Ground states versus low-temperature equilibria in random field Ising chains

Random walk arguments and exact numerical computations are used to study one-dimensional random field chains. The ground state structure is described with absorbing and non-absorbing random walk excursions. At low temperatures, the local magnetization follows the ground state except at regions where a local random field fluctuation makes thermal excitations easier. This is explained by the random walk picture, implying that the magnetization lengthscale is a product of the domain size and the thermal excitation scale. PACS. 05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion – 05.50.+q Lattice theory and statistics (Ising, Potts, etc.) – 75.50.Lk Spin glasses and other random magnets In statistical mechanics of random systems the search for universal properties has a geometric interpretation. If the introduction of disorder is relevant, the real-space properties of the physical states can be understood through scaling exponents. These describe the fluctuations of a domain wall, or the behavior of a spin-spin correlation function, and the configurational energy is coupled to the geometry. A domain wall in a magnet wanders in space, described by a roughness exponent ζ and there is an exponent θ for the free or ground state energy fluctuations (see [1]). Assuming that the ‘zero temperature fixed point’ scenario is true and that the entropy is irrelevant at low enough temperatures, this is all what is needed. The system evolves via Arrhenius-like dynamics so that the cost of moving in the energy landscape is given by the exponential factor exp(∆Eβ), where β = 1/T and T is the temperature, and ∆E ∼ l relates the cost to the scale of the perturbation l. A simple toy model, the random energy model, attempts to describe the landscape but does not have the spatial structure that is crucial in finite-dimensional systems [2]. Similar physics arises in systems with structural frustration, only, like glasses. A random magnet has a ground state (GS), described exactly by the positions and arrangement of the domain walls. Non-trivial examples abound in the form of Ising spin glasses and random field Ising magnets [3]. Here we develop a novel random walk (RW) picture, that allows us to solve exactly for the groundstate of one-dimensional random field chains, generalizing earlier applications of a e-mail: [email protected] RW ideas to this system [4]. We compare, by considering the local magnetizations, this solution to exact numerical computations at T > 0 to discuss the relation of the low-temperature physics to the GS of a random magnet. Our work is analogous to RW arguments in disordered quantum spin chains, where one can also compute exactly physical properties [5]. The GS structure can be constructed for arbitrary field distributions via the random walk ‘algorithm’ (see below). At finite temperatures we present a scaling argument based on the zero-temperature description of the energy landscape, and confirm it by numerical studies of the GS and the local magnetization on a sample-to-sample basis. The second main finding is then the existence of two length scales. These are the zero-temperature length scale of the domains and the typical size of ’easy’ excitations at a given temperature. The latter changes the correlation length of the magnetization from the GS. The emerging picture should be applicable to more general situations than the 1D RFIM. This is the simplest random magnet where a non-trivial GS is mixed with thermal excitations (e.g. random bond Ising magnets have a trivial GS). The non-equilibrium properties of the RFIM chain have recently received attention [6,7] since decimationtype real space renormalization can be applied to domain wall dynamics: each DW undergoes logarithmic Sinai diffusion [8]. The asymptotic state of e.g. coarsening (on which the related RG procedure is to be stopped) is given by our findings. The zeroes of the magnetization profile simply denote the equilibrium positions of domain walls 102 The European Physical Journal B