Analogy Between the “Hidden Order” and the Orbital Antiferromagnetism in ${\mathrm{URu}}_{2-x}{\mathrm{Fe}}_x{\mathrm{Si}}_2$

We study ${\mathrm{URu}}_{2-x}{\mathrm{Fe}}_x{\mathrm{Si}}_2$, in which two types of staggered phases compete at low temperature as the iron concentration $x$ is varied: the nonmagnetic “hidden order” (HO) phase below the critical concentration $x_c$, and unconventional antiferromagnetic (AFM) phase above $x_c$. By using polarization resolved Raman spectroscopy, we detect a collective mode of pseudovectorlike $A_{2g}$ symmetry whose energy continuously evolves with increasing $x$; it monotonically decreases in the HO phase until it vanishes at $x=x_c$, and then reappears with increasing energy in the AFM phase. The mode’s evolution provides direct evidence for a unified order parameter for both nonmagnetic and magnetic phases arising from the orbital degrees-of-freedom of the uranium-$5f$ electrons.

Figure 1

(a) The upper panel shows the phase diagram of ${\mathrm{URu}}_2{\mathrm{Si}}_2$ system, where the black lines show the phase boundaries. The measurements on the iron substituted ${\mathrm{URu}}_{2-x}{\mathrm{Fe}}_x{\mathrm{Si}}_2$ crystals from neutron diffraction [22] (blue triangle), electrical resistivity [2] (green square), magnetic susceptibility [2] (purple triangle), and heat capacity [3] (yellow diamond), are overlaid with the neutron diffraction results for ${\mathrm{URu}}_2{\mathrm{Si}}_2$ under hydrostatic pressure [4] (open square) to show the similarity between the two tuning parameters. The lower panel shows the dependence of the $A_{2g}$ collective mode energy on the Fe concentration, $x$ [Fig. 2]. At the critical concentration, $x=0.1$, the mode maximum is below the accessible energy cutoff. Therefore, the data point is placed at zero energy, with the error bar reflecting the instrumental cutoff. (b)–(g) Schematics of the Ginzburg-Landau free energy in Eq. (1) at various special points in the phase diagram [solid gray circles in (a)]. ${\psi }_{\mathrm{HO}}$ and ${\psi }_{\mathrm{AF}\mathrm{M}}$ are the real and imaginary part of the hexadecapole order parameter, respectively [26, 27].

Figure 2

Low temperature Raman response in the $A_{2g}$ symmetry channel, ${\chi }_{A2g}^{\prime \prime }(\omega ,T)$ [28]. The upper panels show intensity plots, where the intensities are color coded in logarithmic scale. The lower panels show the spectra at about half the transition temperature to emphasize the collective mode, where the error bars represent one standard deviation, and the red solid lines are guides to the eye. The energies of this mode as function of the Fe concentration $x$ are shown in Fig. 1.

Figure 3

The static Raman susceptibility in the $A_{2g}$ symmetry channel (open squares) ${\chi }_{A2g}(0,T)$, compared with the magnetic susceptibility with field applied along the $c$ axis [3] (solid line).

Figure 4

The crystal structure of ${\mathrm{URu}}_{2-x}{\mathrm{Fe}}_x{\mathrm{Si}}_2$ in (a) the HO and (b) the LMAF phases. Illustrations capturing the symmetries of the charge distributions of the ground state wave functions are placed at the uranium atomic sites. On the right are illustrations showing the in-plane structures of the wave functions. In the HO phase, the crystal field state with the lowest energy has $A_{2g}$ symmetry with 8 nodal lines, $|A_{2g}⟩$, which mixes with the first excited state with $A_{1g}$ symmetry, $|A_{1g}⟩$, to form the local wave functions in the HO phase, $|{\mathrm{HO}}^{\pm }⟩\approx \cos \theta |A_{2g}⟩\pm \sin \theta |A_{1g}⟩$. In the LMAF phase, the ordering of the crystal field states switches, and the new wave functions in the LMAF phase are, $|{\mathrm{AF}\mathrm{M}}^{\pm }⟩\approx \cos {\theta }^{\prime }|A_{1g}⟩\pm i\sin {\theta }^{\prime }|A_{2g}⟩$. Here, $\theta \equiv \arcsin (V/{\omega }_0)$ and ${\theta }^{\prime }\equiv \arcsin (V^{\prime }/{\omega }_0)$, respectively. ${\omega }_0$ is the splitting between the lowest lying crystal field states in the minimal model. $V$ and $V^{\prime }$ are the order parameter strength in the HO and LMAF phases, respectively.