Bond-density-wave orders induced by geometric frustration in the kagome metal CeRu3Si2

Abstract

Geometric frustration gives rise to vast manifolds of degenerate ground states and competing orders in spin and charge systems. Typically, classical ground states are governed by a local ``zero-sum constraint" that relieves frustrated antiferromagnetic interactions or Coulomb repulsion. To date, the paradigm of geometric frustration has yielded a rich landscape of emergent phases, from spin ices and quantum spin liquids to charge glasses. However, an analogous phase rooted in chemical bonding has yet to be firmly demonstrated. Here we report the discovery of bond-density-wave orders induced by geometric frustration in the kagome metal CeRu$_3$Si$_2$ above room temperature. Through synchrotron X-ray diffraction, real-space transmission electron microscopy, and model calculations, we observe two distinct long-period superlattices with harmonic and anharmonic structural modulations. Crucially, interlayer bonds between kagome planes modulate in a sublattice-selective manner to fulfill the zero-sum constraint on the kagome lattice. We demonstrate the potential of kagome metals to host complex bond-ordered states constrained by geometric frustration and establish chemical bonding as a distinct pathway to frustration physics in quantum materials even above room temperature.

Figures (5)

• Figure 1: Bond-ordered states satisfying a zero-sum constraint on a geometrically frustrated lattice.a,b,c, Frustration of spin, charge, and interlayer bond on the triangular lattice with antiferromagnetic interactions or Coulomb repulsion. The red (blue) sphere represents up (down) spin, positive (negative) charge, or a site bonded with the upper (lower) layer. The navy (black) arrow marks the spins (interlayer bond direction), while the colored cylinders represent chemical bonding. d, Representative two-in–two-out spin configuration on the pyrochlore lattice, a local motif of spin ice that satisfies a zero-sum constraint. e, Kagome spin ice with one-in-two-out (positive "charge", red) and two-in-one-out (negative "charge", blue) clusters. f,g, Partial bond order on the kagome lattice with stripe and zigzag patterns of interlayer bonding. Two of the three sublattices form bonds with opposite layers, while the other does not, to realize neutral bonding on each triangle. h,i, Sublattice-selective harmonic and anharmonic square bond-density-wave (BDW) orders on the kagome lattice. As in partially bond-ordered states, the local bond zero-sum constraint is satisfied by two out-of-phase waves (colored mesh), minimizing frustrated Coulomb repulsion.
• Figure 2: Observation of harmonic and anharmonic BDW orders in the kagome metal CeRu$_3$Si$_2$.a,b, Top and side views of the parent orthorhombic structure of CeRu$_3$Si$_2$ (stripe, space-group: $Ibmm$). Ru$_1$ and Ru$_2$ form a $d{_z{^2}}$ molecular orbital with opposite kagome layers, whereas Ru$_3$ remains unbonded, realizing partial bond order. c, Reciprocal-space maps in the $hk6$ plane reconstructed from synchrotron X-ray diffraction (XRD) data at $500\,$K (stripe). d, Temperature dependence of superlattice reflection intensities corresponding to $\bm{q}_1 = (2/9, 0, 0)$ (cyan), $\bm{q}_2 = (1/3, 0, 0)$ (orange), and $\bm{q}_1+\bm{q}_2$ (purple). The intensities for $\bm{q}_1$ are scaled by a factor of $0.5$. Superlattice reflections of BDW-I and BDW-II emerge from $T_\text{BDW-I} \sim 360\,$K and $T_\text{BDW-II} \sim 300\,$K, respectively. A cyan line is a mean-field fit to the data and is proportional to $\sqrt{T-T_\text{BDW-I}}$. e, Line cut along the $h06$ line at $300\,$K (cyan) and $50\,$K (purple). The $\bm{q}_1$ superlattice reflection exists at $300\,$K, while its higher-order harmonics $\bm{q}_2$ and $\bm{q}_1+\bm{q}_2$ are clearly observed at $50\,$K. f,g, Reciprocal-space maps in the $hk6$ plane at $300\,$K (BDW-I, harmonic), and $50\,$K (BDW-II, anharmonic).
• Figure 3: Real-space observation of the BDW order at room temperature.a, Real-space image of the (001) surface obtained by transmission electron microscopy (TEM). b,c, Fast Fourier transform (FFT) patterns of two domains A and B in panel a. The arrows indicate reciprocal-space vectors in each orthorhombic domain. Inset: close-up view around a Bragg reflection with superlattice reflections existing only next to extinct Bragg reflections ($2\,1\,0$ and $2\,\hbox{-}1\,0$ for panel b). d,e, Inverse FFT patterns of the domains A and B reconstructed from superlattice reflections (Methods). The amplitude of the real part is displayed after application of a Gaussian filter.
• Figure 4: Sublattice-selective BDW orders with zero net modulation per triangle.a,b, Top and side views of the refined $9\times 1\times 1$ superstructure at $300\,$K (BDW-I). c,e, Interlayer bond length $d_\mathrm{inter}$ of Ru atoms, divided by the lattice constant $c$, as a function of the position $x$ along the $a_\mathrm{BDW}$ axis ($a_\mathrm{BDW}=9a$) in the $9 \times 1 \times 1$ supercell, obtained from superstructure refinement of BDW-I and BDW-II with the synchrotron XRD data. Orange, cyan, and purple denote Ru$_1$, Ru$_2$, and Ru$_3$ atoms, respectively. Dashed lines indicate $d_\mathrm{inter}/c$ in the parent stripe phase, with the purple dashed line marking the unbonded reference position ($d_\mathrm{inter}/c = 1/2$). Solid lines represent fitted sinusoidal modulations in panel c and their square-wave approximations in panel e with the primary modulation vector $\bm{q}_1 = (2/9, 0, 0)$. The Ru$_3$ sublattice forms the out-of-phase wave whose amplitude is twice larger than that of the other two waves. d,f, Triangle-averaged $d_\mathrm{inter}/c$ (net modulation per triangle) assigned at the triangle centroids (gray spheres in the inset). The distance from the dashed line quantifies the degree of net bond neutrality on each triangle. The inset illustrates the sublattice-selective bond modulation relative to the reference $d_\mathrm{inter}$. Error bars are estimated to be several orders of magnitude smaller than the modulation amplitude and are therefore omitted (Methods).
• Figure 5: Anisotropic nearest-neighbor soft-Ising model on the kagome lattice.a, Schematic of the anisotropic pseudospin model derived from an expansion of the Coulomb repulsion. $J$ ($J'$) denote the nearest-neighbor pseudo-exchange interactions between sublattices $1$–$2$ ($1$–$3$ and $2$–$3$). The parent stripe phase with bond modulations of $\pm\delta, 0$ is illustrated, and the inequivalence of $J$ and $J'$ arises from the difference in bond lengths. The interlayer bond modulation $\bm{u}_i(\bm{r})$ on sublattice $i$, defined relative to the stripe phase, is regarded as a pseudospin variable. b, Sublattice-selective harmonic BDW order induced by anisotropic exchange interactions and geometric frustration. The three sublattices on the kagome lattice exhibit the sinusoidal bond modulations $\bm{u}_1(\bm{r})=\bm{u}_2(\bm{r}) =-1/2\bm{u}_3(\bm{r}) = u_0\bm{e}_z\cos(qx)$ to satisfy the local bond zero-sum constraint. c, Spin spiral arising from competing nearest-neighbor ferromagnetic ($J_1<0$) and next-nearest-neighbor antiferromagnetic ($J_2>0$) interactions. Geometric frustration enforces a sublattice-selective texture, leading to effective frustration of exchange interactions: $E_{\text{ex}} = u_0^2\left(- 4J'\cos{q/4} + J\cos{q/2}\right)$. d, Phase diagram of the anisotropic pseudospin model. The color gradient in the harmonic BDW phase indicates its wavenumber $q$, with the thickest shading corresponding to $q = 0$ and the thinnest to $q = 1/2$. The parameter regime that reproduces the observed $q = 2/9$ harmonic BDW order is highlighted by a broken line. The ground state is determined within the variational manifold of bond-ordered states shown in Fig. \ref{['fig:fig1']}f–i, all of which are observed experimentally. Inclusion of higher-order interactions stabilizes the square BDW order (see Methods).