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 <(<[theta(r) - theta(0)](2)>)over bar> similar or equal to Lambda(tau) ln(2)(r/a), where A(tau) = 2 tau(2) -2 tau(3) + O(tau(4)) near the transition and t = 1 - T/T-c. We calculate all higher cumulants and show that they grow as <(<[theta(r) - theta(0)](2n)>(c)> over bar> similar or equal to [2(-1)(n+1)(2n)!zeta (2n - 1)tau(2) + O(tau 3)] ln(r/a), n >= 2, where zeta is the Riemann zeta function. By summation, we obtain the decay of the exponential correlation function as <(< e(iq[theta(r)-theta(0)])>)over bar> similar or equal to (a/r)(eta(q)) exp (-1/2A(q) ln(2)(r/a)), where eta(q) and A(q) are obtained for arbitrary q <= 1 to leading order in tau. The anomalous exponent is eta(q) = cq(2) - tau(2)q(2)[2 gamma(E) +psi(q) + psi(-q)] in terms of the digamma function psi, where c is nonuniversal 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.