Correlated charge order intertwined with time-reversal symmetry-breaking nodal superconductivity in the dual flat band kagome superconductor CeRu$_{3}$Si$_{2}$

Abstract

Kagome materials provide a powerful platform for exploring how flat electronic bands promote symmetry-breaking quantum states, yet studies have so far focused mainly on kagome-derived $d$-electron flat bands. In this paper, we introduce CeRu$_{3}$Si$_{2}$, a kagome superconductor in which our first-principles calculations show the coexistence of Ru $d$-orbital kagome flat bands and heavy-fermion flat bands derived from Ce$^{4+}$ $4f$-states. X-ray diffraction reveals a dominant 1/2 charge order with a much weaker 1/3 component persisting up to room temperature. Theoretical calculations further highlight the correlated nature of these charge-order states. Deep within the charge-ordered state, magnetoresistance emerges below 80 K and strengthens further below 30 K. Zero-field muon spin-rotation measurements show no time-reversal symmetry (TRS) breaking in the normal state, in contrast to LaRu$_{3}$Si$_{2}$ and YRu$_{3}$Si$_{2}$. However, an applied magnetic field induces weak magnetism. Across the $A$Ru$_{3}$Si$_{2}$ family ($A$ = La, Y, and Ce), the superconducting transition temperature $T_{\rm c}$ scales linearly with the onset temperature of normal-state TRS breaking $T_{\rm {TRSB}}$ and the magnitude of the field-induced magnetic response, revealing a direct positive correlation between normal-state symmetry breaking and superconductivity. Furthermore, we identify that CeRu$_{3}$Si$_{2}$ is the first 132-type kagome compound to host nodal superconductivity together with spontaneous internal magnetic fields, providing clear evidence for intrinsic TRS breaking in the superconducting state. These results establish CeRu$_{3}$Si$_{2}$ as a unique platform where intertwined kagome $d$- and heavy fermion $f$-electron flat bands generate a rich hierarchy of electronic orders.

Figures (6)

• Figure 1: Synchrotron X-ray diffraction experiments on a single crystal of CeRu$_{3}$Si$_{2}$. (a-f) Reconstructed reciprocal space along the (0 0 1) direction at 8 r.l.u. (reciprocal lattice units), obtained at various temperatures between 10 K and 300 K. The main reflections are split, due to the formation of orthorhombic domains. Arrows indicate the reciprocal lattice vectors. White, red and green circles mark the Bragg peak, primary charge order CO peak, and the secondary charge order CO-II peak, respectively.
• Figure 2: Correlated charge-ordered states in CeRu$_{3}$Si$_{2}$. (a) Atomic structure of the parent phase ($P6/mmm$). The dashed line indicates the primitive unit cell. (b) Orbital-projected band structure and density of states (DOS) of the parent structure. The sizes of the differently colored open circles overlaid on the band structure are proportional to the orbital-projected weights of the corresponding band states. (c) Phonon dispersion of the parent structure. (d,e) Atomic structures of the CO ($Pmma$) (d) and CO-II ($Imma$) (e) phases. (f) Orbital-projected DOS for the three structures. (g) Schematic energy diagram of the CO states obtained by treating the Ce $4f$ electrons as core (left) and valence (right). In the left panel, CO-II is not stabilized (relaxing into a different structure), and CO* denotes a charge order with $q=1/4$. In the right panel, the energies are plotted as a function of the Hubbard $U$ on the Ce $4f$ orbitals. Non-CO refers to a distorted structure with negligible Ru distortions.
• Figure 3: Magnetotransport experiments on CeRu$_{3}$Si$_{2}$. (a-c) Temperature dependence of the normal state resistivity $\rho$ (a), its first derivative (d$\rho$/dT, left axis) and magnetoresistance (MR, right axis) (b), and Hall resistance measured at 14 T (c). Dashed vertical lines mark two characteristic temperatures, $T_{1}^{*}$ and $T_{2}^{*}$. $T_{1}^{*}$ corresponds to the onset of magnetoresistance and the sign reversal of the Hall resistance, while $T_{2}^{*}$ marks the peak in d$\rho$/dT. (d) Color plot of the normal state MR, drawn as a function of temperature and applied field (up to 14 T). (e) Temperature dependence of the resistivity across the superconducting transition, measured down to 0.6 K under magnetic fields up to 0.8 T. Arrows indicate temperatures corresponding to the onset of the superconducting transition and the zero-resistance state. (f) Field–temperature color map of the resistivity derived from the data in (e). Pink squares indicate the estimated onset of the superconducting transition, $\textit{T}_{\rm{c},\rm{onset}}$, while red squares mark the zero-resistance superconducting state at $\textit{T}_{\rm{c},\rm{zero}}$. Dashed lines are guides to the eye.
• Figure 4: Zero- and high-field $\mu$SR experiments on CeRu$_{3}$Si$_{2}$. (a) The ZF-$\mu$SR time spectra obtained at three different temperatures, $T$= 0.03 K, 1.3 K, and 40 K, covering the temperature range both below and above $T_{\rm c}$. The solid curves represent fits to the recorded time spectra, using the Gaussian Kubo Toyabe (GKT) function. (b) Temperature dependence of the zero-field $\mu$SR relaxation rates, $\Delta$$\Gamma$$_{\rm{ZF}}$ and $\Delta$$\sigma$$_{\rm{ZF}}$, shown after subtraction of the high-temperature contribution. Because the zero-field ${\mu}$SR spectra of the Y and La samples exhibit an exponential form, with the corresponding relaxation rate $\Gamma$ increasing at low temperatures, we present $\Delta$$\Gamma$$_{\rm{ZF}}$. In contrast, the Ce sample shows Gaussian depolarization in zero field, and thus the Gaussian relaxation rate $\Delta$$\sigma$$_{\rm{ZF}}$ is presented. (c) Temperature dependence of the transverse-field $\mu$SR relaxation rate $\Delta$$\sigma_{\rm{HTF}}$, recorded under various applied magnetic fields up to 8 T, shown after subtraction of the high-temperature contribution. Arrows mark two characteristic temperatures $T_{1}^{*}$ and $T_{2}^{*}$. (d) Field dependence of $\Delta$$\sigma_{\rm{HTF}}$ at four selected temperatures below $T_{1}^{*}$.
• Figure 5: Scaling between superconducting and normal state properties across the ARu$_{3}$Si$_{2}$ series (A = Ce, Y, La). (a,b) Superconducting transition temperature $T_{\rm c}$ plotted as a function of the zero-field normal-state TRS breaking onset temperature, $T_{\rm TRSB}$, (a) and the absolute value of the field-induced relaxation rate measured at 6 T and 8 T (b).