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 nematoelasticity. While both quantities are rank-2 tensors, the strain tensor is constrained through the Saint Venant compatibility relations. These constraints, which take the form of three coupled second-order partial differential equations, arise from the lattice displacement vector's role as the potential field. The compatibility relations 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 nematoelasticity that incorporates elastic compatibility explicitly through a corotating helical basis. Within our formalism, we show that elasticity bestows tensor compatibility upon the electronic nematic order parameter by suppressing incompatible electronic nematic fluctuations. As a result, nematoelasticity is markedly different from bare nematicity. In ideal media devoid of defects, we show that the suppression of incompatible nematicity underlies direction-selective criticality, even in the absence of crystalline anisotropy. In realistic systems with defects, meanwhile, we show that elastic pinning fields emanate from distributions of quenched defects, generating random longitudinal and transverse conjugate fields experienced by the local nematic order parameter. The coexistence of direction-selective nematic criticality with pinning effects from random fields is naturally 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- and short-ranged helical nematic modes. Because the compatibility relations are gauge constraints endowed in the isotropic medium, our results constitute universal features of nematoelastic criticality present in all crystalline systems.

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• Nematic order
• Nematic phase transition
• Strongly correlated systems
• Critical phenomena
• Effective field theory
• Landau-Ginzburg theory