VOLUME 30, NUMBER 16 PHYSICAL REVIEW LETTERS 16 APRIL 1973 Variational Approximation to the Ising/Model in a Magnetic Field* S. Krinsky and C. Tracy Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook, New York 11790 and M. Blumet Department of Physics, State University of New York at Stony Brook, Stony Brook, New York 11790, and Brookhaven National Laboratory, Upton, New York 11973 (Received 15 January 1973) A variational approximation to the Ising model in a magnetic field (T^ T c ), based on the zero-field solution, yields an upper bound on the free energy, an approximate equation of state, and a lower bound on the magnetization, all having the correct critical indices. In two dimensions, these are numerically evaluated, and compared with the results of series expansions. We have performed a variational calculation of the magnetization of the ferromagnetic Ising mod- el in an external magnetic field, for T ^T c . Our approach is based on the solution to the Ising model in zero field, so the critical indices a' and j3 are put in exactly. The resulting approxi- mate equation of state has the interesting proper- ty that the indices y' and 6 are related to a' and ]8 by the linear scaling relations. 1 The calculation makes use of the variational principle 2 for F(H) = - /? -1 In Tr exp(- $H), the free energy, which states that F(H)^$=F(H 0 ) + ?r[(H-H n )exp(-$H 0 )] Trexp(-0# o ) (1) where H 0 is arbitrary and J3 = (kT) _1 . The Hamil- tonian of the ferromagnetic Ising model is N H = - T/ViVj-hT/Vi, (2) where (ij) indicates nearest neighbors, cr i =±l, and N is the total number of spins. The exchange constant has been set equal to unity. A standard procedure for the approximate calculation of the free energy is to choose H 0 such that all the trac- es in Eq. (1) can be performed. If H 0 depends on one or more parameters, these are chosen to minimize $, thus providing the best free energy. The mean-field approximation is obtained by tak- ing i = l and determining h' by minimizing $. 3 Mean-field theory does not provide a good quantitative de- scription of the Ising model in the critical region. The mean-field critical indices satisfy the linear scaling relations, but they do not have the cor- rect values. The procedure we adopt is applicable whenev- er the zero-field free energy and the spontane- ous magnetization are known. In mean-field the- ory, the exchange interaction is approximated by an average field of the neighbors, while in our calculation the presence of an external field is approximated by a zero-field system with a larg- er effective exchange. We take 4 (3) and determine J~Jifi,h) by minimizing <3>. Defin- ing/(J3, h) =F/N, il><fi,h,J)=$/N, andM = -d//a/z, we find 4 /(/3, h) < ip@, h, J) by substituting (2) and (3) into (1). Choosing J to minimize the right- hand side of this inequality, we obtain the varia- tional free energy per spin, cp@, h) = ip0, h, J$, h)), and the variational magnetization M^ = - Bcp/dh. This procedure can be applied to more general problems, e.g., the Heisenberg model, by vary- ing a coupling constant multiplying the zero-field piece of the Hamiltonian. Provided we assume knowledge about the zero-field solution, we ob- tain approximate information about the model in a field. We obtain the following results for T ^T c : (i) As h~0, the variational free energy and magnetiza- tion continuously approach the true values. We denote the true values of the critical indices by a', 0, y', and 6. The critical indices of cpifi,h) are a', p, y / , and 6^, where y / , 6^ are given by the linear scaling relations 1 a' + 2j3 +y/ = 2, 06^ = 2-a'-j3. (ii) We find the following lower bound on the 750