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Exact Ground State Calculation of the Interface Morphology in the Two‐Dimensional Random‐Field Ising Model
Author(s) -
Jost M.,
Esser J.,
Usadel K. D.
Publication year - 1997
Publication title -
physica status solidi (b)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.51
H-Index - 109
eISSN - 1521-3951
pISSN - 0370-1972
DOI - 10.1002/1521-3951(199708)202:2<r11::aid-pssb999911>3.0.co;2-i
Subject(s) - mercator projection , ising model , physics , philosophy , statistical physics , cartography , geography
We study the morphology of an interface between spin-up and spin-down domains for the random-eld Ising model (RFIM) in D = 2 at zero temperature. The Hamiltonian for this system is given by 1] where l = (x; y) denotes the lattice points of a square lattice of size L L. The Ising spins s l are coupled ferromagnetically (J > 0) and can have values s l = 1. The rst sum is over all nearest-neighbor pairs hl; l 0 i while the second sum represents the coupling of the spins to an external homogeneous eld H as well as to the quenched random elds h l. Here we choose J = 1 and the random elds are drawn with equal probability from an interval between ? and. Furthermore, periodic boundary conditions in the x-direction are assumed. An interface is imposed to the system by setting the random elds hi of the row y = 1 to a value larger than 4J and for the random elds of the row y = L to a value smaller than ?4J. To calculate the ground states of these systems exactly the Ising-system Eq.(1) is mapped on an equivalent transport network and the maximum ow is calculated using the Ford-Fulkerson algorithm 3]. Due to computer limitations the maximum system size we have simulated is L = 100. It is well known 1] that for H 0 the ground state of the RFIM in D = 2 is a domain state. The domains have a fractal shape and their size distribution follows a power law 2]. Here, the situation is changed due to the imposed boundary conditions in the y-direction. For < C = 1 we always obtained a ground state where two domains of opposite magnetization are separated by a single valued interface proole. This is in agreement with Ji and Robbins 4] who investigated the interface morphology in the RFIM within a zero-temperature single-spin-ip dynamics. That only one interface develops is due to the nite system size. For very large systems it is expected that the RFIM in D = 2 breaks into a domain state regardless of the boundary conditions imposed. To analyze the morphology of the interface in nite systems we measured the height correlation function C(r) = h(x + r) ? h(x)] 2 (2) where the angular brackets denote an average over all lattice sites at positions x …