Ground states of a frustrated spin-$\frac{1}{2}$ antiferromagnet: ${\mathrm{Cs}}_2\mathrm{Cu}{\mathrm{Cl}}_4$ in a magnetic field

We present detailed calculations of the magnetic ground state properties of ${\mathrm{Cs}}_2\mathrm{Cu}{\mathrm{Cl}}_4$ in an applied magnetic field, and compare our results with recent experiments. The material is described by a spin Hamiltonian, determined with precision in high field measurements, in which the main interaction is antiferromagnetic Heisenberg exchange between neighboring spins on an anisotropic triangular lattice. An additional, weak Dzyaloshinskii-Moriya interaction introduces easy-plane anisotropy, so that behavior is different for transverse and longitudinal field directions. We determine the phase diagram as a function of field strength for both field directions at zero temperature, using a classical approximation as a first step. Building on this, we calculate the effect of quantum fluctuations on the ordering wave vector and components of the ordered moments, using both linear spinwave theory and a mapping to a Bose gas which gives exact results when the magnetization is almost saturated. Many aspects of the experimental data are well accounted for by this approach.

Figure 1

(Color online) The magnetic sites and exchange couplings within a single layer of ${\mathrm{Cs}}_2\mathrm{Cu}{\mathrm{Cl}}_4$. Layers are stacked along the $a$ direction, with interlayer spacing $a∕2$ and a relative displacement in the $c$-direction.

Figure 2

(Color online) The phase diagram in the classical limit, with a schematic representation of the different phases. Transitions between the cone states and the ferromagnetic states are second order. The distorted cycloid and the tilted cone states are separated by a first-order transition.

Figure 3

(Color online) The incommensuration $ϵ$ as a function of transverse magnetic field strength $B^a$. Full line: result from $1∕S$ expansion. Long dashed line: result from the classical theory. Short dashed line: linear variation of $ϵ$ with $B^a$, from calculation for dilute Bose gas of spin flips. Symbols are the experimental results taken from Fig. 3(c) of Ref. 3 taken at $T=0.20\mathrm{K}$: ▵ from magnetic Bragg peaks at $\mathbf{Q}=(0,1.5-ϵ,0)$, and ◻ from peaks at $\mathbf{Q}=(0,$ $0.5-ϵ,1)$.

Figure 4

(Color online) (a) The ordered moment as a function of transverse magnetic field strength. Dashed line: classical behavior; full line: with leading quantum corrections, from Eq. (23). (b) Eq. (24) is plotted as a function of the transverse field in full line, whereas $\sin {\theta }_0$ is plotted as a dashed line.

Figure 5

(Color online) (a) The magnetization as a function of transverse magnetic field strength. The dashed line shows the result of classical theory. The thick, full line includes the $1∕S$ correction from Eq. (26). The thin, full line gives experimental data (Ref. 34) measured at $T=60\mathrm{mk}$. (b) Calculated ground state energy as a function of transverse magnetic field. The dashed and thick lines are the results of classical theory and the $1∕S$ correction, Eq. (22), respectively.

Figure 6

(Color online) The component of ordered moment $S^T$ in the plane perpendicular to the field direction, as a function of transverse magnetic field strength. Dashed line: classical theory. Full line: result including $1∕S$ corrections, from Eq. (29). ◻: experimental data at $T<0.1\mathrm{K}$ [taken from Fig. 3(b) of Ref. 3].

Figure 7

(Color online) The incommensuration $ϵ$ as a function of longitudinal field strength. Solid line: result from $1∕S$ expansion, from Eq. (38). Dashed line: result from the classical result. Symbols are the experimental results taken from Ref. 2, Fig. 1(e): ▵ from magnetic peaks at $\mathbf{Q}=(2,1∕2+ϵ,0)$ and ◻ from peaks at $\mathbf{Q}=(1,1∕2+ϵ,0)$. Experimentally, a distorted cycloid occurs for $B^c<1.4\mathrm{T}$ and a second incommensurate phase occurs for fields in the range $1.4\mathrm{T}<B^c<2.1\mathrm{T}$.

Figure 8

(Color line) The incommensuration $ϵ$ as a function of longitudinal magnetic field strength $B^c$. Solid line: result from $1∕S$ expansion. Dashed line: result from calculation for dilute Bose gas of reversed spins. ◻: experimental data (Ref. 5) (no incommensurate ordering is observed for $2.1\mathrm{T}<B^c<7.1\mathrm{T}$).

Figure 9

(Color online) The eccentricity of ellipse $I=S_b∕S_c$ as a function of the transverse magnetic field strength. Line: result of classical analysis. ◻: experimental data (Ref. 5).