition in the Quantum Sine-Gordon Model and the Corresponding Classical Film System

Satoshi Takada, T. Sakaguchi, Susumu Misawa

Published Sep 1, 1981• 3 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.66.820

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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 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 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.
Wilson's renormalization-group method is used for investigating the critical properties and the role of quantum fluctuations for a d-dimensional quantum sine-Gordon model. In the classical regime a phase transition is found in d=2-\ensuremath{\epsilon} (\ensuremath{\epsilon}\ensuremath{\ge}0) dimensions and the static critical exponents are evaluated to leading order in \ensuremath{\epsilon}>0. Within the same formalism, the dynamic critical exponent z is also obtained as a characteristic of the intrinsic dynamics of the model. The quantum regime is briefly discussed, and no phase transition is found for d>1. We argue that a quantum-classical crossover occurs for 1<d\ensuremath{\le}2 in the low-temperature limit as a consequence of a dimensional crossover d\ensuremath{\rightarrow}d+1 which shifts the upper critical dimensionality from dcu{d}_{\mathrm{cu}}dcu​=2 to dcu{d}_{\mathrm{cu}}dcu​=1.
Using the recently obtained exact results for the expectation values of operators in the sine- and sinh-Gordon models [A. B. Zamolodchikov and S. Lukyanov, Nucl. Phys. B 493, 571 (1997); V. Fateev, S. Lukyanov, A. B. Zamolodchikov, and Al. B. Zamolodchikov, Phys. Lett. B 406, 83 (1997)] we calculate the specific heat of the corresponding two-dimensional Euclidean (classical) models. We show that the temperature dependence of the specific heat of the sine-Gordon model, in the commensurate phase, has a maximum well below the Kosterlitz-Thouless transition and that the sinh-Gordon model is thermodynamically unstable in the strong-coupling regime. We give also the temperature dependence of the specific heat in the incommensurate phase of the sine-Gordon model.
A number of two-dimensional (2D) critical phenomena can be described in terms of the 2D sine-Gordon model. With bosonization, several 1D quantum systems can be transformed to the same model. However, the transition of the 2D sine-Gordon model, the Berezinskii-Kosterlitz-Thouless (BKT) transition, is essentially different from a second-order transition. The divergence of the correlation length is more rapid than any power law, and there are logarithmic corrections. These pathological features make it difficult to determine the BKT transition point and critical indices from finite-size calculations. In this paper we calculate correlation functions of this model using a real-space renormalization technique. It is found that several correlation functions, or eigenvalues of the corresponding transfer matrix for a finite system, become degenerate on the BKT line, including the logarithmic corrections. By the use of this degeneracy, which reflects the hidden SU(2) symmetry on the BKT line, it is possible to determine the BKT critical line with high precision from a small amount of data and to identify the universality class. In addition, new universal relations are found. These results shed light on the relation between Abelian and non-Abelian bosonization.
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.