Quadrupolar correlations and spin freezing in $S=1$ triangular lattice antiferromagnets

Motivated by experiments on ${\text{NiGa}}_2{\text{S}}_4$, we discuss characteristic (finite-temperature) properties of spin $S=1$ quantum antiferromagnets on the triangular lattice. Several recent theoretical studies have suggested the possibility of quadrupolar (spin-nematic) ground states in the presence of sufficient biquadratic exchange. We argue that quadrupolar correlations are substantially more robust than the spin-nematic ground state and give rise to a two-peak structure of the specific heat. We characterize this behavior by a $T>0$ semiclassical approximation, which is amenable to efficient Monte Carlo simulations. Turning to low temperatures, we consider the effects of weak disorder on incommensurate magnetic order, which is present when interactions beyond nearest-neighbor exchange are substantial. We show that nonmagnetic impurities act as random fields on a component of the order parameter, leading to the disruption of long-range magnetic order even when the defects are arbitrarily weak. Instead, gradual freezing phenomena are expected on lowering the temperature, with no sharp transition but a rapid slowing of dynamics and the development of substantial spin-glass-like correlations. We discuss these observations in relation to measurements of ${\text{NiGa}}_2{\text{S}}_4$.

Figure 1

(Color online) Finite-temperature phase diagram of the nearest-neighbor only model with an antiferromagnetic $J_1>0$ and a ferronematic $K>0$ obtained from numerical simulations in the sSU(3) approximation. The location of the phase boundaries are obtained from the position of the specific-heat peaks in our numerical simulations. Approaching zero temperature the semiclassical sSU(3) approximation recovers the ground states obtained in the mean-field theory. We note that the phase boundaries indicate a rapid growth of the ferroquadrupolar (FQ) and the magnetic (AFM) correlation lengths, but not true long-range order excluded by the Mermin-Wagner theorem.

Figure 2

(Color online) Finite-temperature calculations of (a) the specific heat and (b) the spin and the quadrupolar correlation lengths ${\xi }_S$ and ${\xi }_Q$ for the nearest-neighbor model in the semiclassical sSU(3) approximation. The two temperature scales associated with the rapid growth of magnetic and ferroquadrupolar correlations result in a characteristic two-peak structure of the specific heat. The data shown are for couplings $K/J_1=1.5$ and the system sizes are chosen to commensurate with the low-temperature spin ordering.

Figure 3

(Color online) Zero-temperature phase diagram of the model with antiferromagnetic third-nearest-neighbor coupling $J_3$ and nearest-neighbor ferromagnetic $J_1$ and ferronematic $K$ couplings obtained from numerical simulations in the sSU(3) approximation. The three phases present at zero temperature have a ferroquadrupolar, $120°$ spiral, or incommensurate spiral ground state, with continuous transitions (dashed lines) between the ferroquadrupolar and the spiral states and a first-order transition between the $120°$ and the incommensurate spiral states. The cross-hatched area marks the region where our finite-temperature calculations indicate two temperature scales. Such a separation is observed both in specific-heat (filled diamonds) and correlation length measurements (open circles) which are expected to coincide in the thermodynamic limit. The red triangle marks the location of the representative system, for which we show finite-temperature properties in Fig. 4.

Figure 4

(Color online) Finite-temperature calculations of (a) the specific heat, (b) the spin and the quadrupolar correlation lengths ${\xi }_S$ and ${\xi }_Q$, and (c) the nematic order parameter ${\left|\psi \right|}^2$ for the model with third-nearest-neighbor interaction in the semiclassical sSU(3) approximation. The two temperature scales associated with the rapid growth of magnetic and ferroquadrupolar correlations again result in a characteristic two-peak structure of the specific heat. The data shown are for couplings $K/J_3=1.1$ and $J_1/J_3=-1.14855$ and the system sizes are chosen to commensurate with the low-temperature spin ordering. In contrast to the nearest-neighbor model, we see a divergent lower peak corresponding to the breaking of lattice rotational symmetry by anisotropic spin fluctuations.

Figure 5

(Color online) Finite-size scaling of the upper peak in the specific heat of the representative system with couplings $K/J_3=1.1$ and $J_1/J_3=-1.14855$. While the peak slightly shifts with increasing system size, there is clearly no divergence.

Figure 6

(Color online) Finite-size scaling of the lower peak in the specific heat of the representative system with $K/J_3=1.1$ and $J_1/J_3\approx -1.14855$. The trend with increasing system size clearly suggests a divergence in the thermodynamic limit.

Figure 7

(Color online) Histogram of the energy density for our representative system in the vicinity of the low-temperature transition. The bimodal character of this distribution proliferates with increasing system size, which is indicative of a first-order transition. Data for different system sizes and temperatures are shown; in particular, $L=84$ $\left(T/J_3=0.2377\right)$, $L=98$ $\left(T/J_3=0.2374\right)$, and $L=112$ $\left(T/J_3=0.2361\right)$.