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arXiv:0909.0526 · Physical Review B

Pokrovsky-Talapov model at finite temperature: A renormalization-group analysis

Achilleas Lazarides, O. Tieleman, C. Morais Smith

Published Dec 15, 2009• 9 citations

Abstract

We calculate the finite-temperature shift of the critical wave vector ${Q}_{c}$ of the Pokrovsky-Talapov model using a renormalization-group analysis. Separating the Hamiltonian into a part that is renormalized and one that is not, we obtain the flow equations for the stiffness and an arbitrary potential. We then specialize to the case of a cosine potential, and compare our results to well-known results for the sine-Gordon model, to which our model reduces in the limit of vanishing driving wave vector $Q=0$. Our results may be applied to describe the commensurate-incommensurate phase transition in several physical systems and allow for a more realistic comparison with experiments, which are always carried out at a finite temperature.

DOI: 10.1103/physrevb.80.245418

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We calculate the finite-temperature shift of the critical wave vector

{Q}_{c}

of the Pokrovsky-Talapov model using a renormalization-group analysis. Separating the Hamiltonian into a part that is renormalized and one that is not, we obtain the flow equations for the stiffness and an arbitrary potential. We then specialize to the case of a cosine potential, and compare our results to well-known results for the sine-Gordon model, to which our model reduces in the limit of vanishing driving wave vector

Q=0

. Our results may be applied to describe the commensurate-incommensurate phase transition in several physical systems and allow for a more realistic comparison with experiments, which are always carried out at a finite temperature.

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.

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.

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.