Magnetic structure and dispersion relation of the $S=\frac{1}{2}$ quasi-one-dimensional Ising-like antiferromagnet ${\mathrm{BaCo}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ in a transverse magnetic field

${\mathrm{BaCo}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ consists of Co chains in which a ${\mathrm{Co}}^{2+}$ ion carries a fictitious spin $\frac{1}{2}$ with Ising anisotropy. We performed elastic and inelastic neutron scattering experiments in ${\mathrm{BaCo}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ in a magnetic field perpendicular to the $c$ axis which is the chain direction. With applying magnetic field along the $a$ axis at 3.5 K, the antiferromagnetic order with the easy axis along the $c$ axis, observed in zero magnetic field, is completely suppressed at 8 T, while the magnetic field gradually induces an antiferromagnetic order with the spin component along the $b$ axis. We also studied magnetic excitations as a function of transverse magnetic field. The lower boundary of the spinon excitations splits gradually with increasing magnetic field. The overall feature of the magnetic excitation spectra in the magnetic field is reproduced by the theoretical calculation based on the spin $\frac{1}{2} XXZ$ antiferromagnetic chain model, which predicts that the dynamic magnetic structure factor of the spin component along the chain direction is enhanced and that along the field direction has clear incommensurate correlations.

Figure 1

(a) ${\mathrm{Co}}^{2+}$ ions in a unit cell projected onto the $ab$ plane. “$+$” and “$-$” represent the spin directions along the $c$ axis in zero field. Horizontal arrows represent the staggered magnetic field induced perpendicular to the magnetic field. (b) A schematic view of the simplified Co chain in ${\mathrm{BaCo}}_2{\mathrm{V}}_2{\mathrm{O}}_8$. $X$ and $Z$ are defined along the magnetic field and chain directions, respectively. $Y$ is perpendicular to both $X$ and $Z$. When the magnetic field is applied along $X$, effective magnetic fields with $+-+-$ ($h_{\mathrm{eff}}^Y$) and $+--+$ ($h_{\mathrm{eff}}^Z$) are induced along $Y$ and $Z$, respectively.

Figure 2

(a) Magnetic Bragg peak intensities as a function of magnetic field measured at 3.5 K on CTAX. Filled and open symbols represent intensities measured while ramping up and down the magnetic field, respectively. (b) Observed and calculated staggered magnetic moment along the $b$ axis as a function of magnetic field.

Figure 3

Energy cuts of the inelastic neutron scattering intensity at 0 T (a) and 10 T (b) in the $\left(0kl\right)$ plane measured at $T=3.5$ K with $E_{\mathrm{i}}=13.1$ meV on DCS. The scattering intensity is integrated in an energy range of $1.5\le E\le 2.5$ meV. The arc-shaped scatterings are background signal from the aluminum sample holder.

Figure 4

(a)–(f) Contour maps of the inelastic neutron scattering intensity $S(Q,E)$ for ${\mathrm{BaCo}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ single crystal measured at ${\mu }_{0}H=0$, 6, and 10 T and $T=3.5$ K with $E_{\mathrm{i}}=13.1$ meV on DCS. The energy resolution is 0.8 meV at the elastic position. The spectra of (a), (c), and (e) and (b), (d), and (f) are integrated in the range of $-1\le k\le 1$ and $5\le k\le 6.5$, respectively. The intensity is shown in logarithmic scale. The solid curves in (a) and (b) are the lower boundary of the spinon excitations calculated for the $S=\frac{1}{2} XXZ$ AF chain model with $J=6.25$ meV and $\epsilon =0.44$. $l=0$, 2, and 4 correspond to $Q=0, \pi$, and $2\pi$, respectively. (g)–(j) $\tilde{S}(Q,E)$ calculated by the DMRG method using the spin Hamiltonian [Eq. (2)]. To directly compare with the experimental data, appropriate fractions of spin components are summed up: 100% of $S_X$ and $S_Y$ components for $(0,0,2)$ [(g) and (i)]; 100%, 20%, and 80% of $S_X, S_Y$, and $S_Z$ components, respectively, for $(0,6,2)$ [(h) and (j)]. (g) and (h) and (i) and (j) are for 0 and 10 T, respectively.

Figure 5

Magnetic excitation spectra as a function of magnetic field measured at $T=3.5$ K with $E_{\mathrm{f}}=5$ meV on CTAX. The energy resolution is 0.25 meV at the elastic position. The spectra of (a)–(e) and (f)–(j) were measured at (0, 0, 2) and (0, 3, 1), respectively. The arrows indicate the excitation peak positions. The solid lines are guides to the eye. Note that the logarithmic scale is used for vertical axis in (i) and (j).

Figure 6

Magnetic excitation gap energies at the magnetic zone center measured at 3.5 K as a function of magnetic field, obtained from the magnetic excitation spectra shown in Fig. 5. Only the split peaks of the lowest energy mode is plotted. The filled and open symbols represent the data points observed at (0, 3, 1) and (0, 0, 2), respectively.

Figure 7

$\tilde{S}(Q,E)$ calculated by the DMRG method using the spin Hamiltonian [Eq. (2)]. (a)–(c) and (d)–(f) are for 0 and 10 T, respectively. (g)–(i) are for 10 T with no staggered field ($h_Y=h_Z=0)$.