Consequences of critical interchain couplings and anisotropy on a Haldane chain

Effects of interchain couplings and anisotropy on a Haldane chain have been investigated by single-crystal inelastic neutron scattering and density functional theory (DFT) calculations on the model compound ${\mathrm{SrNi}}_2{\mathrm{V}}_2{\mathrm{O}}_8$. Significant effects on low-energy excitation spectra are found where the Haldane gap (${\Delta }_0\approx 0.41J$, where $J$ is the intrachain exchange interaction) is replaced by three energy minima at different antiferromagnetic zone centers due to the complex interchain couplings. Further, the triplet states are split into two branches by single-ion anisotropy. Quantitative information on the intrachain and interchain interactions as well as on the single-ion anisotropy is obtained from the analyses of the neutron scattering spectra by the random-phase approximation method. The presence of multiple competing interchain interactions is found from the analysis of the experimental spectra and is also confirmed by the DFT calculations. The interchain interactions are two orders of magnitude weaker than the nearest-neighbor intrachain interaction $J=8.7$ meV. The DFT calculations reveal that the dominant intrachain nearest-neighbor interaction occurs via nontrivial extended superexchange pathways Ni-O-V-O-Ni involving the empty $d$ orbital of V ions. The present single-crystal study also allows us to correctly position ${\mathrm{SrNi}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ in the theoretical $D\text{−}{J}_{\perp }$ phase diagram [T. Sakai and M. Takahashi, Phys. Rev. B 42, 4537 (1990)], showing where it lies within the spin-liquid phase.

Figure 1

(a) A schematic representation of the singlet ground state of a Haldane chain in terms of a valence-bond solid state. (b) A schematic representation (color plot) of the one-magnon excitation spectrum of an isolated Haldane chain over two Brillouin zones. (c) The dispersion relation (solid line) and the intensity profile (dashed line at the bottom) of the excitation spectrum in (b).

Figure 2

The observed (solid circles) and calculated (solid black line) neutron diffraction patterns for ${\mathrm{SrNi}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ measured at 2 K. The difference between observed and calculated patterns is shown by the thin blue line at the bottom. The vertical bars give the allowed Bragg peak positions.

Figure 3

(a) The crystal structure of ${\mathrm{SrNi}}_2{\mathrm{V}}_2{\mathrm{O}}_8$. The Ni, V, and O ions are shown by green (large), yellow (medium), and red (small) spheres, respectively. The dotted lines indicate the dimensions of the unit cell. For clarity, only two chains (out of four chains per unit cell) are shown. (b) Projection of the crystal structure on to the $ab$ plane. Only the atoms and bonds are shown without polyhedra for clarity. Ni-O and V-O bonds are shown by green and yellow lines, respectively. Red arrows represent the direction of the rotation of the screw chains when propagating along the $c$ axis. Neighboring chains rotate in opposite directions. (c) Arrangement of the Ni, V, and O atoms within a given $ab$ plane (at $c=0$). The direct distances between Ni ions along the diagonal directions are shown. The shortest diagonal corresponds to the exchange interaction $J_4$, which occurs via Ni-$\mathrm{O}\cdots \mathrm{O}$-Ni pathway as indicated.

Figure 4

The measured and calculated magnetic excitation spectra. (left) The measured magnetic excitation spectra along (a) $(1,0,l)$ (b), $(3,0,l)$, and (c) $(2,2,l)$ at 3.5 K using the triple-axis spectrometer PUMA. The excitation spectra were obtained by combining several constant-energy scans with an interval of 2 meV. Black dots represent the points of measurements. (middle) The simulated magnetic excitation spectra along (d) $(1,0,l)$, (e) $(3,0,l)$, and (f) $(2,2,l)$ as per the model discussed in the text and using the best-fit parameters (Table 2). The calculated spectra were convoluted with a Gaussian function and corrected for the magnetic form factor. The intensities are in arbitrary units; r.l.u. stands for reciprocal lattice units. Some additional intensities (sharp and diffuse) in the measured spectra are due to the contributions from phonon scattering. (right) The calculated dispersion relations (solid curves) and intensities (dashed curves at the bottom) of each of the six individual modes along (g) $(1,0,l)$, (h) $(3,0,l)$, and (i) $(2,2,l)$. The six modes appear due to the screw nature of the chains (see text for details). For simplicity dispersion curves are shown without considering the anisotropy-induced splittings. Different colors (blue, red, green, magenta, black, and cyan) are used to represent different modes.

Figure 5

The measured and simulated magnetic excitation spectra perpendicular to the chain axis. Measured excitation patterns (a) and (c) using the PANDA spectrometer at 3 K and (d) using the IN8 spectrometer at 1.5 K. Black dots in (a) and (d) represent the points of measurements. The solid blue curves in (a) and (d) are the fitted dispersion curves. (b) and (e) The simulated spectra, corresponding to (a) and (d), as per the model discussed in the text using the best-fit parameters (Table 2). For the comparison, simulated dispersion curves with $J_4=0$ (see text) are also shown in (b) and (e) by the dashed lines. The thin black lines through the measured points in (c) are the Gaussian fits to the data which are convoluted with instrumental resolution. The big solid circular points are the fitted peak positions, and the thick lines are the fitted dispersion curves as per the model discussed in the text.

Figure 6

The measured energy-minima patterns in the (a) $\left(hk0\right)$ and (b) $\left(hk1\right)$ reciprocal planes, respectively. The big red, medium green, and small blue points correspond to the average values of lowest (2.3 meV), intermediate (3.6 meV), and highest (6.1 meV) energy minima, respectively. The arrows indicate the measured dispersion along the diagonal directions. The measured dispersions (c) between $(1,1,1)$ and $(2,2,1)$ and (d) between $(2,3,0)$ and $(3,2,0)$ reciprocal points. The solid lines are the fitted dispersion curves.

Figure 7

The magnetic interaction topology of ${\mathrm{SrNi}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ (mapped onto equivalent straight chains). $J$: nearest-neighbor intrachain interaction, $J^{\prime }$: next-nearest-neighbor intrachain interaction, $J_1$: between two Ni ions from neighboring chains with the same $c$ values (in the $ab$ plane), $J_2$: between two Ni ions from nei-ghboring chains with offset of $c/4,J_3$: between Ni ions from neighboring chains with offset of $c/2$, and $J_4$: between Ni ions from diagonal chains within the $ab$ plane. The $\mathrm{Ni}\cdots \mathrm{Ni}$ direct distances in the real crystal structure for corresponding interactions are also given.

Figure 8

The $D\text{−}{J}_{\perp }$ phase diagram for the weakly coupled Haldane chains (Ref. [12]). The position of ${\mathrm{SrNi}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ obtained from the present study is shown by the red square and is compared with the previous report (Ref. [14]). Positions of the other experimentally studied Haldane chain compounds are also shown as per Ref. [14].

Figure 9

(a) A schematic representation of the diagonal exchange in ${\mathrm{SrNi}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ by a combination of in-plane diagonal interchain interaction $J_4$ (dotted black line) and the NNN intrachain interaction $J^{\prime }$ (solid yellow line). (b) The NN and NNN intrachain interactions $J$ and $J^{\prime }$, respectively, in a single screw chain.

Figure 10

The valence band of ${\mathrm{SrNi}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ as calculated using GGA. Total and atom-resolved densities of states (DOSs) are shown. The Fermi level is at zero energy. The inset shows the orbital-resolved DOS for the Ni $3d$ states.

Figure 11

The GGA band structure of the $e_g$ states (circles) and the fit using Ni $3d_{x^2-y^2}$- and $3d_{3z^2-r^2}$-centered Wannier functions (dashed lines). The Fermi level is at zero energy.

Figure 12

The Wannier function centered at the Ni $x^2\text{−}{y}^2$ orbital. The dominant exchange coupling $J$ is realized by superexchange along the two inequivalent Ni-O-V-O-Ni paths indicated by arrows.