Emergent Haldane phase in the $S=1$ bilinear-biquadratic Heisenberg model on the square lattice

Infinite projected entangled pair states simulations of the $S=1$ bilinear-biquadratic Heisenberg model on the square lattice reveal an emergent Haldane phase in between the previously predicted antiferromagnetic and three-sublattice ${120}^{\circ }$ magnetically ordered phases. This intermediate phase preserves SU(2) spin and translational symmetry but breaks lattice rotational symmetry, and it can be adiabatically connected to the Haldane phase of decoupled $S=1$ chains. Our results contradict previous studies which found a direct transition between the two magnetically ordered states.

Figure 1

Haldane phase emerging in between the AF and the three-sublattice ${120}^{\circ }$ ordered phases.

Figure 2

(a) iPEPS results (full update) for the local magnetic moment $m$ as a function of inverse bond dimension for different values of $\theta$. (b) Extrapolated values of $m$ as a function of $\theta$.

Figure 3

The phase diagram of the anisotropic bilinear-biquadratic $S=1$ model. The phase boundaries were estimated based on $D=10$ simple update simulations (see Supplemental Material [71]).

Figure 4

(a) Energy per site (full update) of the Haldane and three-sublattice (3-SL) state for two different values of $\theta$, plotted as a function of the truncation error $w$. (b) Local magnetic moment $m$ (triangles) of the three-sublattice state, and the difference in bond energies $\mathrm{\Delta }E$ in the $x$ and $y$ direction in the Haldane state (squares), plotted as a function of $1/D$. Note that $m$ and $\mathrm{\Delta }E$ are zero in the Haldane and three-sublattice states, respectively.