Giant magnetic field dependence of the coupling between spin chains in ${\mathrm{BaCo}}_2{\mathrm{V}}_2{\mathrm{O}}_8$

We use nuclear magnetic resonance to map the complete low-temperature phase diagram of the antiferromagnetic Ising-like spin-chain system ${\mathrm{BaCo}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ as a function of the magnetic field applied along the chains. In contrast to the predicted crossover from the longitudinal incommensurate to the transverse antiferromagnetic ordered phase, we find a sequence of three magnetically ordered phases between the critical fields $3.8$ and $22.8 \mathrm{T}$. Their origin is traced to the giant magnetic field dependence of the total effective coupling between spin chains, found to vary by a factor of 24. We explain this interesting phenomenon as emerging from a combination of nontrivially coupled spin chains and incommensurate spin fluctuations in the chains treated as Tomonaga-Luttinger liquids.

Figure 1

Phase diagram of ${\mathrm{BaCo}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ as a function of temperature $T$ and the magnetic field $B$ applied along the $c$ axis. Phase boundaries are obtained from the $T_1^{-1}$ data ($◯$) displayed in Figs. 2 and 2, and from the neutron diffraction ($\square$) and specific heat data ($▵$) in Ref. [30]. Lines are guides to the eye. The inset shows three views of the structural unit of ${\mathrm{BaCo}}_2{\mathrm{V}}_2{\mathrm{O}}_8$. Edge-sharing ${\mathrm{CoO}}_6$ octahedra (blue) form chains along $c$. Exchange coupling between spin-$1/2 \mathrm{Co}^{2+}$ ions (spheres) in neighboring chains is mediated by ${\mathrm{O}}_4$ plaquettes (thin dashed lines) for the NNN chains (neighbors along the $a-b$ bisectors) and by ${\mathrm{VO}}_4$ tetrahedra (red) for the NN chains (neighbors along $a$ or $b$). The intrachain coupling mediated by shorter ${\mathrm{O}}_2$ bridges is much stronger. Thick lines represent different types of interchain couplings.

Figure 2

(a) Temperature dependence $T_1^{-1}\left(T\right)$ in ${\mathrm{BaCo}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ for different magnetic fields applied along the $c$ axis. Thin lines are guides to the eye. Arrows mark the extracted transition temperatures $T_c$. Thick gray lines are power-law fits in the TLL phase. Crosses mark the $T_1^{-1}$ values extrapolated to $T_c$. (b) Field dependence $T_1^{-1}\left(B\right)$ for different temperatures, with the same presentation as in (a). (c)–(f) Representative ${}^{51}\mathrm{V}$ NMR spectra of various phases recorded after different NMR echo-decay times $2\tau$. The frequency scale is relative to ${\nu }_L=\gamma B$, where $\gamma =11.193$ MHz/T and $B$ is corrected for the demagnetizing effect. The IC1 phase (d) yields two sets of V sites located between the NN chains hosting IC ordered waves that are in phase and in opposite phase, respectively. The spectrum for $2\tau =18\mu \mathrm{s}$ is reproduced (dotted line) by the corresponding narrow (dashed line) and broad (short-dashed line) $\mathsf{U}$-shaped components of the same area. The IC2 phase [(e), (f)] yields a single set of V sites as all IC ordered waves are in phase. The spectrum for $2\tau =32\mu \mathrm{s}$ is reproduced by the narrow $\mathsf{U}$-shaped component (dashed line). The spectrum for $2\tau =18\mu \mathrm{s}$ with extended wings (indicated by arrows) is reproduced by the narrow $\mathsf{U}$-shaped component with a distributed width following a 2D sinusoidal modulation of the IC-wave amplitude in the $a-b$ plane (dotted line).

Figure 3

(a), (b) Dependence of the TLL parameters $\eta$ and $u$ on the magnetic field $B$ in ${\mathrm{BaCo}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ as extracted from the power-law fits in Fig. 2. Thick gray lines are predictions of the spin-$1/2XXZ$ chain model with parameters relevant to ${\mathrm{BaCo}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ [3]. (c) Magnetic field dependence of the effective interchain exchange coupling $J^{\prime }$ ($◯$) and of the exchange coupling $J_1$ ($\square$) between the NN chains as extracted from the phase boundary in Fig. 1. The thin blue line is a constant fit to the $J^{\prime }\left(B\right)$ data between $B_c$ and $B_1$, giving $J_2/k_B=0.042$ K. The thick red line is a fit to the $J_1\left(B\right)$ data above $B_1$ using Eq. (2). The inset shows the proposed schemes of the interchain exchange couplings corresponding respectively to the structural units in the inset of Fig. 1.