--- title: Charge Order in Kagome Superconductors url: https://www.emergentmind.com/topics/charge-order-in-kagome-superconductors type: topic --- # Charge Order in Kagome Superconductors Charge order in kagome superconductors denotes the spontaneous breaking of translational symmetry via modulated charge densities, typically manifesting as charge density waves (CDWs) at distinct commensurate or incommensurate wave vectors determined by electronic, phononic, and correlation-driven instabilities. The kagome lattice, with its inherent geometrical frustration, Dirac crossings, flat bands, and van Hove singularities (vHS), provides a unique platform where charge order can intertwine with additional orders like superconductivity, orbital currents, nematicity, and topological phases. Recent advances have revealed a broad range of charge-ordered states in kagome superconductors, notably the A(V₃Sb₅) (A=K, Rb, Cs) and rare-earth Ru silicides (LaRu₃Si₂, YRu₃Si₂), spanning transition temperatures from below 100 K to over 800 K and exhibiting diverse order parameters and microscopic competition/coexistence with unconventional superconductivity. ## 1. Structural Motifs, Instability Mechanisms, and Phenomenological Models Charge order in kagome superconductors predominantly arises from a combination of Fermi-surface nesting at vHS, enhanced low-energy susceptibilities, phonon softening, and/or strong coupling effects such as local exciton crystallization. The classic tight-binding kagome band structure produces Dirac cones at K, a flat band, and logarithmic vHS at the M points. The Fermi-level tuning near vHS via self-doping or chemical substitution promotes instabilities at $\mathbf{Q}_{\rm CO}$ vectors connecting inequivalent M points. A minimal real-space description of a modulated charge state employs the multi-component order parameter: \[ \rho(\mathbf r,T) = \rho_0 + \sum_{i=1}^N \rho_{\mathbf Q_i}(T)\cos(\mathbf Q_i\cdot \mathbf r + \phi_i), \] where $\mathbf Q_i$ are symmetry-related nesting wave vectors and $\rho_{\mathbf Q_i}$ are the modulation amplitudes, serving as CDW order parameters. The corresponding Landau-Ginzburg free energy for two competing orders reads: \[ F[\{\rho_i\}] = \sum_{i=1,2}[a_i(T)|\rho_i|^2 + b_i|\rho_i|^4] + u|\rho_1|^2|\rho_2|^2 + \ldots \] with $a_i(T)=\alpha_i(T-T_{\rm CO,i})$ and $u>0$ introducing competition (as in LaRu$_3$Si$_2$ [2309.09255]). Phonon-driven CDWs are demonstrated by (i) imaginary phonon modes at the relevant $\mathbf{Q}_{\rm CO}$ in DFT, (ii) high $T_{\rm CO}$ coupling to in-plane atomic displacements, and (iii) robustness against disorder, e.g., Fe-doping in La(Ru$_{1-x}$Fe$_x$)$_3$Si$_2$ ($T_{\rm CO-I}\approx400$ K essentially unchanged with $x$ [2309.09255]). In contrast, correlation-driven/“exciton crystallization” scenarios posit localized Frenkel-type bosonic excitons forming close-packed charge crystals on the kagome sublattice, described by a bosonic Hamiltonian with local formation energies and hard-core inter-exciton repulsion [2510.02289]. The instability criterion, $1-V(\mathbf Q)\chi_0(\mathbf Q,T)=0$, mirrors the random-phase approximation for conventional CDW but arises from strongly bound excitons. ## 2. Experimental Signatures and Characterization Charge order in kagome superconductors is identified using synchrotron X-ray and neutron diffraction, STM/STS, angle-resolved photoemission (ARPES), nuclear quadrupole resonance (NQR), nuclear magnetic resonance (NMR), and muon spin rotation ($\mu$SR). - **Superlattice Peak Patterns and Propagation Vectors:** - LaRu$_3$Si$_2$: (1/4,0,0), with a secondary (1/6,0,0) order at low $T$ [2309.09255, 2402.16219]. - AV$_3$Sb$_5$: 2×2×L triple-Q modulations at (1/2,0,0), (0,1/2,0), and (1/2,1/2,0), as well as 3D stackings that select distinct real-space patterns (Star-of-David, Tri-hexagonal, staggered) [2202.01902, 2112.15288, 2203.05055]. - YRu$_3$Si$_2$: (1/2,0,0)–type order sets a record $T_0\simeq800$ K [2507.06885]. - Sn-doped CsV$_3$Sb$_5$: When the $2\times2$ CDW collapses, emergent short-range $3\times1$ stripe order with $q\approx(1/3,0)$ appears [2509.17467]. - **Order Parameter Behavior and Critical Exponents:** - Onset of the superlattice intensity $I_{\mathbf Q}(T)\propto |\rho_{\mathbf Q}(T)|^2$ shows classic mean-field $\beta=1/2$ exponents near $T_{\rm CO}$, with correlation lengths $\xi_{\rm CO}(T)$ extracted from reciprocal-space HWHM. Diffuse scattering above $T_{\rm CO}$ and suppressed $\xi_{\rm CO}$ with disorder (e.g., Fe-substitution) support variable-strength pinning [2309.09255]. - **Phase Competition and Coexistence:** - Both XRD and $\mu$SR confirm simultaneous presence of competing (or coexisting) CO and SC signatures, with evolution directly linked to composition, temperature, and external perturbations (pressure, disorder, strain). ## 3. Chiral Charge Order, Time-Reversal Symmetry Breaking, and Coupled Orders One of the unique electronic orders in the kagome superconductors is chiral (time-reversal symmetry-breaking; TRSB) charge order. TRSB is established via enhancement of the internal magnetic field width in $\mu$SR and direct detection of anomalous Hall effects: - **KV$_3$Sb$_5$ and RbV$_3$Sb$_5$:** Chiral triple-Q 2×2 CDW (complex order parameters with locked $2\pi/3$ phase differences) [2012.15709, 2106.13443, 2105.00550]. Theoretical and STM studies confirm loop-current patterns on the kagome trimer units, with tunable chirality via small out-of-plane magnetic fields. TRSB fields of $0.3-1.8$ G are observed below $T^*\sim80$ K [2106.13443]. Magnetically-driven NMR and Kerr signatures confirm broken TRS in the same temperature range [2209.07340]. - **LaRu$_3$Si$_2$:** Primary (1/4,0,0) order is non-magnetic, but the secondary (1/6,0,0) order, below $T_{\rm CO,2}\approx80$ K, exhibits a pronounced $\mu$SR internal field broadening ($\sim$0.4 G below $T^*\approx35$ K), Hall sign reversal, and nonzero magnetoresistance, indicating the emergence of a TRSB phase [2402.16219]. - **YRu$_3$Si$_2$:** TRSB sets in at a much lower $T_2^*\approx25$ K deep within the CO phase; field-induced static magnetism appears at $T_1^*\approx90$ K [2507.06885]. Chiral CDWs also couple to nematic or stripe order. Pressure or chemical substitution can drive transitions from triple-Q chiral orders to unidirectional stripe (single-Q) or nematic orders (C$_6\rightarrow$C$_2$), as evident in Sn-doped CsV$_3$Sb$_5$ ($3\times1$ stripes) [2509.17467] and pressure-tuned AV$_3$Sb$_5$ ($4a$ stripes) [2209.07340]. ## 4. Charge Order–Superconductivity Interplay and Topological Phenomena Charge order fundamentally intertwines with superconductivity in kagome systems. The competition/coexistence is governed by Ginzburg-Landau free energies with repulsive coupling ($u>0$), as well as by Fermi surface reconstructions due to CDW-induced band folding and gap opening at van Hove points. - **Suppression and Domes:** In AV$_3$Sb$_5$, pressure or chemical tuning that suppresses the triple-Q (2×2) CDW leads to superconducting (SC) domes reaching $T_c=3-8$ K, with the maximum $T_c$ typically appearing at the border of CDW suppression [2102.10959, 2106.13443, 2203.05055, 2202.01902]. In CsV$_3$Sb$_5$, distinct CO patterns (superimposed Star-of-David, staggered trihexagonal) correlate with distinct superconducting domes, altering both $T_c$ and superfluid density [2203.05055]. - **Gap Evolution and Pairing Symmetry:** In RbV$_3$Sb$_5$ and KV$_3$Sb$_5$, coexistence with TRSB CDW yields nodal pairing with line nodes in the superconducting gap (linear-in-$T$ penetration depth), which evolves toward a nodeless, fully gapped (but TRSB) state as pressure suppresses the CDW [2202.07713]. Multigap superconductivity with strong-coupling characteristics ($2\Delta/k_B T_c\gg3.5$), small superfluid densities (Uemura scaling), and chiral $d_{x^2-y^2}+id_{xy}$-type order parameters are observed/modeled in these materials [2106.13443, 2202.07713, 2302.12377]. - **Impact of Orbital-Current CDW and Gap Anisotropy:** Theoretical analyses demonstrate that chiral (orbital-current, flux) CDW reconstructs the Fermi surface anisotropically, such that even purely $s$-wave pairing develops nodal or deep-minima gap features, producing a $V$-shaped density of states and residual zero-energy spectral weight [2302.12377, 2406.15273]. Incremental coupling to chiral flux order (CFP) in trihexagonal states yields topological superconductivity and Chern-number transitions ($C=0\rightarrow2$), with possible realization of Majorana edge modes in AV$_3$Sb$_5$ [2406.15273]. - **Microscopic Reorganization by Stripe and Nematic Order:** Suppression of the global $2\times2$ CDW can reveal short-range stripe ($3\times1$) orders which partially gap different Fermi surface segments and reduce the density of states at $E_F$ [2509.17467], resonating with analogous features in the cuprates. ## 5. Charge Order Beyond Weak Coupling: Exciton-Crystallization and Strong Correlation Conventional Fermi-surface-nesting CDW mechanisms alone cannot explain several features of kagome charge order, such as high transition temperatures, commensurate patterns, and their robustness against disorder. Alternative frameworks incorporate: - **Crystallization of Frenkel Excitons:** Long-lived bond-centered Frenkel excitons, with strong short-range interactions, crystallize on the kagome net, yielding commensurate $2\times2$ or $\sqrt{3}\times\sqrt{3}$ patterns, as parametrized by condensates of bosonic excitons at specific wave vectors (e.g., M or K) [2510.02289]. The model rationalizes high $T_{\rm co}$, strong local lattice distortions above global $T_c$, and observations of order switching by fine-tuning exciton energy via strain or pressure. Phenomena such as local persistence of the CDW gap above $T_{\rm co}$, spin-$1$ excitonic contributions, and first-order-like transition signatures further support this scenario. - **Cooperative Electronic–Lattice Interactions:** For stripe and uniaxial orders, electronic correlations at the vHS enhance electronic susceptibility at hidden wave vectors (e.g., $q=(1/3,0)$), which couple to soft phonons and lattice distortions, resulting in emergent charge stripes [2509.17467]. This cooperative mechanism enables stabilization of otherwise subdominant or short-range orders under modest chemical/structural pressures. ## 6. Tunable, Intertwined, and Defect-Engineered Charge Order The flexibility of kagome systems allows manipulation of intertwined charge and superconducting states by external parameters: - **Pressure, Doping, and Disorder:** Hydrostatic pressure can drive transitions between CDW and stripe/nematic/SC phases, enabling $2$-dome $T_c$ structures (CsV$_3$Sb$_5$ [2203.05055, 2209.07340, 2202.01902]), while chemical doping (Ti, Sn, Fe) at transition-metal sites can suppress one order and unlock new modulations or competing phases [2110.12651, 2509.17467, 2309.09255]. - **Topological Defects and Surface Engineering:** STM manipulation of Cs atoms at the AV$_3$Sb$_5$ surface demonstrates that stacking-fault/antiphase boundaries in the 2×2×2 CDW can nucleate quasi-2D superconducting condensates and primary $4\times4$ pair-density waves (PDW), tunable on/off and spatially modulated [2408.06174]. These engineered boundaries are associated with emergent in-gap states, altered vortex-core spectra, and potential platforms for Majorana physics. - **Surface Nematicity:** Surface 1×4 superlattices in RbV$_3$Sb$_5$ and CsV$_3$Sb$_5$ emerge via spontaneous or field-induced domain formation, and couple nontrivially to bulk charge order [2105.00550]. Their modulation symmetry and coupling to triple-Q CDW can control in-plane anisotropy of superconducting and normal-state properties. ## 7. Outlook and Future Directions Open directions in the study of charge order in kagome superconductors include: - **Direct Imaging and Spectroscopy:** Detailed ARPES, STM, and inelastic X-ray/neutron scattering are essential to elucidate the reconstructed Fermi surfaces, phonon softening/electron coupling, and gap anisotropy in coexisting/competing phases [2309.09255]. - **Role of Time-Reversal and Topological Phenomena:** $\mu$SR and Kerr-effect probes will further clarify the landscape of TRSB orders and their impact on anomalous transport and topological superconductivity [2309.09255, 2507.06885, 2202.07713]. - **Microscopic Theory:** Development of multi-order-parameter Landau theories, tight-binding models including both phonon and correlation effects, and quantum Monte Carlo simulations will refine understanding of the subtle balance and cascade of density-wave and superconducting orders [2406.15273, 2510.02289]. - **Materials Design:** Engineering multi-vHS band structures (as in YRu$_3$Si$_2$) and controlled manipulation of chemical substitution or external fields offer routes to designing materials with tunable $T_{\rm CO}$ and intertwined topological orders [2507.06885]. In summary, charge order in kagome superconductors is a highly tunable, intertwined phenomenon that emerges from a confluence of lattice geometry, vHS electronic structure, electron-phonon and correlation effects, and is deeply linked to unconventional superconductivity, magnetoelectric and topological properties. The interplay of multiple commensurate and incommensurate charge orders, chiral and nematic phases, and strong coupling to superconductivity establish these materials as model platforms for studying new forms of quantum order and manipulation in correlated quantum matter [2309.09255, 2402.16219, 2202.01902, 2507.06885, 2510.02289, 2509.17467, 2203.05055]. Source: https://www.emergentmind.com/topics/charge-order-in-kagome-superconductors