Ground-State Roughness of the Disordered Substrate and Flux Lines in d=2.

We apply optimization algorithms to the problem of finding ground states for crystalline surfaces and flux-line arrays in the presence of disorder. The algorithms provide ground states in polynomial time, which provides for a more precise study of the interface widths than from Monte Carlo simulations at finite temperature. Using d › 2 systems up to size 420 2 , with a minimum of 2 3 10 3 realizations at each size, we find very strong evidence for a ln 2 sLd super-rough state at low The flux-line arrays formed in a 2D type-II dirty superconductor with the magnetic field parallel to the plane and the surface configurations of a crystalline defect-free material deposited on a disordered substrate (DS) are closely related systems. They have been studied for both the intrinsic interest and because they serve as prototypical models for elastic media in a disordered environment. Both have low temperature glassy phases in which equilibrium and dynamic properties are dominated by the disorder. In the continuum limit they are both described by the random-phase sine-Gordon model (RSGM). However, analytic attempts at understanding the equilibrium properties of the glassy phase based on the RSGM have yielded conflicting results [1,2]. Finite-temperature simulations of the RSGM or the corresponding discrete Gaussian model for the disordered substrate have also been ambiguous [3 ‐7]. Moreover, it is not clear to what extent universality arguments, which yield the RSGM as the continuum limit, can be trusted at temperatures well below the glass transition. The aim of the present work is to address both issues by finding the exact minimum energy configurations in discrete models of the DS surface and that of the fluxline arrays. This yields their T › 0 shapes for any given disorder realization; averaging over disorder allows for the evaluation of their averaged physical properties. In particular, the quantities that theory and simulation have focused on are the height-height correlations in the disordered substrate model and the line-line correlations in the flux-line model. The flux-line system is discretized by its mapping to a surface model (see below), and one may observe the transition by inspecting the roughness scaling of the respective surfaces instead. The transition into the glassy phase is exhibited by a change in these height-height correlations in both models. The predictions from the analytic studies of the RSGM are as follows: Above the transition temperature T › Tg the surface is always logarithmically rough, with a prefactor proportional to T . Below the transition temperature, renormalization group (RG) calculations [1] predict a ln 2 sLd behavior, while variational approaches [2] predict the persistence of the lnsLd behavior but with the prefactor unchanging for T # Tg. Numerical simulations have differed in their results as well. Simulations of the RSGM with weak coupling have shown no transition at all [3]. Others have shown a transition with a lnsLd [4] or a ln 2 sLd behavior [5]. Monte Carlo simulations of the discrete Gaussian version of the model have exhibited the transition, but the behavior of the roughness could better fit the ln sLd [6] or the ln 2 sLd