VOLUME 53, NUMBER 18 PHYSICAL REVIEW LETTERS 29 OCTOBER 1984 Lower Critical Dimension of the Random-Field Ising Model John Z. Imbrie Lyman Laboratory, Harvard University, Cambridge, Massachusetts 02138 (Received 27 August 1984) A new argument is given for a lower critical dimension d/= 2 for the Ising model in a ran- dom magnetic field. It forms the basis for a proof that the three-dimensional model exhibits long-range order at zero temperature and small disorder. This settles the controversy between the values d, = 2 and d, = 3. PACS numbers: 75.10.Hk In this paper I present a new argument that the lower critical dimension, d[, for the Ising model in a random magnetic field is 2. Previous heuristic pro- posals for d 1 = 2 and also for d 1 = 3 have been given. Both cases have adherents. The question hinges on whether or not long-range order occurs in three dimensions, at low or zero temperature in the pres- ence of a small, random magnetic field. This physics issue has been resolved by the find- ing of a new, exact formula for the ground state energy-see (2) below. This formula is used else- where! to prove that d[ 2. The formula for the energy has two important properties: (1) The ener- gy is expressed as a sum of local functions of the magnetic fields in various regions, so that it is amenable to statistical analysis. (2) The sizes of the regions vary through a succession of increasing length scales (associated with an inductive analysis of the ground state), and the probability distribu- tions of the functions scale accordingly. Specifically, it is shown that if the disorder is small, the model in dimension d = 3 exhibits long- range order at zero temperature. At the conceptual level, this argument leads to the same conclusion for low temperatures. Indeed, I expect that a proof for low temperatures will be possible by combining the methods described here with the expansion methods developed for disordered systems by Frohlich and Imbrie. 2 The model is defined by the Hamiltonian for a finite subset A C Z3 with plus boundary conditions: H+ (A) = I +(1- (Tj(Tj) - I +hj(Tj' (i,j) i EA Here O"j= ± 1 for i E Z3, O"j= 1 for i A, and (i,j) denotes a nearest-neighbor pair. The magnetic fields h j are taken to be independent random vari- ables with a common Gaussian distribution with mean zero and width (hl) 1/2 = € (a measure of the disorder). The angular brackets indicate an average over the magnetic fields. We write peE) for the probability of the event E. Let us write (Tmio(A +) for the spin configuration of minimum energy H+ (A). It is unique, with probability 1. At temperature T = 0, the question of long-range order reduces to properties of (Tmin(A +) as A increases to Z3. We have long-range order if (T r in (A +) is more often + 1 than -1 for some fixed i E Z3, and if the disparity is uniform as A in- creases to Z3. This is the content of the following theorem, proved in Ref. 1. Theorem.-Let An be a sequence of cubes cen- tered at the origin 0 E Z3. There exists a constant C > 0 such that for any i E Z3 and any n, P«(TFiO(A:) = -1) exp( C/e 2 ). (1) The limit limn_ooO"Fio(An+)=(Trin exists with probability 1 and satisfies the same bound. Other results of Ref. 1 include a proof of near- exponential decay of correlations between ground- state spins: I((TWin(T jin) - «(TWin) «(Tjin) I exp{- cj exp[ - c' (In lnj) 2]E - 2} . In Ref. 2 it is shown that the model has no long- range order for large e. Hence there is a T = 0 tran- sition from long-range order to absence of long- range order as the disorder parameter € increases. I expect that my methods will be useful in other problems, for example in studying the interface in random-field models. A reasonable conjecture is that the interface in d dimensions is rigid for d > 3, as a result of the similarity with the (d - 1)- dimensional bulk problems studied here. The continuum interface may of course be much rougher. 3 ,4 We recall that the lower critical dimension is de- fined as the dimension above which long-range or- der occurs. Recent numerical workS has indicated ordering in three dimensions, which would imply d 1 = 2. However, neither the domain-wall argument for d[ = 2 nor the dimensional-reduction argument 6 for d l = 3 has been universally accepted. Domain walls are defined as surfaces separating regions of constant (T. According to the domain-wall argu- © 1984 The American Physical Society 1747