Experimental measurement of the isolated magnetic susceptibility

The isolated susceptibility ${\chi }_{\mathrm{I}}$ may be defined as a (nonthermodynamic) average over the canonical ensemble, but while it has often been discussed in the literature, it has not been clearly measured. Here, we demonstrate an unambiguous measurement of ${\chi }_{\mathrm{I}}$ at avoided nuclear-electronic level crossings in a dilute spin ice system, containing well-separated holmium ions. We show that ${\chi }_{\mathrm{I}}$ quantifies the superposition of quasiclassical spin states at these points and is a direct measure of state concurrence and populations.

Figure 1

(a) Schematic showing the frequency-dependent decoupling of the real (black) and imaginary (red) parts of the magnetic susceptibility in a system with well-separated spin-lattice and spin-spin relaxation times. One can expect three plateaus of purely real response corresponding to ${\chi }_T$, ${\chi }_S$, and ${\chi }_{\mathrm{I}}$, respectively. (b) Avoided level crossing (upper blue curves). In this paper we are solely interested in adiabatic evolution (blue). It is illustrated for an initial state on the lower branch, as indicated by a black circle. Also illustrated is diabatic evolution (pink) that could occur if there were a Landau-Zener crossover at frequencies greater than those studied here. The lower curves (same color code) indicate the corresponding isolated susceptibilities for the ensemble of two state systems treated in Sec. 2, where the adiabatic case yields a peak in ${\chi }_{\mathrm{I}}$, while the diabatic case gives zero, as shown in (a).

Figure 2

(a) The four Ising-like $\langle 111\rangle$ spin orientations of ${\mathrm{Ho}}^{3+}$ with respect to the [111] direction of the applied field (blue) and the basal plane (shaded; apical spin is in red, basal spins are in magenta). In the dilute sample only one or zero magnetic sites are likely to be occupied by magnetic ${\mathrm{Ho}}^{3+}$ in any given tetrahedron of the crystal structure. (b) Energy diagram of the hyperfine levels of an apical ${\mathrm{Ho}}^{3+}$ ion as a function of magnetic field with an effective splitting of $\mathrm{\Delta }/k_{\mathrm{B}}=0.013$ K, showing direct and avoided level crossings. (c) The effective electronic spin-$1/2{\mathrm{Ho}}^{3+}$ ion has eight Zeeman split levels which are degenerate in zero field (gray arrows). As the field is increased from zero to positive values, there are four level crossings where only the electron spin reverses. When brought into resonance by applied field, superposed $\pm m_S$ states have finite isolated susceptibility.

Figure 3

Calculated values of the three susceptibilities [Eqs. (1, 2, 3)] at $T=2.1$ K for the apical spin Hamiltonian (13), with spin concentration equal to that of the sample studied.

Figure 4

Temperature dependence of the dc field-cooled susceptibility of ${\mathrm{Y}}_{1.9975}{\mathrm{Ho}}_{0.0025}{\mathrm{Ti}}_2{\mathrm{O}}_7$ measured along the [111] axis (blue points) compared to a Curie law fit summed with a small temperature-independent component (red line). Inset: graph replotted as $1/{\chi }_T$.

Figure 5

Frequency and magnetic field dependence of the real part of the susceptibility measured at 2.1 K with an ac field amplitude of 0.2 mT, demonstrating a gradual reduction to four peaks as the frequency is increased.

Figure 6

(a) The real (blue points) and imaginary (gray points) part of the measured susceptibility at 2.1 K, 10 kHz, and a probe field of 0.2 mT compared to the theory, with a single value of $\mathrm{\Delta }/k_{\mathrm{B}}=0.015\mathrm{K}$ (black line). (b) Experiment versus theory with an empirical distribution of $\mathrm{\Delta }$ [Eq. (18)]. The fitted parameters are $f_1=0.511\left(6\right),f_2=0.352\left(3\right)$ for 2.1 K, 0.2 mT; $f_1=0.455\left(6\right),f_2=0.312\left(3\right)$ for 2.1 K, 1.0 mT; and $f_1=0.52\left(1\right),f_2=0.335\left(4\right)$ for 4 K, 0.2 mT. These parameters affect the peak shape, not peak heights, and are expected to evolve with probe field but not significantly with temperature. It is also possible to fit the data with smooth distributions (see text).

Figure 7

The real part of the susceptibility at 11 Hz as a function of field and decreasing temperature, demonstrating that the isolated response is revealed by low temperature as well as high frequency and that the distribution of peak intensities is no longer that of a Boltzmann distribution (contrast with Fig. 6).