IC/96/144 United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS SEMI-LOCAL INVARIANCE IN ISING MODELS WITH MULTI-SPIN INTERACTION

We examine implications of semi-local invariance in Ising models with multispin interaction. In ergodic models all spin-spin correlation functions vanish and the local symmetry is the same as in locally gauge-invariant models. The d = 3 model with four-spin interaction is nonergodic at low temperature but the magnetic symmetry remains unbroken. The d = 3 model with eight-spin interaction is ergodic but undergoes the phase transition and most likely its low-temperature phase is characterized by a nonlocal order parameter. MIRAMARE TRIESTE August 1996 1 Permanent address: Magnetism Theory Division, Department of Physics, A. Mickiewicz University, Ul. Matejki 48/49, Poznah, Poland. While the physics of Ising models with two-spin interaction is relatively well understood, much less is known about the behaviour of models with multi-spin interactions. When the Ising model is considered as a lattice gas then such terms reflect the multi-atom interactions and in many systems they are likely to play a very important role. Only in very few cases models with multi-spin terms are exactly solvable[?]. In these particular cases the multi-spin terms are responsible for the nonuniversal critical behaviour. In the present Letter we consider a certain class of Ising models defined on the cartesian d-dimensional lattice. First, let us consider the d = 2 model with four-spin interaction. This model is described by the Hamiltonian: H = -JY^SiSjSkSl (1) where S = 1/2 spin operators Si = ± are placed at sites of the square lattice of the linear size L. Summation in (??) is performed over elementary plaquettes formed by the sites i,j, k, l. Results obtained in the present Lettter are related to the following property: Hamiltonian (??) is invariant with respect to a semi-local group of transformations which flip the entire row (or column) of spins. This group of transformations in the following will be referred to as G. Immediate proof follows from the structure of the Hamiltonian (??). Now, let us calculate the correlation function < SiSj >: < SiSj >= E SiSje-W/Z, Z=J2 e"'*, (2) {Sk} {Sk} where i =£ j and β = y^. To calculate (??) let us notice that there must be a row or a column which contains the site i but does not contain the site j . With each configuration {Sk} in which Si = 1 let us associate a configuration {Sk}' which is obtained from {Sk} by flipping all spins along this row or column. Since the contributions to < SiSj > coming from {Sk} cancel with those from {Sk}' (both configurations have the same energy) we obtain that < SiSj > = 0. Since all spin-spin correlation functions vanish we obtain that the system is a perfect paramagnet with the susceptibility χ = k1T a n d n o spontaneous magnetization