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Progress of Theoretical Physics

Phase Transition in 2-D Sine-Gordon System

T. Ohta

Published Oct 1, 1978• 38 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.60.968

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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.
A systematic renormalisation group technique for studying the 2D sine-Gordon theory (Coulomb gas, XY model) near its phase transition is presented. The new results are (a) higher order terms in the flow equations beyond those of Kosterlitz (1974) give rise to a new universal quantity; (b) this in turn gives the universal form as well as the relative coefficient of the next-to-leading term in the correlation function of the XY model; (c) the free energy (1PI vacuum sum) is calculated after the singularity at beta 2 =4 pi is treated; (d) vortices with multiple charges are shown to be irrelevant; (e) symmetry breaking fields are analysed systematically. The main ideas that the sine-Gordon theory can be defined as a double expansion in alpha (fugacity) and delta = beta 2 /8 pi -1 (distance from the critical temperature at alpha =0). Wave-function and coupling constant ( alpha ) renormalisations are necessary and sufficient, around beta 2 =8 pi where cos phi acquires dimension 2, for functions with elementary SG fields. This gives rise to renormalisation of beta . The renormalisability is proved to the order calculated in the context of the SG theory, and in general, by using the equivalence to the Thirring-Schwinger model. The renormalised beta 2 plays a role analogous to the dimension in a phi 4 theory-8 pi being the critical dimension. beta 2 >8 pi gives an infrared asymptotically free theory which leads to the well-known fixed line. The infrared properties are understood by analogy with the non-linear sigma model.
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.
We have developed a quantum-mechanical approach to study the critical properties of the quantum sine-Gordon equation describing a massless scalar field in 1+1 space-time dimensions with interaction density proportional to cos(${\mathrm{\ensuremath{\beta}}}_{0}$\ensuremath{\varphi}). The main feature of the present theory is to find an appropriate starting point for perturbation expansion, which can directly treat the infrared divergences encountered in the conventional perturbation theory. Exact critical conditions for the phase transition of the model are derived; and the quantum sine-Gordon Hamiltonian is found to correspond exactly to a free-field model. Near the critical regime, a gap will open near zero momentum in the excitation spectrum, and the corresponding ground-state wave function is a pairing quasiparticle state, analogous to the BCS superconducting state. It is proved that this ground state will possess quasi-long-range and quasi-off-diagonal long-range correlations. Comparisons with the Kosterlitz-Thouless classical theory of the two-dimensional Coulomb gas are also made.
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.