How correlations change the magnetic structure factor of the kagome Hubbard model

The kagome Hubbard model (KHM) is a paradigmatic example of a frustrated two-dimensional model. While its strongly correlated regime, described by a Heisenberg model, is of topical interest due to its enigmatic prospective spin-liquid ground state, the weakly and moderately correlated regimes remain largely unexplored. Motivated by the rapidly growing number of metallic kagome materials (e.g., ${\mathrm{Mn}}_3\mathrm{Sn}, {\mathrm{Fe}}_3{\mathrm{Sn}}_2$, FeSn, ${\mathrm{Co}}_3{\mathrm{Sn}}_2{\mathrm{S}}_2, {\mathrm{Gd}}_3{\mathrm{Ru}}_4{\mathrm{Al}}_{12}$, and $A{\mathrm{V}}_3{\mathrm{Sb}}_5$ with $A=\mathrm{K}$, Rb, Cs), we study the respective regimes of the KHM by means of three complementary numerical methods: the dynamical mean-field theory, the dynamical vertex approximation, and determinant quantum Monte Carlo. In contrast to the archetypal square lattice, we find no tendencies toward magnetic ordering, as magnetic correlations remain short-range. Nevertheless, the magnetic correlations undergo a remarkable crossover as the system approaches the metal-to-insulator transition. The Mott transition itself does not affect the magnetic correlations. Our equal-time and dynamical structure factors can be used as a reference for inelastic neutron scattering experiments on the growing family of metallic kagome materials.

Figure 1

(a) Unit cell (shaded green) of the kagome lattice described by the basis vectors $a_1$ and $a_2$ comprises three sites labeled 1, 2, and 3. The Wigner-Seitz cell is shaded ochre. (b) High-symmetry points of the reciprocal space, the first and the extended Brillouin zones (BZs). (c) The band structure (left) and the density of states (DOS, right) of the tight-binding kagome model with a positive $t$. The blue line marks the chemical potential ${\mu }_{\text{tb}}\approx 0.47$, for which the noninteracting system is half filled.

Figure 2

Local spectral functions for interaction strength ranging from $U=3$ to 9 at $T=0.33$ in units of the hopping $t\equiv 1$ as obtained by DMFT, DQMC, and $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$. The gray line is the density of states of the underlying tight-binding model, the opaque dotted blue bars show the approximate position of the Hubbard bands calculated by Eq. (9). We show no $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$ data for $U=8$ and 9, because the respective calculations did not converge.

Figure 3

Quasiparticle renormalization factor $Z$ [Eq. (8)] based on DMFT and DQMC data as a function of the interaction strength $U$ for the temperatures specified in the legend box.

Figure 4

Spectral functions of DMFT, $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$, and DQMC on a high-symmetry path through the BZ. The dotted green lines are the tight-binding bands, the white line is the Fermi energy. At weak interaction (left column), there is no sizable renormalization, whereas strong interactions considerably change the spectrum (right column). Spurious momentum-dependent variations of spectral weight in DQMC ($U=3$) stem from variations of the peak widths—an artifact of analytic continuation.

Figure 5

Momentum- and band-resolved quasiparticle renormalization $Z_{\alpha }\left(\mathbf{k}\right)$ at $U=6$ and $T=0.33$ obtained by projection of the $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$ self-energy onto the tight-binding eigenbasis.

Figure 6

Eigenvalues of the static magnetic $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$ susceptibility in the first BZ for $U=3$ at $T=1$ (upper row) and $T=0.1$ (middle row), and $U=6, T=0.1$ (lower row). There are three eigenvalues corresponding to the three bands in Fig. 1: the left column shows the projection of $\chi$ on the lowest-energy eigenstate [energies below the Dirac point in Fig. 1]; the middle column that of the part of the band structure above the Dirac point and below the flat band; finally, the right column corresponds to the flat-band states.

Figure 7

Temperature dependence of the maximal magnetic susceptibility ${\chi }_{\text{max}}$ with respect to the momenta and the three eigenvectors (three panels in Fig. 6) at $\omega =0$ for the weakly ($U=3$) and moderately ($U=7.5$) correlated regimes.

Figure 8

Equal-time structure factors $S_0\left(\mathbf{q}\right)[\mathrm{Eq}$. (11)] in the weakly correlated regime ($U=3$) as a function of temperature (top: $T=0.33$, bottom: $T=0.1$). Left: DQMC, right: $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$. The dashed (solid) hexagons denote the boundary of the (extended) BZs.

Figure 9

Equal-time susceptibility $\chi (\mathbf{R}\equiv (x,y\left)\right)$ in real space for the weakly correlated regime ($U=3$). Units along the $x$ and $y$ axes are unit cell constants. The area of a circle reflects the absolute value of the respective term, blue color denotes positive (ferromagnetic) correlations, red color denotes negative (antiferromagnetic) correlations. Since the magnitude of the long-range correlations is very small, we add a background shading to indicate the sign. Note that weak long-range correlations are antiferromagnetic for ${\mathbf{R}}_{3a}$ and ferromagnetic for ${\mathbf{R}}_2$ and ${\mathbf{R}}_{3b}$ (see Table 1 for the notation of intersite vectors).

Figure 10

Equal-time structure factors $S_0\left(\mathbf{q}\right)[\mathrm{Eq}$. (11)] as a function of $U$ (top to bottom) for DQMC (left), $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$(top right, middle right), and DMFT (bottom right) at $T=0.33$. We used the DMFT susceptibility for $U=10$ to calculate the structure factor, as the respective $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$ calculation did not converge.

Figure 11

Equal-time susceptibility $\chi (\mathbf{R}\equiv (x,y\left)\right)$ as a function of $U$ (top: $U=6$, bottom: $U=10$) as calculated by DQMC (left), $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$(top right), and DMFT (bottom right) at $T=0.33$. The area of a circle reflects the absolute value of the respective term, blue color denotes positive (ferromagnetic) correlations, red color denotes negative (antiferromagnetic) correlations. Note that weak long-range correlations are ferromagnetic for ${\mathbf{R}}_2, {\mathbf{R}}_{3a}$ and antiferromagnetic for ${\mathbf{R}}_{3a}$ (see Table 1 for the notation of intersite vectors).

Figure 12

Dynamical structure factor $S(\mathbf{q},\omega )$ in the weakly correlated regime ($U=3$) at $T=0.33$ (top) and $T=0.1$ (bottom) calculated using DQMC (left) and $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$ (right) on a path through the extended BZ (Fig. 1).

Figure 13

Dynamical structure factor as a function of interaction strength (left to right) as calculated by DQMC (top) and $\mathrm{D}\mathrm{\Gamma }\mathrm{A}$ (bottom, except for DMFT in the right bottom plot) on a path through the extended BZ (Fig. 1) at $T=0.33$. Note that the upper limit of the $\omega$ axis ($y$ axis) is reduced as $1/U$ upon increasing $U$ to resolve the feature-rich range; for large $U$, such a rescaling must hold as the coupling of the Heisenberg model is $J=4t^2/U$. Note that in DMFT the structure factor is enlarged due to the lack of a self-energy feedback.

Figure 14

How certain real-space correlations influence the magnetic susceptibility and structure factor. In the first three columns, we show the projection of the magnetic susceptibility on the lower-band, middle-band, and flat-band eigenstates of the tight-binding Hamiltonian, respectively. The right-most column shows the corresponding structure factor $S\left(\mathbf{q}\right)$ on the extended BZ. The rows correspond to various correlations in real space: Thus, in the first row we show momentum-space correlations arising from on-site correlations. The second row corresponds to nearest-neighbor correlations (${\mathbf{R}}_1$). The third, fourth, and fifth rows correspond to ${\mathbf{R}}_2, {\mathbf{R}}_{3a}$ and ${\mathbf{R}}_{3b}$ correlations.