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arXiv:cond-mat/0511439 · Journal of Statistical Mechanics Theory and Experiment

Numerical simulations of the random phase sine–Gordon model and renormalization group predictions

David Lancaster, Juan Jesus Ruiz-Lorenzo

Published Jan 3, 2007• 1 citation
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Abstract

Numerical simulations of the random phase sine–Gordon model suffer from strong finite size effects preventing the non-Gaussian log 2 r component of the spatial correlator from following the universal infinite volume prediction. We show that a finite size prediction based on perturbative renormalization group (RG) arguments agrees well with new high precision simulations for small coupling and close to the critical temperature.

DOI
10.1088/1742-5468/2007/01/p01003

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Numerical simulations of the random phase sine–Gordon model suffer from strong finite size effects preventing the non-Gaussian log 2 r component of the spatial correlator from following the universal infinite volume prediction. We show that a finite size prediction based on perturbative renormalization group (RG) arguments agrees well with new high precision simulations for small coupling and close to the critical temperature.
At large distances and in the low-temperature phase, the quenched correlation functions in the 2 d random phase sine-Gordon model have been argued to be of the form: , with = ( T − T c ). However, renormalization group computations predict B ≠ 0, while variational approaches (which are supposed to be exact for models with a large number of components) give B = 0. We introduce a large- N version of the random phase sine-Gordon model. Using non-Abelian bosonization and renormalization group techniques, we show that the correlation functions of our models have the above form but with a coefficient B suppressed by a factor 1/ N 3 compared to A .
We consider the two-dimensional random-phase sine-Gordon and study the vicinity of its glass transition temperature Tc{T}_{c}Tc​, in an expansion in small $\ensuremath{\tau}=({T}_{c}\ensuremath{-}T)/{T}_{c}$, where TTT denotes the temperature. We derive renormalization group equations in cubic order in the anharmonicity, and show that they contain two universal invariants. Using them we obtain that the correlation function in the super-rough phase for temperature T<TcT<{T}_{c}T<Tc​ behaves at large distances as $\overline{\ensuremath{\langle}{[\ensuremath{\theta}(x)\ensuremath{-}\ensuremath{\theta}(0)]}^{2}\ensuremath{\rangle}}=\mathcal{A}{\mathrm{ln}}^{2}(|x|/a)+\mathcal{O}[\mathrm{ln}(|x|/a)]$, where the amplitude A\mathcal{A}A is a universal function of temperature $\mathcal{A}=2{\ensuremath{\tau}}^{2}\ensuremath{-}2{\ensuremath{\tau}}^{3}+\mathcal{O}({\ensuremath{\tau}}^{4})$. This result differs at two-loop order, i.e., $\mathcal{O}({\ensuremath{\tau}}^{3})$, from the prediction based on results from the ``nearly conformal'' field theory of a related fermion model. We also obtain the correction-to-scaling exponent.
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 extend the exact multilocal renormalization group (RG) method to study the flow of the effective action functional. This important physical quantity satisfies an exact RG equation which is then expanded in multilocal components. Integrating the nonlocal parts yields a closed exact RG equation for the local part, to a given order in the local part. The method is illustrated on the O(N) model by straightforwardly recovering the eta exponent and scaling functions. Then it is applied to study the glass phase of the Cardy-Ostlund, random phase sine Gordon model near the glass transition temperature. The static correlations and equilibrium dynamical exponent z are recovered and several results are obtained, such as the equilibrium two-point scaling functions. The nonequilibrium, finite momentum, two-time t,t' response and correlations are computed. They are shown to exhibit scaling forms, characterized by exponents lambda(R) not equal lambda(C), as well as universal scaling functions that we compute. The fluctuation dissipation ratio is found to be nontrivial and of the form X[q(z)(t-t'),t/t']. Analogies and differences with pure critical models are discussed.
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.