Introduction

The interaction between electrons in a solid and the vibrations of its nuclei (phonons) can induce a variety of ordered phases1,2,3,4,5,6. This electron-phonon coupling modifies the effective mass of itinerant electrons, and the resulting dressed quasiparticles (polarons) can pair and condense into a superconducting (SC) phase or form a periodic modulation of electron density, i.e., CDW order. At low temperatures, these various phases can compete or potentially coexist. Over the past several decades, studies of model Hamiltonians describing electron-phonon coupling have attempted to capture the interplay between their emergent ordered phases. In particular, the Holstein model7 has been subject to much numerical and analytical study because it incorporates a simplified electron-phonon interaction into a straightforward tight-binding Hamiltonian, yet exhibits a variety of competing ordered ground states.

A key feature of the Holstein model is an on-site momentum-independent electron-phonon coupling, which leads to an effective electron-electron attraction. Phonons are modeled as quantum harmonic oscillators of fixed frequency ω0 situated on each site of a lattice, with their motion independent of their neighbors. At low temperatures and at particular electron filling fractions, numerical studies have revealed the emergence of CDW order on square8,9,10,11,12,13,14,15,16,17,18,19,20,21,22, triangular23, cubic24, and honeycomb lattices25,26, with the transition temperature being sensitive to lattice geometry and dimensionality. A recent study of the Lieb lattice has also established the existence of CDW order in the Holstein model in a flat band system27.

In recent years, kagome lattices have attracted attention as a host of exotic phases owing to their high degree of geometrical frustration, and the presence of a flat band. The spin-1/2 kagome lattice Heisenberg antiferromagnet (KHAF) with nearest-neighbor interactions lacks any magnetic ordering, but the exact nature of the ground state been subject to much debate, with several candidates such as the Dirac spin-liquid, Z2 spin-liquid, and valence bond crystal proposed28,29,30,31. A recent study of the KHAF in the presence of spin-lattice coupling has shown that introducing Einstein phonons on each site can induce a magnetically ordered phase32. For example, a \(\sqrt{3}\times \sqrt{3}\) ordered phase with a 1/3-magnetization plateau emerges in weak magnetic field, breaking a Z3 symmetry, with the transition belonging to the 3-state Potts model universality class. The ordering wavevector for this phase lies at the K-points, i.e., corners, of the hexagonally-shaped Brillouin zone.

The ground state properties of the half-filled kagome lattice Hubbard model are also debated. Dynamical mean field theory (DMFT) and determinant quantum Monte Carlo (DQMC) studies have identified a metal-insulator transition (MIT) in the range Uc/t ~ 7–933,34,35, while variational cluster approximation (VCA) calculations estimate Uc/t ~ 4–536. Recent density-matrix renormalization group (DMRG) calculations find a MIT at Uc/t ~ 5.4, along with strong spin-density wave fluctuations in the translational symmetry breaking insulating phase, signaled by an enhancement in the spin structure factor at the K-points of the Brillouin zone37. CDW formation on the kagome lattice has also been observed in the extended Hubbard model. At an average electron density per site of 〈n〉 = 2/3 or 4/3, or at the van Hove filling 〈n〉 = 5/6, several types of order have been observed38,39,40,41, including CDW, spin-density wave, and bond ordered wave states. In particular, at large V/U (where V is the nearest-neighbor repulsion), a CDW phase with a \(\sqrt{3}\times \sqrt{3}\) supercell has been proposed for 〈n〉 = 2/3 and 5/6, which has been termed CDW-III in previous studies40,41. In the attractive Hubbard model, recent results42 indicate short-ranged charge correlations at 〈n〉 = 2/3 satisfying the triangle rule.

Recent experiments on kagome metals such as AV3Sb5 (A = K, Rb, Cs) also motivate an understanding of CDW formation on this geometry43,44,45,46,47,48,49,50,51,52,53,54,55. In these systems, charge ordering has been observed at the M-points, corresponding to lattice distortions that form a star-of-David or inverse star-of-David CDW pattern. This ordering wavevector coincides with saddle points in the band structure and van Hove singularities where electronic correlations are enhanced. Theoretical studies of these materials56,57,58,59,60,61,62,63,64, including first-principles density functional theory and mean field calculations, have corroborated these findings, where CDW ordering at the M-points has been observed near the van Hove filling.

Finally, kagome lattices have also been achieved in ultracold atom experiments65 where they have been used to examine Bose-Einstein condensation of 87Rb66, and Rydberg atoms with large entanglement entropy and topological order67.

Although the Holstein coupling provides a paradigmatic model of the electron-phonon interaction, the properties of the Holstein model on the kagome lattice are not yet understood, and the possible existence of CDW order remains hitherto unexplored. In this work, we study the kagome lattice Holstein model using a scalable algorithm based upon hybrid Monte Carlo (HMC) sampling68, and measure the charge correlations as a function of temperature, electron density, phonon frequency, and electron-phonon coupling. We present evidence for CDW order appearing at an average electron density per site of 〈n〉 = 2/3, with an ordering wavevector at the K-points of the Brillouin zone, yielding a \(\sqrt{3}\times \sqrt{3}\) supercell. Away from this filling, we find no signatures of CDW order at any ordering momenta for temperatures T ≥ t/20.

Results

Kagome lattice Holstein model

The Holstein model describes electrons coupled to local dispersionless phonon modes in a lattice through an on-site electron-phonon interaction7. Its Hamiltonian is

$$\begin{array}{lll}\hat{H}\,=\,-t\mathop{\sum}\limits_{\langle i,j\rangle ,\sigma }\left({\hat{c}}_{i\sigma }^{{\dagger} }{\hat{c}}_{j\sigma }+h.c.\right)-\mu \mathop{\sum}\limits_{i\sigma }\left({\hat{n}}_{i\sigma }-\frac{1}{2}\right)\\ \,\qquad+\,\frac{1}{2}\mathop{\sum}\limits_{i}{\hat{P}}_{i}^{2}+\frac{{\omega }_{0}^{2}}{2}\mathop{\sum}\limits_{i}{\hat{X}}_{i}^{2}+\lambda \mathop{\sum}\limits_{i\sigma }{\hat{n}}_{i\sigma }{\hat{X}}_{i},\end{array}$$
(1)

where \({\hat{c}}_{i\sigma }^{{\dagger} }\)\(({\hat{c}}_{i\sigma }^{})\) are creation (destruction) operators for an electron at site i with spin σ = {↑↓}, \({\hat{n}}_{i\sigma }={\hat{c}}_{i\sigma }^{{\dagger} }{\hat{c}}_{i\sigma }^{}\) is the electron number operator, and μ is the chemical potential, which controls the overall filling fraction. The first term describes itinerant electrons hopping between nearest-neighbor sites of the lattice, with a fixed hopping parameter t = 1 setting the energy scale. In the non-interacting limit, the electronic bandwidth is W = 6 for the kagome lattice. On each site i are local oscillators of fixed frequency ω0, with \({\hat{X}}_{i}\) and \({\hat{P}}_{i}\) the corresponding phonon position and momentum operators, respectively, with the phonon mass normalized to M = 1. The local electron density \({\hat{n}}_{i\sigma }\) is coupled to the displacement \({\hat{X}}_{i}\) through an on-site electron-phonon interaction λ, which we report here in terms of a dimensionless parameter \({\lambda }_{{{{\rm{D}}}}}={\lambda }^{2}/{\omega }_{0}^{2}\,W\).

The kagome lattice vectors a1 = (1, 0) and \({{{{\bf{a}}}}}_{2}=(\frac{1}{2},\frac{\sqrt{3}}{2})\) are shown in Fig. 1a, with corresponding reciprocal lattice vectors \({{{{\bf{b}}}}}_{1}=(2\pi ,-\frac{2\pi }{\sqrt{3}})\) and \({{{{\bf{b}}}}}_{2}=(0,\frac{4\pi }{\sqrt{3}})\), where we have set the lattice constant a = 1. There are three sites per unit cell with basis vectors uA = (0, 0), \({{{{\bf{u}}}}}_{{{{\rm{B}}}}}=(\frac{1}{2},0)\), and \(\scriptstyle{{{{\bf{u}}}}}_{{{{\rm{C}}}}}=\left(\frac{1}{4},\frac{\sqrt{3}}{4}\right)\), forming a network of corner sharing triangles with three sublattices, as shown in Fig. 1a. Each site i may instead be indexed by unit cell and the sublattice {A, B, C}, such that e.g., ni,α denotes the electron density at the site belonging to sublattice α within the unit cell at position i. In this work, we study finite-size lattices with periodic boundary conditions, with linear dimension L (up to L = 15), N = L2 unit cells, and Ns = 3N total sites. Note that discrete momentum values are given by \({{{\bf{k}}}}=\frac{{m}_{1}}{L}{{{{\bf{b}}}}}_{1}+\frac{{m}_{2}}{L}{{{{\bf{b}}}}}_{2}\) where mi is an integer and 0 ≤ mi < L.

Fig. 1: Kagome lattice and band structure.
figure 1

a Geometry of the kagome lattice for L = 6, with lattice vectors a1 = (1, 0) and \({{{{\bf{a}}}}}_{2}=(\frac{1}{2},\frac{\sqrt{3}}{2})\). Colors denote the three triangular sublattices. b Left: The tight-binding electronic band structure for the kagome lattice showing the three distinct bands. Dashed lines indicate the Fermi energy at specific electron densities. Right: The non-interacting density of states D(E) for the kagome lattice. A delta function at E = 2t is due to the flat band.

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There are multiple ways to break the sublattice symmetry of the kagome lattice. It is, therefore, important to construct an order parameter that will detect charge ordering independent of the charge distribution within the unit cell. For example, at a filling fraction of 1/3, electrons may localize by doubly occupying only one site per unit cell, breaking a Z3 symmetry. We, therefore, define an order parameter ρcdw that with perfect CDW order takes on one of three values \({{{{\rm{e}}}}}^{{{{\rm{i}}}}2\pi (\frac{s}{3})}\), where s = {0, 1, 2} corresponds to which way this symmetry is broken. The order parameter ρcdw should also be zero in the completely disordered state, where for any unit cell i we have \(\langle {\hat{n}}_{{{{\bf{i}}}},{{{\rm{A}}}}}\rangle =\langle {\hat{n}}_{{{{\bf{i}}}},{{{\rm{B}}}}}\rangle =\langle {\hat{n}}_{{{{\bf{i}}}},{{{\rm{C}}}}}\rangle\). Hence we define

$${\rho }_{{{{\rm{cdw}}}}}=\frac{{n}_{{{{\rm{c}}}}}}{2N}\mathop{\sum}\limits_{{{{\bf{i}}}}}{{{{\rm{e}}}}}^{-{{{\rm{i}}}}({{{\bf{q}}}}\cdot {{{\bf{i}}}})}\left(\langle {\hat{n}}_{{{{\bf{i}}}},{{{\rm{A}}}}}\rangle +{{{{\rm{e}}}}}^{{{{\rm{i}}}}\frac{2\pi }{3}}\langle {\hat{n}}_{{{{\bf{i}}}},{{{\rm{B}}}}}\rangle +{{{{\rm{e}}}}}^{{{{\rm{i}}}}\frac{4\pi }{3}}\langle {\hat{n}}_{{{{\bf{i}}}},{{{\rm{C}}}}}\rangle \right)$$
(2)

where i is a unit cell index, N is the total number of unit cells, q is the ordering wavevector, and nc is a normalization constant included to fix ∣ρcdw∣ = 1 in the case of perfect CDW order. A structure factor that scales with system size can then be defined as \({S}_{{{{\rm{cdw}}}}}({{{\bf{q}}}})\propto N\langle | {\hat{\rho }}_{{{{\rm{cdw}}}}}{| }^{2}\rangle\), where again a proportionality constant can be included to fix Scdw(q) = N for the case of perfect CDW order.

For any pair of sites in the kagome lattice, we denote their density-density correlation in position space by

$${c}_{\alpha ,\nu }({{{\bf{r}}}})=\frac{1}{N}\mathop{\sum}\limits_{{{{\bf{i}}}}}\langle {\hat{n}}_{{{{\bf{i}}}}+{{{\bf{r}}}},\alpha }{\hat{n}}_{{{{\bf{i}}}},\nu }\rangle ,$$
(3)

where α and ν label the sublattice {A, B, C} of the two sites, and r is the displacement vector between their unit cells. The Fourier transform of cα,ν(r) gives a generic charge structure factor

$${S}_{\alpha ,\nu }({{{\bf{q}}}})=\mathop{\sum}\limits_{{{{\bf{r}}}}}{e}^{{{{\rm{i}}}}{{{\bf{q}}}}\cdot {{{\bf{r}}}}}{c}_{\alpha ,\nu }({{{\bf{r}}}}),$$
(4)

which provides information about the nature of an emergent CDW phase, where q is a discrete momentum value within the first Brillouin zone. For an ideal CDW pattern with ordering wavevector q, Sα,α(q) will reach a maximal value proportional to the number of sites, while for α ≠ ν the structure factor will vanish.

In the following section, we show evidence of CDW ordering on the kagome lattice where electrons localize on only one site per unit cell but alternates cyclically between the {A, B, C} sublattices from one unit cell to the next. To study the onset of this phase, we set nc = 1 in Eq. (2) and define a charge structure factor

$$\begin{array}{lll}{S}_{{{{\rm{cdw}}}}}({{{\bf{q}}}})\,=\,3N\langle | {\hat{\rho }}_{{{{\rm{cdw}}}}}{| }^{2}\rangle \\ \qquad\qquad=\,\frac{3}{4}\mathop{\sum}\limits_{\alpha }\left({S}_{\alpha ,\alpha }({{{\bf{q}}}})-\frac{1}{2}\mathop{\sum}\limits_{\nu \ne \alpha }{S}_{\alpha ,\nu }({{{\bf{q}}}})\right).\end{array}$$
(5)

Additional details are given in Supplementary Discussion. Note that we employ a μ-tuning algorithm69 to determine the chemical potential for any desired target density.

Measurements of charge order

For the kagome lattice, the non-interacting tight-binding electronic structure with t > 0 has three separate bands, including one flat band at the highest energy (E = 2t). The lower bands touch at two inequivalent Dirac points in the Brillouin zone, which we denote \(K=(\frac{2\pi }{3},\frac{2\pi }{\sqrt{3}})\) and \({K}^{{\prime} }=(\frac{4\pi }{3},0)\). The lower band is completely occupied at an average electron density per site of 〈n〉 = 2/3 (i.e., an overall filling fraction of f = 1/3), while the upper band is fully occupied at 〈n〉 = 4/3 (f = 2/3). There are also saddle points in the band structure at the point \(M=(\pi ,\frac{\pi }{\sqrt{3}})\), which produce singularities in the density of states and sit at the Fermi level for average electron densities of 〈n〉 = 1/2 (f = 1/4) and 〈n〉 = 5/6 (f = 5/12). Fig. 1b plots the non-interacting band structure and density of states for the kagome lattice, illustrating these features.

To begin, we study the variation of local quantities as a function of electron density, at fixed ω0 and λD. We set ω0/t = 0.1 to facilitate CDW ordering in the Holstein model, as bipolarons should localize more readily in the limit ω0/t → 0 due to reduced quantum fluctuations. We also fix a moderate value of the electron-phonon coupling λD = 0.4. We will discuss the rationale for this choice of parameters shortly.

In Fig. 2a, we show the average electron density per site 〈n〉 as a function of chemical potential μ for an L = 12 lattice, as the inverse temperature is varied from β = 2–14. We observe the formation of a plateau at 〈n〉 = 2/3 as the temperature is lowered, signaling the opening of a gap. No signatures of CDW ordering is observed at fillings away from 〈n〉 = 2/3 for these parameters. We also calculate the average electron kinetic energy as a function of electron density as shown in Fig. 2b. We observe a sharp change at 〈n〉 = 2/3, where the magnitude of the electron kinetic energy becomes maximal. This is a signature of a CDW phase transition, since a configuration of doubly-occupied sites surrounded by empty nearest-neighbor sites maximizes the number of bonds along which electron hopping is permitted (and corresponds to an average electron density per site 〈n〉 = 2/3 on the kagome lattice). Note that since the kagome lattice is not bipartite, particle-hole symmetry is not present and thus both the kinetic energy and average filling are not symmetric about half-filling.

Fig. 2: Electron density and kinetic energy.
figure 2

a Average electron density per site 〈n〉 as a function of the tuned chemical potential μ, for an L = 12 lattice with ω0 = 0.1 and λD = 0.4 fixed. Results are shown for β = 2, 8, and 14, with a dashed line indicating the filling 〈n〉 = 2/3. b Electron kinetic energy as a function of the electron density 〈n〉, for the same set of parameters. Error bars correspond to the standard deviation of the measured mean.

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To further study the opening of a CDW gap as the temperature is lowered, we calculate the momentum integrated spectral function A(ω), which is related to the imaginary time dependent Green’s function through the integral equation

$$G({{{\bf{k}}}},\tau )=\langle \hat{c}({{{\bf{k}}}},\tau ){\hat{c}}^{{\dagger} }({{{\bf{k}}}},\tau )\rangle =\int\,d\omega A({{{\bf{k}}}},\omega )\frac{{{{{\rm{e}}}}}^{-\omega \tau }}{1+{{{{\rm{e}}}}}^{-\beta \omega }},$$
(6)

which we invert using the maximum entropy method to obtain A(ω)70. In Fig. 3, we show the momentum integrated spectral function for an L = 15 lattice (Ns = 775) for a range of temperatures down to β = 24, again fixing ω0 = 0.1, λD = 0.4, and an average electron density per site of 〈n〉 = 2/3. We observe three peaks in the spectral function corresponding to the three-band structure. As the temperature is lowered, A(ω) reaches zero and a finite gap begins to open at β ≳ 18, as shown in the bottom panel, indicating a transition to an insulating CDW phase.

Fig. 3: Spectral function.
figure 3

Top: Momentum integrated spectral function A(ω) shown for a range of inverse temperatures from β = 2 to β = 24, at filling fraction 〈n〉 = 2/3 (with ω0 = 0.1, λD = 0.4). The linear lattice dimension is L = 15 i.e., Ns = 775. Bottom: a close-up view of the finite gap opening for β ≳ 18 where A(ω) = 0.

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At an average electron density per site of 〈n〉 = 2/3, the lower energy band is completely filled and touches the upper band at the Dirac points K and \({K}^{{\prime} }\). To study the onset of CDW order at this filling, we, therefore, calculate the charge structure factor Scdw [Eq. (5)] evaluated at q = K, as a function of phonon frequency, electron-phonon coupling, and temperature.

In Fig. 4, we show the variation of Scdw(K) as the phonon frequency ω0 is increased from 0.1 to 1.0. In the antiadiabatic limit (ω0 → ∞), deformation of the lattice is weakened as sites respond more quickly to electron hopping and bipolarons do not readily localize, inhibiting the formation of a stable CDW pattern22. In addition, quantum fluctuations are enhanced at large ω0, further suppressing CDW order23. For ω0 ≳ 0.4 we observe no significant growth in Scdw(K) as the temperature is lowered from β = 2 to β = 20. However, for ω0 ≲ 0.3, the structure factor begins to increase in magnitude as the temperature is reduced, growing more rapidly with β as ω0 → 0. We therefore fix ω0 = 0.1, and vary the dimensionless electron-phonon coupling λD, in order to determine the region in which CDW order at 〈n〉 = 2/3 is most enhanced and subsequently estimate Tc for these parameters.

Fig. 4: Charge structure factor vs. phonon frequency.
figure 4

Charge structure factor Scdw(K) as a function of phonon frequency, for a range of temperatures from β = 2 to β = 20. Results are shown for an L = 6 lattice at 〈n〉 = 2/3, λD = 0.4. Error bars correspond to the standard deviation of the measured mean.

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At small values of λD, we find no enhancement in Scdw(K) from β = 2 to β = 20 i.e., for λD ≲ 0.3 there is no sign of CDW order in this temperature range, as shown in Fig. 5. This may be due to the critical temperature becoming exponentially suppressed as λD → 0. However, another possibility is a finite λD is necessary for CDW formation, as is the case in the honeycomb lattice Holstein model at half-filling25, which similarly has Dirac cones and a vanishing density of states at the Fermi surface. As λD increases, the effective electron-electron attraction is enhanced, and we observe an increase in the charge structure factor as pairs of electrons arrange themselves into a periodic CDW. As the temperature is reduced, we find that there is a maximum in Scdw(K) at approximately λD ≈ 0.4. At larger λD, the CDW structure factor is smaller, and eventually no significant growth is observed as the temperature is lowered from β = 2 to β = 20. This behavior might originate from the higher effective bipolaron mass at large λD, which will hinder their arrangement into an ordered CDW phase, as the energy barrier associated with moving from site to site is proportional to λD, thus promoting self-trapping. Consequently, Tc rapidly decreases as λD becomes much larger than its optimal value. We note that similar behavior has been observed in the honeycomb, square, and Lieb lattice Holstein models25,27.

Fig. 5: Charge structure factor vs. λD.
figure 5

Charge structure factor Scdw(K) as a function of the dimensionless electron-phonon coupling λD, for a range of temperatures from β = 2 to β = 20. Results are shown for an L = 6 lattice at 〈n〉 = 2/3, ω0 = 0.1. Error bars correspond to the standard deviation of the measured mean.

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The momentum dependence of Scdw(q) is shown in Fig. 6, where the charge structure factor at 〈n〉 = 2/3 is evaluated over the first Brillouin zone for an L = 12 lattice. An enhancement in the structure factor is observed at the Dirac points as the temperature is lowered, corresponding to the onset of an ordered CDW phase, with the magnitude of Scdw increasing rapidly around β ≳ 17. For all other momentum values, including at the M and Γ-points, we find no enhancement in charge correlations with inverse temperature β, at this filling.

Fig. 6: Charge structure factor in momentum space.
figure 6

Charge structure factor Scdw(q) shown across the Brillouin zone of the kagome lattice with L = 12, shown for β = 14, 17 and 20. The locations of high-symmetry points in momentum space at \(K=(2\pi /3,2\pi /\sqrt{3})\), \({K}^{{\prime} }=(4\pi /3,0)\), \(M=(\pi ,\pi /\sqrt{3})\), and Γ = (0, 0) are indicated.

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A real-space depiction of the CDW correlations at 〈n〉 = 2/3 is shown in Fig. 7, which plots density-density correlations \(\langle \hat{n}({{{\bf{r}}}})\hat{n}({{{\bf{0}}}})\rangle\) over an L = 12 lattice with periodic boundary conditions. Here r = 0 is the position of a fixed reference site belonging to the A sublattice. Hence Fig. 7 depicts cα,ν(r) with the origin fixed at this reference site. The CDW pattern is characterized by the localization of electron pairs on only one site per unit cell, which belongs to either the A, B, or C sublattice, alternating cyclically between these from one unit cell to the next (in both the a1 and a2 directions). The fact that K and \({{{{\bf{K}}}}}^{{\prime} }\) are the ordering wavevectors for this pattern can be understood as follows. In terms of the reciprocal lattice vectors, we have \({{{\bf{K}}}}=\frac{1}{3}({{{{\bf{b}}}}}_{1}-{{{{\bf{b}}}}}_{2})\) and \({{{\bf{{K}}}^{{\prime} }}}=\frac{1}{3}(2{{{{\bf{b}}}}}_{1}+{{{{\bf{b}}}}}_{2})\). If the doubly-occupied sites are separated by a displacement r = n1a1 + n2a2, then the Fourier transform of the density-density correlation function will have peaks at K or \({{{\bf{{K}}}^{{\prime} }}}\) if K ⋅ r = 2mπ or \({{{\bf{{K}}}^{{\prime} }}}\cdot {{{\bf{r}}}}=2m\pi\), where \(m\in {\mathbb{Z}}\). This is satisfied if \(({n}_{1}-{n}_{2})\,{{\mathrm{mod}}}\,\,3=0\) (for K) or \((2{n}_{1}+{n}_{2})\,{{\mathrm{mod}}}\,\,3=0\) (for \({{{\bf{{K}}}^{{\prime} }}}\)), which are equivalent conditions. In other words, moving along either the a1 or a2 directions, density-density correlations will repeat with a periodicity of three unit cells, i.e., for each unit cell, the site on which the electron pairs localize will alternate cyclically between the {A, B, C} sublattices. For any given unit cell, the onset of this type of CDW order therefore breaks a Z3 symmetry, and the phase transition should belong to the 3-state Potts model universality class.

Fig. 7: Real space density-density correlations.
figure 7

Real space density-density correlations \(\langle \hat{n}({{{\bf{0}}}})\hat{n}({{{\bf{r}}}})\rangle\), where \(\hat{n}({{{\bf{0}}}})\) denotes the electron density at a reference site located at the origin (gray region). For each site at position r, the color of its Voronoi cell indicates the magnitude of \(\langle \hat{n}({{{\bf{0}}}})\hat{n}({{{\bf{r}}}})\rangle\). Results are shown for an L = 12 lattice with periodic boundary conditions, for β = 16 (left) and β = 20 (right) at filling 〈n〉 = 2/3 (with λD = 0.4 and ω = 0.1).

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Estimation of T c for CDW phase

For an approximate estimate of the critical temperature we can examine when the correlations become long-ranged on a finite-size lattice. As shown in Fig. 7, at β = 16 the charge order is emerging, but it is not quite long-ranged. At β = 20 however, we see the periodic density correlations persist over the whole lattice. This suggests Tc should lie at an intermediate temperature between these two values; however, for a more accurate estimate, we must study the onset of charge order for several different lattice sizes.

In Fig. 8, we show the variation of the charge structure factor Scdw(K) with inverse temperature β, for lattices with linear dimension L = 6, 9, 12 and 15, for a range of temperatures down to β = 24. At high temperatures, Scdw(K) is relatively small and independent of lattice size. However, as the temperature is reduced, Scdw(K) grows and becomes dependent on the lattice size for β ≳ 18. This signals that correlations are becoming long-ranged and thus sensitive to system size on a finite lattice, and suggests a critical temperature of βc ≈ 18. A more accurate determination of Tc can be made by studying the correlation ratio

$${R}_{{{{\rm{c}}}}}=1-\frac{{S}_{{{{\rm{cdw}}}}}({{{\bf{q}}}}+d{{{\bf{q}}}})}{{S}_{{{{\rm{cdw}}}}}({{{\bf{q}}}})},$$
(7)

where the ordering wavevector q = K here, and ∣dq∣ is the spacing between discrete momentum values for a lattice of linear dimension L. For the kagome lattice we average over the six nearest neighbors of the K-point in momentum space to obtain S(K + dq). The correlation ratio Rc is defined such that in the CDW phase, Rc → 1 as L → ∞, (since Scdw(q) will diverge with L if there is long-range order), while Rc → 0 if there is no long-range order. When plotted for different lattice sizes, the crossing of Rc curves gives an estimate of the critical point. In Fig. 9, we plot Rc for lattices with L = 6, 9, 12, and 15, for the same parameters as in Fig. 8 (〈n〉 = 2/3, ω0 = 0.1, λD = 0.4). There is a crossing at βc ≈ 18, which is consistent with our previous estimates of βc obtained from observing the opening of a finite gap in A(ω), the onset of long-ranged density-density correlations, and the temperature at which Scdw becomes dependent on lattice size.

Fig. 8: Charge structure factor vs. inverse temperature.
figure 8

Charge structure factor Scdw(K) as a function of inverse temperature β, for lattice sizes L = 6, 9, 12 and 15, at filling 〈n〉 = 2/3. A lattice size dependence in the order parameter emerges at β ≳ 18, indicating the onset of CDW order. Here, we fix λD = 0.4 and ω0 = 0.1. Error bars correspond to the standard deviation of the measured mean.

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Fig. 9: Correlation ratio crossing.
figure 9

Correlation ratio Rc as a function of β, showing a crossing at βc ≈ 18. Data is shown for lattice sizes L = 6, 9, 12 and 15, for the same parameters as in Fig. 8.

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Thus far, we have studied the emergence of CDW order on the kagome lattice at a fixed electron density of 〈n〉 = 2/3 per site. This choice was motivated by the observation of a CDW gap at 〈n〉 = 2/3 and a sharp change in electron kinetic energy during sweeps of μ and 〈n〉, and the fact that this filling corresponds to a completely filled lower band, which meets the middle band at the Dirac points K and \({K}^{{\prime} }\). However, we also considered fillings of 〈n〉 = 1/2 and 〈n〉 = 5/6, i.e., densities at which the saddle points in the non-interacting band structure (at the M-points) and their van Hove singularities are at the Fermi energy. We also consider 〈n〉 = 4/3, which corresponds to completely filled lower and middle bands, with a quadratic touching at the Γ-point between the flat and middle bands (see Fig. 1). In all of these cases, we find no evidence for the formation of a CDW. For example, there are no anomalous features in components of the total energy, or any indications of a plateau in the 〈n〉 vs. μ plots near these fillings, as shown in Fig. 2. Moreover, as the temperature is lowered (β increases) the charge structure factor Scdw(q) does not grow significantly and remains relatively small in magnitude, as shown in Fig. 10 for several high-symmetry points q in the Brillouin zone [Γ = (0, 0), \({{{\bf{K}}}}=(\frac{2\pi }{3},\frac{2\pi }{\sqrt{3}})\), and \({{{\bf{M}}}}=(\pi ,\frac{\pi }{\sqrt{3}})\)]. We fix ω0 = 0.1 here to avoid suppression of any potential CDW order, which occurs in the antiadiabatic limit. These results thus suggest an absence of any charge ordering at these fillings, at least for inverse temperatures β < 20. In other words, our results show no evidence for other varieties of CDW order in the kagome lattice Holstein model other than at the K-points at 〈n〉 = 2/3.

Fig. 10: Scdw(q) at 〈n〉 = 1/2, 5/6, and 4/3.
figure 10

Charge structure factor Scdw(q) as a function of inverse temperature β at several fixed electron densities: (a) 〈n〉 = 1/2, (b) 〈n〉 = 5/6, and (c) 〈n〉 = 4/3, for an L = 6 lattice. Data is shown for λD = 0.25 (solid line) and λD = 0.40 (dashed line) for several momenta q: Γ = (0, 0), \({{{\bf{K}}}}=(\frac{2\pi }{3},\frac{2\pi }{\sqrt{3}})\), and \({{{\bf{M}}}}=(\pi ,\frac{\pi }{\sqrt{3}})\). The phonon frequency is fixed at ω0 = 0.1. Error bars correspond to the standard deviation of the measured mean.

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Discussion

We performed hybrid Monte Carlo simulations of the Holstein model on the kagome lattice on systems of up to Ns = 775 sites, and studied the onset of CDW order while varying the electron filling, phonon frequency, electron-phonon coupling, and temperature. Our HMC algorithm allows us to simulate larger system sizes and access lower, more realistic phonon frequencies than in previous DMQC studies of the Holstein model. We observe evidence of CDW order at an average electron density of 〈n〉 = 2/3 per site (i.e., an overall filling fraction of f = 1/3), signaled by the opening of a gap in A(ω) at the Fermi surface, long-ranged density-density correlations, and the extensive scaling of the charge structure factor Scdw(K) below the critical temperature. From our analysis of the correlation ration Rc, we estimate a CDW transition at Tc ≈ t/18 = W/108, where W is the non-interacting electronic bandwidth.

This value of Tc is notably lower than the CDW transition temperatures found in the Holstein model on alternative geometries, e.g., at λD = 0.4, Tc ≈ t/6 on the honeycomb and Lieb lattices, while Tc ≈ t/4 on the square lattice25,27. Moreover, the CDW order appears only for a narrow range of electron-phonon coupling strengths in the kagome lattice, peaked at λD ≈ 0.4 (for ω0/t = 0.1). In contrast, previous Holstein model studies on square, honeycomb, and Lieb lattices have found CDW transitions across a broad range λD ∈ [0.25, 1]22,25,27. On bipartite geometries with equal numbers of A and B sites, such as the square and honeycomb lattices, CDW formation in the Holstein model occurs at half-filling i.e., 〈n〉 = 1. However, on the Lieb lattice, for which NA ≠ NB, when CDW order forms the density shifts away from half-filled to either 〈n〉 = 2/3 or 〈n〉 = 4/3, corresponding to completely filled lower and flat bands, respectively27. Although the kagome lattice similarly exhibits a three-band structure, the geometry is frustrated, unlike the Lieb case, and we find that charge order emerges only at 〈n〉 = 2/3 for temperatures T ≥ t/20 with an ordering wavevector at the K-points and a \(\sqrt{3}\times \sqrt{3}\) supercell. Our simulations did not reveal CDW order at other ordering momenta or electron densities, including at the van Hove filling.

The CDW order we find is analogous to the \(\sqrt{3}\times \sqrt{3}\) long-range magnetic order observed in the kagome lattice Heisenberg antiferromagnet, when it is coupled to local site-phonon modes32. The same CDW phase has also been proposed as the ground state in certain regimes of the extended Hubbard model40,41, i.e., at fillings of 〈n〉 = 2/3 and 〈n〉 = 5/6 for large V/U, where U is the on-site Hubbard term and V is the nearest-neighbor repulsion, and has been termed CDW-III in these studies.

It should be noted that the CDW order we observe does not correspond to the star-of-David or inverse star-of-David patterns observed recently in kagome metals such as AV3Sb5 (A = K, Rb, Cs), which exhibit ordering at the M-points. A recent work41 showed that such a CDW ordering is observed in the kagome lattice Hubbard model when a Su–Schrieffer–Heeger electron-phonon coupling is introduced. Here, the electron-phonon coupling modulates the electron hopping term, and is conceptually distinct from Holstein model, in which electrons and phonons interact on a single site, rather than on the bonds of the lattice.

Methods

Hybrid Monte Carlo simulation

Previous finite temperature studies of the Holstein model have typically employed DQMC71,72. In this method, the inverse temperature β = LtΔτ is discretized along an imaginary time axis with Lt intervals of length Δτ, and the partition function is expressed as \(Z={{{\rm{Tr}}}}\,{{{{\rm{e}}}}}^{-\beta \hat{H}}={{{\rm{Tr}}}}\,{{{{\rm{e}}}}}^{-\Delta \tau \hat{H}}{{{{\rm{e}}}}}^{-\Delta \tau \hat{H}}\ldots {{{{\rm{e}}}}}^{-\Delta \tau \hat{H}}\). Since Eq. (1) is quadratic in fermionic operators, these can be traced out, giving an expression for Z in terms of the product of two identical matrix determinants \(\det M({x}_{i,\tau })\), which are functions of the space and time-dependent phonon displacement field only. Monte Carlo sampling using local updates to the phonon field {xi,τ} is performed and physical quantities can be measured through the fermion Green’s function \({G}_{ij}=\langle {c}_{i}^{{\dagger} }{c}_{j}\rangle ={[{M}^{-1}]}_{ij}\). Although there is no sign problem73 for the Holstein model, these studies have been limited for two main reasons. First, the computational cost of DQMC scales as \({N}_{{{{\rm{s}}}}}^{3}{L}_{{{{\rm{t}}}}}^{}\), where Ns is the total number of lattice sites, prohibiting the study of large system sizes. Secondly, the restriction to local updates results in long autocorrelation times at small phonon frequencies. This aspect has limited simulations to phonon frequencies of ω0 ≳ t, which is unrealistic for most real materials, and is far from the regime where CDW order in the Holstein model is typically the strongest (ω0 ≪ t).

Significant efficiency gains are possible by using a dynamical sampling procedure that updates the entire phonon field at each time-step74,75. In this work, we use a recently developed collection of techniques to perform finite temperature simulations on extremely large clusters68. Our HMC-based approach achieves a near-linear scaling with system size74,76,77, allowing us to study lattices of up to Ns = 775 sites at temperatures as low as T = t/24. Our algorithm efficiently updates the phonon field simultaneously, allowing study of a realistic phonon frequency ω0/t = 0.1.

Near-linear scaling is achieved by rewriting each matrix determinant \(\det M\) as a multi-dimensional Gaussian integral involving auxiliary fields Φσ that will also be sampled. Here, the partition function becomes

$${{{\mathcal{Z}}}}\approx {\left(2\pi \right)}^{{N}_{{{{\rm{s}}}}}{L}_{{{{\rm{t}}}}}}\int{{{\mathcal{D}}}}{\Phi }_{\uparrow }{{{\mathcal{D}}}}{\Phi }_{\downarrow }{{{\mathcal{D}}}}x\,{{{{\rm{e}}}}}^{-S(x,{\Phi }_{\sigma })},$$
(8)

where the total action is

$$S(x,{\Phi }_{\sigma })={S}_{{{{\rm{B}}}}}(x)+{S}_{{{{\rm{F}}}}}(x,{\Phi }_{\sigma })$$
(9)

with the fermionic (F) and bosonic (B) contributions

$${S}_{{{{\rm{F}}}}}\left(x,{\Phi }_{\sigma }\right)=\frac{1}{2}\mathop{\sum}\limits_{\sigma }{\Phi }_{\sigma }^{T}{\left({M}^{T}M\right)}^{-1}{\Phi }_{\sigma }$$
(10)
$${S}_{{{{\rm{B}}}}}(x)=\frac{\Delta \tau }{2}\mathop{\sum}\limits_{i,\tau }\left[{\omega }_{0}^{2}{x}_{i,\tau }^{2}+{\left(\frac{{x}_{i,\tau +1}-{x}_{i,\tau }}{\Delta \tau }\right)}^{2}\right].$$
(11)

A Gibbs sampling procedure is then adopted where Φσ and x are alternately updated. The auxiliary field Φσ may be directly sampled. Using HMC, global updates to the phonon fields x can be performed by introducing a conjugate momentum p and evolving a fictitious Hamiltonian dynamics using a symplectic integrator68.