Two-Step Restoration of SU(2) Symmetry in a Frustrated Ring-Exchange Magnet

We demonstrate the existence of a spin-nematic, moment-free phase in a quantum four-spin ring-exchange model on the square lattice. This unusual quantum state is created by the interplay of frustration and quantum fluctuations that lead to a partial restoration of SU(2) symmetry when going from a four-sublattice orthogonal biaxial Néel order to this exotic uniaxial magnet. A further increase of frustration drives a transition to a fully gapped SU(2) symmetric valence bond crystal.

Figure 1

Schematic phase diagram of the cyclic four-spin exchange Hamiltonian (1) on the square lattice as a function of the parameter $\theta$.

Figure 2

(a) The orthogonal Néel state. It has a coplanar, four-sublattice structure. The vector chirality on a plaquette possesses a $(\pi ,\pi )$ structure. (b) Tower of states in the orthogonal four-sublattice Néel state for $J=-2$, $K=1$ ($\theta \approx 0.85\pi$). The eigenstates between the dashed lines form the ground-state manifold of the orthogonal Néel state. Taking into account exact degeneracies of some of the states, we find $2S+1$ low-lying levels in each spin-$S$ sector, consistently with a complete SU(2)-symmetry-breaking order parameter. Inset: finite size scaling of the spin gap, indicating a vanishing spin gap in the thermodynamic limit.

Figure 3

(a) Real-space vector chirality correlations $⟨[{\mathbf{S}}_0\wedge {\mathbf{S}}_1{]}^z[{\mathbf{S}}_i\wedge {\mathbf{S}}_j{]}^z⟩$ for a $N=40$ sample in the spin-nematic phase at $\theta =0.3\pi$. The black bond denotes the oriented reference bond. The width of the lines is proportional to the correlation strength. (b) Tower of states in the spin-nematic state. Inset: finite size scaling of the spin gap, indicating a vanishing spin gap in the thermodynamic limit.

Figure 4

The evolution of the finite size spectral gaps within the QDJS of the orthogonal Néel state on a $N=32$ sample. The bold lines denote levels that remain in the QDJS of the spin-nematic state. The other levels detach from the QDJS as $\theta \lesssim \pi /2$.

Figure 5

Staggered valence bond crystal phase: (a) Dimer correlations in the ground state of a 36 site sample at $\theta \approx 0.17\pi$. Full, red (dashed, blue) lines denote negative (positive) correlations. The linewidth is proportional to the correlation. (b) Low energy spectrum of a 36 site sample in the same phase. The required four singlets [the singlet at $(\pi ,\pi )$ marked with a star is twofold degenerate] with the correct quantum numbers corresponding to a staggered VBC are found below the first triplet.