Quantum phase transitions in a quasi-one-dimensional Ising quantum magnet in transverse fields

We report neutron scattering and thermodynamic measurements of the quasi-one-dimensional Ising magnet $\mathrm{Sr}{\mathrm{Co}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ under transverse fields along the tetragonal $a/b$ direction. Our experiments reveal a Néel-type magnetic order in zero field, which successively changes into a coplanar antiferromagnetic order in applied fields. The detailed evolution of the noncollinear magnetic order in magnetic fields can be understood by considering the competition between the intrinsic magnetic interactions, uniform transverse field, and effective staggered field. Moreover, a series of gapped discrete confined-spinon modes are observed in zero field. With increasing field, the spin gap progressively softens, reaching a minimum value at $\sim 7\mathrm{T}$ where the Néel-type magnetic ordering moment is completely suppressed. This corresponds to the three-dimensional quantum critical point (QCP). The implications for multiple QCPs in this class of materials are discussed.

Figure 1

Schematic of the lattice and magnetic structure for $\mathrm{Sr}{\mathrm{Co}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ at zero field. Only two chains in the unit cell are plotted for clarity, and the arrows indicate the magnetic moments below $T_N$.

Figure 2

Magnetic susceptibility measurements on $\mathrm{Sr}{\mathrm{Co}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ single crystal under magnetic field $H=1\mathrm{T}$ applied along the $b$ and $c$ directions. The inset shows the magnetic susceptibility curves near the transition temperature. The vertical line at 5 K indicates the transition temperature.

Figure 3

Heat capacity measurements on $\mathrm{Sr}{\mathrm{Co}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ single crystal under magnetic fields along the $b$ direction. An offset has been added in each heat capacity curve for clarity, and the heat capacity data are raw data without background subtraction.

Figure 4

Field dependence of the magnetization of $\mathrm{Sr}{\mathrm{Co}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ single crystal measured at various temperatures. The field is applied along the $b$ direction. The red lines denote the field derivations of the magnetization curves.

Figure 5

Phase diagram of $\mathrm{Sr}{\mathrm{Co}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ under transverse fields along the $b$ direction. The solid black circles represent the Néel temperature $T_N$ determined by the heat capacity measurements, and the hollow blue circles represent the critical fields determined by the magnetization curves. The red line is a function fit to $T\sim {(H-H_c)}^{0.5}$ with the critical field of $H_c\approx 6.96\pm 0.08\mathrm{T}$.

Figure 6

(a) Field dependence of the neutron diffraction intensity at the indicated Q positions at 1.5 K. The red line is a fit to $I\sim {(H-H_c)}^{2\beta }$ with the critical field of $H_c\approx 7.001\pm 0.006\mathrm{T}$ and the critical exponent of $2\beta \approx 0.18\pm 0.02$. The data were collected at SPINS. (b) Q-cuts across (2, 0, 1) along the $L$ direction at 4 and 10 T at $E=0\mathrm{meV}$. (c) Q-cuts across (1, 0, 1) along the $L$ direction at 4 and 10 T at $E=0\mathrm{meV}$. (d), (e) Elastic constant-energy slice obtained at 1.5 K at 4 and 10 T, respectively. The color bar indicates scattering intensity in arbitrary units in linear scale, r.l.u., reciprocal lattice units. From (b) to (e), the data were collected at DCS.

Figure 7

(a) Schematic of the magnetic structure of $\mathrm{Sr}{\mathrm{Co}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ under transverse fields of 4 T. Only two chains in the unit cell are plotted for clarity, and the blue balls represent the ${\mathrm{Co}}^{2+}$ ions. (b) Schematic of the magnetic structure of $\mathrm{Sr}{\mathrm{Co}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ under transverse fields of 10 T.

Figure 8

Comparison of the squared magnetic structure factors between observations and calculations. (a) Nuclear peaks measured at zero field. (b) Magnetic peaks measured at zero field. (c) Magnetic peaks measured at $H=4\mathrm{T}$. The inset shows stronger magnetic peaks at the wave vector k = (0, 0, 1). (d) Magnetic peaks measured at $H=10\mathrm{T}$. The agreement factor is indicated on each panel.

Figure 9

Energy scans of the spin excitations in $\mathrm{Sr}{\mathrm{Co}}_2{\mathrm{V}}_2{\mathrm{O}}_8$ at indicated transverse fields at Q = (0, 0, 2) and 1.5 K. The solid lines are the Gaussian fits of the data. The dashed lines indicate the elastic incoherent scattering. The data were collected at SPINS.

Figure 10

The energy ratios of high energy excitation modes with respect to the lowest energy mode in $\mathrm{Sr}{\mathrm{Co}}_2{\mathrm{V}}_2{\mathrm{O}}_8$. The black, red, and blue dots indicate the energy ratios of the first, second, and third peaks fitted from the energy scan measured at each field, respectively. The dashed lines indicate the theoretically expected ratios for the $E_8$ excitations adopted from Refs. [16, 18].