Spatially Modulated Superfluid State in Two-Dimensional ${}^4\mathrm{He}$ Films

The second layer of ${}^4\mathrm{He}$ films adsorbed on a graphite substrate is an excellent experimental platform to study the interplay between superfluid and structural orders. Here, we report a rigid two-frequency torsional oscillator study on the second layer as a function of temperature and ${}^4\mathrm{He}$ atomic density. For the first time, we show experimentally that the superfluid density is independent of frequency, which can be interpreted as unequivocal evidence of genuine superfluidity. The phase diagram established in this work reveals that a superfluid phase coexists with hexatic density-wave correlation and a registered solid phase. This suggests the second layer as a candidate for hosting two exotic quantum ground states: the spatially modulated superfluid and supersolid phases resulting from the interplay between superfluid and structural orders.

Figure 1

Superfluid measurement in two-dimensional ${}^4\mathrm{He}$ films. (a) Rigid two-frequency torsional oscillator (TO). (b) ${\mathrm{N}}_2$ pressure isotherm for the determination of the surface area of substrate. (c) Layer-by-layer growth of ${}^4\mathrm{He}$ films manifested by discontinuous increases in vapor pressure. (d),(e) Temperature dependence of the TO period, and (f),(g) amplitude in the low (in-phase) and high (out-of-phase) modes at different rim velocities. Empty TO responses (gray symbols) obtained at different driving velocities are superimposed over each other, indicating that the TO operates in the linear response regime.

Figure 2

Temperature evolution of second layer superfluidity. Superfluid density ${\rho }_s$ as a function of temperature $T$ in a coverage region of $17.01–18.23\mathrm{atoms}/{\mathrm{nm}}^2$ for the (a) low- and (b) high-frequency modes, and $18.23–18.83\mathrm{atoms}/{\mathrm{nm}}^2$ for the (c) low- and (d) high-frequency modes. The dashed lines are linear fits of ${\rho }_s$ as a function of log($T$).

Figure 3

Coverage dependence of the second layer superfluidity. False-color map of superfluid density ${\rho }_s$ in the second layer as a function of coverage $n$ and temperature $T$ measured at (a) the in-phase and (b) the out-of-phase modes. The dashed lines are contours of ${\rho }_s$ from 0.1 (outermost) to 0.7 (innermost) with a step of 0.1. (c) Superfluid onset temperature $T_s$ as a function of $n$. The gray solid line is the prediction from BKT theory $T_c(n)=0.15(n-n_0)$ [38]. (d) ${\rho }_s$ isotherm at 22 mK plotted against $n$. (e) Testing the linear relation between ${\rho }_s$ and $T_s$. The solid gray line is a linear fit fixing the $y$ intercept to zero. The shaded region indicates $1\sigma$ bound (or 68% confidence) of the fitting.

Figure 4

Phase diagram of the second layer of ${}^4\mathrm{He}$ films. $T_s$ from this work (cyan) and the temperature of heat capacity loci (open circles and solid triangles) are plotted against reduced coverage (see text) with $n_2=21.2\mathrm{atoms}/{\mathrm{nm}}^2$ [20, 21]. A theoretical phase diagram [24] is drawn above the figure for comparison under the assumption of $n_2=20.4\mathrm{atoms}/{\mathrm{nm}}^2$. The uncertainty of $n_r$ is smaller than 0.01.