One dimensionalization in the spin-1 Heisenberg model on the anisotropic triangular lattice

We investigate the effect of dimensional crossover in the ground state of the antiferromagnetic spin-1 Heisenberg model on the anisotropic triangular lattice that interpolates between the regime of weakly coupled Haldane chains $\left(J^{\prime }\ll J\right)$ and the isotropic triangular lattice $\left(J^{\prime }=J\right)$. We use the density-matrix renormalization group (DMRG) and Schwinger boson theory performed at the Gaussian correction level above the saddle-point solution. Our DMRG results show an abrupt transition between decoupled spin chains and the spirally ordered regime at ${(J^{\prime }/J)}_c\sim 0.42$, signaled by the sudden closing of the spin gap. Coming from the magnetically ordered side, the computation of the spin stiffness within Schwinger boson theory predicts the instability of the spiral magnetic order toward a magnetically disordered phase with one-dimensional features at ${(J^{\prime }/J)}_c\sim 0.43$. The agreement of these complementary methods, along with the strong difference found between the intra- and the interchain DMRG short spin-spin correlations for sufficiently large values of the interchain coupling, suggests that the interplay between the quantum fluctuations and the dimensional crossover effects gives rise to the one-dimensionalization phenomenon in this frustrated spin-1 Hamiltonian.

Figure 1

Geometry of the AF exchange interactions. The interpolation among the different systems is explained in the text.

Figure 2

DMRG results for the unfrustrated case $J^{\prime \prime }=0$: scaling of the spin gap $\mathrm{\Delta }$ (top panel) and the squared staggered magnetization $m^2$ (bottom panel) versus $1/L$ for different values of the interchain coupling $J^{\prime }$. The system sizes shown are $L=4,6,8,10$, and 12.

Figure 3

DMRG results for the unfrustrated case $J^{\prime \prime }=0$: $L\times \mathrm{\Delta }$ versus interchain coupling $J^{\prime }$ for different system sizes. Inset: extrapolation of $J_c^{\prime }$ as the different crossing points for successive lattice sizes versus $1/L$. See the text.

Figure 4

DMRG results for the frustrated case $J^{\prime }=J^{\prime \prime }$: spin gap $\mathrm{\Delta }$ versus interchain coupling $J^{\prime }$. The system sizes shown are $L=6,8$, and 10.

Figure 5

DMRG results for the frustrated case $J^{\prime }=J^{\prime \prime }$: intra- and interchain spin-spin correlations between nearest neighbors as a function of $J^{\prime }$.

Figure 6

Magnetic wave vector $Q_0$ predicted by the Schwinger bosons as a function of $J^{\prime }=J^{\prime \prime }$. Red squares and blue circles correspond to the saddle-point solution and Gaussian corrections, respectively. The evolution of ${\mathbf{Q}}_0$ from the isotropic triangular to the decoupled chain regime is shown in the top left (see the text).

Figure 7

Spin stiffness predicted by the Schwinger bosons as a function of $J^{\prime }=J^{\prime \prime }$. Red squares and blue circles correspond to the saddle-point solution and Gaussian corrections, respectively.