Complex Field-Induced States in Linarite ${\mathrm{PbCuSO}}_4(\mathrm{OH}{)}_2$ with a Variety of High-Order Exotic Spin-Density Wave States

Low-temperature neutron diffraction and NMR studies of field-induced phases in linarite are presented for magnetic fields $H\parallel b$ axis. A two-step spin-flop transition is observed, as well as a transition transforming a helical magnetic ground state into an unusual magnetic phase with sine-wave-modulated moments $\parallel H$. An effective ${\tilde{J}}_1\text{−}{\tilde{J}}_2$ single-chain model with a magnetization-dependent frustration ratio ${\alpha }_{\mathrm{eff}}=-{\tilde{J}}_2/{\tilde{J}}_1$ is proposed. The latter is governed by skew interchain couplings and shifted to the vicinity of the ferromagnetic critical point. It explains qualitatively the observation of a rich variety of exotic longitudinal collinear spin-density wave, ${\mathrm{SDW}}_p$, states ($9\ge p\ge 2$).

Figure 1

(a) Phase diagram of linarite with $H\parallel b$. For a description of the phases I–V, see the text. Red diamonds mark new transition temperatures into phase V. Green balls indicate the ($H$,$T$) points of Fig. 3. (b) Neutron scattering scans along $k$ at 1.7 K as function of the magnetic field, crossing from phase I via III into IV. Solid lines are Gaussian fits for peak position determination.

Figure 2

(a) The magnetic Bragg peak position $(0,k,0.5)$ shift in phase V. (b) Field dependence of the propagation vector in phase V compared to the 1D model ${\mathrm{SDW}}_p$ states with $2\le p\le 9$. Theory lines include the magnetization obtained from $M(H,T)$ scans [24].

Figure 3

${}^1\mathrm{H}\text{−}\mathrm{NMR}$ spectra of linarite in phase V (a)–(e) and in the paramagnetic polarized state at 7.5 T (f) close to saturation.

Figure 4

(a) Lattice model of weakly coupled $J_1\text{−}{J}_2$ chains. (b) Fitting of $S(\mathbf{q},\omega )$ at $M/M_s=0.2$, between $\mathbf{q}=(\pi ,\pi )$ for two coupled chains with $\alpha =0.36$, $J_{\mathrm{ic}}=0.1|J_1|$, and $q=\pi$ for a single chain with ${\alpha }_{\mathrm{eff}}=0.257$. (c) Phase diagram of the multimagnon bound states and the estimated effective frustration ratio ${\alpha }_{\mathrm{eff}}$ for linarite. (d) Suggested propagation vector using (c). (e),(f) Local spin densities $⟨S_i^z⟩$ at ${\alpha }_{\mathrm{eff}}=0.26$ and $M/M_s=0.95$ for (e) periodic and (f) open chains.