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

Abstract

PACS. 05.40.-a -Fluctuation phenomena, random processes, noise, and Brownian motion. PACS. 05.50+q -Lattice theory and statistics (Ising, Potts, etc.). PACS. 75.50.Lk -Spin glasses and other random magnets.Abstract. -We discuss with the aid of random walk arguments and exact numerical computations the magnetization properties of one-dimensional random field chains. The ground state structure is explained in terms of absorbing and non-absorbing random walk excursions. At low temperatures, the magnetization profiles follow those of the ground states except at regions where a local random field fluctuation makes thermal excitations feasible. This follows also from the non-absorbing random walks, and implies that the magnetization length scale is a product of these two scales. It is not simply given by the Imry-Ma-like ground state domain size nor by the scale of the thermal excitations.In statistical mechanics of random systems the search for universality can be interpreted geometrically. That is, if the introduction of disorder into a system is relevant, the real-space properties of the physical states can be understood in terms of scaling exponents. These describe the fluctuations of a domain wall, or the behavior of a spin-spin correlation function. The central ingredient is that the configurational energy is coupled to geometric fluctuations. Consider a domain wall in a magnet. If the spatial fluctuations are described by a roughness exponent ζ, then there is an associated exponent θ describing the free or ground state energy fluctuations. Assuming that the 'zero temperature fixed point' scenario is true or that the entropy is irrelevant at low enough temperatures, this is all what is needed to describe the physics. The system evolves via Arrhenius-like dynamics so that the cost of moving in the energy landscape is given by the usual exponential factor exp(∆Eβ), where β = 1/T and T is the temperature, and ∆E ∼ l ζ relates the cost to the scale length of the perturbation l. Consider now a random magnet. It has a ground state (GS) which is described exactly by the positions and arrangement of the domain walls. Examples abound in particular in Ising systems, where non-trivial GSs exist for spin glasses and random field systems [1]. In this work we investigate with random walk arguments and exact numerical computations how the aforementioned picture applies in the case of one-dimensional random field chains. We find that for arbitrary field distributions [2] the GS structure can be understood via the c EDP Sciences

Numerical study of the three-dimensional random-field Ising model at zero and positive temperature

“…Similar correlations between ground states and thermal states were found in one dimension. 37,38 We illustrate the above statement for one 32 3 realization ͑ ⌬ 0 = 2.0, seed 1003͒ whose specific heat and susceptibility are shown in Figs. 10͑g͒ and 10͑h͒, respectively.…”

Low temperature properties of the random field Potts chain

“…We have earlier presented a way to divide the sequence of random fields h i for the RFIC in such a way so that one can understand the ensuing domain structure [7]. The idea is to look at trial random walks: start from a test site, and follow the sum of the random fields left/right (an arbitrary choice).…”

“…Finally we investigate how the local regions of the magnetization evolve at finite T [7]. We calculate the average length l m (l m = l T ) from the finite temperature configurations.…”

Optimization in random field Ising models by quantum annealing

“…The second order phase transition of the 2d Ising model is destroyed by the random fields, though residual ordering persists in finite systems [10]. Though there is thus no long range order either in one or two dimensions, systems of a finite size may have ordered ground states when the typical cluster size exceeds the system size, which is true also at zero temperature [10,11,12]. However, in three dimensions one has a temperature dependent critical strength of the random field h c (T ) below which the system is ordered [13].…”