Low temperature expansion of the gonihedric Ising model

We investigate a model of closed $(d-1)$-dimensional soft-self-avoiding random surfaces on a $d$-dimensional cubic lattice. The energy of a surface configuration is given by $E=J(n_{2}+4k n_{4})$, where $n_{2}$ is the number of edges, where two plaquettes meet at a right angle and $n_{4}$ is the number of edges, where 4 plaquettes meet. This model can be represented as a $\Z_{2}$-spin system with ferromagnetic nearest-neighbour-, antiferromagnetic next-nearest-neighbour- and plaquette-interaction. It corresponds to a special case of a general class of spin systems introduced by Wegner and Savvidy. Since there is no term proportional to the surface area, the bare surface tension of the model vanishes, in contrast to the ordinary Ising model. By a suitable adaption of Peierls argument, we prove the existence of infinitely many ordered low temperature phases for the case $k=0$. A low temperature expansion of the free energy in 3 dimensions up to order $x^{38}$ ($x={e}^{-\beta J}$) shows, that for $k>0$ only the ferromagnetic low temperature phases remain stable. An analysis of low temperature expansions up to order $x^{44}$ for the magnetization, susceptibility and specific heat in 3 dimensions yields critical exponents, which are in agreement with previous results.