We present a theoretical method for deriving the stress tensor and elastic response of ordered systems within a Ginzburg-Landau-type density field theory in the linear regime. This is based on spatially coarse graining the microscopic stress which is determined by the variation of a free energy with respect to mass displacements. We find simple expressions for the stress tensor for phase field crystal models for different crystal symmetries in two and three dimensions. Using tetradic product sums of reciprocal lattice vectors, we calculate elastic constants and show that they are directly related to the symmetries of the reciprocal lattices. We also show that except for bcc lattices there are regions of model parameters for which the elastic response is isotropic. The predicted elastic stress-strain curves are verified by numerical strain-controlled bulk and shear deformations. Since the method is independent of a reference state, it extends also to defected crystals. We exemplify this by considering an edge and screw dislocation in the simple cubic lattice.

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We derive a phase field crystal model that couples the diffusive evolution of a microscopic structure with the fast dynamics of a macroscopic velocity field, explicitly accounting for the relaxation of elastic excitations. This model captures better than previous formulations the dynamics of complex interfaces and dislocations in single crystals as well as grain boundary migration in poly-crystals where the long-range elastic field is properly relaxed. The proposed model features a diffusivity that depends non-linearly on the local phase. It induces more localized interfaces between a disordered phase (liquid-like) and an ordered phase, e.g., stripes or crystal lattices. For stripes, the interface dynamics is shown to be strongly anisotropic. We also show that the model is able to evolve the classical PFC at mechanical equilibrium. However, in contrast to previous approaches, it is not restricted to a single-crystal configuration or small distortions from a fixed reference lattice. To showcase the capabilities of the model, we consider a few examples, from the annihilation of dislocation loops in a single crystal at mechanical equilibrium to the relaxation of a microstructure including crystalline domains with different orientations and grain boundaries. During the self-annihilation of a mixed type dislocation loop (i.e. not shear or prismatic), long-range elastic effects cause the loop to move out of plane before the annihilation event.

We use the phase field crystal model to study the nucleation of edge dislocations under an applied stress field. Under quasi-static loading of a perfect lattice, a dislocation dipole nucleates (due to Burgers vector conservation) with Burgers vector and slip plane chosen according to the lattice orientation with respect to the applied stress. The phase field crystal correctly accounts for elastic energy storage prior to nucleation, and for dissipative relaxation during the nucleation event. We show that both a lattice incompatibility field and the resolved stresses are sensitive diagnostics of the fluctuations prior to nucleation, of the location of the nucleation event, and of the Burgers vector and slip direction of the dislocations that will be nucleated upon further increases in the stress. A direct calculation of the phase field energy accurately correlates with the nucleation event as signaled by the lattice incompatibility field. We finally show that a Schmid-like criterion predicts the critical nucleation stress.

Topological defects and smooth excitations determine the properties of systems showing collective order. We introduce a generic non-singular field theory that comprehensively describes defects and excitations in systems with O(n) broken rotational symmetry. Within this formalism, we explore fast events, such as defect nucleation/annihilation and dynamical phase transitions where the interplay between topological defects and non-linear excitations is particularly important. To highlight its versatility, we apply this formalism in the context of Bose-Einstein condensates, active nematics, and crystal lattices.