Solitonic excitations in the Ising anisotropic chain ${\mathrm{BaCo}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ under large transverse magnetic field

We study the dynamics of the quasi-one-dimensional Ising-Heisenberg antiferromagnet ${\mathrm{BaCo}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ under a transverse magnetic field. Combining inelastic neutron scattering experiments and theoretical analyses by field theories and numerical simulations, we mainly elucidate the structure of the spin excitation spectrum in the high-field phase, appearing above the quantum phase transition point ${\mu }_0{H}_c\approx 10\mathrm{T}$. We find that it is characterized by collective solitonic excitations superimposed on a continuum. These solitons are strongly bound in pairs due to the effective staggered field induced by the nondiagonal $g$ tensor of the compound and are topologically different from the fractionalized spinons in the weak-field region. The dynamical susceptibility numerically calculated with the infinite time-evolving block decimation method shows an excellent agreement with the measured spectra, which enables us to identify the dispersion branches with elementary excitations. The lowest-energy dispersion has an incommensurate nature and has a local minimum at an irrational wave number due to the applied transverse field.

Figure 1

Canted antiferromagnetic structure of the ${\mathrm{Co}}^{2+}$ screw chains of ${\mathrm{BaCo}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ in a 12 T transverse magnetic field applied along the $\mathbf{b}$ axis (shown by the dark green arrow): (a) perspective view of one chain (the Co and O atoms are white and red, respectively; the ${\mathrm{CoO}}_6$ octahedra are materialized in light blue; and the ${\mathrm{Co}}^{2+}$ spins are represented by the blue arrows); (b) projection in the $(\mathbf{a},\mathbf{b})$ plane of the two types of Co chains, with red (blue) spins for the chains having a $4_1$ ($4_3$) screw axis symmetry. Two chains of the same type correspond to each other by the lattice centering $\frac{1}{2}(\mathbf{a}+\mathbf{b}+\mathbf{c})$. In both (a) and (b), the labeling of the Co atoms is the same as in Refs. [15, 23]. At 12 T, the 0.92-${\mu }_{\mathrm{B}}$ antiferromagnetic component of the magnetic moment is aligned along the $\mathbf{a}$ axis with a field-induced 0.55-${\mu }_{\mathrm{B}}$ ferromagnetic component along the $\mathbf{b}$ direction. (c) Phase diagram as a function of the applied transverse magnetic field with a sketch of the soliton excitations and their time evolution on both sides of the critical field. At zero field, the spinon excitations carry a spin $S_c=\pm 1/2$, corresponding to the topological index of the excitation, and hop by two sites along the chain axis when time evolves. At 12 T in the high-field phase, the elementary excitations carry a topological index $S_a=\pm 1$, hop by one site when time evolves, and are dual from the low-field spinons [23].

Figure 2

Inelastic scattering intensity maps showing the intrachain dispersion of the magnetic excitations along (a)–(c) $\mathbf{Q}=(2,0,Q_L)$ and (d)–(f) $\mathbf{Q}=(0,0,Q_L)$, in a transverse field of 12 T applied along the $\mathbf{b}$ axis: (a) and (d) maps obtained experimentally on ThALES and IN12, respectively, from a series of constant-$\mathbf{Q}$ energy scans, compared with the numerically calculated neutron scattering cross section [48] (b) and (e) of a tetramerized chain with $t=0.3$ meV and (c) and (f) of a uniform chain ($t=0$), as explained in the text. In (a), the yellow dotted lines show the energy scans presented in Figs. 3–3, and the first three modes are labeled in (a) and (b) by the orange numbers. The white lines in (c) correspond to the same three modes plus a fourth one, all sketched in Fig. 6 with the same symbols (solid, dotted, dashed, and dash-dotted lines for modes 1, 2, 3, and 4, respectively). Constant-Q energy scans along $(2,0,Q_L)$ and $(0,0,Q_L)$ comparing experimental and calculated data are shown in Appendix pp3.

Figure 3

Constant-$\mathbf{Q}$ energy scans measured on IN12 using polarized neutrons in the spin-flip (SF) and non-spin-flip (NSF) channels at (a) and (b) $\mathbf{Q}=(2,0,1)$, (c) $\mathbf{Q}=(2,0,1.2)$, and (d) $\mathbf{Q}=(2,0,1.5)$. These scans are materialized by the yellow dotted lines on the map of Fig. 2. Note the about 16 times larger intensity scale for (a) as compared with (b)–(d). The SF intensity corresponds to fluctuations occurring in the $(\mathbf{a},\mathbf{c})$ plane and perpendicular to the scattering vector $\mathbf{Q}$, while the NSF one corresponds to fluctuations along the $\mathbf{b}$ axis. Here, neutron monitor (mon).

Figure 4

(a)–(c) $S_{aa}, S_{bb}$, and $S_{cc}$ components of the intensity color map calculated at 12 T [48] and shown in Fig. 2. (d)–(f) show the same quantities but calculated in a higher magnetic field of 15.7 T. The dashed red lines show the shift of the incommensurate minimum, from $Q_L\simeq 1.26$ to $Q_L\simeq 1.32$, with increasing magnetic field.

Figure 5

Dynamical susceptibility calculated for the simplified model of Eq. (4) describing a linear spin chain with $\epsilon =0.53, {\mu }_{0}H=12\mathrm{T}$. $(g_{ba},g_{bb})$ are equal to (0.94,2.35) for (a)–(c) and equal to (0.94,0) for (d)–(f). The components of dynamical susceptibility (a) and (d) $S_{aa}$, (b) and (e) $S_{bb}$, and (c) and (f) $S_{cc}$ are shown. In contrast to all other theoretical results, the squared neutron magnetic form factor $f^2\left(Q\right)$ of ${\mathrm{Co}}^{2+}$ was not included in these calculations, which are not directly compared with experiments.

Figure 6

Schematic drawing of the excitation spectrum of ${\mathrm{BaCo}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ with the same labeling of the modes as in Fig. 2 without (a) and with (b) the replica due to the presence of four ions in the lattice unit cell describing the screw chains (see Sec. 3c).

Figure 7

Constant-$\mathbf{Q}$ scans at chosen $(2,0,Q_L)$ values as a function of the energy: those measured on ThALES (left panels) and the calculated ones [48] (right panels). Note that the same scaling ratio is used in all panels between experiment and theory.

Figure 8

Constant-$\mathbf{Q}$ scans at chosen $(0,0,Q_L)$ values as a function of the energy: those measured on IN12 (left panels) and the calculated ones [48] (right panels). Note that the same scaling ratio is used in all panels between experiment and theory.