Detection of a Disorder-Induced Bose-Einstein Condensate in a Quantum Spin Material at High Magnetic Fields

The coupled spin-1 chains material ${\mathrm{NiCl}}_2\text{−}4\mathrm{SC}({\mathrm{NH}}_2{)}_2$ (DTN) doped with Br impurities is expected to be a perfect candidate for observing many-body localization at high magnetic field: the so-called “Bose glass,” a zero-temperature bosonic fluid, compressible, gapless, incoherent, and short-range correlated. Using nuclear magnetic resonance, we critically address the stability of the Bose glass in doped DTN, and find that it hosts a novel disorder-induced ordered state of matter, where many-body physics leads to an unexpected resurgence of quantum coherence emerging from localized impurity states. An experimental phase diagram of this new “order-from-disorder” phase, established from nuclear magnetic resonance $T_1^{-1}$ relaxation rate data in the $13\pm 1\%$ Br-doped DTN, is found to be in excellent agreement with the theoretical prediction from large-scale quantum Monte Carlo simulations.

Figure 1

(a) Doping Br to replace Cl atoms modifies locally both the affected bond, $J\to J^{\prime }$, and the single-ion anisotropy of the nearest spin, $D\to D^{\prime }$, denoted, respectively, by red bonds and dots in the three-dimensional structure shown in (b). (c) The canted spin polarization in the BEC phase is marginally perturbed by doping. (d) In the BEC* phase, the order is formed among partially polarized doped sites, on a fully polarized background of regular spins. (e) In the Bose-glass regime, the canted polarization at doped sites is uncorrelated and fluctuating. The $c$-axis direction is horizontal in (a) and vertical in (b)–(e).

Figure 2

Sketch of the global phase diagram of (a) pure DTN and (b) 13% doped $\mathrm{DTN}X$, where colors denote the BEC (blue) and BEC* (red) phases, and the Bose-glass (BG, yellow) and gapped (green) regimes. (c) Focus on the main BEC* phase. The $T_c$ determined from QMC simulations for 12.5% doping (blue open diamonds) is compared to $T_c$ estimates from the $T_1^{-1}$ NMR data in a $(13\pm 1)\%$ doped sample that are shown in Fig. 3: solid red dots and diamonds denote, respectively, the maximum of $T_1^{-1}(T)$ and $T_1^{-1}(H)$ dependence, reflecting the maximum of the critical spin fluctuations, while orange squares provide the lowest estimate for the $T_c$, taken to be the point where the $T_1^{-1}(T)$ data turn into their BEC regime (see the text). The gray small dots and diamonds are, respectively, the experimental points and the QMC simulation of the BEC phase boundary in the pure DTN, as reported in [23]. Lines are guide to the eyes.

Figure 3

(a) Temperature dependence of $T_1^{-1}$ in 13% doped $\mathrm{DTN}X$ at selected field values. The strong decrease of $T_1^{-1}$ at low temperature is a signature of the ordered BEC phase, while the relatively broad maximum corresponds to the critical spin fluctuations at the phase transition into this phase. The error bars are less than the symbol size, except for one point. (b) Magnetic field dependence of $T_1^{-1}$ in the vicinity of the phase transition into the ordered phase. Several positions of the 13% doped $\mathrm{DTN}X$ phase diagram (solid dots) are compared to the case of much less, 4% doped $\mathrm{DTN}X$ (open squares), to show how strongly is the $T_1^{-1}$ peak broadened by increasing the doping-induced disorder.