Location of the Lee-Yang zeros and absence of phase transitions in some Ising spin systems

Joel L. Lebowitz, David Ruelle, Eugene R. Speer

Published 2012-04-02, updated 2012-08-06Version 2

We consider a class of Ising spin systems on a set \Lambda of sites. The sites are grouped into units with the property that each site belongs to either one or two units, and the total internal energy of the system is the sum of the energies of the individual units, which in turn depend only on the number of up spins in the unit. We show that under suitable conditions on these interactions none of the |\Lambda| Lee-Yang zeros in the complex z = exp{2\beta h} plane, where \beta is the inverse temperature and h the uniform magnetic field, touch the positive real axis, at least for large values of \beta. In some cases one obtains, in an appropriately taken \beta to infinity limit, a gas of hard objects on a set \Lambda'; the fugacity for the limiting system is a rescaling of z and the Lee-Yang zeros of the new partition function also avoid the positive real axis. For certain forms of the energies of the individual units the Lee-Yang zeros of both the finite- and zero-temperature systems lie on the negative real axis for all \beta. One zero-temperature limit of this type, for example, is a monomer-dimer system; our results thus generalize, to finite \beta, a well-known result of Heilmann and Lieb that the Lee-Yang zeros of monomer-dimer systems are real and negative.