Monolayer Kagome metals AV3Sb5

Recently, layered kagome metals AV3Sb5 (A = K, Rb, and Cs) have emerged as a fertile platform for exploring frustrated geometry, correlations, and topology. Here, using first-principles and mean-field calculations, we demonstrate that AV3Sb5 can crystallize in a mono-layered form, revealing a range of properties that render the system unique. Most importantly, the two-dimensional monolayer preserves intrinsically different symmetries from the three-dimensional layered bulk, enforced by stoichiometry. Consequently, the van Hove singularities, logarithmic divergences of the electronic density of states, are enriched, leading to a variety of competing instabilities such as doublets of charge density waves and s- and d-wave superconductivity. We show that the competition between orders can be fine-tuned in the monolayer via electron-filling of the van Hove singularities. Thus, our results suggest the monolayer kagome metal AV3Sb5 as a promising platform for designer quantum phases.

ii based kagome metals, current experiments are mainly focused on the 3D layered structures [1-5, 11-16, 18, 19, 21-26], while their theoretical analysis largely relies on an effective kagome model in two dimensions [6][7][8][9]20]. The dimensionality has been tacitly assumed as an irrelevant parameter, but this assumption has been generically refuted in layered systems [32,33]. A few research groups have made pioneering efforts to tackle this issue by successfully exfoliating thin films of AV 3 Sb 5 [34][35][36]. However, the importance of dimensionality in this family of kagome metals has remained elusive to date.
In this work, we theoretically demonstrate that the AV 3 Sb 5 monolayer is different from the 3D layered bulk by performing density-functional theory (DFT) and mean-field theory (MFT) calculations. At the crux of our results is the absence of dimensional crossover between the monolayer and the layered bulk. We argue that the symmetry-lowering is inevitable in the monolayer, enforced by the stoichiometry of AV 3 Sb 5 . The reduced symmetries give rise to significant changes in the formation of VHSs. Notably, unconventional VHSs appear, referred to as type-II VHSs. As a consequence, enhanced electronic instabilities are observed, leading to the emergence of competing orders such as CDW doublets, time-reversal breaking CDWs, s-and d-wave superconductivity.
Our calculations predict that the AV 3 Sb 5 monolayer can be thermodynamically stable. In connection with future experiments, we calculate the anomalous Hall conductivity that can probe the correlated orders. Possible experimental schemes are discussed to tune the electron-filling based on mechanical and chemical treatment.

Crystal structure and Symmetry
We begin by elucidating the similarities and differences between the crystal structures of the bulk and monolayer AV 3 Sb 5 (A = K, Rb, Cs). As delineated in Figs. 1a and b, both systems comprise multiple sub-layers. Most importantly, a 2D kagome sub-layer is formed from V atoms, coexisting with Sb sub-layers. While these are similar in both systems, differences arise from the alkali atoms A. In the monolayer (bulk) system, alkali metals energetically favor to form rectangular (triangular) sub-layers shown in Fig. 1b Most importantly, the lowered symmetry of the monolayer AV 3 Sb 5 leads to rearrangement of the VHS in energy-momentum space. The mechanism of the rearrangement is illustrated in Fig. 1c, in which we trace the k-points that host the VHS. Hereafter, we refer to these momenta as the VHS points. The pristine BZ with the T 1×1 translational symmetry initially hosts three inequivalent VHS points at M i (i = 1, 2, 3), as in the case of the bulk AV 3 Sb 5 (left panel in Fig. 1c). Upon the zone folding by lowering T 1×1 to T √ 3×1 , the M 1 VHS point is folded to Γ and the M 1 and M 2 VHS points are merged to the M point of the reduced BZ (middle panel in Fig. 1c). The symmetry-lowering further hybridizes the two states at M (M 2 and M 3 ), such that it annihilates the VHS at M and creates four new VHS points off M , marked as P i (i = 1, 2, 3, 4) in the right panel of Fig. 1c.
The rearrangement of VHS points is observed in our first-principles calculations. Figures 2a   and b show exemplary DFT bands of KV 3 Sb 5 with archetypal kagome bands distilled by our tight-binding theory (see Methods for the details of the TB model). In Fig. 2b, the divergence of the density of states (DOS) is clearly observed at E = −6 (+9) meV. A close inspection reveals that the diverging DOS at E = −6 meV arises from off high-symmetry momenta at P i = M + (±0.054, ±0.021)Å −1 (Fig. 2d), while the divergence at E = 9 meV arises from the exact highsymmetry Γ point (Fig. 2c). In this respect, the VHS points at E = 9 meV (E = −6 meV) belong to the type-I (type-II) class, where the type-I (II) refers to a class of VHSs that originates from (off) time-reversal invariant momenta [37][38][39]. Our calculations further reveal that the type-II VHSs generically occur in the kagome metals regardless of A = K, Rb, and Cs (see Supplementary Note 2 for the DFT results of the RbV 3 Sb 5 and CsV 3 Sb 5 monolayers).
The emergence of the type-II VHS is one of the key features of the monolayer AV 3 Sb 5 . Few remarks are as follows. First, the type-II VHS points P i (i = 1, 2, 3, 4) ( Fig. 2d) consist of a mixed contribution from both the B and C sublattices ( Supplementary Fig. 3), referred to as a mixed-type flavor [12,20,40]. This is in contrast to the type-I VHS point at Γ (Fig. 2c), which is purely contributed from the A sublattice, referred to as a pure-type flavor. Second, the increased number of VHS points quantitatively changes the characteristic of the diverging DOS at −6 meV.
Namely, the peak at −6 meV is significantly enhanced than that of the type-I VHS at 9 meV ( Fig. 2b), contributed from the quartet VHS points at P i (i = 1, 2,3, 4). Such quantitative changes are of immediate impact on the electronic properties, such as instabilities driven by the type-II vi

Competing orders
The rearranged VHSs manifest their properties in competing orders of correlated electronic states. In monolayer AV 3 Sb 5 , we employ the standard mean-field theory with the constructed TB model and uncover phase diagrams with CDWs and superconductivity (SC). The onsite and nearest-neighbor Coulomb interactions are introduced, where U (V ) describes the on-site (nearest-neighbor) density-density type interaction and R, α, and σ represent the lattice site, sublattice, and spin, respectively. We consider two classes of order Remarkably, any CDW order parameter in the monolayer AV 3 Sb 5 forms a doublet, as illustrated in Fig. 3c. The doublet formation of CDWs is one of the key characteristics of the monolayer AV 3 Sb 5 , originating from the reduced symmetry. The lowered T √ 3×1 -translational symmetry of AV 3 Sb 5 monolayer plays a crucial role to double the CDW channels of the higher T 1×1 -symmetry, and the number of multiplets is solely determined by their quotient group, T 1×1 /T √ 3×1 = Z 2 . For example, the two SD-CDW phases, dubbed SD-1 and SD-2, are the Z 2 members, distinguished by the alkali chains hosted on and off the center of SD, respectively, as illustrated in Fig. 3c.
The corresponding phenomenological Landau theory of the doublet CDWs becomes exotic.
Introducing a bosonic real two-component spinor, Ψ T SD ≡ (ψ SD−1 , ψ SD−2 ) with order parameters of SD-1 (ψ SD−1 ) and SD-2 (ψ SD−2 ), the Landau functional for the SD-CDW phases is given by with phenomenological constants r SD and s SD . Here, the Pauli matrixρ z describes the spinor space and higher order terms are omitted for simplicity. The s SD -term describes a free-energy difference between SD-1 and SD-2, which is nonzero when the T 1×1 -translational symmetry is broken. Depending on s SD , the system energetically favors one of the doublet CDWs, enriching phase diagrams of the monolayer AV 3 Sb 5 .
Our mean-field analysis indeed finds enriched phase diagrams of the AV 3 Sb 5 monolayer. We consider six configurations of CDWs (see Fig. 3c) and nine spin-singlet channels of SCs (see Ta- Tables I and II. viii ble II). In Figs. 3a, b, we illustrate representative mean-field phase diagrams of KV 3 Sb 5 in U -V space obtained at two different chemical potentials µ 1 = −6 meV and µ 2 = 35 meV, where the zero chemical potential is set to the neutral filling. In what follows, we point out key observations made from the phase diagrams.
First, five distinct CDW orders can be accessible by fine-tuning the chemical potential µ. For example, in the vicinity of type-II VHS at µ 1 = −6 meV (Fig. 3a), SD-2 and ISD-2 dominantly occur with sizable regions of ISD-1 in the energy ranges of -0.6 eV < U < 0.4 eV and -20 meV < V < 40 meV. Similarly, a TRSB-2 CDW phase is uncovered under the condition µ ≥ 30 meV (Fig. 3b) in a wide range of U and V values. Moreover, the SD-1 phase appears near the type-I VHS at µ 3 = 9 meV (see Supplementary Note 6). A small variation of chemical potential -6 meV < µ < 40 meV can tune the types and flavors of the VHSs, which should enable an on-demand onset of a variety of CDW phases, ranging from ISD-1/2, SD-1/2, to TRSB-2.
Second, competition between CDWs and SC is generically observed. A conventional s-wave SC phase is observed near µ 1 = −6 meV, which competes with ISD-1/2 and SD-2 at negative U and positive V as shown in Figs. 3a, b. Similarly, an unconventional d-wave SC phase is observed near µ 2 = 35 meV. This competes with ISD-1 at positive U and negative V interactions as shown in Fig. 3b. Regarding the d-wave SC, the corresponding chemical potential is quite higher than the energy of the type-II VHS µ 1 , closer to the edge of the high energy band. Thus, we conclude that the dominant CDW and SC phases cannot be solely explained as a results of VHS. Instead, the interplay between the VHS, filling, and interaction should be crucial for complete understanding of the competing mechanism.
The final important observation from our mean-field study is nontrivial topology of the correlated CDW gaps. Notably, a non-zero Chern number C is induced in the energy spectra when gapped by the two time-reversal symmetry broken CDW phases TRSB-1 and TRSB-2. For example, the lowest unoccupied and highest occupied energy spectra of TRSB-1 (TRSB-2) host the Chern number C = 2 and C = 1 (C = 3 and C = −2), respectively. As shown in Fig. 4a, the different Chern numbers between the two phases arise due to the concurrent sign-change of the Berry curvature at high-symmetry momenta M 1 , K 1 , and K 2 . These topological CDW gaps with distinct Chern numbers motivate us to calculate the anomalous Hall conductivity σ xy shown in

DISCUSSION AND CONCLUSION
We have so far investigated unique features of the AV 3 Sb 5 monolayer. Our system is unlike the bulk, hosted in distinct symmetry class. The contrast is even more evident in the phase diagrams that we calculated with the bulk D 6h and the monolayer D 2h symmetries, respectively (see Supplementary Fig. 7 for the D 6h phase diagrams). The lowered D 2h symmetry in the monolayer features a tendency to foster the CDW orders. This is in line with the previous experiments, in which a CDW order is observed to suppress SC as the thickness of the AV 3 Sb 5 film decreases [34,35].
The exotic orders should be accessible in monolayer AV 3 Sb 5 in a controlled fashion. Our x mean-field diagrams (Figs. 3a and b) show that the occurrence of a specific electronic order is highly contingent upon the correct filling of electrons, which can be fine-tuned via mechanical and chemical means. For example, by applying a uniform biaxial strain in a range of ±2% variations, µ can be tuned from 50 meV to -100 meV (Fig. 4c). Moreover, the light doping of alkali atoms A (see Fig. 4d) or substituting Sb with Sn [41] can be a fine knob to adjust the chemical potential. Owing to the 2D geometry of the AV 3 Sb 5 monolayer, we believe that there exist more opportunities (such as ionic gating) to tailor the competing orders hosted therein.
Finally, we argue that the AV 3 Sb 5 monolayer should be possible to synthesize. The cohesive energy of the monolayer is calculated as 3.8 eV/atom for all three alkali atoms. This value is comparable to the bulk value of ∼ 3.9 eV/atom. Similarly, the exfoliation energies of the monolayer are calculated as 42, 45, and 45 meV/Å 2 for A = K, Rb, and Cs, respectively. These values are amount to existing two-dimensional materials, such as graphene (∼ 21 meV/Å 2 ) [42], hBN (∼ 28 meV/Å 2 ) [42], and Ca 2 N (∼ 68 meV/Å 2 ) [43]. In addition, our DFT phonon bands of the ISD-1 phase are clean of imaginary frequencies (see Supplementary Note 8), implying dynamical stability of the monolayer structure. Encouragingly, the recent experiments have successfully exfoliated thin layers of AV 3 Sb 5 up to five layers using the taping methods [35]. Current developments of the chemical solution reaction method could be an appropriate technique to weaken the interlayer interaction of the bulk system and separate the monolayer [44].
In the 2 × 2 unit cell, the mean-field Hamiltonian is written as, and with the twelve-component spinor,Ψ T k = (Ã T k ,B T k ,C T k ) andα T k = (α 1,k , α 2,k , α 3,k , α 4,k ). The indices i ∈ {1, 2, 3, 4} and σ ∈ {↑, ↓} denote the orbital sites and spins, respectively. We note that the order parameter φ ∈ (Φ, ∆) is set as a real value. The specific forms of 4 × 4 bond matrices O a (k) and 12 × 12 pairing gap functionsΓ(k) are tabulated in Table I TRSB-2 (Φ = 0) method [47]. For the exchange-correlation energy, the generalized-gradient approximation functional of Perdew-Burke-Ernzerhof [48] is employed. The van der Waals correction is included within the zero damping DFT-D3 method of Grimme [49]. The kinetic energy cutoff for the plane wave basis is 300 eV. The force criteria for optimizing the structures is set to 0.01 eV/Å. The monolayer AV 3 Sb 5 is simulated using a periodic supercell with a vacuum spacing of ∼ 20Å. The k-space integration is done with 10×17 k-points for the √ 3×1 structure. For the DOS calculation, we use 46 × 69 k-points. The exfoliation energy is calculated by using the Jung-Park-Ihm method [42]. The phonon dispersions are calculated using the finite difference method implemented in the Phonopy software [50].
Berry curvature and anomalous Hall conductivity. -The berry curvature Ω n (k) associated with the n-th energy band of the TB Hamiltonian H(k) is given by Here, E n (k) and |u nk are nth eigenvalue and eigenstate of H(k) and the derivative in the momentum space ∂ i ≡ ∂ ∂k i is adopted. The anomalous Hall conductivity is calculated as a function of energy E by integrating the n-band Berry curvatures for E n < E over the BZ σ xy (E) = 1 2π En<E Ω n (k).