Spontaneous trimerization in a bilinear-biquadratic $S=1$ zig-zag chain

Recent theoretical studies raised the possibility of a realization of spin nematic states in the $S=1$ triangular lattice compound ${\mathrm{NiGa}}_2{\mathrm{S}}_4$. We study the bilinear-biquadratic spin 1 chain in a zig-zag geometry by means of the density matrix renormalization group method and exact diagonalization. We present the phase diagram focusing on antiferromagnetic interactions. Adjacent to the known Haldane–double Haldane and the extended critical phase with dominant spin nematic correlations we find a trimerized phase with a nonvanishing energy gap. We discuss results for different order parameters, energy gaps, correlation functions, and the central charge, and make connection to field theoretical predictions for the phase diagram.

Figure 1

Zig-zag geometry of the model with $J_1$ $\left(J_2\right)$ the nearest (next nearest) neighbor coupling strength.

Figure 2

(Color online) The phase diagram of the spin 1 BLBQ zig-zag chain. The area of the black dots in the trimerized phase scales with the magnitude of the trimer order parameter. We verified that the central charge is 2 in the critical phase for several points (red diamonds). The green triangles mark the phase boundary of the trimerized phase obtained from a level spectroscopy analysis from the ED data.

Figure 3

(Color online) Extrapolation of the bond energy oscillation amplitude in the middle of the chain leading to a finite trimer order parameter in the trimerized phase ($\theta =0.28\pi$, $J_2∕J_1=1$). Inset: The local bond energies form a pattern of period 3 ($L=48$ in this example).

Figure 4

(Color online) Energy gaps of spin excitations along cuts in the parameter space with $J_2∕J_1=1$ (left-hand side) and $\theta =\pi ∕4$ (right-hand side).

Figure 5

(Color online) Energy gap between the ground state of singlet and triplet sector (upper plot) and spin correlation length (lower plot) for different system sizes for fixed $J_2∕J_1=1.25$. The finite energy gap at the transition and the weak size dependence of the correlation length indicate a first order phase transition. The correlation length was obtained by a fit to the exponential decay of the spin-spin correlations.

Figure 6

(Color online) Spin (stars) and quadrupolar (crosses) correlation functions (normalized) for $\theta =0.42\pi$, $J_2=0.25$. The two solid lines are fits to the functional form predicted by field theory. Logarithmic corrections to the power law lead to dominant quadrupolar correlations.