Phase diagram of the frustrated spatially-anisotropic $S=1$ antiferromagnet on a square lattice

We study the $S=1$ square lattice Heisenberg antiferromagnet with spatially anisotropic nearest-neighbor couplings $J_{1x}$ and $J_{1y}$ frustrated by a next-nearest-neighbor coupling $J_2$ numerically using the density-matrix renormalization-group (DMRG) method and analytically employing the Schwinger-Boson mean-field theory (SBMFT). Up to relatively strong values of the anisotropy, within both methods we find quantum fluctuations to stabilize the Néel-ordered state above the classically stable region. Whereas SBMFT suggests a fluctuation-induced first-order transition between the Néel state and a stripe antiferromagnet for $1/3\le J_{1x}/J_{1y}\le 1$ and an intermediate paramagnetic region opening only for very strong anisotropy, the DMRG results clearly demonstrate that the two magnetically ordered phases are separated by a quantum-disordered region for all values of the anisotropy with the remarkable implication that the quantum paramagnetic phase of the spatially isotropic $J_1\text{−}{J}_2$ model is continuously connected to the limit of decoupled Haldane spin chains. Our findings indicate that for $S=1$ quantum fluctuations in strongly frustrated antiferromagnets are crucial and not correctly treated on the semiclassical level.

Figure 1

(Color online) The model studied in this paper consists of a $S=1$ Heisenberg system with spatially anisotropic nearest-neighbor couplings $J_{1x}$ and $J_{1y}$ and isotropic next-nearest-neighbor couplings $J_2$. Pending the balance of these couplings one finds either a simple (a) staggered or (b) “stripe” antiferromagnetic order. In the limit of isolated chains a (c) stack of isolated Haldane spin chains is formed and based on our DMRG calculations we could conclude that these survive all the way to the isotropic limit in the vicinity of the point of maximal frustration $\left(J_2/J_1=0.5\right)$.

Figure 2

Phase diagram for $S=1$ as a function of and anisotropy $\alpha =\left(J_{1y}-J_{1x}\right)/J_{1y}>0$ between the nearest-neighbor exchange couplings and relative strength of the next-nearest-neighbor coupling $J_2/J_{1y}$ obtained within SBMFT. Up to an anisotropy $\alpha \approx 0.66$ the $\left(\pi ,\pi \right)$ and $\left(0,\pi \right)$ antiferromagnetic orders are separated by a first-order transition (solid line). At the tricritical point the first-order line splits into two second-order lines (dashed and dotted) separating the two magnetic phases from a gapped nonmagnetic phase.

Figure 3

(Color online) (a) The ground-state energy per site $E_0/N$ at different system size $N=4\times 4$ (blue triangle), $6\times 6$ (red circle), and $8\times 8$ (black square) for the isotropic case with $J_{1x}=J_{1y}=J_1$. (b) Ground-state energy per site $E_0/N$ at $N=8\times 8$ as a function of the truncation error. The dashed line is the extrapolation to zero error limit. Inset: the spin-1 gap $\Delta E$ as a function of the truncation error.

Figure 4

(Color online) The evolution of the peak values of the structure factors, (a) $S_z\left(\pi ,\pi \right)/N$, (b) $S_z\left(0,\pi \right)/N$, and (c) of the spin-1 gap, at three different size $N=4\times 4$, $6\times 6$, and $8\times 8$, as well as their thermodynamic limit extrapolations in the isotropic case $J_{1x}=J_{1y}=J_1=1$. The finite-size scaling for the spin-1 gap is shown in (d).

Figure 5

(Color online) The peak values, (a) $S_z\left(\pi ,\pi \right)/N$, (b) $S_z\left(0,\pi \right)/N$, and (c) the spin-1 gap, versus $J_2/J_{1y}$ at $J_{1x}/J_{1y}=0.5$ for different sizes. The finite-size scaling for the spin-1 gap is shown in (d).

Figure 6

(Color online) (a) $S_z\left(\pi ,\pi \right)/N$, (b) $S_z\left(0,\pi \right)/N$, and (c) the spin-1 gap, versus $J_{1x}/J_{1y}$ at $J_2/J_{1y}=0.25$ for different sample sizes. In (d), the finite-size scaling for the spin-1 gap is given.

Figure 7

(Color online) (a) $S_z\left(\pi ,\pi \right)/N$, (b) $S_z\left(0,\pi \right)/N$, and (c) the spin-1 gap, versus $J_{1x}/J_{1y}$ at $J_2=0$ at different sizes. In (d), the finite-size scaling for the spin-1 gap is given.

Figure 8

(Color online) At $J_{1x}=0$, the structure factors (a) $S_z\left(\pi ,\pi \right)/N$ and (b) $S_z\left(0,\pi \right)/N$ as well as the (c) spin-1 gap versus $J_2/J_{1y}$ are shown at different sizes, including their thermodynamic extrapolations.

Figure 9

(Color online) (a) The size dependence of the structure factor [for a given $N_y=6$ (six-leg systems)] at $J_{1x}=0$ and small $J_2/J_{1y}$ which quickly saturates at $N_x>N_y$, indicating that $x=1/N_y$ in the scaling function $f\left(x\right)$ for the structure factor is more appropriate in this extreme limit. The corresponding finite-size scaling of the structure factors for square samples are illustrated in (b) and (c), respectively. For comparison, the spin-1 gap at different $J_2$ is still well scaled by $x=1/N$ in the scaling function as shown in (d).

Figure 10

(Color online) Phase diagram for the anisotropic $J_1\text{−}{J}_2$ model determined by DMRG. The Néel order and stripe order phases are separated by a paramagnetic regime with the boundaries denoted by the red line with full circles for the upper and the blank line with full squares for the lower transition points, respectively. In the middle of the paramagnetic region lies in a dotted line with crosses which marks the maximal spin-1 gap $\Delta E_{\text{max}}$. Note that the symbols of circle, square, and triangular denote the phase-transition points determined by DMRG in the previous figures.