Session T07: Kagome and Flat Band Materials

Evolution of charge correlations in CsV3Sb5-x Snx studied via synchrotron x-ray diffraction

The kagome metal CsV₃Sb₅ hosts a complex charge density wave state that is proposed to intertwined with its lower temperature superconducting phase in a nontrivial manner. Of particular interest is the relationship between charge correlations and the stability of the superconducting state in this material. The behavior of the charge density wave (CDW) state can be studied via synchrotron x-ray scattering measurements, as the CDW manifests as a structural distortion at around 94K in the parent structure in the form of 2x2x4, 2x2x2, and 2x2x1 superstructures. Furthermore, the CsV₃Sb₅ structure allows for chemical doping of holes via Sn atoms onto the Sb sites, exhibiting a suppression of the charge density wave at small hole-doping levels and a double-dome type superconducting phase diagram with increasing Sn doping. In our study, we probe the evolution of charge correlations as a function of hole-doping in  CsV₃Sb₅ via synchrotron x-ray diffraction measurements. With increasing Sn doping in CsV₃Sb_5-xSn_X, the 2x2x4 distortion is suppressed, giving rise to short-range correlations that further weaken at higher doping levels. The creation of intermediate incommensurate charge correlations and eventual formation of quasi-1D charge correlations is discussed.   

Interplay of strong correlations and band topology in pyrochlore compound

Correlated phases derived from topological flat bands are of extensive current interest. Flat bands

in three dimensions (3D) occur from destructive interference and motivate the consideration of

compact localized states. Recently, correlated phases and non-Fermi liquid behavior have been

discovered in various families of flat band pyrochlore materials [1,2]. Here we construct

exponentially localized Wannier orbitals to write low-energy, interacting tight-binding models

that faithfully capture the band structure and topology of the active bands [3]. Using these

models, we provide insights into correlated orders [4] and quantum fluctuations [5] and that

might emerge in pyrochlore compounds. Our results point toward 3D topological flat band

materials as an ideal platform to realize and elucidate the interplay of band topology and strong

correlation effects in three dimensional quantum matter.

[1] Huang, J., Setty, C., Deng, L., You, J. Y., Liu, H., Shao, S., ... & Yi, M. (2023). Three-

Dimensional Flat Bands and Dirac Cones in a Pyrochlore Superconductor. arXiv preprint

arXiv:2304.09066.

[2] Huang, J., et al Under Review (2023)

[3] Hu, H., Setty, C., Si, Q., (unpublished).

[4] C. Setty et al., arXiv:2105.15204.

[5] L. Chen et al., arXiv:2307.09431.   

Visualizing many body phases in a partially filled kagome flat band

The kagome lattice with spin-orbit coupling exhibits a topologically non-trivial flat band in which the effect of Coulomb interactions between the localized charge carriers is believed to be strong. Hence, material realizations of the kagome lattice provide a promising platform to search for new quantum phases of matter at the confluence of topology and strong electronic correlations. We previously showed that the kagome metal CoSn exhibits a quasi-two-dimensional flat band whose occupied electronic states are strongly localized in real space [1]. Here, we study the low energy density of states of Co_1-xFe_xSn in which partial flat band fillings are realized by hole-doping with Fe.

We will present results from temperature-dependent scanning tunneling microscopy measurements on Co_1-xFe_xSn. Combining high-resolution spectroscopy with spectroscopic imaging on samples with different doping levels x, we observe a rich sequence of electronic states appearing at the Fermi energy that cannot be explained within a single-particle picture. We will discuss our findings in the context of electron-electron interaction-induced many body states at partial flat band fillings.   

Polaron spectra and Orthogonality Catastrophe for correlated flat bands

The spectrum of a single immobile impurity embedded in a Fermi sea, which is related to the Anderson Orthogonality Catastrophe (OC), can be computed exactly and is characterized by a divergent power law. We generalize this canonical problem by replacing the impurity with a single particle moving in a flat band with non-trivial Bloch band geometry. We show that, when lattice effects are sufficiently weak or the Fermi momentum is small enough, there is a universal correction to the power law exponent which is proportional to the Fermi energy times the quantum metric of the heavy band. We illustrate this numerically for the Lieb lattice and draw connections to polaron-type ultracold gas experiments.   

Instabilities of the Lieb Lattice

The Lieb lattice is a bipartite lattice which has three sites per unit cell that are either doubly or four-fold coordinated. Due to the four-fold roational symmetry about the four-fold coordinated sites, the Lieb lattice has states which are localized on the two-fold coordinated sites and produce a perfectly a flat band. The presence of strong interactions is expected to lead to instabilities of the lattice, as has been established by Lieb's Theorem for the half-filled state. .Assuming that the instabilities are of second-order, the boundaries of the paramagnetic phase of the Lieb lattice have been determined by examining the zero frequency limit of the random phase approximation for the susceptibilities. An analysis of the site-dependent components of the susceptibilities, indicates that the paramagnetic state becomes unstable at an infinitesimally small values of the screened Coulomb interaction, U, for particular band fillings. The predicted phase transition include a transition to a partially compensated ferromagnetic state at half-filling, which is in accordance with Lieb's Theorem. The inferred magnetic state is predicted to be stable by Hartree-Fock total energy calculations and is shown to persist in the large U limit of the model. In addition, the RPA calculations show antiferrmagnetic ordering in which the size of the unit cell is quadrupled..   

Electronic structures of modulated charge density waves in a kagome metal FeGe

The discovery of a charge density wave (CDW) phase in the A-type antiferromagnetic order of the kagome metal FeGe solidifies the unique standing of FeGe among kagome systems. Our earlier studies revealed characteristic electronic structures from the kagome lattice, namely flat bands, Dirac crossings, and Van Hove singularities using angle-resolved photoemission spectroscopy (ARPES). We have also identified the temperature dependence of a Van Hove singularity and moderate electron-phonon interactions, providing insights into the correlated characteristics of FeGe.

Recently, studies have reported that the CDW in FeGe can be modulated by post-annealing processes, realizing the tuning of the CDW order between long-range order and complete suppression. This newly discovered tunability has opened up opportunities for further in-depth investigations of the CDW in FeGe. We have successfully reproduced long-range CDW order and its suppression by changing post-annealing conditions. Physical property characterizations including magnetic susceptibility agree with existing reports. In this presentation, we will present electronic structure characterizations of both the long-range and suppressed CDW phases in FeGe by ARPES. Temperature-dependent electronic structures will be discussed as well, highlighting the discovery of multiple temperature scales that either differ or coincide in both phases. Our systematic comparison between the two phases helps understanding the nature of CDW phases in terms of electronic structures, opening ways to examine novel mechanism that intertwines CDW and other orders in kagome lattices.   

Fractional Chern Insulating States in flat bands derived from systems with quadratic band crossing points

Topological flat bands in two-dimensional lattice systems are known for their potential to host strongly correlated electronic quantum states and fractional Chern insulators. We focus on flat bands induced by periodic strain potentials over systems with quadratic band crossing points (QBCP). In such systems, there are exactly flat bands at different strain potential strengths, each having very different Berry curvature distributions [Wan et al. Phys. Rev. Lett. 130, 216401 (2023)]. Utilizing exact diagonalization, we investigate the many body instabilities in the systems with Coulomb repulsion projected into the flat bands. We identify the fractional Chern insulating ground states at various fractional electron fillings. Moreover, at half-filling, the composite Fermi liquid (CFL) is stabilized.  We will discuss the role of the quantum geometry and particle-hole symmetry on the fractional Chern insulating state and composite Fermi liquid.   

Intervalley magnetism and superconductivity in sign-problem-free models of topological flat bands

Recent experiments in magic angle twisted bilayer graphene have shown a complex phenomenology driven by the interplay between electron correlation and band topology. Inspired by these experiments, we use numerically exact Quantum Monte Carlo calculations to study a topological flat band model involving both spin and valley degrees of freedom in the presence of electron-electron interactions. The model is sign-problem-free at arbitrary filling of the flat bands and for a range of interaction parameters. In the limit with only projected flat-band degrees of freedom, we find a robust insulating state at half-filling of the flat bands, and a competing superconducting phase when doped away from half-filling. Our findings can account for and provide a route to understand various aspects of the rich phenomenology in topologically non-trivial moiré materials.   

Strong correlations and band topology in the kagome flat band material CsCr3Sb5

Flat band kagome materials present an ideal platform to study the cooperation between strong

correlations and band topology. A flat band develops from destructive interference, which

motivates the consideration of compact localized states and, more precisely, compact molecular

orbitals. This has led to a theoretical description of the correlations in coupled flat and dispersive

bands in terms of a Kondo lattice in the basis of the molecular orbitals; a phase diagram is found

featuring strong quantum fluctuations and metallic quantum criticality [1]. Such a phase diagram

has recently been realized in the chromium based kagome material CsCr3Sb5 which contains a

flat band near the Fermi energy [2], Here, using first principles, we examine the electronic

structure and band topology in this system. We study properties of Wannier orbitals formed from

the flat bands. Using these Wannier orbitals, we construct minimal interacting models and

address both the quantum fluctuations [1] and electronic orders [3].

[1] L. Chen et al., arXiv:2307.09431.

[2] Y. Liu et al., arXiv:2309.13514.

[3] C. Setty et al., arXiv:2105.15204.   

Intertwined van-Hove Singularities as a Mechanism for Loop Current Order in Kagome Metals AV3Sb5

Recent experiments on Kagome metals AV₃Sb₅ (A=Cs,Rb,K) indicated spontaneous time-reversal symmetry breaking in the charge density wave state in the absence of static magnetization. The loop current order (LCO) is proposed as its cause, but a microscopic model explaining the emergence of LCO through electronic correlations has not been firmly established. We show that the coupling between van-Hove singularities (vHS) with distinct mirror symmetries is a key ingredient to generate LCO ground state. By constructing an effective model, we find that when multiple vHS with opposite mirror eigenvalues are close in energy, the nearest-neighbor electron repulsion favors a ground state with coexisting LCO and charge bond order. It is then demonstrated that this mechanism applies to the Kagome metals AV₃Sb₅. Our findings provide an intriguing mechanism of LCO and pave the way for a deeper understanding of complex quantum phenomena in Kagome systems.   

Nature of charge density wave in kagome metal ScV6Sn6

Kagome lattice materials offer a fertile ground to discover novel quantum phases of matter, ranging from unconventional superconductivity and quantum spin liquids to charge orders of various profiles. However, understanding the genuine origin of the quantum phases in kagome materials is often challenging, owing to the intertwined atomic, electronic, and structural degrees of freedom. Here, we combine angle-resolved photoemission spectroscopy, phonon mode calculation, and chemical doping to elucidate the driving mechanism of the √3×√3 charge order in a newly discovered kagome metal ScV₆Sn₆. In contrast to the case of the archetype kagome system AV₃Sb₅ (A= K, Rb, Cs), the van Hove singularities in ScV₆Sn₆ remain intact across the charge order transition, indicating a marginal role of the electronic instability from the V kagome lattice. Instead, we identified a three-dimensional band with dominant planar Sn character opening a large charge order gap of 260 meV and strongly reconstructing the Fermi surface. Our complementary phonon dispersion calculations further emphasize the role of the structural components other than the V kagome lattice by revealing the unstable planar Sn and Sc phonon modes associated to the √3×√3 phase. Finally, in the constructed phase diagram of Sc(V_1-xCr_x)₆Sn₆, the charge order remains robust in a wide doping range x ≈ 0 ~ 0.10 against the Fermi level shift up to ≈ 120 meV, further making the electronic scenarios such as Fermi surface or saddle point nesting unlikely. Our multimodal investigations demonstrate that the physics of ScV₆Sn₆ is fundamentally different from the canonical kagome metal AV₃Sb₅, uncovering a new mechanism to induce symmetry-breaking phase transition in kagome lattice materials.   

Discovery of electronic nematicity in titanium-based kagome metal CsTi3Bi5

Rotation symmetry breaking of the electronic structure can in general be generated by structural anisotropy or electronic correlations via Coulomb repulsion. In kagome superconductors AV₃Sb₅ (A = Cs, Rb, K), rotation symmetry breaking appears to be intricately linked to the rotation symmetry breaking of the 2a0×2a0 charge density wave, leading to the formation of an orthorhombic phase. We use spectroscopic-imaging scanning tunneling microscopy (SI-STM) to discover an electronic nematic phase in the cousin kagome metal CsTi₃Bi₅. This electronic nematic state is characterized by rotation symmetry breaking in the electronic signal in the absence of any charge density wave phase reported in AV₃Sb₅. A comparison between the measured band structure and density functional theory calculations reveals notable differences at low energies, hinting at electronic correlations in CsTi₃Bi₅. The analysis of quasiparticle interference (QPI) patterns uncovers rotational symmetry breaking associated with titanium-derived d orbitals. Our work reveals a rare purely nematic electronic state on a hexagonal lattice that may be described by a three-Potts nematic order parameter.    

Infrared probe of charge density wave gap in ScV6Sn6

The V-based kagome metals AV₃Sb₅ (A = K, Rb, Cs) exhibit a cascade of exotic quantum phenomena including charge-density wave (CDW) order and superconductivity. Considerable effort has been made to understand the nature of the CDW phase of AV₃Sb₅, but the origin remains elusive. A new family of the V-based kagome metals RV₆Sn₆ (R = rare earth ions) has attracted recent interest. Among RV₆Sn₆, only ScV₆Sn₆ shows a CDW order. Thus, RV₆Sn₆ can be a new platform for investigating the nature of the CDW phase of the V-based kagome metals. In this talk, we will discuss the electronic response of RV₆Sn₆ (R = Y, Sc) investigated by the infrared spectroscopy and DFT calculations. While the optical conductivity spectra of YV₆Sn₆ show no anomaly from 10 K to 300 K, those of ScV₆Sn₆ exhibit drastic changes below the CDW transition temperature T_CDW ≈ 92 K: the suppression of the Drude responses and the appearance of the absorption peaks at about 34 and 270 meV. Our DFT calculations demonstrate that the CDW gaps corresponding to the absorption peak at 270 meV open most clearly on the k_z = 1/3 and 1/2 planes. The calculated phonon dispersions of the pristine phase of ScV₆Sn₆ reveal that the structural instability with the imaginary phonon frequencies on the A-H-L plane (k_z = 1/2) and along the M(bar) - K(bar)  line (k_z = 1/3) induces the out-of-plane charge modulation, indicating that the CDW transition of ScV₆Sn₆ is associated with its structural phase transition.   

Exactly Solvable Magnetic Phase Transition and Spin Stiffness in (Generalized) Kagome lattices

The kagome lattice Hubbard model exhibits a strongly interacting gapless flat band with divergent quantum geometry and is an ideal platform for the theoretical and experimental study of correlated topological materials. We show that a large family of similar Hubbard models have exactly solvable ground states up to half filling of the flat bands, generalizing a result of Mielke and Tasaki. We prove a lower bound on the critical density at which these models exhibit a magnetic phase transition, with possible relevance to the onset of ferromagnetism observed in twisted MoTe₂. Lastly, we reveal the importance of band geometry to the spin wave stiffness at half filling, where interactions regularize a naively divergent quantum geometric contribution. We explain a duality relating this result to the Cooper pair mass in flat band superconductors.