Possible line of critical points for a random field Ising model in dimension 2

2014 We study a particular random field Ising model in dimension 2 at 0 temperature. On each site the random field is either + ~ with probability p/2, ~ with probability p/2 or 0 with probability 1 2014 p. Using finite size scaling arguments, we show that for small p, the average correlation function between two spins at distance R decreases like R-~(p) where the exponent ~(p) = 2 03C0p + O(p2). The assumptions made to obtain this result and the possible generalizations to other random field models are discussed Tome45 N° 12 15 JUIN 1984 LE JOURNAL DE PHYSIQUE-LETTRES J. Physique Lett. 45 (1984) L-577-L-581 15 JUIN 1984, Classification Physics Abstracts 05.70 64.60H The random field Ising model [1-5] (RFIM) has been for a long time a controversial subject. The question of the lower critical dimensionality de (above which ferromagnetic order can exist) has been much debated between those [2] who claim that de = 2 according to the Imry-Ma [1] picture and those [3] who assert that de = 3 is a direct consequence of the dimensionality shift d -+ d 2 (a ferromagnetic spin model in a random magnetic field in dimension d has the same exponents as the same model without a random field in dimension d 2). The controversy is now beginning to be resolved because recent studies [4] of the properties of an interface in presence of a random field give more confidence in the fact that dp = 2 whereas some difficulties [5] have been found in the arguments which gave ~ -~ 2013 2. Anyhow, even if one accepts that df = 2, the properties of the RFIM right at d = 2 are not at all clear. The purpose of the present Letter is to give arguments in favour of a Kosterlitz-Thouless phase Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyslet:019840045012057700 L-578 JOURNAL DE PHYSIQUE LETTRES at zero temperature in dimension 2 for a random field Ising model. The model is defined by the Hamiltonian where J is the ferromagnetic interaction, L denotes the sum over nearest neighbours on a ij > square lattice, the spins (Ji are Ising spins and hi is the random field on the ith site. The peculiarity of this model lies in the probability distribution p(h) of the field hi : the random field can only be 0, + oo or oo : This model is in several aspects simpler than other random field models : for example it can be solved exactly in one dimension [6] at any temperature and for any value of p whereas other random field models remain unsolved in ID at finite temperature. Our study of the 2D case at 0 temperature consists of two steps. First, we calculate the correlation length çn(P) for p 1 on an infinite strip of finite width n. We obtain that ~(/?) increases linearly with n for large n and we calculate the constant A for small p. Then we use finite size scaling [7-9] which tells us that if çn(P) increases linearly with n, the 2D system is at criticality. We shall calculate the critical exponent ~ which characterizes the power law decrease of the average spin-spin correlation function at criticality by using a relation [8, 9] between the exponent r¡ and the amplitude A of equation (3). Let us start by calculating çn(P) for a strip of width n with periodic boundary conditions in the limit p 1. çn(P) is defined here by the exponential decrease of the average [10] correlation function (10 (1L > between two spins at distance L along the strip Consider a strip of width n with L + 1 columns numbered from 0 to L. Let us fix once and for all, the spins of column 0 to be +. We define F+ (L) the number of unsatisfied bonds of the strip in its ground state if we take all the spins of column L to be +. F+ (L) is simply related to the ground state energy. Similarly, let us denote by F_ (L) the number of unsatisfied bonds in the ground state if all the spins of column L are -. Since on each site between columns 1 and L 1, the field is randomly distributed according to (2), the difference A (L) = F + (L) F _ (L) has a certain probability distribution 6~(J). When L -~ oo, QL(d ) converges exponentially towards a limit probability distribution 6~(~) and this exponential convergence gives the correlation length ~(/?) L-579 ISING MODEL IN A RANDOM FIELD It is easy to see that (5) and (6) indeed define the same length Çn. The reason is that the correlation function (10 (1 L > is just given by We shall use (6) to calculate the correlation length çn(P) in the limit p 1. To do so, we need to make some remarks about the structure of the ground state. Because p 1, the distance between two spins in an infinite field is very large. The consequence is that the ground state is composed of a succession of positive and negative domains along the strip. The frontier between 2 successive domains is always a straight interface which cuts only n bonds a cross the strip. In addition to that, each spin with an infinite field which belongs to a domain with a wrong sign. costs only 4 unsatisfied bonds. So in the limit p 1 the ground state of an infinite strip is composed of a succession of very long domains separated by straight interfaces perpendicular to the strip and inside the domains there are isolated spins with an infinite field opposite to the sign of the domain. Keeping this structure in mind, one can write the recursion relation for J (L). With probability 1 np, all sites at column L have a zero field, therefore A (L + 1) = A (L). With probability A~/2, there is one site at column L with hi = + oo, and thus A (L + 1 ) = max (d (L) 4, n). This comes from the fact that F+ (L + 1) = F + (L) and F _ (L + 1) = min (F_(L) + 4, F+(L) + n) because the system has to choose the lowest energy between an isolated spin + in a domain which costs 4 unsatisfied bonds or a frontier between column L and column L + 1 which costs n unsatisfied bonds. Similarly with probability ~p/2, there is one site at column L with hi = oo and then J(L + 1) = min (A(L) + 4, n). We see that J makes a random walk constrained to remain between n and n and the only allowed values of A are ± (n 4 K) with K integer. One can easily write the recursion relation for Q~(J) : where e = 1 or 8 = 2013 1. From (8) and (9) we calculate the correlation length Çn given by (6). For an odd width n > 3, one finds