Elastic properties of hidden order in ${\mathrm{URu}}_2{\mathrm{Si}}_2$ are reproduced by a staggered nematic

We develop a phenomenological mean-field theory describing the hidden-order phase in $\mathrm{U}{\mathrm{Ru}}_2{\mathrm{Si}}_2$ as a nematic of the $B_{1g}$ representation staggered along the $c$ axis. Several experimental features are reproduced by this theory: the topology of the temperature-pressure phase diagram, the response of the elastic modulus $(C_{11}-C_{12})/2$ above the transition at ambient pressure, and orthorhombic symmetry breaking in the high-pressure antiferromagnetic phase. In this scenario, hidden order is characterized by broken rotational symmetry that is modulated along the $c$ axis, the primary order of the high-pressure phase is an unmodulated nematic, and the triple point joining those two phases with the high-temperature paramagnetic phase is a Lifshitz point.

Figure 1

Phase diagrams for (a) $\mathrm{U}{\mathrm{Ru}}_2{\mathrm{Si}}_2$ from experiments (neglecting the superconducting phase) [1], (b) mean-field theory of a one-component ($B_{1g}$ or $B_{2g}$) Lifshitz point, and (c) mean-field theory of a two-component ($E_g$) Lifshitz point. Solid lines denote second-order transitions, while dashed lines denote first-order transitions. Later, when we fit the elastic moduli predictions for a $B_{1g}$ OP to data along the ambient pressure line, we will take $\mathrm{\Delta }\tilde{r}=\tilde{r}-{\tilde{r}}_c=a(T-T_c)$.

Figure 2

RUS measurements of the elastic moduli of $\mathrm{U}{\mathrm{Ru}}_2{\mathrm{Si}}_2$ at ambient pressure as a function of temperature from recent experiments [22] (blue solid line) alongside fits to theory (magenta dashed and black dashed lines). The solid yellow region shows the location of the HO phase. (a) $B_{2g}$ modulus data and a fit to the standard form [35]. (b) $B_{1g}$ modulus data and a fit to (18) (magenta dashed line) and a fit to (A21) (black dashed line). The fit gives $C_{B_{1g}}^0\simeq \left[71-\left(0.010{\text{K}}^{-1}\right)T\right]\text{GPa}$, $b^2/D_{\perp }{q}_{*}^4\simeq 6.28\text{GPa}$, and $b^2/a\simeq 1665\text{GPa}{\text{K}}^{-1}$. Addition of a quadratic term in $C_{B_{1g}}^0$ was not needed here for the fit [35]. (c) $B_{1g}$ modulus data and the fit of the bare $B_{1g}$ modulus. (d) $B_{1g}$ modulus data and the fits transformed by ${\left[C_{B_{1g}}^0(C_{B_{1g}}^0/C_{B_{1g}}-1)\right]}^{-1}$, which is predicted from (18) to equal $D_{\perp }{q}_{*}^4/b^2+a/b^2|T-T_c|$, e.g., an absolute-value function.