Field-induced incommensurate ordering in Heisenberg chains coupled by Ising interaction: Model for ytterbium aluminum perovskite ${\mathrm{YbAlO}}_3$

We study isotropic antiferromagnetic Heisenberg chains coupled by antiferromagnetic Ising interaction as an effective spin model for the ytterbium aluminum perovskite ${\mathrm{YbAlO}}_3$. Using the density-matrix renormalization group (DMRG) method we calculate the magnetization curve, local spin, central charge, and dynamical spin structure factors in the presence of magnetic field. From the fitting of the experimental magnetization curve, the effective intrachain and interchain couplings are estimated as $J=2.3$ K and $J_{\mathrm{ic}}=0.8$ K, respectively. We can quantitatively explain the experimental observations: (i) phase transition from antiferromagnetic to incommensurate order at field 0.35 T, and (ii) quantum critical behaviors at the saturation field of 1.21 T. Furthermore, the low-energy excitations in the experimental inelastic neutron scattering spectra can be well described by our DMRG results of the dynamical structure factors.

Figure 1

Lattice structure of the effective spin model (1) for ${\mathrm{YbAlO}}_3$. Red and blue chains denote two sublattices, where a bipartite square lattice is formed in the $a-b$ plane. Intrachain $J$ is isotropic Heisenberg and interchain $J_{\mathrm{ic}}$ is Ising interactions.

Figure 2

(a) Fitting of the low-temperature experimental magnetization curve for $H\parallel a$ by DMRG result with the system (1), where the experimental magnetization is normalized by $g^a(=g^z)=7.6$. (b) DMRG results of magnetization curve for several kinds of interchain couplings with different XXZ anisotropy.

Figure 3

$z$ component of local spin $\langle S_{i,1}^z\rangle$ for (a) $H=0-0.35$ T, (b) 0.36 T, (c) 0.44 T, and (d) 0.89 T. (e) $\langle S_{n,\lambda }^z\rangle$ with $\lambda =\frac{3+{(-1)}^n}{2}$ for $H=0.89$ T (see text). The red line is a fitting by Eq. (2).

Figure 4

(a) Finite-size scaling analysis for the amplitude of AFM modulation at $M=0$ for several strengths of the Ising interchain coupling $J_{\mathrm{ic}}$. (b) $L\to \infty$ extrapolated values of the amplitude as a function of $J_{\mathrm{ic}}$. (c) Finite-size scaling analysis for the amplitude of IC oscillation for several strengths of magnetic field $H$ with our ${\mathrm{YbAlO}}_3$ parameters ($J=2.3$ K, $J_{\mathrm{ic}}=0.8$ K). (d) $L\to \infty$ extrapolated values of the amplitude as a function of $H$ for ${\mathrm{YbAlO}}_3$.

Figure 5

Central charge for (a) the isotropic SU(2) Heisenberg chain and (c) the system (1) as a function of the magnetization. (b),(d) Finite-size scaling analysis of the central charge to the saturation limit $M\to 0.5$. The line in (b) is a linear fit.

Figure 6

Dynamical spin structure factors $S^{zz}(q_c,\omega )$ (left panel), $S^{+-}(q_c,\omega )$ (center panel), and $S^{-+}(q_c,\omega )$ (right panel) for several strengths of the magnetic field $H\parallel z$. Open circle in the left-top panel denote a large $\delta$ peak indicating the AFM order. The dotted lines in the top panels are magnon dispersion obtained within the spin-wave analysis.

Figure 7

Dynamical spin structure factor $S^{zz}(q_c,\omega )$ for the spin chain (A1) with (a) $\mathrm{\Delta }=1.43$, (b) $\mathrm{\Delta }=1$, and (c) $\mathrm{\Delta }=0.77$ using a periodic 24-site cluster. The other parameters are set to be $J=2.3$ K and $h_z=0.8$ K, which were obtained in Sec. 3e.

Figure 8

Fitting of the low-temperature experimental magnetization curve for $H\parallel a$ by DMRG result with our effective spin model (1), where the experimental magnetization is normalized by $g^a(=g^z)=7.6$.

Figure 9

Static spin structure factors for (a),(b) $H=0-0.35$ T, (c),(d) $H=0.44$ T, (e),(f) $H=0.67$ T, (g),(h) $H=0.89$ T, and (i),(j) $H=1.03$ T using an open cluster with $50\times 2$ sites.