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Physical review. B, Condensed matter

Renormalization-group approach to commensurate-incommensurate transitions in two dimensions

M. W. Puga, Eugen Šimánek, H. Beck

Published Sep 1, 1982• 11 citations
Publisher

Abstract

We present a renormalization-group-approach analysis of the commensurate-incommensurate transition of the two-dimensional classical sine-Gordon system which includes incommensurability as a new renormalizable parameter. The critical behavior of various quantities shows corrections with respect to the simpler self-consistent harmonic approximation (SCHA). However, our result for the phase boundary still has a SCHA-type $T$ dependence, suggesting that incommensurability tends to suppress the role of fluctuations.

DOI
10.1103/physrevb.26.2673

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We present a renormalization-group-approach analysis of the commensurate-incommensurate transition of the two-dimensional classical sine-Gordon system which includes incommensurability as a new renormalizable parameter. The critical behavior of various quantities shows corrections with respect to the simpler self-consistent harmonic approximation (SCHA). However, our result for the phase boundary still has a SCHA-type TTT dependence, suggesting that incommensurability tends to suppress the role of 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 commensurate-incommensurate transition of a two-dimensional classical sine-Gordon system is studied by the equivalent one-dimensional quantum system. The latter is expanded around its classical limit, corresponding to a low-temperature expansion of the two-dimensional system. It is found that both mass and wavefunction renormalisation are required. The critical exponent for incommensurability is beta =0, but if the momentum cut-off is kept finite one obtains beta =1. The classical limit is a singular point and the results are reliable when they are not too close to the transition.
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.
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.
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.
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.