Elementary excitations of the $S=\frac{1}{2}$ one-dimensional antiferromagnet ${\text{KCuGaF}}_6$ in a magnetic field and quantum sine-Gordon model

Elementary excitations of the $S=\frac{1}{2}$ one-dimensional Heisenberg antiferromagnet ${\text{KCuGaF}}_6$ with exchange constant $J/k_B=103\text{K}$ were investigated by high-frequency electron spin resonance (ESR) measurements combined with a pulsed high magnetic field. When an external magnetic field $H$ is applied in ${\text{KCuGaF}}_6$, a staggered magnetic field $h$ is induced perpendicular to $H$ owing to the staggered $\mathit{g}$ tensor and the Dzyaloshinsky-Moriya (DM) interaction with an alternating $\mathit{D}$ vector. Consequently, ${\text{KCuGaF}}_6$ in a magnetic field is represented by the quantum sine-Gordon (SG) model. We observed many resonance modes including a soliton resonance, breathers, interbreather transitions, and two-breather excitation. Their resonance conditions are beautifully described by the quantum SG field theory with one adjustable parameter $c_s=h/H$. To investigate the relationship between the Curie term due to the DM interaction and the proportionality coefficient $c_s$, magnetic susceptibility measurements were also performed varying the external field direction.

Figure 1

(Color online) Perspective view along the $a$ axis of the crystal structure of ${\text{KCuGaF}}_6$. Dotted lines denote the chemical unit cells.

Figure 2

(Color online) Temperature dependence of magnetic susceptibilities $\chi$ in ${\text{KCuGaF}}_6$ measured at $H=0.1\text{T}$ for $H\parallel a$, $H\parallel b$, and $H\parallel c$. The susceptibilities for $H\parallel b$ and $H\parallel c$ are shifted upward by $5\times {10}^{-4}$ and $2\times {10}^{-3}\text{emu}/\text{mol}$, respectively. Solid and dashed lines denote the uniform component ${\chi }_u$ and the Curie term of the magnetic susceptibility for $H\parallel a$, respectively.

Figure 3

(Color online) Dependence of magnetic susceptibility on field direction in the $ab$ plane. Susceptibility data were collected at $T=1.8$ and 30 K. Solid lines are fits by sine curves with a period of $180°$. $\theta$ denotes the angle between the external field and the normal of the cleavage (1,1,0) plane.

Figure 4

(Color online) Illustration of low-energy excitations of model (1) around $q=0$. Soliton, antisoliton, soliton resonance, and three breathers are labeled as $S$, $\overline{S}$, $E_s$, and $M_1\sim M_3$, respectively.

Figure 5

Examples of ESR absorption spectra in ${\text{KCuGaF}}_6$ measured at $T=0.5\text{K}$ for (a) $H\parallel c$ and (b) $H\perp \left(1,1,0\right)$. Arrows labeled $E_s$, $M_n$, and $M_n-M_{n^{\prime }}$ denote the fields for soliton resonance, $n\text{th}$ breather, and interbreather transition, respectively. Resonances labeled $U_n$ are those whose origins are unclear. Sharp absorption labeled DPPH is the marker of $g=2.00$.

Figure 6

(Color online) Frequency vs field diagrams of ESR modes in ${\text{KCuGaF}}_6$ measured at $T=0.5\text{K}$ for (a) $H\parallel c$, (b) $H\parallel b$, (c) $H\perp \left(1,1,0\right)$, and (d) $H\parallel a$. Symbols denote experimental resonance fields. Thick and thin solid lines and dashed lines labeled as $E_s$, $M_n$, $M_n-M_{n^{\prime }}$, and $2M_1$ are resonance conditions calculated from Eqs. (13, 14, 15) with $J/k_B=103\text{K}$ and the proportionality coefficient $c_s$ shown in Table .