Random Phase sine-Gordon Model Research Articles

Exact form of the exponential correlation function in the glassy super-rough phase

We consider the random-phase sine-Gordon model in two dimensions. It describes two-dimensional elastic systems with random periodic disorder, such as pinned flux-line arrays, random field XY models, and surfaces of disordered crystals. The model exhibits a super-rough glass phase at low temperature $T<T_{c}$ with relative displacements growing with distance $r$ as $\bar{\langle [\theta(r)-\theta(0)]^2\rangle} \simeq A(\tau) \ln^2 (r/a)$, where $A(\tau) = 2 \tau^2- 2 \tau^3 +\mathcal{O}(\tau^4)$ near the transition and $\tau=1-T/T_{c}$. We calculate all higher cumulants and show that they grow as $\bar{\langle[\theta(r)-\theta(0)]^{2n}\rangle}_c \simeq [2 (-1)^{n+1} (2n)! \zeta(2n-1) \tau^2 + \mathcal{O}(\tau^3) ] \ln(r/a)$, $n \geq 2$, where $\zeta$ is the Riemann zeta function. By summation, we obtain the decay of the exponential correlation function as $\bar{\langle e^{iq\left[\theta(r)-\theta(0)\right]}\rangle} \simeq (a/r)^{\eta(q)} \exp\boldsymbol(-\frac{1}{2}\mathcal{A}(q)\ln^2(r/a)\boldsymbol)$ where $\eta(q)$ and ${\cal A}(q)$ are obtained for arbitrary $q \leq 1$ to leading order in $\tau$. The anomalous exponent is $\eta(q) = c q^2 - \tau^2 q^2 [2\gamma_E+\psi(q)+\psi(-q)]$ in terms of the digamma function $\psi$, where $c$ is non-universal and $\gamma_E$ is the Euler constant. The correlation function shows a faster decay at $q=1$, corresponding to fermion operators in the dual picture, which should be visible in Bragg scattering experiments.

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

Numerical simulations of the random phase sine–Gordon modelsuffer from strong finite size effects preventing the non-Gaussianlog2 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 tothe critical temperature.

Exact multilocal renormalization of the effective action: application to the random sine Gordon model statics and nonequilibrium dynamics.

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.

Hybrid simulated annealing using Tsallis statistics

We use the Hybrid Monte Carlo method to sample the Tsallis configurational probability p(x) which depends on the parameter q. We use here a homogeneous Markov chain which satisfies the detailed balance condition for all values of q and we apply this method to Simulated Annealing optimization problems. In two examples, a test function with many minima and the discrete Random phase sine-Gordon Model, we show the dependence on the q parameter of the ground state configuration energy obtained after annealing. We conclude that the performance of the simulated annealing method is improved by using q>1 values.

Numerical simulations of the random phase sine-Gordon model

We have performed comprehensive numerical simulations of the Random Phase Sine Gordon Model, studying both statics and dynamics for various values of the coupling. The glass transition can be seen both in static and dynamic signals at a temperature that depends on the coupling. Our results agree qualitatively (statics) and quantitatively (dynamics) with Renormalisation Group predictions.