Dynamics of ${}^3\mathrm{He}$ in one dimension in the Luttinger liquid limit

NMR techniques have been used to investigate the dynamics of highly degenerate ${}^3\mathrm{He}$ confined to the interior of ${}^4\mathrm{He}$ plated nanotubes of MCM-41 (Mobil Composition Matter No. 41). The ${}^3\mathrm{He}$ line density was $0.93\pm 0.10{\mathrm{nm}}^{-1}$, corresponding to a Fermi degeneracy temperature of $T_F\approx 0.1$ K. Experiments conducted down to 10 mK reveal a linear temperature dependence of the nuclear spin-lattice relaxation for $T<T_F$, following a peak at $2T_F$, which are hallmarks of Luttinger liquid physics.

Figure 1

Illustration showing a cross section of one hexagonal nanotube of MCM-41 with ${}^4\mathrm{He}$ plating (blue) and internal ${}^3\mathrm{He}$ atoms (green). $D$, the largest inscribed circle, is 2.1–2.4 nm, and following the arguments of Matsushita et al. [8, 9], the diameter accessible to ${}^3\mathrm{He}$ atoms is 0.8–1.1 nm. The inset at the lower right-hand side illustrates the typical stacking of the nanochannels as large bundles of nanotubes with a given orientation.

Figure 2

Schematic representation (not to scale) of the low-temperature NMR cell and link to a solid silver cold finger extending from a nuclear refrigerator [41, 42]. The copper nuclear refrigerator was used in this experiment for temperatures below 11 mK. The high-purity annealed Cu and high-purity silver post provided an excellent thermal link between the thermometers and the NMR cell.

Figure 3

Comparison of the observed relaxation of the nuclear magnetization $M\left(t_W\right)$ (solid black circles) with a stretched exponential decay: $exp[-{\left(\frac{t_W}{T_1}\right)}^{\beta }]$ for $\beta =0.7$ (solid blue line, best fit) and with pure exponential decay (solid red line) for $\beta =1.0$ for a sample at 25 mK. For this sample the pure exponential decay does not provide a good fit to the data. Additional examples for other temperatures are given in the Supplemental Material. Note that while ideally $M\left(0\right)=0$, small nonzero values occur due to the limitations on rf pulse amplitude to prevent unacceptable heating. ($t_W$ is the waiting time between pulse sequences.)

Figure 4

Variation of the exponent ($\beta$) for the stretched exponential fit to the magnetization relaxation as a function of temperature. The solid black circles are for the sample with Ag powder and the solid red triangles are for an earlier sample [18] without Ag powder.

Figure 5

Observed temperature dependence of the nuclear spin-lattice relaxation times for the new sample with Ag powder (solid blue circles) compared to the earlier sample without Ag powder (solid red triangles). The dashed blue line depicts the limiting behavior $T_1\propto T$ at low temperatures and the dashed red line is $T_1\propto T^{-1/2}$ for high temperatures. The best overall fit to the data is for $T_F=95\pm 5$ mK. The solid green line is the theoretical dependence from Polini and Vignale [21] that does not include transverse excitations, and the solid black line is the dependence if excitations to the lowest transverse mode with ${\mathrm{\Delta }}_{01}/k_B=0.77$ K are included. Typical error bars are shown for the two sets of data.