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Superconductivity involving finite-momentum pairing 1 can lead to spatial-gap and pair-density modulations, as well as Bogoliubov Fermi states within the superconducting gap. However, the experimental realization of their intertwined relations has been challenging. Here we detect chiral kagome superconductivity modulations with residual Fermi arcs in KV 3 Sb 5 and CsV 3 Sb 5 using normal and Josephson scanning tunnelling microscopy down to 30 millikelvin with a resolved electronic energy difference at the microelectronvolt level. We observe a U-shaped superconducting gap with flat residual in-gap states. This gap shows chiral 2 a  × 2 a spatial modulations with magnetic-field-tunable chirality, which align with the chiral 2 a ×  2 a pair-density modulations observed through Josephson tunnelling. These findings demonstrate a chiral pair density wave (PDW) that breaks time-reversal symmetry. Quasiparticle interference imaging of the in-gap zero-energy states reveals segmented arcs, with high-temperature data linking them to parts of the reconstructed vanadium d -orbital states within the charge order. The detected residual Fermi arcs can be explained by the partial suppression of these d -orbital states through an interorbital 2 a ×  2 a PDW and thus serve as candidate Bogoliubov Fermi states. In addition, we differentiate the observed PDW order from impurity-induced gap modulations. Our observations not only uncover a chiral PDW order with orbital selectivity but also show the fundamental space–momentum correspondence inherent in finite-momentum-paired superconductivity. Using normal and Josephson scanning tunnelling microscopy, chiral kagome superconductivity modulations with corresponding residual Fermi arcs are detected in KV 3 Sb 5 and CsV 3 Sb 5 .

The recent discovery of a variety of intricate electronic order in kagome metals has sprouted significant theoretical and experimental interest. From an electronic perspective on the potential microscopic origin of these phases, the most basic model is given by a Hubbard model on the kagome lattice. We employ functional renormalization group (FRG) to analyze the kagome Hubbard model. Through our methodological refinement of FRG both within its N-patch and truncated unity formulation, we resolve previous discrepancies of different FRG approaches (Wang et al., 2013 vs. Kiesel et al., 2013), and analyze both the pure (\\(p\\)-type) and mixed (\\(m\\)-type) van Hove fillings of the kagome lattice. We further study the RG flow into symmetry broken phases to identify the energetically preferred linear combination of the respective order parameter without any need for additional mean field analysis. Our findings suggest some consistency with recent experiments, and underline the richness of electronic phases already found in the kagome Hubbard model. We also provide a no-go theorem for a complex charge bond ordered phase in the single orbital kagome Hubbard model, suggesting that this model cannot capture aspects of orbital current phases.

We identify a superconducting order featuring spatial pair modulations on the kagome lattice subject to onsite Hubbard U and nearest neighbor V interactions. Within our functional renormalization group analysis, this state appears with a concomitant d-wave superconducting (SC) instability at zero lattice momentum, where it distinguishes itself through intra-unit cell modulations of the pairing function thus breaking the discrete space group symmetry. The relative weight of the sublattice modulated superconductor (SMS) and d-wave SC is influenced by the absolute interaction strength and coupling ratio V /U . Parametrically adjacent to this domain at weak coupling, we find an intra-unit cell modulated vestigial charge density wave and an s-wave SC instability. Our study provides a microscopic setting and thorough description of this novel SMS arising within a translation symmetry broken background.

Translation symmetry-breaking order is assumed to be suppressed by the lack of Fermi surface nesting near certain higher-order Van Hove singularities (HOVHS). We show the anisotropic band-flattening inherent to such HOVHS, combined with broadening of the Fermi surface due to elevated critical temperatures, results in the Fermi surface becoming approximately nested at a wavevector unrelated to the precise shape of the Fermi surface - leading to a \\(\\sqrt{3}\\times\\sqrt{3}\\) Kekulé density wave formation. The effect is demonstrated using unbiased renormalization group calculations for a model of the breathing kagome lattice. Our mechanism - termed diffuse nesting - represents an entirely new notion in the study of Fermi surface instabilities.

We analyze the phase transition between a symmetric metallic parent state and itinerant altermagnetic order. The underlying mechanism we reveal in our microscopic model of electrons on a Lieb lattice does not involve orbital ordering, but derives from sublattice interference.

Translation symmetry-breaking order is assumed to be suppressed by the lack of Fermi surface nesting near certain higher-order Van Hove singularities (HOVHS). We show the anisotropic band-flattening inherent to such HOVHS, combined with broadening of the Fermi surface due to elevated critical temperatures, results in the Fermi surface becoming approximately nested at a wavevector unrelated to the precise shape of the Fermi surface - leading to a \\(33\\) Kekulé density wave formation. The effect is demonstrated using unbiased renormalization group calculations for a model of the breathing kagome lattice. Our mechanism - termed diffuse nesting - represents an entirely new notion in the study of Fermi surface instabilities.

Recent experiments have identified fascinating electronic orders in kagome materials, including intriguing superconductivity, charge density wave (CDW) and nematicity. In particular, some experimental evidence for AV\\(_3\\)Sb\\(_5\\) (A = K,Rb,Cs) and related kagome metals hints at the formation of orbital currents in the charge density wave ordered regime, providing a mechanism for spontaneous time-reversal symmetry breaking in the absence of local moments. In this work, we comprehensively explore the competitive charge instabilities of the spinless kagome lattice with inter-site Coulomb interactions at the pure-sublattice van Hove filling. From the analysis of the charge susceptibility, we find that, at the nesting vectors, while the onsite charge order is dramatically suppressed, the bond charge orders are substantially enhanced owing to the sublattice texture on the hexagonal Fermi surface. Furthermore, we demonstrate that nearest-neighbor and next nearest-neighbor bonds are characterized by significant intrinsic real and imaginary bond fluctuations, respectively. The 2\\(\\)2 loop current order is thus favored by the next nearest-neighbor Coulomb repulsion. Interestingly, increasing interactions further leads to a nematic state with intra-cell sublattice density modulation that breaks the \\(C_6\\) rotational symmetry. We further explore superconducting orders descending from onsite and bond charge fluctuations, and discuss our model's implications on the experimental status quo.

Recent discoveries in kagome materials have unveiled their capacity to harbor exotic quantum states, including intriguing charge density wave (CDW) and superconductivity. Notably, accumulating experimental evidence suggests time-reversal symmetry breaking within the CDW, hinting at the long-pursued loop current order (LCO). Despite extensive research efforts, achieving its model realization and understanding the mechanism through unbiased many-body simulations have remained both elusive and challenging. In this Letter, we develop a microscopic model for LCO on the spinless kagome lattice with nonlocal interactions, utilizing unbiased functional renormalization group calculations to explore ordering tendencies across all two-particle scattering channels. At the Van Hove filling, we identify sublattice interference to suppress onsite CDW order, leaving LCO, charge bond order, and nematic CDW state as the main competitors. Remarkably, a \\(2\\times2\\) LCO emerges as the many-body ground state over a significant parameter space with strong second nearest-neighbor repulsion, stemming from the unique interplay between sublattice characters and lattice geometry. The resulting electronic model with LCO bears similarities to the Haldane model and culminates in a quantum anomalous Hall state. We also discuss potential experimental implications for kagome metals.

Recent discoveries in kagome materials have unveiled their capacity to harbor exotic quantum states, including intriguing charge density wave (CDW) and superconductivity. Notably, accumulating experimental evidence suggests time-reversal symmetry (TRS) breaking within the CDW, hinting at the long-pursued loop current order (LCO). Despite extensive research efforts, achieving its model realization and understanding the mechanism through unbiased many-body simulations have remained both elusive and challenging.In this work, we develop a microscopic model for LCO on the spinless kagome lattice with non-local interactions, utilizing unbiased functional renormalization group calculations to explore ordering tendencies across all two-particle scattering channels. At the van Hove filling, we identify sublattice interference to suppress onsite CDW order, leaving LCO, charge bond and nematic CDW state as the main competitors. Remarkably, a \\(22\\) LCO emerges as the many-body ground state over a significant parameter space with strong second nearest-neighbor repulsion, stemming from the unique interplay between sublattice characters and lattice geometry. The resulting electronic model with LCO bears similarities to the Haldane model and culminates in a quantum anomalous Hall state. We also discuss potential experimental implications for kagome metals.

Superconductivity involving finite momentum pairing can lead to spatial gap and pair density modulations, as well as Bogoliubov Fermi states within the superconducting gap. However, the experimental realization of their intertwined relations has been challenging. Here, we detect chiral kagome superconductivity modulations with residual Fermi arcs in KV3Sb5 and CsV3Sb5 by normal and Josephson scanning tunneling microscopy down to 30mK with resolved electronic energy difference at microelectronvolt level. We observe a U-shaped superconducting gap with flat residual in-gap states. This gap exhibits chiral 2 by 2 spatial modulations with magnetic field tunable chirality, which align with the chiral 2 by 2 pair density modulations observed through Josephson tunneling. These findings demonstrate a chiral pair density wave (PDW) that breaks time-reversal symmetry. Quasiparticle interference imaging of the in-gap zero-energy states reveals segmented arcs, with high-temperature data linking them to parts of the reconstructed V d-orbital states within the charge order. The detected residual Fermi arcs can be explained by the partial suppression of these d-orbital states through an interorbital 2 by 2 PDW and thus serve as candidate Bogoliubov Fermi states. Additionally, we differentiate the observed PDW order from impurity-induced gap modulations. Our observations not only uncover a chiral PDW order with orbital-selectivity, but also illuminate the fundamental space-momentum correspondence inherent in finite momentum paired superconductivity.