Impact of elastic compatibility relations in electronic nematic phenomena

A defining property of electronic nematic order is its inevitable coupling to inter-unit-cell atomic displacements of the crystal, whose quantization gives rise to acoustic phonons. In the long wavelength regime, the continuum theory of elasticity captures the low-energy physics of such macroscopic deformations of the lattice. As a result, long-wavelength nematic phenomena must inherit the same constraints of classical elasticity as those imposed by the Saint Venant Compatibility Relations (SVCR). These SVCR arise since only three out of the six strain components are independent. Here, we show that the SVCR induce direction-selective criticality in electronic nematic transitions, a behavior that was previously attributed to acoustic phonons. We exemplify this direction-selective criticality in two- and three-dimensional electronic nematics based on energetic considerations of the three independent classical elastic strain fields permitted by the SVCR. Since our result is founded on tensor compatibility only, it also applies to other transitions described by an order parameter bilinearly coupled to fluctuating compatible tensor fields.