Spin gap in the quasi-one-dimensional $S=\frac{1}{2}$ antiferromagnet ${\mathrm{K}}_2{\mathrm{CuSO}}_4{\mathrm{Cl}}_2$

Electron spin resonance (ESR) experiments in the quasi-one-dimensional (quasi-1D) $S=\frac{1}{2}$ antiferromagnet ${\mathrm{K}}_2{\mathrm{CuSO}}_4{\mathrm{Cl}}_2$ reveal the opening of a gap in the absence of magnetic ordering, as well as an anisotropic shift of the resonance magnetic field. These features of the magnetic excitation spectrum are explained by a crossover between a gapped spinon-type doublet ESR formed in a 1D antiferromagnet with uniform Dzyaloshinskii-Moriya interaction and a Larmor-type resonance of a quasi-1D Heisenberg system.

Figure 1

Sketch of crystallographic structure of ${\mathrm{K}}_2{\mathrm{CuSO}}_4{\mathrm{Cl}}_2$.

Figure 2

Temperature evolution of the ESR line (top panels) and temperature dependence of the ESR field (bottom panels) for $\mathbf{H}\parallel a$ at $\nu$ = 4.66 GHz (left panels) and for $\mathbf{H}\parallel b$ at $\nu$ = 26.27 GHz (right panels). Transmission is normalized to have a linear dependence on the imaginary part of the susceptibility (see text). The records of transmission are vertically offset for clarity. DPPH is a $g=2.00$ marker.

Figure 3

Examples of normalized ESR records taken at $T=0.45\mathrm{K}$ for $\mathbf{H}\parallel a$. The zoom of the 0.73-GHz ESR line is fourfold for vertical and horizontal scale. DPPH is a $g=2.00$ marker.

Figure 4

Examples of normalized ESR records taken at $T=0.45\mathrm{K}$ for $\mathbf{H}\parallel b$. The zoom of the 0.71-GHz ESR line is threefold for vertical and horizontal scale. DPPH is a $g=2.00$ marker.

Figure 5

Frequency-field diagram for ESR at $T=0.45$ K for $\mathbf{H}\parallel a$, $\mathbf{H}\parallel b$, $\mathbf{H}\parallel c$.

Figure 6

Shift of the resonance frequencies relative to paramagnetic resonance for $\mathbf{H}\parallel a$ (top panel) and $\mathbf{H}\parallel b$ (bottom panel). Dashed lines are theoretical dependencies for the gapped mode and lower-frequency component of the spinon doublet according to (6) and (5) with $\mathrm{\Delta }=\pi D/2$ = 2 GHz. Shaded areas present the range of absorption exceeding half of the maximum absorption value.

Figure 7

Temperature evolution of the ESR line for $\mathbf{H}\parallel a$ at $\nu$ = 0.73 GHz (left panel) and for $\mathbf{H}\parallel b$ at $\nu$ = 0.71 GHz (right panel). The solid horizontal line segment indicates the area of ESR absorption exceeding $50\%$ of its maximum magnitude for the ESR line taken at the lowest temperature. Transmission is normalized and transformed to have a linear dependence on the imaginary part of the susceptibility (see text). The records of transmission are vertically offset for clarity.

Figure 8

Linewidth vs frequency dependencies of ESR at $T=0.45$ K for $\mathbf{H}\parallel a$ and $\mathbf{H}\parallel b$.

Figure 9

Temperature dependencies of the ESR integral intensity at different frequencies in the range $\nu$ = 2.4–36.4 GHz for $\mathbf{H}\parallel a$ (top panel) and $\mathbf{H}\parallel b$ (bottom panel). Dashed lines in both panels represent normalized static magnetic susceptibility of the $S=\frac{1}{2}$ Heisenberg chain according to the Bonner-Fisher calculation for $J=3.2$ K [17].

Figure 10

Temperature dependencies of the ESR half width at half maximum $\delta H_{1/2}$ at different frequencies in the range $\nu$ = 2.4–36.4 GHz for $\mathbf{H}\parallel a$ (top panel) and $\mathbf{H}\parallel b$ (bottom panel).

Figure 11

Normalized ESR absorption curves at frequency $\nu$ = 20.7 GHz for different orientations of the magnetic field at $T=1.3$ K. Left: rotation of the magnetic field within the $bc$ plane. Right: rotation of the magnetic field within the $ac$ plane.

Figure 12

Angular dependencies of the resonance field (circles) and half width at half maximum of the ESR line $\delta H_{1/2}$ (squares). Top: angular dependencies for the $\nu$ = 27.5 GHz ESR line at $T=80$ K for field in the $bc$ plane. The dashed line represents theoretical dependence according to (4). The solid line is a fit to the experimental HWHM using (7). Middle and bottom: dependencies for the $\nu$ = 20.7 GHz ESR line taken at $T=1.3$ K for field in the $bc$ and $ac$ planes, respectively.