Anisotropic field-induced gap in the quasi-one-dimensional antiferromagnet ${\mathrm{KCuMoO}}_4\left(\mathrm{OH}\right)$

We investigated magnetic and thermodynamic properties of $S=\frac{1}{2}$ quasi-one-dimensional antiferromagnet ${\mathrm{KCuMoO}}_4\left(\mathrm{OH}\right)$ through single-crystalline magnetization and heat capacity measurements. At zero field, it behaves as a uniform $S=\frac{1}{2}$ Heisenberg antiferromagnet with $J=238\mathrm{K}$, and exhibits a canted antiferromagnetism below $T_{\mathrm{N}}=1.52\mathrm{K}$. In addition, a magnetic field $H$ induces the anisotropy in magnetization and opens a gap in the spin-excitation spectrum. These properties are understood in terms of an effective staggered field induced by staggered $g$ tensors and Dzyaloshinsky-Moriya (DM) interactions. Temperature dependencies of the heat capacity and their field variations are consistent with those expected for quantum sine-Gordon model, indicating that spin excitations consist of soliton, antisoliton, and breather modes. From field dependencies of the soliton mass, the staggered field normalized by the uniform field $c_{\mathrm{s}}$ is estimated as 0.041, 0.174, and 0.030, for $H\parallel a$, $b$, and $c$, respectively. Such a large variation of $c_{\mathrm{s}}$ is understood as the combination of staggered $g$ tensors and DM interactions which induce the staggered field in the opposite direction for $H\parallel a$ and $c$ but almost the same direction for $H\parallel b$ at each Cu site.

Figure 1

(a) Schematic view of a one-dimensional antiferromagnet with staggered $g$ tensors (blue circles) and DM interactions (red arrows). A dashed line represents principal axes for each $g$ tensor. Directions of its principal axes and DM vectors are chosen so that they match those of ${\mathrm{KCuMoO}}_4\left(\mathrm{OH}\right)$. (b) Cu and O atoms of ${\mathrm{CuO}}_6$ octahedra in ${\mathrm{KCuMoO}}_4\left(\mathrm{OH}\right)$. Green ellipses represent $x^2-y^2$ orbitals carrying $S=\frac{1}{2}$ spins.

Figure 2

Schematic view of soliton, breather, antisoliton excitations expected in a one-dimensional antiferromegnetic chain with staggered $g$ tensors and DM interactions. Red (blue) arrows indicate that transverse spin components are twisted counterclockwise (clockwise).

Figure 3

(a) Temperature dependencies of the magnetic susceptibility above 2 K for $H\parallel a$, $b$, and $c$ in a field of 1 T. Dashed curves represent a fit to the Eqs. (4). (b) Double-logarithmic plot of the magnetic susceptibility between 2–30 K. The dashed curve represents a fit to the Eqs. (5). (c) The magnetic susceptibility below 2 K and magnetization curves (shown in the inset) of randomly oriented single crystals.

Figure 4

Temperature dependencies of the magnetic heat capacity divided by temperature. Magnetic fields of 0–9 T are applied along $H\parallel a$, $b$, and $c$ axes. Solid curves represent a fit to the calculated curves from Bethe ansatz [38].

Figure 5

(a) Field dependencies of the soliton mass $M_0$ for $H\parallel a$, $b$, and $c$. The solid curve represents a fit to Eq. (8), and $c_{\mathrm{s}}$ estimated from the fit is shown close to the fitting curve. A dashed curve is a fit to Eq. (9), which is derived from a mean field theory. The former and the latter fits are performed in a field region of 0–3 T and 3–9 T (filled symbols), respectively. (b) A $M_0-H^{2/3}$ plot with the fit to Eq. (8). (c) A $M_0-H^{1/2}$ plot with the fit to Eq. (9).

Figure 6

The GGA band structure (circles) in comparison with the Fourier-transformed Cu-centered Wannier functions (WF) and the tight-binding model (TB) comprising the five leading transfer integrals (Table 3). The Fermi level is at zero energy. The notation of $k$ points: $\mathrm{\Gamma }$ (000), X ($\frac{\pi }{a}00$), S ($\frac{\pi }{a}\frac{\pi }{b}0$), Y ($0\frac{\pi }{b}0$), Z ($00\frac{\pi }{c}$), U ($\frac{\pi }{a}0\frac{\pi }{c}$), R ($\frac{\pi }{a}\frac{\pi }{b}\frac{\pi }{c}$), and T ($0\frac{\pi }{b}\frac{\pi }{c}$).

Figure 7

(a) Direction of DM vectors in ${\mathrm{KCuMoO}}_4\left(\mathrm{OH}\right)$. Dashed arrows indicate the order of Cu atoms ($i$, $i+1$, $i+2$, $\cdots$) to define DM vectors. There are four different DM vectors which are related to each other by symmetry. (b) Schematic view of of staggered fields ${\mathbf{h}}_{\mathrm{s}}\left(g\right)$ and ${\mathbf{h}}_{\mathrm{s}}\left(D\right)$ caused from staggered $g$ tensor and DM interactions, respectively. The chains 1 and 2 are equivalent to those in Fig. 7. (c) Effective magnetic fields as a sum of a uniform and staggered fields on each Cu site.

Figure 8

Schematic view of two different Cartesian coordinate systems $xyz$ and $abc$ defined at Cu atom ($\frac{1}{2}$, $\frac{1}{2}$, 0). The $x$, $y$, and $z$ axes are defined based on the local environment of the Cu site: $x$ axis is defined along Cu–O2 bonds, $y$ axis is defined along Cu–O4 bonds, and $z$ axis is defined perpendicular to the both axes. The $a$, $b$, and $c$ axes represent crystallographic axes of ${\mathrm{KCuMoO}}_4\left(\mathrm{OH}\right)$. Green ellipses schematically illustrate $d_{x^2-y^2}$ orbitals responsible for $S=\frac{1}{2}$.