Phase transitions and spin dynamics of the quasi-one dimensional Ising-like antiferromagnet ${\mathrm{BaCo}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ in a longitudinal magnetic field

By combining inelastic neutron scattering and numerical simulations, we study the quasi-one-dimensional Ising-like quantum antiferromagnet ${\mathrm{BaCo}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ in a longitudinal magnetic field applied along the Ising anisotropy, which is also the chain direction. The external field closes the excitation gap due to the magnetic anisotropy, inducing a transition from the Néel ordered state to an incommensurate longitudinal spin density wave phase. If the field is increased further, another transition into a transverse antiferromagnetic phase takes place at 9 T due to the competition between longitudinal and transverse correlations. We numerically and experimentally show that the model of XXZ chains connected by a weak interchain interaction well reproduces this transition. We also calculate the dynamical susceptibility and demonstrate that it agrees quantitatively with inelastic neutron scattering measurements. In contrast to the abrupt change of magnetic ordering, the spectra do not change much at the transition at 9 T, and the spin dynamics can be described as a Tomonaga-Luttinger liquid. We also refine the modeling of ${\mathrm{BaCo}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ by including a four-site periodic term arising from the crystal structure which enables one to account for an anomaly of the magnetic susceptibility appearing at 19.5 T, as well as for the anticrossing observed in the inelastic neutron scattering spectra.

Figure 1

(a) Phase diagram of ${\mathrm{BaCo}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ in the parameter space of the longitudinal magnetic field and temperature determined from neutron diffraction, specific heat, and NMR measurements. The phases correspond to longitudinal antiferromagnetic (LAF), longitudinal spin density wave (LSDW), and transverse antiferromagnetic (TAF) orderings as determined by neutron diffraction [15, 18], while the phase labeled “?” was inferred to be another incommensurate phase from NMR [10]. (b)–(d) Schematic pictures for the spin arrangement in the magnetic phases LAF, LSDW, and TAF appearing in the field region below the field $H_{1/2}$ corresponding to half of the saturated magnetization. These sketched components should be added to a ferromagnetic component, i.e., uniform magnetization, which increases with the field above $H_{\mathrm{c}}$ [see Fig. 2, right axis].

Figure 2

(a) Field dependence of the component along ${\mathbf{c}}^{*}$ of the magnetic reflection determined by neutron diffraction (open symbols, left axis) [18]. It scales with the calculated field-induced ferromagnetic component, i.e., uniform magnetization, $M_z$ (black solid stars, right axis) [see Fig. 3 and main text]. The arrows show the $l$ positions for the incommensurate modulation $\delta$ in reciprocal lattice units (r.l.u.), which is probed by neutron diffraction up to 9.3 T, then is extrapolated above for magnetic fields corresponding to the IN5 measurements presented in (b), which shows $l$-cuts around the $(3,2,0)$ position at zero energy, $T=50\mathrm{mK}$, and different magnetic fields. The neutron intensity is given in arbitrary units (a.u.). (c)–(e) Intensity maps at constant energy equal to $0.0\pm 0.1\mathrm{meV}$ in the $(h,k,0)$ and $(h,k,\delta )$ scattering planes, at 50 mK for different magnetic fields: (c) 0 T, (d) 8 T where $\delta =0.26$ was previously determined in the LSDW phase, and (e) 10 T where one would expect incommensurate magnetic Bragg peaks with $\delta =0.35$ if the system would still be in the LSDW phase. The blue dotted circles highlight the nuclear Bragg peaks and the purple solid circles highlight the magnetic Bragg peaks corresponding either to the $(1,0,0)$ or $(1,0,\delta )$ propagation vector.

Figure 3

(a) Calculated staggered transverse magnetization $M_x$ vs the magnetic field for two different interchain interactions $J^{\prime }$. The spin-flop transition is obtained at 9 T for $J^{\prime }=0.17$ meV. (b) $M_x$ and uniform magnetization $M_z$ vs the magnetic field calculated up to saturation for $J^{\prime }=0.17$ meV.

Figure 4

Inelastic neutron scattering spectra (dynamical susceptibility) calculated numerically for the model given by Eq. (4) at various magnetic fields ranging from 6 up to 20.75 T. The total (left), transverse (middle), and longitudinal (right) components are shown. The spin-flop transition ${\mu }_0{H}^{*}$ detected by neutron diffraction and the transition at half the saturated magnetization ${\mu }_0{H}_{1/2}$ detected by NMR are indicated by dashed lines.

Figure 5

Inelastic neutron scattering intensity maps measured on IN5 at 50 mK, showing the dispersion of the excitations around $\mathbf{Q}=(3,2,0)$ along the ${\mathbf{c}}^{*}$ direction, at (a) 0 T, (b) 6 T, (c) 8 T, and (d)–(f) 10 T. The incident wavelength is 3.4 Å for (a)–(c) and (e), 4.8 Å for (d), and 2.3 Å for (f), which allows one to explore the different energy ranges. The blue arrows indicate the two-string signal in (e) and (f). (g) The total dynamical susceptibility, calculated numerically for the model given by Eq. (4) at 10 T, to be compared to the inelastic neutron scattering spectrum from (d). (h) An energy cut at $\mathbf{Q}=(2,2,\delta )$ with $\delta =0.35$, measured at 10 T and $\lambda =4.8$ Å (blue circles with line), where the dispersion along the ${\mathbf{c}}^{*}$ direction takes an energy minimum [see the blue arrow in (d)]. Note that the excitation gap is slightly larger and thus more visible for the cut $(h,2,\delta )$ with $h=2$ than $h=3$; hence, we adopt the value $h=2$ in (h). The result of numerical calculation (red line) is compared with the measured data in (h). (i) Energy cuts at $\mathbf{Q}=(3,2,0)$ for all measured magnetic fields at 3.4 Å. These cuts agree well with the calculated results presented in (j).

Figure 6

(a) Magnetic susceptibility $d{M}_z/dH$ and (b) transverse staggered magnetization $M_x$ calculated numerically for the model of Eq. (5) as a function of $M_z$ normalized by its saturated value $M_z^{\mathrm{sat}}$. The three green arrows locate the corresponding field values at which the various anomalies occur. These values are in good agreement with the experimental ones for $H^{*}, H_{1/2}$, and $H_s$. The interchain coupling $J^{\prime }=0.17\mathrm{meV}$ and two different kinds of four-site periodic perturbations, $J_{\pi /2}=J_{\pi }=0.61\mathrm{meV}$ (solid purple line) and $J_{\pi /2}=J_{\pi }=1.16\mathrm{meV}$ (dotted orange line), are included in the model. The value of $J_{\pi /2}=J_{\pi }=0.61\mathrm{meV}$ reproduces well the bump of $d{M}_z/dH$ at half saturation, contrary to the value of $J_{\pi /2}=J_{\pi }=1.16\mathrm{meV}$ used in Ref. [17].

Figure 7

(a) Inelastic neutron scattering spectra (dynamical susceptibility) numerically calculated at zero field $H=0$ with an interchain coupling $J^{\prime }=0.17\mathrm{meV}$ and $J_{\pi /2}=J_{\pi }=0.61\mathrm{meV}$ for scattering vectors $(2,0,l)$ ($0\le l\le 4$). The anticrossing of the dispersion is seen at $l=(\mathrm{integer}+1/2)$. (b) An expansion of (a) in the $2\le l\le 3$ interval.

Figure 8

(a) Particle-hole band at zero field obtained through the Jordan-Wigner transformation. Two specific particle-hole excitations are pointed out by the black and yellow arrows, which correspond to zero-energy and maximum-energy excitation, respectively. The Fermi level is denoted by $E_{\mathrm{F}}$, which is set to zero. The fermions are located below $E_{\mathrm{F}}$ (thick line), while the holes are located above (thin line). (b) The applied longitudinal field $H$ is equivalent to a chemical potential which splits the degeneracy of the electron-hole bands. The intraband and interband zero-energy excitations, which, respectively, correspond to longitudinal $S_{\parallel }$ and transverse $S_{\perp }$ fluctuations, are pointed out by the blue and pink arrows. The incommensurability arises from the splitting of the bands. (c) Two-spinon continuum expected for the spin-1/2 $\mathit{XY}$ chain at zero field. Black and yellow points are the excitations corresponding to the black and yellows arrows. (d),(e) Dynamical susceptibility spectrum of the longitudinal ($S_{\parallel }$) and transverse ($S_{\perp }$) fluctuations expected for a spin-1/2 $\mathit{XY}$ chain in the case of $H>0$. We point out some of the incommensurate positions where the gap closes by the blue and pink points, which are associated to the blue and pink arrows. The evolution of excitation spectra shown in (c)–(e) looks similar even in the presence of nearest-neighbor interaction, in particular for the Ising anisotropic system in the TLL region, $H_{\mathrm{c}}<H<H_{\mathrm{sat}}$, but the energy, which depend on the anisotropy, is then renormalized, e.g., multiplied by $\pi /2$ for the Heisenberg case. The black dashed lines in (d) and (e) show the effect of folding from the crystal structure of ${\mathrm{BaCo}}_2{\mathrm{V}}_2{\mathrm{O}}_8$.