Disorder-Induced Revival of the Bose-Einstein Condensation in $\mathrm{Ni}({\mathrm{Cl}}_{1-x}{\mathrm{Br}}_x{)}_2\text{−}4\mathrm{SC}({\mathrm{NH}}_2{)}_2$ at High Magnetic Fields

Building on recent NMR experiments [A. Orlova et al., Phys. Rev. Lett. 118, 067203 (2017).], we theoretically investigate the high magnetic field regime of the disordered quasi-one-dimensional $S=1$ antiferromagnetic material $\mathrm{Ni}({\mathrm{Cl}}_{1-x}{\mathrm{Br}}_x{)}_2\text{−}4\mathrm{SC}({\mathrm{NH}}_2{)}_2$. The interplay between disorder, chemically controlled by Br-doping, interactions, and the external magnetic field, leads to a very rich phase diagram. Beyond the well-known antiferromagnetically ordered regime, an analog of a Bose condensate of magnons, which disappears when $H\ge 12.3\mathrm{T}$, we unveil a resurgence of phase coherence at a higher field $H\sim 13.6\mathrm{T}$, induced by the doping. Interchain couplings stabilize the finite temperature long-range order whose extension in the field—temperature space is governed by the concentration of impurities $x$. Such a “minicondensation” contrasts with previously reported Bose-glass physics in the same regime and should be accessible to experiments.

Figure 1

(a) Schematic temperature $T$—magnetic field $H$ phase diagram of the clean DTN compound, showing the BEC dome surrounded by a quantum paramagnet (QPM) and a polarized ferromagnet (FM). (b)–(c) The two types (clean and Br-doped) of $S=1$ dimers and their arrangement in a three-dimensional array of coupled chains modeling $\mathrm{DTN}X$, see Eq. (1). (d) Field-induced energy level crossing of a Br-doped $S=1$ dimer. (e) Schematic $T$—$H$ phase diagram for $\mathrm{DTN}X$ at low doping, with an impurity-induced ${\mathrm{BEC}}^{*}$ dome revival in the vicinity of the crossover field $H^{*}$. Yellow baselines show the regions where a BG is expected.

Figure 2

Finite size scaling analysis for the disorder average spin stiffness ${\rho }_s(L)$ (top panels) and transverse AF order parameter $m_x(L)$ (bottom panels) following the scaling forms given by Eq. (4). QMC results obtained for the $\mathrm{DTN}X$ Hamiltonian Eq. (1) on $L\times L/10\times L/10$ lattices of various sizes at $H=H^{*}=13.6\mathrm{T}$ with $x=10\%$ of impurities.

Figure 3

Critical ordering temperature for the impurity-induced LRO at $H^{*}=13.6\mathrm{T}$ plotted against the impurity concentration $x$ for the $S=1$ model Eq. (1) (hexagon) and for the effective HCB toy-model Eq. (3) at half filling (circle) plotted against $(5\delta {)}^{-1}$, suggesting LRO for all finite $x$ values. The average effective pairwise coupling between impurities in the transverse direction is also shown (square) for comparison. Lines are guides to the eyes.

Figure 4

Field $H$—temperature $T$ phase diagram for $\mathrm{DTN}X$ obtained by QMC simulations for both the $S=1$ model Eq. (1) with $x=0$ (square), $x=10\%$ (hexagon), $x=12.5\%$ (diamond), $x=16.67\%$ (triangle), and the HCB effective Hamiltonian Eq. (3) with $\delta =2$ (circle). The HCB ${\mathrm{BEC}}^{*}$ dome has been centered around 13.5 T. Dashed lines are guides to the eyes.