Continuous elastic phase transitions in pure and disordered crystals

Abstract

We review the theory of second-order (ferro-)elastic phase transitions, where the order parameter consists of a certain linear combination of strain tensor components, and the accompanying soft mode is an acoustic phonon. In threedimensional crystals, the softening can occur in one-or two-dimensional soft sectors. The ensuing anisotropy reduces the effect of fluctuations, rendering the critical behaviour of these systems classical for a one-dimensional soft sector, and classical with logarithmic corrections in case of a two-dimensional soft sector. The dynamical critical exponent is z = 2, and as a consequence the sound velocity vanishes as c s ∝ |T − T c | 1/2 , while the phonon damping coefficient is essentially temperature-independent. Even if the elastic phase transition is driven by the softening of an optical mode linearly coupled to a transverse acoustic phonon, the critical exponents retain their mean-field values. Disorder may lead to a variety of precursor effects and modified critical behaviour. Defects that locally soften the crystal may induce the phenomenon of local order parameter condensation. When the correlation length of the pure system exceeds the average defect separation n −1/3 D , a disorder-induced phase transition to a state with non-zero average order parameter can occur at a temperature T c (n D ) well above the transition temperature T 0 c of the pure crystal. Near T 0 c , the order-parameter curve, susceptibility, and specific heat appear rounded. For T < T c (n D ) the spatial inhomogeneity induces a static central peak with finite q width in the scattering cross section, accompanied by a dynamical component that is confined to the very vicinity of the disorder-induced phase transition. † Present address:

Critical elasticity at zero and finite temperature

“…For classical phase transitions, the coupling of critical fluctuations to the elastic degrees of freedom, following early work by Rice [17] and Domb [18], were extensively studied in the 1970ies [19,20,21,22,23,24,25,26,27,28,29,30,31]. Some of the results presented here are straightforward generalizations of these previous works to the case of quantum phase transitions.…”

“…Such an approach was chosen in Ref. [32] where the quantum critical signatures of continuous elastic quantum phase transitions were discussed building on previous work by Cowley [26] and Schwabl and collaborators [27,28,29,30]. At such an instability an eigenvalue of the elastic constant matrix C ijkl vanishes.…”

Driven surface diffusion with detailed balance and elastic phase transitions

“…In this section, we essentially follow the early work of Cowley [17], and Folk, Iro, and Schwabl (FIS) [18,19]. For background information on elastic phase transitions, we refer to these original papers as well as reviews by Bruce and Cowley [20], Rao and Rao [21], and Schwabl and Täuber [22].…”

“…The crystallographic unit cell undergoes an elastic deformation with a certain linear combination of components of the strain tensor as an order parameter. In the case that no third order invariants of the order parameter are possible (or vanish by chance in the cubic case), the transition to the structurally distorted crystal is continuous, and the order parameter can be reduced to the spatial derivatives of the displacement field u along the crystallographic symmetry axis [17][18][19]22]. Augmenting the stretching energy with stabilizing bending terms and then reducing it to its parts that are relevant in the RG sense, we set up a generalized elastic energy that is of the same form as the aforementioned quasistatic Hamiltonian H with the displacement field u(z, r ⊥ ) taking on the role of the height field h(z, r ⊥ ).…”