Collective spin resonance excitation in the gapped itinerant multipole hidden order phase of ${\mathrm{URu}}_2{\mathrm{Si}}_2$

An attractive proposal for the hidden order (HO) in the heavy electron compound ${\mathrm{URu}}_2{\mathrm{Si}}_2$ is an itinerant multipole order of high rank. It is due to the pairing of electrons and holes centered on the zone center and boundary, respectively, in states that have maximally different total angular momentum components. Due to the pairing with a commensurate zone boundary ordering vector the translational symmetry is broken and a HO quasiparticle gap opens below the transition temperature $T_{HO}$. Inelastic neutron scattering (INS) has demonstrated that for $T<T_{HO}$ the collective magnetic response is dominated by a sharp spin exciton resonance at the ordering vector Q that is reminiscent of spin exciton modes found inside the gap of unconventional superconductors and Kondo insulators. We use an effective two-orbital tight-binding model incorporating the crystalline-electric-field effect to derive closed expressions for quasiparticle bands reconstructed by the multipolar pairing terms. We show that the magnetic response calculated within that model exhibits the salient features of the resonance found in INS. We also use the calculated dynamical susceptibility to explain the low-temperature NMR relaxation rate.

Figure 1

(a) Fermi surface of the paramagnetic phase downfolded with Q to the st BZ of the HO phase consisting of two closed interpenetrating sheets. (b) FS cuts in the $k_z=0$ plane for paramagnetic (PM) phase ($\varphi =0$; blue dashed line) and HO phase ($\varphi =1$ with red solid line). Momentum range in (b) is given by $-\pi \le k_i\le \pi$. (c) Fermi surface sheets of ${\mathrm{URu}}_2{\mathrm{Si}}_2$ in the $E_{-}(1,1)$ HO phase ($\varphi =1$) in the st BZ. The FS is reconstructed in k-space regions connected by the nesting vector $\mathbf{Q}=(0,0,1)$ and breaks up into four smaller sheets without $C_4$ symmetry (for $|k_z|>0$).

Figure 2

(a) Density of states (DOS) showing the evolution of HO gap for small $\omega$. (b) Temperature dependence of DOS at the Fermi energy ($\omega =0$). Here we used a mean-field-type interpolation $\varphi \left(T\right)=\varphi tanh\left(1.76\sqrt{T_{HO}/T-1}\right)$ for the temperature dependence of the HO parameter.

Figure 3

Temperature evolution of the different tensor components of static susceptibility ${\widehat{\chi }}_0(q=0,\omega =0)$: (a,b) $xx$ and $zz$ components exhibit suppression below $T_{HO}$. Note the scale difference due to larger magnetic moment matrix elements for the $z$ direction. (c) $xy$ component vanishes above $T_{HO}$ because of tetragonal symmetry and becomes finite below due to fourfold symmetry breaking induced by the $E_{-}(1,1)$ HO phase.

Figure 4

Bare (noninteracting) dynamical susceptibility ${\chi }_0^{zz}(\mathbf{Q},\omega )$ at the bct $Z$ point $(1,0,0)$ (r.l.u.) below $(\varphi =1)$ and above $(\varphi =0)$ the HO temperature: (a) the imaginary part and (b) the real part. The singular behavior in the low-energy region is due to the HO gap opening and leads to the resonance appearance in Fig. 5.

Figure 5

(a) Collective magnetic $(zz,\perp )$ excitation spectrum: the imaginary part of the collective RPA susceptibility at the bct $Z$ point versus energy. The position of the pronounced peak determines the (almost $T$-independent) resonance energy ${\omega }_r$ in the HO phase. Here and in the following figures, $J_{\perp }\left(\mathbf{Q}\right)=0.156t_0=1.04$ meV. (b) Evolution of the imaginary part of the RPA susceptibility at resonance energy ${\omega }_r\left(T\right)$ in $q$ scan around bct $Z$ point for different temperatures.

Figure 6

(a) Contour plot of the imaginary part of the bare susceptibility ${\chi }_0^{zz} \perp$ to the tetragonal plane, (b) the RPA susceptibility ${\chi }_{RPA}^{\perp }$, (c) corresponding plots of the imaginary part of the bare and (d) the RPA spectrum of ($yy,\parallel$) dynamical susceptibility with a large $J_{\parallel }\left(\mathbf{Q}\right)=0.91t_0=5.8$ meV. For isotropic $J_{\parallel }\left(\mathbf{Q}\right)=J_{\perp }\left(\mathbf{Q}\right)$, no resonance appears in this channel. Here we set $T=3$ K in the HO phase with $\varphi =1$. The resulting resonance peak in (b),(d) is at ${\omega }_r=0.18$ meV. The dispersive V-shaped tails are remnants of the quasi-1D features of particle-hole continuum due to the nesting.

Figure 7

3D plot of the imaginary part of the perpendicular component of (a) the bare (noninteracting) ${\chi }_0^{zz}$ and (b) RPA susceptibility ${\chi }_{RPA}^{\perp }$ in hidden order phase with with $\varphi =1$ and $T=3$ K [same as Fig. 6]. In the latter the main resonance peak and V-shaped dispersive features at higher energies can be identified.

Figure 8

Temperature dependency of the normalized NMR relaxation rate ${\left(T_1\right)}_{\perp }$ at NMR resonance frequency ${\omega }_0\ll {\mathrm{\Delta }}_{HO}$. The relaxation rate ${\left(T_1\right)}_{\parallel }$ has a similar $T$ dependence but is larger due to the uniaxial anisotropy of the magnetic matrix elements.