Theory of Electronic Nematic Criticality Constrained by Elastic Compatibility

Abstract

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.

Figures (10)

• Figure 1: Two-dimensional elastic compatibility relation enforced in momentum space. Each panel shows the dilatation, $\varepsilon_{xx} + \varepsilon_{yy} = \partial_xu_x + \partial_yu_y$ induced by a symmetry-breaking static strain wave with well-defined wavevector, $\boldsymbol{q}$. The direction-dependence of these constraints are present in systems of any crystal symmetry. The colorbar represents the form factors that relate the symmetry-breaking (a) deviatoric, $\varepsilon_{x^2 - y^2}=\varepsilon_{xx} - \varepsilon_{yy}$, and the (b) shear, $\varepsilon_{2xy} = 2\varepsilon_{xy}$, strains to the symmetry-preserving dilatations, as enforced by the single compatibility condition in 2D.
• Figure 2: Linear transformation from the $d$-orbital nematic basis, $\boldsymbol{\varphi}$, into the helical basis, $\boldsymbol{\Phi}$, as functions of the wavevector, $\boldsymbol{q}$. The $\Phi_{1,2,4}$ panels are also shown in Ref. ShortPaper. Each panel visualizes the square-amplitude $(\hat{Q}_{a,b})^2$ to determine the contribution to the $\Phi_a$ helical order parameter from the $\varphi_b$$d$-orbital order parameter. For $\Phi_3$, the $\varphi_{z^2}$ and $\varphi_{x^2 - y^2}$ ($\varphi_{2xy}$) amplitudes have coincident maximum magnitude along, for example, the $[101]$ ($[111]$) directions.
• Figure 3: Elastic deformations of a lattice in the presence of a compatible planar $\varphi_{2xy}$ nematic order parameter with well-defined momentum, $\boldsymbol{q}$. In all figures, the modulation is sinusoidal, with identical amplitudes that are greatly exaggerated for visualization purposes. The equilibrium positions are indicated by the gray points, and the displaced locations are shown in blue and orange, obtained from solving the nemato-elastic equations of state, Eq. (\ref{['eq:elastic_equations_of_state']}). The contrast of blue and orange is solely to emphasize displacements of volume elements of linear size $\ell$. Top-down (a) and side (b) views of the strained three-dimensional isotropic medium undergoing a critical $\varphi_{2xy}$ fluctuation mode, proportional to the helical $\Phi_2$ nematic amplitude with $\hat{q} \propto [010]$. Top-down (c) and side (d) views of a non-critical $\varphi_{2xy}$ fluctuation mode, proportional to the helical $\Phi_1$ nematic amplitude with $\hat{q}\propto [110]$. The suppression of the $\varphi_{2xy}$ wave in (c,d) is apparent from the accompanying dilatation strain, whereas the critical wave in (a,b) does not induce local-volume changes.
• Figure 4: Schematic of universal direction-selective nematic criticality induced by elastic compatibility. Each panel shows different nematic momenta directions, $\hat{q}$, as the red arrow. The orange and blue ellipsoids represent the possible critical $(\Phi_2, \Phi_3)$ helical nematic order parameters for a given momentum direction. The semi-major axes of the ellipsoids are oriented parallel to the critical directors in the $d$-orbital basis selected by the specified momenta. Note that the critical director can point in any direction along the cone spanned by the blue and orange ellipsoids, shown explicitly in (a). Nematic directors that lie in the plane transverse to the momentum (red) are incompatible $(\Phi_4, \Phi_5)$. (a) When the critical momentum is polar, $\hat{q} \propto [001]$, the $(\varphi_{2yz},\varphi_{2zx})$$d$-orbital order parameters span the critical nematic manifold, with directors aligned along the $[011]$ and $[101]$ axes, respectively. (b,c) When the critical momentum is in the plane, there is an in-plane director and an out-of-plane director. (b) For $\hat{q}\propto [100]$ (or $[010]$), the in-plane critical nematic order parameter is $\varphi_{2xy}$. (c) When $\hat{q}\propto [110]$ (or $[1\bar{1}0]$), the critical in-plane nematic order parameter is $\varphi_{x^2 - y^2}$. (d) By rotating the critical momentum in the full 3D space, while the corresponding nematic orientations change in the $d$-orbital basis, it is always the $(\Phi_2, \Phi_3)$ helical doublet which spans the critical manifold.
• Figure 5: Defect-gauge transformation acting on a straight edge dislocation. In each figure, the dislocation core is the green star. The Burgers vector is represented by the red vertical arrows and the crystal planes in the dislocated medium are numbered. (a) The Volterra surface is taken along the leftward horizontal axis, $S$, creating a discontinuous upward jump in the 3, 2, and 1, crystal planes, as read from right-to-left. (b) An equivalent Volterra surface is drawn vertically, $S^\prime$, now leading to a downward jump of planes $3$-$7$, read from right-to-left. Both $S$ and $S^\prime$ are anchored to the dislocation core (green star) and have the same Burgers vector. (c) The appropriate defect-gauge transformation, as described in the text, amounts to a redefinition of the crystal planes around the defect core, effectively displacing all sites within the volume enclosed by $S$ and $S^\prime$. This physically equivalent redefinition of the displacement vector exemplifies defect gauge redundancy.