Kinetic frustration induced supersolid in the $S=\frac{1}{2}$ kagome lattice antiferromagnet in a magnetic field

We examine instabilities of the plateau phases in the spin-$\frac{1}{2}$ kagome lattice antiferromagnet in an applied field by means of degenerate perturbation theory, and find some emergent supersolid phases below the $m=\frac{5}{9}$ plateau. The wave functions of the plateau phases in a magnetic field have the particular construction based on the building blocks of resonating hexagons and their surrounding sites. Magnon excitations on each of these blocks suffer from a kinetic frustration effect, namely, they cannot hop easily to the others since the hopping amplitudes through the two paths destructively cancel out with each other. The itinerancy is thus weakened, and the system is driven toward the strong coupling regime, which together with the selected paths allowed in real space bears a supersolid phase. This mechanism is contrary to that proposed in lattice Bose gases, where the strong competing interactions suppress with each other, allowing a small kinetic energy scale to attain the itinerancy. Eventually, we find a supersolid state in which the pattern of resonating hexagons is preserved from the plateau crystal state and only one third of the originally polarized spins outside the hexagons dominantly join the superfluid component, or equivalently, participate in the magnetization process.

Figure 1

(a) $\sqrt{3}\times \sqrt{3}$ structure in the reported $m=\frac{1}{3},\frac{5}{9},\frac{7}{9}$ plateau phases [44, 48], where hexagons with dashed circles represent localized magnons and the rest of the spins are fully polarized. A and B label those two types of inequivalent sites in the crystal state. (b) Hexamerized kagome lattice used in the perturbation theory approach, which becomes the uniform kagome lattice at $J^{\prime }=J$. The two lattices have the same space-group symmetry.

Figure 2

(a) Lower-energy levels of the Heisenberg model on an isolated hexagon in the total magnetization sectors $S_{⎔}^z=0,1,2$, with the corresponding eigenstates denoted as $|S_{⎔}^z,l\rangle$, with $l=0,\cdots$. The eigenstates are labeled by $({\sigma }_{⎔},k_{⎔})$, where ${\sigma }_{⎔}$ is the reflection symmetry eigenvalue and, when applicable, the momentum eigenvalues $k_{⎔}$. (b) Magnetization staircases of the model in Fig. 1 at $J^{\prime }=0$, with the three plateaus also present in the Heisenberg kagome lattice $J^{\prime }=J$.

Figure 3

(a) Evolution of the one-magnon excitation energy $e_{\mathrm{ex}}\left(J^{\prime }\right)$, as a function of $J^{\prime }$, measured against the energy of the plateau $|{\mathrm{\Psi }}_{5/9}\rangle$. A magnetic field contributing as a chemical potential of one magnon $h$ is also subtracted. The lower critical phase boundary is hence given by $h_c$ that yields $e_{\mathrm{ex}}\left(J^{\prime }\right)=0$ ($e_{\mathrm{ex}}$ includes $h$ by construction). Here, $|{\mathrm{\Psi }}_c^{\mathrm{ex}}{\rangle }_I$ and $|{\mathrm{\Psi }}_{a/b}^{\mathrm{ex}}{\rangle }_I$ are the seeds of the flat mode and the dispersive modes, respectively. The energy range of the four different bands of the dispersive modes are shown. (b) Part of the phase diagram we obtained by the perturbation. The broken lines are the boundaries of the $m=\frac{5}{9}$ plateau phase evaluated with the flat-mode magnon instability. The supersolid phase appears when the dispersive mode in (a) replaces the flat mode and becomes the lowest one-magnon excited state. Our scheme cannot identify the lower boundary of the supersolid phase which is left blank, and the other plateaus $m=\frac{1}{3}$ and $\frac{7}{9}$, are not shown for clarity. The stars at $J^{\prime }=1$ mark the region of the $\frac{5}{9}$ plateau as obtained by DMRG [48].

Figure 4

(a) Effective decorated triangular lattice: the hexagon sites (big empty circles) at the centers of A hexagons form a triangular lattice hosting the $a_I^{†}$ magnons, and the three sublattices of B sites (filled colored circles) with the $b_i^{†}$ magnons give decorating sites in the center of every triangular-lattice bond. In gray is the unit cell of the effective lattice. (b) Hopping connections of the effective model (7). (c) Hopping amplitudes (top) and chemical potentials (bottom) as functions of $J^{\prime }$. (d) Spatial geometry of the two dominant transfer integrals $t_{\text{AB};1}$ and $t_{\text{BB};2}$. The paths of $t_{\text{BB};2}$ form three decoupled large kagome lattices, of which only one is drawn here.

Figure 5

Left panel: excitation energy spectra of model (7) for the couplings $J^{\prime }=0.84$ (top) and $J^{\prime }=1$ (bottom). The applied magnetic field is chosen so that the lowest-energy excitation closes the gap. The solid black lines denote the three lowest bands $e_{n,\mathit{k}}$ of the model (7) along the path $\mathrm{\Gamma }KM\mathrm{\Gamma }$. The dashed red lines are the flat band coming from the first one-magnon states localized in each hexagon. The minima of these energy bands are the candidates for the one-magnon instability below the $m=\frac{5}{9}$ plateau state, expected to give the lower critical field $h_c$. Right panel: first Brillouin zone of the hexamerized lattice. The coordinates of the high-symmetry points are ${\mathit{K}}_1=(2\pi /3,2\pi /\sqrt{3}), {\mathit{K}}_2=(4\pi /3,0), {\mathit{M}}_{1,2}=(\pi ,\pm \pi /\sqrt{3})$, and ${\mathit{M}}_3=(0,2\pi /\sqrt{3})$. The path used in the left panel is shown in red.

Figure 6

Supersolids induced by the one-magnon instabilities below the $m=\frac{5}{9}$ plateau at (a) the ${\mathit{M}}_1$ and (b) $\mathbf{\Gamma }$ wave vectors. The gray regions correspond to the sites with finite densities of one-magnon species. Inset: the three degenerate configurations of the B-site spins with a three-sublattice structure at the kagome lattice $J^{\prime }=1$ ($\mathbf{\Gamma }$ instability).

Figure 7

(a) Illustration of all the two-magnon processes appearing in the effective Hamiltonian (14). The top row represents the assisted hoppings, the middle row the pair hoppings, and the bottom row the density-density interaction terms. Each red hexagon denotes the closed loop of the six B sites surrounding one A hexagon [see Fig. 4]. Green and blue dots represent the magnons (the latter is used for the assisting magnons of top row). (b) $J^{\prime }$ dependence of the amplitudes of the five most dominant processes.

Figure 8

Local magnetizations in the supersolid phase at $\frac{1}{3}<m\le \frac{5}{9}$ for $J^{\prime }=0.84$. These values are evaluated from the MF approximation of the effective Hamiltonian (15). Top: magnetization per site in each resonating hexagon. Bottom: sublattice magnetizations of the three B-site sublattices. The star symbol indicates the plateau lower critical field $h_c=1.70$.

Figure 9

Local magnetizations in the supersolid phase at $\frac{1}{3}<m\le \frac{5}{9}$ for the kagome antiferromagnet $J^{\prime }=1$. These values are calculated from the MF approximation of the effective Hamiltonian (15). Top: magnetization per site in each resonating hexagon. Bottom: sublattice magnetizations of the three B-site sublattices. The star symbol indicates the plateau lower critical field $h_c=2.33$.

Figure 10

Magnetization per spin $m$ between the two plateaus at $m=\frac{1}{3}$ and $\frac{5}{9}$ for $J^{\prime }=1$, as evaluated from the MF approximation of the effective Hamiltonian (15). Inset: representation of the supersolid spin structure (V state) around a resonating hexagon.

Figure 11

(a) Diagonal second-order perturbative process on a ${|0,1\rangle }_{\text{A};I}$ hexagon, or A-hexagon magnon, which contributes to the chemical potential ${\mu }_{\text{A}}$. Blue color indicates the presence of a magnon on the A hexagon or a B site, and striped filling excited intermediate states. The perturbation is separated between its diagonal and off-diagonal terms $V_{\mathrm{diag}}$ and $V_{\mathrm{off}}$. (b) Off-diagonal second-order process responsible for the hopping $t_{\text{AB};1}$.