Interplay of charge and spin fluctuations of strongly interacting electrons on the kagome lattice

We study electrons hopping on a kagome lattice at third filling described by an extended Hubbard Hamiltonian with on-site and nearest-neighbor repulsions in the strongly correlated limit. As a consequence of the commensurate filling and the large interactions, each triangle has precisely two electrons in the effective low-energy description, and these electrons form chains of different lengths. The effective Hamiltonian includes the ring exchange around the hexagons as well as the nearest-neighbor Heisenberg interaction. Using large-scale exact diagonalization, we find that the effective model exhibits two phases: If the charge fluctuations are small, the magnetic fluctuations confine the charges to short loops around hexagons, yielding a gapped charge-ordered phase. When the charge fluctuations dominate, the system undergoes a quantum phase transition to a resonating plaquette phase with ordered spins and gapless spin excitations. We find that a peculiar conservation law is fulfilled: the electron in the chains can be divided into two sublattices, and this division is conserved by the ring exchange term.

Figure 1

(a) A configuration satisfying the constraint of zero or one electron per site and two electrons of arbitrary spin per triangle. (b) Quantum fluctuations allow tunneling between different degenerate configurations of spins and charge: ring exchange with amplitude $g$ and spin-exchange with an amplitude of $J$. The ring exchange processes change the number of electrons on the starred (blue) sublattice shown in (c), always by two. This is crucial for gauging away the sign of $g$ explained in the text. (d) All allowed configurations can be expressed in terms of a two-colored fully packed loop model on the honeycomb lattice; as an example, we show a representation of the configuration presented in (a).

Figure 2

Values of the parameter $\alpha$ defined in Eq. (7) as contour plots on the plane of $\left|t\right|/U$ and $\left|t\right|/V$. (a) For $t>0$ the exchange is always antiferromagnetic, while (b) for $t<0$ the exchange becomes ferromagnetic [shaded (pink) region] for large on-site repulsion $U$, as it follows from the perturbative expansion up to third order in the hopping, Eq. (6). The effective Hamiltonian describes the region close to the origin in the unshaded (white) region $U>V$ and $U>0$.

Figure 3

Dynamic sublattice structure defined on a loop of charges, consisting of an even number of sites. The effective ring exchange denoted by arrows preserves the magnetization on each of the dynamic sublattices $A$ and $B$ (see Appendix pp1 for details).

Figure 4

Energy spectrum of the $N=36$ cluster at $\alpha =1$. We set one of the two dynamic sublattices to be fully polarized as $S_B=6$ while varying the spin $S_A$ of the other sublattice. The ground state is realized at the $\Gamma$ point for $S_A=6$, indicating that both dynamic sublattices hold maximal spins.

Figure 5

(a) Excitation energies of the spin Hamiltonian $\left({\mathcal{H}}_{\mathrm{spin}}\right)$ for two loop configurations of the 36-site cluster: one consisting of four short loops of length 6, and the other consisting of one 18-site loop and one 6-site loop. Numbers next to the levels indicate in which state each loop is; e.g., 0001 means that three loops are in the ground state and one is in the first excited spin state.

Figure 6

Anderson tower for the $N=36$ cluster for (a) $\alpha =0.9$ and (b) $\alpha =0.8$. The energies of the two low-lying two series of states belonging to different irreducible representations, IR01 and IR05 (see Appendix pp2), which are even and odd with respect to $P_{AB}$, behave proportionally to $S(S+1)$. As $\alpha$ increases and/or the system size increases, their gradient decreases toward 0, at which all these points fall onto the same horizontal line (become degenerate). Such behavior is a strong indication of antiferromagnetic ordering.

Figure 7

Energy spectra for two clusters, $N=27$ in (a) and $N=36$ in (b). The level crossing occurs at ${\alpha }_c\approx 0.6$ between the energy levels of the same quantum number for both clusters, which indicates a possible quantum phase transition. All energies are measured from the lowest energy singlet state.

Figure 8

Ground-state phase diagram of Eq. (2). With increasing $\alpha$, the short-loop phase undergoes a first-order quantum phase transition into the plaquette phase at around $\alpha \sim 0.6$. The purple hexagons in the left panel denote the short loops formed by neighboring electrons, and blurred blue hexagons in the right panel indicate the presence of resonating plaquettes.

Figure 9

(a), (b) Charge-charge correlation function in real space for $\alpha =0$ and 1, respectively. Correlations are calculated with respect to the marked site ${\mathbf{r}}_{\mathbf{0}}$ at the center. The radius of the dots is proportional to the absolute value of the correlation, while the color encodes the sign (red corresponds to a positive value; blue, to a negative value). (c), (d) The same data in momentum space in the extended Brillouin zone. The maximum in (c) is found at $\mathbf{Q}=(8\pi /3,0)$ and symmetry-related points (high-symmetry points denoted $K_2$ and $K_2^{\prime })$.

Figure 10

(a), (b) Plaquette-plaquette correlation function in real space for $\alpha =0$ and 1, respectively. Correlations are calculated with respect to the marked site ${\mathbf{r}}_{\mathbf{0}}$ at the center. The radius of the dots is proportional to the absolute value of the correlation, while the color encodes the sign (red corresponds to a positive value; blue, to a negative value). (c), (d) The same data in momentum space in the first Brillouin zone. The peaks in (d) are found at the $K$ and $K^{\prime }$ points, with $\mathbf{Q}=(2\pi /3,2\pi /\sqrt{3})$ and $\mathbf{Q}=(4\pi /3,0)$, respectively.

Figure 11

(a), (b) Spin-spin correlation function in real space for $\alpha =0$ and 1, respectively. Correlations are calculated with respect to the marked site ${\mathbf{r}}_{\mathbf{0}}$ at the center. The radius of the dots is proportional to the absolute value of the correlation, while the color encodes the sign (red corresponds to a positive value; blue, to a negative value). (c), (d) The same data in momentum space in the extended Brillouin zone.

Figure 12

Amplitudes of the (a) charge, (b) plaquette, (c) and spin structure factors, shown in Figs. 9–11, at the corresponding ordering wave vectors $\mathbf{Q}$. The gradual changes in all panels indicate the melting of different kinds of ordering as the parameter $\alpha$ is varied.

Figure 13

Description of a cluster with $N=36$ sites and periodic boundary conditions. The symmetries of the point group are shown in dark red: the hexagon denotes the rotation center of order 6 $\left(C_6\right)$; the triangle, the rotation center of order 3 $\left(C_3^{\prime }\right)$; and the oval, the rotation center of order 2 $\left(C_2^{\prime }\right)$. Dashed lines are the reflections $\sigma$ and ${\sigma }^{\prime }$. Dark-blue lines with half-arrows denote glide reflections.