Schwinger boson theory of the $J_1, J_2=J_3$ kagome antiferromagnet

We study the kagome antiferromagnet for quantum spin-1/2 with first $J_1$, second $J_2$, and third $J_3$ neighbor exchanges, along the $J_2=J_3\equiv J$ line. We use Schwinger boson mean-field theory for the precise determination of the phase diagram, and two different rewritings of the Hamiltonian to build an intuition about the origin of the transitions. The spin liquid obtained at $J=0$ remains essentially stable over a large window, up to $J\approx 1/3$, because it is only weakly frustrated by the $J$ term. Then, at $J\approx 1/2$, the intermediate $Z_2$ spin liquid condenses into a long-range chiral order because of the change of nature of the local magnetic fluctuations. As a side benefit, our Hamiltonian rewriting offers an exact solution for the ground state of our model on a Husimi cactus.

Figure 1

Left: The 12-site unit cell of the kagome lattice used in the Schwinger boson theory with the first $J_1$, second $J_2$, and third $J_3,J_{3d}$ neighbor couplings. In this Letter, we set the energy scale with $J_1=1$ and consider $J_2=J_3\equiv J$ and $J_{3d}=0$. The arrows show the bond orientations of the $A$ and $B$ parameters for nearest neighbors. The orientations for the $J_2$ and $J_3$ bonds are not displayed. Right: The first Brillouin zone (BZ) of the 12-site unit cell (dashed hexagon) is shown, as well as the first and second BZs of the kagome lattice. Relevant high-symmetry points are displayed.

Figure 2

(a) SBMFT phase diagram of the kagome antiferromagnet along the $J_2=J_3=J$ line. Three phases are identified: a chiral spin liquid (cuboc1) [3, 30], a $Z_2$ spin liquid ($Z_2^{(1,0)}$) originating from the $q=0$ Ansatz, and a magnetically ordered phase (${\mathrm{LRO}}^{(1,1)}$) coming from the Bose condensation of its spin-liquid counterpart, the $Z_2^{(1,1)}$. For each ground state, the sign structure of the nearest neighbors $A_{ij}$ is given, with thin (thick) lines corresponding to positive (negative) values of $A_{ij}$. For all phases, nearest neighbors $|A_{ij}|=\mathcal{A}$. For the cuboc1, the arguments of $A_{ij}$ are different between up and down triangles. Phases are also characterized by the flux ${\varphi }_r$ on rhombii and ${\varphi }_h$ on hexagonal Wilson loops. (b)–(d) Finite-size scaling of the energy gap $\mathrm{\Delta }$ above the ground state for $J=\{0.2,0.45,0.6\}$ with $n_s=6l^2$. The $y$ axis in (d) has been multiplied by 10. The spin length in SBMFT is $S=(\sqrt{3}-1)/2$, which gives the good quantum number $\langle {\widehat{\mathbf{S}}}^2\rangle =3/4$ of a quantum spin-1/2 [3].

Figure 3

Correlations in real space (top) and Fourier space $S(\mathbf{q},\omega )$ [Eq. (4)] (bottom) for the (a) cuboc1 at $J=0.2$, (b) $Z_2^{(1,0)}$ at $J=0.4$, and (c) ${\mathrm{LRO}}^{(1,1)}$ at $J=0.6$. The reference site is the blue circle at the center. The strength and sign of the correlations are, respectively, given by the radius and color of the circles (red is negative). The path in Fourier space is given in Fig. 1. System size is $n_s=1728$.

Figure 4

Structure factor $S\left(\mathbf{q}\right)$ of the cuboc1, $Z_2^{(1,0)}$ and ${\mathrm{LRO}}^{(1,1)}$ Ansätze for increasing $J$ (given in each panel). The $Z_2^{(1,0)}$ structure factor is qualitatively the same for $0.33<J\le 0.5$, but when the $Z_2^{(1,0)}$ Ansatz becomes an excitation for $J>0.50$ (gray boxed panels), half-moon patterns appear in the second BZs. The color scale is normalized on all panels, except for ${\mathrm{LRO}}^{(1,1)}$ where a cutoff is imposed to emphasize the low-intensity scattering. System size is $n_s=1728$.