Experimental observation of quantum entanglement in low-dimensional spin systems

We report macroscopic magnetic measurements carried out in order to detect and characterize field-induced quantum entanglement in low-dimensional spin systems. We analyze the pyroborate $\mathrm{Mg}\mathrm{Mn}{\mathrm{B}}_2{\mathrm{O}}_5$ and the warwickite $\mathrm{Mg}\mathrm{Ti}\mathrm{O}\mathrm{B}{\mathrm{O}}_3$, systems with spin $5∕2$ and $1∕2$, respectively. By using the magnetic susceptibility as an entanglement witness we are able to quantify entanglement as a function of temperature and magnetic field. In addition, we experimentally distinguish a system exhibiting a random singlet-phase state from one with the Griffiths phase. This analysis opens the possibility of a more detailed characterization of low-dimensional materials.

Figure 1

(Color online) Upper: magnetic susceptibility versus temperature. (a) Experimental data for $\mathrm{Mg}\mathrm{Ti}\mathrm{O}\mathrm{B}{\mathrm{O}}_3$ (open circles) and fitting using the susceptibility expression of a RSP (solid line). In panel (b), the experimental data for $\mathrm{Mg}\mathrm{Mn}{\mathrm{B}}_2{\mathrm{O}}_5$ (open circles) and for high temperatures a Curie-Weiss fitting (solid line). Lower: exponent $\alpha$ of $\chi \propto T^{-\alpha }$ as a function of $\ln T$ for (c) $\mathrm{Mg}\mathrm{Ti}\mathrm{O}\mathrm{B}{\mathrm{O}}_3$ and (d) $\mathrm{Mg}\mathrm{Mn}{\mathrm{B}}_2{\mathrm{O}}_5$, the solid line in (c) is the theoretical curve for the exponent $\alpha$ for the RSP (see text).

Figure 2

(Color online) (a) Magnetization normalized by the saturation magnetization $M_s=g{\mu }_{B}Ns$ and (b) ac magnetic susceptibility as a function of the applied magnetic field for $\mathrm{Mg}\mathrm{Ti}\mathrm{O}\mathrm{B}{\mathrm{O}}_3$ (open squares) and $\mathrm{Mg}\mathrm{Mn}{\mathrm{B}}_2{\mathrm{O}}_5$ (open circles), both for $T=2\mathrm{K}$. The solid lines represent a power law fitting for the $\mathrm{Mg}\mathrm{Mn}{\mathrm{B}}_2{\mathrm{O}}_5$ data and a logarithmic fitting for the $\mathrm{Mg}\mathrm{Ti}\mathrm{O}\mathrm{B}{\mathrm{O}}_3$ data. (c) ac magnetic susceptibility ($H_{ac}=10\mathrm{Oe}$, $f=1\mathrm{kHz}$) as a function of temperature for different values of an applied dc field $\left(\mathrm{Mg}\mathrm{Ti}\mathrm{O}\mathrm{B}{\mathrm{O}}_3\right)$. The solid line indicates a Curie-law fitting at high temperatures.

Figure 3

(Color online) (a) Experimental data of magnetic susceptibility for $\mathrm{Mg}\mathrm{Ti}\mathrm{O}\mathrm{B}{\mathrm{O}}_3$ (solid circles) and the limit for the EW (solid line). The system presents entanglement below the solid line. In panel (b) we show the same analysis for $\mathrm{Mg}\mathrm{Mn}{\mathrm{B}}_2{\mathrm{O}}_5$. The insets show the quantity $E$ as defined in Eq. (1) which quantifies the entanglement detected by the EW as a function of temperature.

Figure 4

(Color online) The sum of the total spin variance in three directions $i=x,y,z{\sum }_i{\Delta }^2{J}_i$ (solid line), the spin variance in the $z$ direction ${\Delta }^2{J}_z$ (dot-dashed line), the sum ${\Delta }^2{J}_z+⟨J_z⟩$ (dashed line), and expectation value of the $z$ component of the total spin $⟨J_z⟩$ (dotted line) for (a) a pair of spins $S=1∕2$, (b) a pair of spins $S=5∕2$, and (c) a pair of spins $S=1∕2$ interaction with a random interaction $I_i$ (power law distribution and $T=0.05$) as a function of magnetic field. This is normalized by the exchange interaction and in the random case by the cutoff of the distribution.

Figure 5

(Color online) Experimental data for $E$ using Eq. (3) for $B\ne 0$ for $\mathrm{Mg}\mathrm{Ti}\mathrm{O}\mathrm{B}{\mathrm{O}}_3$ and $\mathrm{Mg}\mathrm{Mn}{\mathrm{B}}_2{\mathrm{O}}_5$. The insets show the extrapolation of $E$ for very high values of $B$. The dashed lines are guides for the eyes for value of $B$ where we do not have experimental results for $E$.