Twelve sublattice ordered phase in the $J_1-J_2$ model on the kagomé lattice

Motivated by recent experiments on an $S=1∕2$ antiferromagnet on the kagomé lattice, we investigate the Heisenberg $J_1-J_2$ model with ferromagnetic $J_1$ and antiferromagnetic $J_2$, Classically the ground state displays Néel long-range order with 12 noncoplanar sublattices. The order parameter has the symmetry of a cuboctahedron, it fully breaks $SO\left(3\right)$ as well as the spin-flip symmetry, and we expect from the latter a ${\mathbb{Z}}_2$ symmetry breaking pattern. As might be expected from the Mermin-Wagner theorem in two dimensions, the $SO\left(3\right)$ symmetry is restored by thermal fluctuations while the ${\mathbb{Z}}_2$ symmetry breaking persists up to a finite temperature. A complete study of $S=1∕2$ exact spectra reveals that the classical order subsists for quantum spins in a finite range of parameters. First-order spin wave calculations give the range of existence of this phase and the renormalizations at $T=0$ of the order parameters associated to both symmetry breakings. This phase is destroyed by quantum fluctuations for a small but finite $J_2∕\mid J_1\mid \simeq 3$, consistently with exact spectra studies, which indicate a gapped phase.

Figure 1

(Color online) The first Brillouin zone of the kagomé lattice with its points of high symmetry: the center of zone $\Gamma$ $(\mathbf{q}=\mathbf{0})$, the two zone corners $M_1$ and $M_2$, and the three edge centers $X_1$, $X_2$, and $X_3$.

Figure 2

(Color online) Classical order parameter for $J_1<0$ and $J_2>\mid J_1\mid ∕3$ with the associated lattice symmetry breaking. The 12 sublattices and the 4 different types of hexagons are numbered. Also shown is the plane containing the six spins of hexagon 1 (dashed).

Figure 3

(Color online) Classical phase diagram of the Heisenberg $J_1-J_2$ model on the kagomé lattice at $T=0$. The transition between the ferromagnetic state and the antiferromagnetic cuboc state occurs at $J_2=-J_1∕3$.

Figure 4

(Color online) An order parameter and its image by a spin flip. After a global rotation of the sublattice directions, it is clear that the two order parameters are related one to the other by a mirror symmetry.

Figure 5

The normalized scalar chirality ${\sigma }_{\Delta }$ computed on the order parameter of Fig. 2. The spin-flip trivially permutes the + and −.

Figure 6

(Color online) Specific heat $C∕k_B$ (red dashed line) and chiral susceptibility $k_B{\chi }_{\sigma }$ (blue solid line) versus temperature on a 1200 spins sample for $J_2∕\mid J_1\mid =0.5$: both quantities diverge at $T_c=0.24$, simultaneously with the vanishing of the chiral order parameter $m_{\sigma }$ (not shown).

Figure 7

The $N=36$ sample.

Figure 8

Top: Exact spectrum of the Hamiltonian (1), with $J_2∕\mid J_1\mid =0.5$, for the $N=36$ sample shown in Fig. 7. The exact energies per spin are displayed versus $S(S+1)$. The + (resp. −) symbols are eigenstates with even (resp. odd) parity under the ${\mathcal{R}}_{\pi }$ lattice rotation. Other symbols are eigenstates without the ${\mathcal{R}}_{\pi }$ symmetry. The full symmetries of the lowest eigenstates are given in Table . The expected QDJS for the 12-sublattice Néel order appear between the two dashed lines. In particular, even and odd parity eigenstates come in an equal number in each spin sector, as expected for the ${\mathbb{Z}}_2$ symmetry breaking. Bottom: Zoom on the QDJS (energies rescaled).

Figure 9

(Color online) Kagomé lattice with nearest (solid) and next-nearest neighbor exchange (dashed). Note that we span all the exchange paths by considering the 12 links inside an empty hexagon ⎔ located at $R_{⎔}$ and then iterate over the whole Bravais lattice. $\mathbf{u}$ and $\mathbf{v}$ are the basis vectors of the Bravais lattice of length 2 and the three sites $\alpha$, $\beta$, $\gamma$ per Bravais cell define the three unit vectors ${\mathbf{e}}_{\alpha }=\mathbf{u}∕2$, ${\mathbf{e}}_{\beta }=(\mathbf{v}-\mathbf{u})∕2$ and ${\mathbf{e}}_{\gamma }=-\mathbf{v}∕2$.

Figure 10

(Color online) Extrapolated values of the magnetization in the local basis $m^{\infty }$ (solid line) and the alternate scalar chirality $m_{\xi }^{\infty }$ (dashed line) as a function of $J_2∕\mid J_1\mid$.

Figure 11

An exact spectrum of the $N=36$ sample for $J_2∕\mid J_1\mid =5.0$ versus total spin $S$, The symbol convention is the same as that of Fig. 8.

Figure 12

(Color online) Top: Finite size scaling of the spin gap $\Delta E$ for $J_2∕\mid J_1\mid =5.0$, in the supposed to be gapped phase. Bottom: The same gap in the 12-sublattice Néel phase, for $J_2∕\mid J_1\mid =0.5$, closes as $1∕N$, as expected for the à la Néel $SU\left(2\right)$ symmetry breaking (Sec. 3A).