Exact form of the exponential correlation function in the glassy super−rough phase
Pierre Le Doussal1, Zoran Ristivojevic2, Kay Jörg Wiese1
1CNRS−Laboratoire de Physique Théorique de
l'Ecole Normale Supérieure,
24 rue Lhomond, 75005 Paris, France,
2Centre de Physique Théorique, Ecole Polytechnique, CNRS, 91128 Palaiseau, France.
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 for temperature T < Tc with relative displacements growing with distance r as ⟨[θ(r)−θ(0)]2⟩ ≃ A(τ) ln2 (r/a),
where A(τ) = 2 τ2− 2 τ3 + O(τ4) near the transition and τ = 1−T/Tc. We calculate all higher cumulants and show that
they grow as ⟨[θ(r)−θ(0)]2n⟩c ≃ [2 (−1)n+1 (2n)! ζ(2n−1) τ2 + O(τ3)]ln(r/a), n ≥ 2, where ζ is the Riemann zeta function. By summation we
obtain the decay of the exponential correlation function as ⟨eiq[θ(r)−θ(0)]⟩ ≃ (a/r)η(q)
e−A(q)/2 ln2(r/a) where η(q) and A(q) are given for arbitrary q ≤ 1 to leading order
in τ. The anomalous exponent is η(q) = c q2 − τ2 q2 [2 γE+ψ(q)+ψ(−q)]
in terms of the digamma function ψ, where c is non−universal and γE is the Euler constant. The correlation function strongly decays at q = 1, corresponding to fermion operators in the dual picture; it should be visible in Bragg scattering experiments.
arXiv:1304.4612 [pdf]
Phys. Rev. B 87 (2013) 214201 [pdf]
Copyright (C) by Kay Wiese. Last edited April 18, 2013.