Unexpected $z$-Direction Ising Antiferromagnetic Order in a Frustrated Spin-$1/2$ $J_1-J_2$ $XY$ Model on the Honeycomb Lattice

Using the density matrix renormalization group on wide cylinders, we study the phase diagram of the spin-$1/2$ $XY$ model on the honeycomb lattice, with first-neighbor ($J_1=1$) and frustrating second-neighbor ($J_2>0$) interactions. For the intermediate frustration regime $0.22\lesssim J_2\lesssim 0.36$, we find a surprising antiferromagnetic Ising phase, with ordered moments pointing along the $z$ axis, despite the absence of any $S_z{S}_z$ interactions in the Hamiltonian. Surrounding this phase as a function of $J_2$ are antiferromagnetic phases with the moments pointing in the $x-y$ plane for small $J_2$ and a close competition between an $x-y$ plane magnetic collinear phase and a dimer phase for large values of $J_2$. We do not find any spin-liquid phases in this model.

Figure 1

(a) Local magnetic moments for an XC8 cylinder with $J_2$ varying along the length of the cylinder. The dashed lines show the locations of particular $J_2$ values. In (a), we show a cylinder with $J_2$ varying from 0.12 to 0.30. For $J_2\sim 0.22$, the $xy$-plane Néel order rotates to $z$-direction Ising order. In (b), $J_2$ is varied from 0.28 to 0.46. The second phase transition point is located at $J_2\sim 0.36$.

Figure 2

(a) The ground state on the XC8 cylinder at $J_2=0.5$, showing strong dimer correlations aligned horizontally. We call this state a dimer state. (b) The ground state on YC6 cylinder at $J_2=0.5$. It has $xy$-plane collinear magnetic order, with antiferromagnetic chains along the horizontal zigzag direction together with ferromagnetic first-neighbor correlations between these zigzag chains.

Figure 3

(a) Local magnetization $|⟨S_z⟩|$ versus distance from the end of an XC8 cylinder for various $J_2$ values. (b) The magnetization at the cylinder center versus $J_2$. (c) The derivative of the central $|⟨S_z⟩|$ versus $J_2$. The peaks of the derivative at $J_2=0.23$ and 0.36 indicate the two phase transition points.

Figure 4

(a) The absolute value of the local magnetization $|⟨S_z⟩|$ for various XC and YC cylinders versus distance along the cylinder. The inset shows the extrapolated magnetization (extrapolated versus the truncation error) at the cylinder center, with error bars, versus the inverse of cylinder circumference $C$. (b) The entanglement entropy versus circumference for XC and YC cylinders in the intermediate Ising ordered phase at $J_2=0.3$. The intercepts are consistent with zero.

Figure 5

(a) The extrapolation of the ground state energy per spin for $J_2=0.3$ versus truncation error for the XC12 cylinder. The black curve (circles) is the energy per spin from subtracting the energies of two XC12 cylinders with lengths $L_x=20$ and $L_x=16$. The red curve (squares) is from subtracting two cylinders with $L_x=24$ and $L_x=20$. These two subtractions extrapolate to the same energy per spin of $-0.29445(1)$. (b) The ground state energy per spin for $J_2=0.3$ versus the inverse of cylinder circumference from our DMRG calculations, compared with the variational Monte Carlo (VMC) result from Table III in the Supplemental Material of Ref. 22, and exact diagonalization [22].