Nuclear magnetic resonance study of the magnetic-field-induced ordered phase in the ${\text{NiCl}}_2\text{−}4\text{SC}{\left({\text{NH}}_2\right)}_2$ compound

Nuclear magnetic resonance study of the high magnetic field $\left(H\right)$ part of the Bose-Einstein condensed (BEC) phase of the quasi-one-dimensional (quasi-1D) antiferromagnetic quantum spin-chain compound ${\mathrm{NiCl}}_2\text{−}4\mathrm{SC}{\left({\mathrm{NH}}_2\right)}_2$ was performed. We precisely determined the phase boundary, $T_{\mathrm{c}}\left(H\right)$, down to 40 mK; the critical boson density, $n_{\mathrm{c}}\left(T_{\mathrm{c}}\right)$; and the absolute value of the BEC order parameter $S_{\perp }$ at very low temperature $(T=0.12\mathrm{K})$. All results are accurately reproduced by numerical quantum Monte Carlo simulations of a realistic three-dimensional (3D) model Hamiltonian. Approximate analytical predictions based on the 1D Tomonaga-Luttinger liquid description are found to be precise for $T_{\mathrm{c}}\left(H\right)$, but less so for $S_{\perp }\left(H\right)$, which is more sensitive to the strength of 3D couplings, in particular close to the critical field. A mean-field treatment, based on the Hartree-Fock-Popov description, is found to be valid only up to $n_{\mathrm{c}}\cong 4\%$ $(T<0.3$ K), while for higher $n_{\mathrm{c}}$ boson interactions appear to modify the density of states.

Figure 1

Evolution of ${}^{14}\mathrm{N}$ NMR spectrum when entering the BEC phase in DTN at $T=0.12$ K. In the spectrum taken at $H_{\mathrm{c}}2=12.32$ T (bottom), the contributions from the two crystallographic sites, orange-colored N(2) and gray-colored N(1), are clearly separated, and the quadrupolar splittings, indicated by blue horizontal lines, are easy to identify. Within the ordered phase, at 12.22 and 11.27 T, the AF order splits each line into four, as shown on the lowest frequency N(2) line by the dotted curves. Orange-gray hatching denotes the region where the N(2) and N(1) lines overlap.

Figure 2

The order parameter $\left(S_{\perp }\right)$ in the BEC phase of DTN at $T=0.12$ K determined by NMR (circles) and by QMC simulations (crosses). NMR points are overlapped by neutron diffraction data from Ref. [10] downscaled by $-25\%$ (squares). The orange solid line and dashed-dotted black curve are the $T=0$ prediction by DMRG+MF and by TLL+MF, respectively. The inset shows a zoom close to $H_{\mathrm{c}}2$.

Figure 3

NMR data (red circles) for (a) $T_{\mathrm{c}}$ and (b) the critical boson density $n_{\mathrm{c}}$ at $T_{\mathrm{c}}$ (see the text) are compared with theoretical predictions: QMC data points are shown as crosses, HFP predictions are given by the dotted and solid lines (for the parabolic and true magnon dispersion, respectively), and the TLL prediction for $T_{\mathrm{c}}$ [Eq. (4), with $k=0.67]$ by the dash-dotted line (see the text). In (a) squares correspond to the magnetocaloric effect data from Ref. [7] (with the field values downscaled by $-2.3\%$ to overlap the slightly different $H_{\mathrm{c}}2$ values). The lower inset in (b) explains the determination of the H(2) line-shift frequency (right scale in the main panel) that measures $n_{\mathrm{c}}$. Other insets are zooms close to $H_{\mathrm{c}}2$.