Spin excitations and quantum criticality in the quasi-one-dimensional Ising-like ferromagnet ${\mathrm{CoCl}}_2·2{\mathrm{D}}_2\mathrm{O}$ in a transverse field

We present experimental evidence for a quantum phase transition in the easy-axis $S=3/2$ anisotropic quasi-one-dimensional ferromagnet ${\mathrm{CoCl}}_2·2{\mathrm{D}}_2\mathrm{O}$ in a transverse field. Elastic neutron scattering shows that the magnetic order parameter vanishes at a transverse critical field ${\mu }_0{H}_c=16.05$(4) T, while inelastic neutron scattering shows that the gap in the magnetic excitation spectrum vanishes at the same field value, and reopens for $H>H_c$. The field dependence of the order parameter and the gap are well described by critical exponents $\beta =0.45\pm 0.09$ and $z\nu$ close to $1/2$, implying that the quantum phase transition in ${\mathrm{CoCl}}_2·2{\mathrm{D}}_2\mathrm{O}$ differs significantly from the textbook version of a $S=1/2$ Ising chain in a transverse field. We attribute the difference to weak but finite three-dimensionality of the magnetic interactions.

Figure 1

The zero field magnetic unit cell of ${\mathrm{CoCl}}_2·2{\mathrm{D}}_2\mathrm{O}$. ${\mathrm{Co}}^{2+}$ spins align ferromagnetically along $\mathbf{b}$ in chains along $\mathbf{c}$. Spins on neighboring chains are antiferromagnetically aligned. $\mathbf{x}$ and $\mathbf{y}$ represent Co-Cl bond directions, and the easy and hard axes of magnetization in the ($\mathbf{a},\mathbf{c}$) plane, respectively [13]. ${\mathrm{D}}_2\mathrm{O}$ molecules are omitted for clarity. In our experiments, the magnetic field was applied along $\mathbf{x}$.

Figure 2

(a) Examples of raw data for the ${\mathbf{Q}}_{\mathrm{AFM}}=\left(211\right)$ magnetic Bragg peak. The solid lines represent fits of two Gaussians, as explained in the main text. (b) Square root of the integrated intensity of the Bragg peak as a function of applied field. The solid line is a power law fit. The dashed line is a guide to the eye. All data were taken at $T=1.5\mathrm{K}$.

Figure 3

Raw inelastic neutron scattering data from RITA-II measured at ${\mathbf{Q}}_{\text{AFM}}=\left(211\right)$ in transverse fields between $13\mathrm{T}$ and $14.9\mathrm{T}$ at $T=1.5\mathrm{K}$. The blue solid line at each field represents the sum of a Gaussian low-energy magnetic peak and a broad and field-independent contribution that was estimated mainly from the $14.9\mathrm{T}$ data. The background estimate is shown as red solid lines that overlap with the blue lines at higher energy transfers.

Figure 4

(a)–(e) Raw inelastic neutron scattering data from FLEXX for transverse fields close to $H_c$ and $T=1.5\mathrm{K}$. The solid lines are Gaussian fits to the peaks at ${\mathbf{Q}}_{\mathrm{AFM}}=\left(211\right)$, assuming a background given by the incoherent scattering at $\mathbf{Q}=\left(1.91.60.9\right)$. The latter is shown in panel (e) and represented by solid red lines in panels (a)–(e). In panels (f)–(j), this background model is subtracted from the raw data. (k) Field dependence of the gap extracted from Gaussian fits to the RITA-II and FLEXX data. The line is calculated using the four-spin cluster model [18]. The fields in the model calculation were rescaled by about $12\%$ to match the experimentally measured value.

Figure 5

Data from the MARI experiment, showing the $\mathbf{Q}$-integrated intensity vs energy for two temperatures: 5 K and 100 K. The black circles are the results obtained when integrating the scattering intensity with respect to $Q$ between 0 and 4 ${\AA }^{-1}$, whereas the green triangles are the results obtained, when integrating from 0 to 9 ${\AA }^{-1}$. The red solid lines are the calculated results obtained by the mean-field/random-phase approximation theory using a cluster basis of 3 spins along the Co chains and assuming the background scattering shown by the dashed blue lines [18].

Figure 6

Magnetic phase diagram of ${\mathrm{CoCl}}_2·2{\mathrm{D}}_2\mathrm{O}$ in a transverse magnetic field. At low temperatures and fields, ${\mathrm{CoCl}}_2·2{\mathrm{D}}_2\mathrm{O}$ orders antiferromagnetically, as shown in Fig. 1.

Figure 7

Data from the CNCS experiment, showing the magnetic dispersion in a $\left(\mathbf{Q},\hslash \omega \right)$-slice with the momentum axis along the (h 0 h/2) reciprocal space direction and the neutron energy transfer to the sample, $\hslash \omega$, vectical. The left panels show the measured data in zero field (top) and $12\mathrm{T}$ (bottom) transverse field. The right panels shows the corresponding correlation functions at $H/H_c=0$ and 0.75 calculated with in the mean-field/random-phase approximation theory with a cluster basis of four spins [18]. To simulate the effect of experimental resolution, the calculated spectra were convoluted with a Lorentzian with an energy-independent width $\mathrm{\Gamma }=0.15\mathrm{meV}$. From this data, it is observed that the dispersion softens exactly at $h=2$.