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Statistical mechanics of phase transitions in elastic media with vanishing thermal expansion

We consider the elastic theory for Ising transitions in an isotropic elastic medium in the zero thermal expansion (ZTE) limit. We use this theory to study the nature of the fluctuations in the system near the second phase transitions at $T_c$ in the ZTE limit given by $dT_c/dV=0$, where $V$ is the system volume, and explore anomalous elasticity. Allowing for the local strain to couple {\em asymmetrically} with the states of the order parameter, we uncover the dramatic effects of these couplings on the fluctuations of the local displacements near $T_c$, and also on the nature of the phase transition itself. Near second order phase transitions and with weak asymmetry in the order parameter - strain couplings, the variance of the displacement fluctuations in two dimensions scale with the system size $L$ in a universal fashion as $[\ln (L/a_0)]^{2/3}$; $a_0$ is a small-scale cutoff. Likewise, the correlation functions of the difference of the local displacements at two different points separated by $r$ scale as $[\ln (r/a_0)]^{2/3}$ for large $r$. For stronger asymmetry, this variance diverges as $L$ exceeds beyond a (nonuniversal) size, determined by the model parameters, signaling a transition to a phase with only short range order or the loss of the positional order of the elastic medium. At dimensions higher than two, for sufficiently weak selectivity, the variance of the displacement fluctuations is $L$-independent corresponding to long range order. However, if the selectivity parameters rise beyond a dimension-dependent threshold values, again the positional order is lost with a concomitant transition to a phase with short range order. Large values of the order parameter - strain couplings can turn the phase transition into a first order as well.