Theory of electronic nematic criticality constrained by elastic compatibility

The defining property of electronic nematicity -- the spontaneous breaking of rotational symmetry -- implies an unavoidable coupling between the nematic order parameter and elastic strain fields, known as nemato-elasticity. While both quantities are rank-2 tensors, the strain tensor is constrained through the Saint Venant compatibility relations. These three coupled second-order partial differential equations arise from the lattice displacement vector's role as a potential field, and they reflect the underlying gauge invariance of geometric deformations which are violated only in the presence of crystalline defects. In this work, we develop a theory of nemato-elasticity that incorporates elastic compatibility explicitly through a co-rotating helical basis. With our formalism, we show elasticity bestows tensor compatibility upon the nematic order parameter by suppressing incompatible nematic fluctuations. As a result, nemato-elasticity is markedly different from bare nematicity. In ideal media devoid of defects, we show the suppression of incompatible nematicity underlies direction-selective criticality, even in the absence of crystalline anisotropy. In systems with defects, meanwhile, we show that elastic pinning fields emanate from quenched defects, generating random longitudinal and transverse conjugate fields for the local nematic order parameter. The coexistence of direction-selective nematic criticality with pinning effects from random fields is explained within our theory from the transformation to the helical basis, implying that local experimental probes of nematicity will be influenced by a linear -- but nonlocal -- combination of long-ranged and short-ranged helical nematic modes. Because the compatibility relations are gauge constraints endowed in the isotropic medium, our results constitute universal features of nemato-elastic criticality present in all crystalline systems.