Resonating quantum three-coloring wave functions for the kagome quantum antiferromagnet

Motivated by the recent discovery of a macroscopically degenerate exactly solvable point of the spin-$1/2XXZ$ model for $J_z/J=-1/2$ on the kagome lattice [H. J. Changlani et al. Phys. Rev. Lett. 120, 117202 (2018)]—a result that holds for arbitrary magnetization—we develop an exact mapping between its exact ”quantum three-coloring” wave functions and the characteristic localized and topological magnons. This map, involving ”resonating two-color loops,” is developed to represent exact many-body ground state wave functions for special high magnetizations. Using this map we show that these exact ground state solutions are valid for any $J_z/J\ge -1/2$. This demonstrates the equivalence of the ground-state wave function of the Ising, Heisenberg, and $XY$ regimes all the way to the $J_z/J=-1/2$ point for these high magnetization sectors. In the hardcore bosonic language, this means that a certain class of exact many-body solutions, previously argued to hold for purely repulsive interactions $\left(J_z\ge 0\right)$, actually hold for attractive interactions as well, up to a critical interaction strength. For the case of zero magnetization, where the ground state is not exactly known, we perform density matrix renormalization group calculations. Based on the calculation of the ground state energy and measurement of order parameters, we provide evidence for a lack of any qualitative change in the ground state on finite clusters in the Ising $\left(J_z\gg J\right)$, Heisenberg $\left(J_z=J\right)$, and $XY$ $\left(J_z=0\right)$ regimes, continuing adiabatically to the vicinity of the macroscopically degenerate $J_z/J=-1/2$ point. These findings offer a framework for recent results in the literature and also suggest that the $J_z/J=-1/2$ point is an unconventional quantum critical point whose vicinity may contain the key to resolving the spin-$1/2$ kagome problem.

Figure 1

Two representative three-colorings on the kagome lattice corresponding to the $q=0$ and $\sqrt{3}\times \sqrt{3}$ solutions. The colors red, blue, and green represent the classical ${120}^{\circ }$ states or their quantum equivalents.

Figure 2

Representative example of a single magnon state with amplitudes $1,\omega ,{\omega }^2$ in the three-coloring basis, written as a many-body coloring wave function with a projection operator.

Figure 3

Comparison of ground state energies from exact diagonalization and diagonalization in the three-color basis as a function of $J_z$ (in units of $J=1$) for the (left panel) 36d cluster for $S_z=14$ ($m=7/9$ or 1/9 filling of bosons) and (central panel) for $S_z=12$ ($m=2/3$ or 1/6 filling of bosons) in the range $-1\le J_z\le 0$. For a thin torus such as the $4\times 2\times 3$ torus for $S_z=8$ ($m=2/3$ or 1/6 filling of bosons) shown in the rightmost panel, the exact solution holds. In cases where the Hamiltonian is frustration free for $J_z\ge -1/2$ (here, leftmost and rightmost panels), the exact ground state solution holds for arbitrary $J_z\ge -1/2$.

Figure 4

Definition of resonating color loops on a kagome lattice. Each RCL is obtained by taking a difference of two three-colorings, which differ only on a single two-color loop. In the top panel, the RCL is located on a hexagon and in the bottom panel it is located on a topological (noncontractible) loop, here winding along the horizontal direction. The RCLs when projected to a single spin-down (magnon) sector are exactly equal to localized or topological magnons on the kagome lattice up to a (projective) phase and an innocuous normalization.

Figure 5

A representative example of projection on to the $k=2$ spin-down sector on a configuration with two RCLs. The projection properties of RCLs ensure localization of bosons/spin downs to localized hexagons.

Figure 6

Representative locations of localized and topological single particle modes as resonating color loops are shown, including a 10 site loop that may be thought of as a composition of two hexagonal localized modes. Figure 4 shows how to transcribe the above RCL representation into the magnon modes. Apart from the single RCL at a chosen representative location, the rest of lattice is the same valid three-coloring, which makes the cancellation at all other sites exact.

Figure 7

Many-body ground state wave function for magnons represented in a three-coloring basis. The top panel shows the case of four magnons (bosons) on the $4\times 2\times 3$ torus. The construction generalizes to $L$ magnons on the $L\times 2\times 3$ lattice, i.e., 1/6 filling. Each magnon is confined to a strip and the many-body wave function is simply a product state of corresponding RCLs. Similar constructions apply at 1/9 filling to the infinite kagome and any finite cluster that accommodates the $\sqrt{3}\times \sqrt{3}$ pattern (middle panel). For 1/6 filling the construction also generalizes to two dimensions on the “squagome” lattice built up of triangular motifs (lowest panel).

Figure 8

Ground state energy per site from DMRG for the XC-8 cylinder in the limit of infinite length for the range $-1\le J_z\le 1$. The red dashed line indicates the energy ($=J_z/2$) of pure ferromagnetic states. The inset zooms into a narrow range around $J_z=-1/2$. The error bars are presented but smaller than the symbol sizes. The dotted lines in the inset indicate the exact energy $-1/4$ at $J_z=-1/2$.

Figure 9

First and second derivative of the energy per site as a function of $J_z$. The error bars are presented but smaller than the symbol sizes. The discontinuity in the first derivative and the peak in the second derivative at $J_z=-1/2$ signal the occurrence of a quantum phase transition.

Figure 10

Spatial profile of spin moments $\langle S_i^z\rangle$ and valence bond energies $\langle {\mathbf{S}}_i·{\mathbf{S}}_j\rangle$ on a representative XC-8 cylinder for $J_z=-0.495$ and $J_z=-0.505$. The maximum spin moment for $J_z=-0.495$ is $\sim 5\times {10}^{-4}$. The solid (dashed) bonds represent the negative (positive) valence bond energies.

Figure 11

Extrapolation of the energy per site with $1/L_x$ for $J_z=-0.51$, $-0.505$, and $-0.495$ from the DMRG data. The lengths used for $J_z=-0.51$ are 10, 12, 14, 16, 18, 20, 22, 24, 26, 30, for $J_z=-0.505$ are 8,10,12,14, and for $J_z=-0.495$ are 4, 8, 10, 12. Quadratic extrapolations are performed for $J_z=-0.51$ and $-0.495$, and linear extrapolation is performed for $J_z=-0.505$. Notice the resolution of the $y$-axis tic marks for $J_z=-0.505$.

Figure 12

The spin liquid (upper panel) and the VBS (lower panel) states found in the DMRG simulations for $J_z=-0.35$. The widths of the bonds are proportional to the valence bond energies $\langle \overline{S_i}·\overline{S_j}\rangle$ and the lengths of the arrows proportional to the spin moments $\langle {S_i}^z\rangle$. The maximum magnitude of the spin moment for the spin liquid is $\approx 9\times {10}^{-6}$.

Figure 13

Extrapolation of the total energy (left panel) and the “$J_2$ energy” (center panel) per site with the truncation error in DMRG for the spin liquid (SL) and the VBS states as shown in Fig. 12. The rightmost panel shows the estimated energy with finite $J_2$, more explicitly $E(J_1,J_2)\equiv E\left(J_1\right)+J_2\langle H\left(J_2\right)\rangle$. The crossing shows the suggested transition between SL and VBS with finite $J_2$. The light colors show the error bars of $E(J_1,J_2)$.