Excitations in the quantum paramagnetic phase of the quasi-one-dimensional Ising magnet ${\mathrm{CoNb}}_2$${\mathrm{O}}_6$ in a transverse field: Geometric frustration and quantum renormalization effects

The quasi-one-dimensional (1D) Ising ferromagnet ${\mathrm{CoNb}}_2$${\mathrm{O}}_6$ has recently been driven via applied transverse magnetic fields through a continuous quantum phase transition from spontaneous magnetic order to a quantum paramagnet, and dramatic changes were observed in the spin dynamics, characteristic of weakly perturbed 1D Ising quantum criticality. We report here extensive single-crystal inelastic neutron scattering measurements of the magnetic excitations throughout the three-dimensional (3D) Brillouin zone in the quantum paramagnetic phase just above the critical field to characterize the effects of the finite interchain couplings. In this phase, we observe that excitations have a sharp, resolution-limited line shape at low energies and over most of the dispersion bandwidth, as expected for spin-flip quasiparticles. We map the full bandwidth along the strongly dispersive chain direction and resolve clear modulations of the dispersions in the plane normal to the chains, characteristic of frustrated interchain couplings in an antiferromagnetic isosceles triangular lattice. The dispersions can be well parametrized using a linear spin-wave model that includes interchain couplings and further neighbor exchanges. The observed dispersion bandwidth along the chain direction is smaller than that predicted by a linear spin-wave model using exchange values determined at zero field, and this effect is attributed to quantum renormalization of the dispersion beyond the spin-wave approximation in fields slightly above the critical field, where quantum fluctuations are still significant.

Figure 1

Lattice of ${\mathrm{Co}}^{2+}$ ions and relevant (Ising) exchange paths. (a) Zigzag chains running along $c$ (and buckled along $b)$ have a dominant ferromagnetic nearest-neighbor coupling $J_z$ and weaker next-nearest-neighbor exchange $J_z^{\prime }$. (b) The buckled magnetic chains form two inequivalent isosceles triangular lattices in the basal $ab$ plane with interchain interactions $J_1$ (along the $\pm \mathit{b}$ bonds) and $J_2$ [along the $(\pm \mathit{a}\pm \mathit{b})/2$ bonds]. For clarity, only the blue triangular lattice is shown.

Figure 2

Dispersion along the chain direction $l$ in the paramagnetic phase at 9 T and $\sim$0.03 K, measured on LET with an incident energy $E_i=10$ meV. (a) A single dispersive mode is observed for small interchain wave-vector component $k$, and (b) a second, weaker-intensity mode becomes visible for finite $k$. The plots show the averaged neutron scattering intensity for $\left|k\right|$ in the range [0.0,0.2] in (a) and [0.28,0.38] in (b).

Figure 3

(a) Inelastic neutron scattering intensity at constant energy transfer $E$ as a function of wave vector in the $\left(h0l\right)$ plane measured at 7 T and 0.5 K on MACS. Rows 1 and 3 are data, rows 2 and 4 are calculations using the one-sublattice spin-wave model in Eq. (3) with intensities given in Eq. (6) and an overall single intensity scaling factor for all the panels. The calculations include the neutron polarization factor, the spherical magnetic form factor for ${\mathrm{Co}}^{2+}$, and instrumental resolution effects. At low energies, near the dispersion minimum, strong dispersion is seen along both $h$ and $l$. As the energy increases, the dispersion along the interchain direction $h$ becomes flatter, suggesting that interchain couplings become less relevant at higher energies. (b) Wave-vector scans along (2,0,$l)$ and (3,0,$l)$ at constant energy transfer $E=0.8$ meV showing displacement of the spin-wave peaks upon changing $h$, data (green circles) and spin-wave model (blue line). Error bars represent $\pm 1$ standard deviation.

Figure 4

Inelastic neutron scattering intensity as a function of wave vector and energy transfer measured on LET at 7 T and $\sim$$0.06$ K (left), calculated intensity using the four-sublattice spin-wave model (center) described in Sec. 4b2 and comparison between dispersion data points (magenta filled circles) with the spin-wave dispersion relation Eq. (3) (LSWT, green square) and the hopping model Eq. (1) (HM, black cross) (right). Horizontal gray dashed lines are guides to the eye to emphasize the bandwidth of the dispersion along various directions. The calculated intensities include the neutron polarization factor, the spherical magnetic form factor for ${\mathrm{Co}}^{2+}$, and instrumental resolution effects. (a) Dispersion along the chain direction $l$ showing the full bandwidth of 2.7 meV $(E_i=10$ meV). (b) Zoom into the low-energy part of the dispersion along $l$ showing the energy gap at 0.48 meV $(E_i=4$ meV). (c) Dispersion along the interchain direction $k$ for wave vectors near $(1,k,0)$ with a bandwidth of 0.05 meV. Note in the data (left panel) the presence at large $\left|k\right|$ and energies above the main mode of additional scattering intensity that decreases rapidly as $k\to 0$. This is attributed (middle panel) to the shadow mode that appears because of the unit cell doubling in the $ab$ plane due to the alternate rotation of Ising axes. (d) Dispersion along the interchain direction $h$ for wave vectors near $(h,0.025,0)$ showing minima at odd $h$ and a bandwidth of 0.16 meV.

Figure 5

(a) Observed neutron scattering intensity in constant energy slices as a function of wave vector in the $\left(hk0\right)$ plane at 7 T and 1.8 K, extracted from a four-dimensional Horace scan on LET. Regions of strong intensity show the constant-energy contours of the interchain dispersion in the triangular $ab$ plane. Each slice shows data averaged for energies within $\pm 0.03$ meV of the nominal value. Solid lines show the edges of the hexagonal Brillouin zone of the triangular lattice. (b) Surface plot of the interchain dispersion using Eqs. (3, 4, 5, 6, 7, 8) and (c) projection contour plot in the $\left(hk0\right)$ plane. Solid black circles show the Brillouin zone centers (nuclear Bragg peak positions) of the triangular lattice, and solid red circles indicate minimum gap positions of the dispersion surface. In (c) the highlighted area shows the momentum space probed experimentally in (a).

Figure 6

Intensity of the main (shadow) mode [filled (open) circles] as a function of the interchain wave vector $k$, fitted to the functional form for a buckled magnetic chain [green solid (blue dashed) line], as described in the text. The intensities were extracted at the ferromagnetic zone boundary $l=-1$ from the same data set as in Fig. 2. Error bars represent $\pm 1$ standard deviation.