Parasitic Small-Moment Antiferromagnetism and Nonlinear Coupling of Hidden Order and Antiferromagnetism in ${\mathrm{URu}}_2{\mathrm{Si}}_2$ Observed by Larmor Diffraction

We report for the first time simultaneous microscopic measurements of the lattice constants, the distribution of the lattice constants, and the antiferromagnetic moment in high-purity ${\mathrm{URu}}_2{\mathrm{Si}}_2$, combining Larmor and conventional neutron diffraction at low temperatures and pressures up to 18 kbar. Our data demonstrate quantitatively that the small moment in the hidden order (HO) of ${\mathrm{URu}}_2{\mathrm{Si}}_2$ is purely parasitic. The excellent experimental conditions we achieve allow us to resolve that the transition line between HO and large-moment antiferromagnetism (LMAF), which stabilizes under pressure, is intrinsically first order and ends in a bicritical point. Therefore, the HO and LMAF must have different symmetry, which supports exotic scenarios of the HO such as orbital currents, helicity order, or multipolar order.

Figure 1

(a) Schematic of Larmor diffraction [24, 25]; C1 through C4 are radio frequency spin flipper coils, $\overrightarrow{G}$ is the reciprocal lattice vector; ${\Theta }_B$ is the Bragg angle; AD is the polarization analyzer and detector. See text for details. (b) Typical variation of the polarization $P$ as a function of the total Larmor phase $\Phi$. (c) Pressure dependence of the distribution of lattice-constants for the $a$ and $c$ axis. With increasing pressure the width is essentially unchanged.

Figure 2

Temperature dependence of the lattice constants and thermal expansion at various pressures. The HO and LMAF transitions are indicated by empty and filled arrows, respectively. (a) Data for the $a$ axis; (b) thermal expansion of the $a$ axis derived from the data in (a). (c) Data for the $c$ axis; (d) thermal expansion of the $c$ axis derived from the data in (c).

Figure 3

Key features and pressure versus temperature phase diagram of ${\mathrm{URu}}_2{\mathrm{Si}}_2$. (a) Temperature dependence of the magnetic peak height at various pressures. (b) Pressure dependence of the low-temperature magnetic moment $m_s$. (c) Phase diagram based on Larmor diffraction and conventional magnetic diffraction data. The onset of LMAF and HO in our data is marked by full and empty symbols, respectively ($x$ marks a transition near base temperature). For better comparison data of $T_N$ and $T_0$ from Refs. 15, 17, 18, 19 are shown, where symbols with bright contours refer to $T_N$ and symbols with dark contours to $T_0$. (d) Specific-heat jump derived from thermal-expansion data via the Ehrenfest relation (full circles). Heat capacity data is taken from Refs. 1 (square) and 17 (empty circle).

Figure 4

Extended phase diagram of ${\mathrm{URu}}_2{\mathrm{Si}}_2$ based on the data presented here (diamonds) and signatures in resistivity (bright [34] and dark [17] circles along the PM-LMAF boundary). $T_{\max ,\rho }$ and $T_{\max ,M_c}$ denote coherence maxima in the resistivity [34] and magnetization [28], respectively. The HO might either mask a QCP (I) or replace LMAF (II) near quantum criticality.