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

Abstract

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 $\overline{⟨[\theta (r)-\theta (0){]}^2⟩}\simeq A(\tau ){\ln }^2(r/a)$, 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 ${\overline{⟨[\theta (r)-\theta (0){]}^{2n}⟩}}_c\simeq [2(-1{)}^{n+1}(2n)!\zeta (2n-1){\tau }^2+\mathcal{O}({\tau }^3)]\ln (r/a)$, $n\ge 2$, where $\zeta$ is the Riemann zeta function. By summation, we obtain the decay of the exponential correlation function as $\overline{⟨e^{iq\left[\theta (r)-\theta (0)\right]}⟩}\simeq (a/r{)}^{\eta (q)}\exp (-\frac{1}{2}\mathcal{A}(q){\ln }^2(r/a))$ where $\eta (q)$ and $\mathcal{A}(q)$ are obtained for arbitrary $q\le 1$ 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.

19 pages, 9 figures

References in corpus (6)

• Long-range correlated random field and random anisotropy O(N) models: A functional renormalization group study
• Super-Rough Glassy Phase of the Random Field XY Model in Two Dimensions
• Disordered free fermions and the Cardy Ostlund fixed line at low temperature
• Super-rough phase of the random-phase sine-Gordon model: Two-loop results
• Smectic-glass transition in a liquid crystal cell with a "dirty" substrate
• Smectic order, pinning, and phase transition in a smectic liquid crystal cell with a random substrate

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