Pressure-induced rotational symmetry breaking in ${\mathrm{URu}}_2{\mathrm{Si}}_2$

Phase transitions and symmetry are intimately linked. Melting of ice, for example, restores translation invariance. The mysterious hidden order (HO) phase of ${\mathrm{URu}}_2{\mathrm{Si}}_2$ has, despite relentless research efforts, kept its symmetry breaking element intangible. Here, we present a high-resolution x-ray diffraction study of the ${\mathrm{URu}}_2{\mathrm{Si}}_2$ crystal structure as a function of hydrostatic pressure. Below a critical pressure threshold $p_c\approx 3$ kbar, no tetragonal lattice symmetry breaking is observed even below the HO transition $T_{\text{HO}}=17.5$ K. For $p>p_c$, however, a pressure-induced rotational symmetry breaking is identified with an onset temperatures $T_{\text{OR}}\sim 100$ K. The emergence of an orthorhombic phase is found and discussed in terms of an electronic nematic order that appears unrelated to the HO, but with possible relevance for the pressure-induced antiferromagnetic (AF) phase. Existing theories describe the HO and AF phases through an adiabatic continuity of a complex order parameter. Since none of these theories predicts a pressure-induced nematic order, our finding adds an additional symmetry breaking element to this long-standing problem.

Figure 1

Crystal structure and mosaicity of ${\mathrm{URu}}_2{\mathrm{Si}}_2$. (a) Ambient-pressure conventional unit cell of ${\mathrm{URu}}_2{\mathrm{Si}}_2$ with tetragonal $I4/mmm$ structure. (b) Transverse diffraction scans (sample rotation $\omega$) through $(h,h,0)$ Bragg peaks with $h$ being an integer as indicated. Blue (yellow) symbols indicate data from this work (Ref. [22]). The Lorentzian peak width ${\eta }_T$ results from a combination of crystal mosaicity and instrument resolution.

Figure 2

Pressure-induced orthorhombicity in ${\mathrm{URu}}_2{\mathrm{Si}}_2$. (a) Possible domain formation for $Immm$ and $Fmmm$ orthorhombic structures. (b) Corresponding Bragg peak along the orthorhombic $(h,\overline{h},0)$ and $(h,h,0)$ directions projected to a horizontal axis. The different domains lead to specific Bragg peak splittings as indicated by the color code. (c)–(j) Transverse and longitudinal scans through $(2,\overline{2},0)$ and (2, 2, 0) for $p<p_c$ and $p>p_c$ with $p_c=3$ kbar. Transverse and longitudinal scans are fitted with Lorentzian and Voigt profiles, respectively (solid lines). The transverse splitting of the (2, 2, 0) reflection is modeled by fitting two Lorentzians (solid and dashed lines).

Figure 3

Temperature and pressure dependence of orthorhombicity. (a), (b) Transverse scans through the (2, 2, 0) Bragg reflection for hydrostatic pressures and temperatures as indicated. Dashed gray lines indicate the results of fitting with two Lorentzian peaks and solid lines are their sum. (c) Hydrostatic-pressure and (d) temperature-dependent orthorhombic order parameter $\delta$, derived from the Lorentzian fits shown in (a) and (b). Solid lines are guides to the eye. Error bars on $\delta$ are set by the standard deviation of the relevant fitting parameters. Systematic errors of the pressure are indicated by horizontal bars.

Figure 4

Temperature-pressure phase diagram of ${\mathrm{URu}}_2{\mathrm{Si}}_2$. The hidden order, long-range antiferromagnet order, and superconducting (SC) phases are displayed with gray, yellow, and blue shadings, respectively. In addition, the tetragonal-to-orthorhombic phase boundary is indicated by the hashed area. The emergence of orthorhombicity is given both as a function of pressure and in-plane lattice parameter. By contrast, HO, LRAF, and SC are indicated only as a function of pressure [14]. Notice that the in-plane lattice parameter $a$ is an absolute scale whereas the pressure axis remains relative without an ambient-pressure lattice parameter $a_0$ determination. The orthorhombic transition (circular markers) was only measured for a subset of pressures applied. The triangular marker indicates the orthorhombic $Fmmm$ onset found by Tonegawa et al. [12]. Due to the logarithmic temperature scale, the absence of orthorhombicity at pressures of 0 and 1.3 kbar is not displayed.