Stiffening or softening of elastic media: Anomalous elasticity near phase transitions

TL;DR

This paper develops a comprehensive theory describing how elastic properties of isotropic media change anomalously near Ising phase transitions, revealing conditions for stiffening, destabilization, and first-order transitions due to spin-lattice couplings.

Contribution

It introduces a minimal model linking spin-lattice interactions to anomalous elasticity near phase transitions, highlighting new behaviors in two and three dimensions.

Findings

Elastic media can significantly stiffen or destabilize near phase transitions.

In two dimensions, elasticity shows logarithmic enhancement of positional order.

In three dimensions, certain couplings lead to long-range positional order and first-order transitions.

Abstract

We present the general theory of Ising transitions in isotropic elastic media with vanishing thermal expansion. By constructing a minimal model with appropriate spin-lattice couplings, we show that in two dimensions near a continuous transition the elasticity is anomalous in unusual ways: the system either significantly stiffens with a hitherto unknown unique positional order logarithmically stronger than quasi-long range order, or, as the inversion-asymmetry of the order parameter in its coupling with strain increases, it destabilizes when system size $L$ exceeds a finite threshold. At three dimensions, stronger inversion-asymmetric couplings induce instability to the long-range positional order for all $L$. Sufficiently strong order parameter-displacement couplings can also turn the phase transition first order at all dimensions, concomitant with finite jumps in the elastic modulii across the transition. Our theory establishes a {\em one-to-one correspondence} between the order of the phase transitions and anomalous elasticity near the transitions.