Hidden-order symmetry and superconductivity in ${\mathrm{URu}}_2{\mathrm{Si}}_2$ investigated by quasiparticle interference

The hidden order (HO) in ${\mathrm{URu}}_2{\mathrm{Si}}_2$ has been determined as a high-rank multipole formed by itinerant $5f$ electrons with distinct orbital structure imposed by the crystalline electric field. Because this can lead to a considerable number of different multipoles, it is of great importance to use microscopic techniques that are sensitive to their subtle physical differences. Here, we investigate whether the quasiparticle interference (QPI) method can distinguish between the two most frequently proposed HO parameter models: the even rank-4 hexadecapole and the odd-rank-5 dotriacontapole model. We obtain the quasiparticle dispersion and reconstructed Fermi surface in each HO phase adapting an effective two-orbital model of $5f$ bands that reproduces the main Fermi surface sheets of the para phase. We show that the resulting QPI spectrum reflects directly the effect of fourfold symmetry breaking in the rank-5 model, which is absent in the rank-4 model. Therefore we suggest that the QPI method should give the possibility of a direct discrimination between the two most investigated models of HO in ${\mathrm{URu}}_2{\mathrm{Si}}_2$. Furthermore, the signature of proposed chiral $d$-wave superconducting (SC) order parameter in QPI of the coexisting HO + SC phase is investigated.

Figure 1

(a) Fermi surface sheets of ${\mathrm{URu}}_2{\mathrm{Si}}_2$ in the extended simple tetragonal (st) BZ with an electron sheet at $\Gamma (0,0,0)$ and a hole sheet around $Z(0,0,2\pi /c)$ $\left(c\equiv 1\right)$. The two sheets are nested by $\mathbf{Q}=(0,0,1)$. FS cuts with $k_z=0$ (b) and $k_y=0$ (c) in the reduced st BZ (Z-point folded onto $\Gamma )$. The momentum range in (b) and (c) is given by $-\pi \le k_i\le \pi$.

Figure 2

Dispersion along the st BZ path $\Gamma (0,0,0),M(\frac{1}{2},\frac{1}{2},0)$; $X(0,\frac{1}{2},0)$; $Z(0,0,\frac{1}{2})$; $A(\frac{1}{2},\frac{1}{2},\frac{1}{2})$; and $R(0,\frac{1}{2},\frac{1}{2})$. (a) Effective $f$ bands in the para phase. (b) and (c) Reconstructed quasiparticle bands in the $E_{-}(1,1)$ HO phase for $\varphi =0.8$ and $1.7$. The gapping of quasiparticle dispersion at k points connected by the nesting vector $\mathbf{Q}=(0,0,1)$ (r.l.u.) can be clearly seen. (d) DOS with evolution of HO gap with $\varphi$ [also Fig. 4]. Here and in subsequent figures, we define $\varphi =|{\varphi }_z^{\mathbf{Q}}|$ or $|{\mathit{\varphi }}^{\mathbf{Q}}|$ in units of $t_0=6.66$ meV.

Figure 3

(a) Fermi surface sheets of ${\mathrm{URu}}_2{\mathrm{Si}}_2$ in the $E_{-}(1,1)$ HO phase in the reduced BZ (folded by Q). The FS is reconstructed in k-space regions connected by the nesting vector $\mathbf{Q}=(0,0,1)$ and breaks up into four larger and four smaller sheets (partly hidden). FS cuts in (b) the $k_x\text{−}{k}_y$ plane and (c) the $k_x\text{−}{k}_z$ plane. We note that the fourfold symmetry breaking of the $E_{-}$ HO does not appear in (b) where $k_z=0$ (cf. Fig. 4). The momentum range in (b) and (c) is given by $-\pi \le k_i\le \pi$.

Figure 4

Comparison of reconstructed Fermi surface sheets of ${\mathrm{URu}}_2{\mathrm{Si}}_2$ for different HO symmetry in the $k_x\text{−}{k}_y$ plane with $|k_z|=0.35\pi$. (a) In the $E_{-}(1,1)$ phase, the breaking of fourfold rotational $C_4$ symmetry for $k_z\ne 0$ is obvious. Changing to $E_{-}(1,\overline{1})$ domain corresponds to $\pi /2$ rotation. (b) For $A_{2+}$, HO fourfold symmetry is preserved. (c) Zoomed DOS in the HO gap region for various $\varphi$. Charge carrier DOS is reduced to small values for large $\varphi$. Momentum range in (a) and (b) is given by $-\pi \le k_i\le \pi$.

Figure 5

(a) Reconstructed HO Fermi surface $\left(\omega =0\right)$ for $\varphi =0.7$ (in units of $t_0)$ in the $k_x\text{−}{k}_y$ plane with $k_z=0$ where $E_{-},A_{2+}$ are equivalent. The characteristic scattering wave vectors ${\mathbf{q}}_{1-3}^{a-c}$ for the QPI spectrum are indicated. (b) Partial QPI (absolute value) spectrum $(\omega =0)$ for $k_z=0$ slice of the Fermi surface. All ${\mathbf{q}}_i^{\alpha }$ are present and the image of HO FS sheets that have doubled ‘$2k_F$’ dimension is clearly visible. The momentum range is given by $-\pi \le k_i\le \pi$.

Figure 6

Partial QPI (absolute value) spectra $\left(\omega =0.124t_0\right)$ for different $|k_z|>0$ slices for $E_{-}$ (first row) and $A_{2+}$ (second row) HO. For larger $k_z$, distinct $C_4$ symmetry breaking due to the ${\zeta }_{\mathbf{k}}$ function in the dispersion of Eq. (14) appears for $E_{-}$, while $C_4$ is preserved for $A_{2+}$. (The momentum range is given by $-\pi \le q_{x,y}\le \pi$.)

Figure 7

Constant energy surfaces $\left({\epsilon }_{i\mathbf{k}}=\omega \right)$ for $k_z=0$ (top row, identical for both HO) and total QPI (absolute value) spectra $\left(\omega =0,-0.124t_0,0.124t_0\right)$ for $E_{-}$ (center row) and $A_{2+}$ (bottom row) HO parameter. $C_4$ symmetry is preserved for $A_{2+}$ while a distinct rotational symmetry breaking is still visible in the integrated QPI for $E_{-}$. Note that the latter is due to the $|k_z|>0$ contributions not depicted in the top row. Dashed arrows denote an image folded back into the first BZ. The momentum range is given by $-\pi \le k_i\le \pi$.

Figure 8

(a) Size of the gap function [Eq. (35)] on the HO Fermi surface in $k_x\text{−}{k}_z$ plane. The dashed red curves show the spectral function at energy $\omega =0$ for hidden ordered phase and blue thick curves show the quasi-2D spectral function at energy $\omega =0.3{\Delta }_0$ in HO + SC phase. Note that the node line ${\Delta }_{\mathbf{k}}=0$ is in $k_z=0$ plane. (b) The DOS shows the evolution of the SC pseudogap on top of the larger HO gap. Comparison of total QPI (absolute value) image of (c) $E_{-}$ HO phase $\left(\varphi =0.7\right)$ [equivalent to Fig. 7] with (d) HO $(E_{-}$ or $A_{2+})$ + chiral $d$-wave SC phase $\left(\varphi =0.7,{\Delta }_0=0.1\right)$. In the HO phase (c), the $k_z$ summation leads to the appearance of $C_4$ symmetry breaking. In the HO + SC phase, the gapping of states with $|k_z|>0$ in (a) reduces (d) to a quasi-2D QPI spectrum that contains only the $C_4$ symmetric $k_z=0$ slice and therefore is equivalent for $E_{-}$ or $A_{2+}$ HO. The momentum range is given by $-\pi \le k_i\le \pi$.