Kosterlitz-Thouless transition in uniformly confined $^4$He

Abstract

This study investigates the Kosterlitz-Thouless (KT) transition in superfluid $^4$He confined within uniform nanochannels. While the universal jump in superfluid density is a well-established phenomenon, predicting the absolute transition temperature ($T_{KT}$) based on film geometry has remained a long-standing challenge, often relying on empirical fits. Using on-chip nanofluidic Helmholtz resonators with channel heights of 10, 15, and 20 nm, we probe the transition using 4th sound resonant modes.We demonstrate that the observed shift in the transition temperature relative to the bulk lambda point ($T_λ$) is accurately accounted for by including two-dimensional thermal excitations, specifically 2D rotons. By incorporating these roton-like excitations into the static KT theory, we can predict absolute transition temperatures that align with our experimental measurements and historical data without invoking traditional coherence length scaling arguments. Furthermore, we show that the dynamical extension of the KT theory (AHNS) fully describes the dissipation peaks observed near the transition without requiring ad-hoc free vortex contributions. These results provide compelling evidence that roton excitations, rather than correlation length scaling, govern the finite-size behaviour of confined superfluid $^4$He

Figures (9)

• Figure 1: (a) Sketch of the Helmholtz resonator used for the experiment with all relevant dimensions, $D_c$ is the vertical confinement of the nanochannel. (b) Finite elements method numerical simulation of 4th sound pressure for the first two resonance modes in the nanofluidic cavity. (c,d) Heat maps of Direct and Cross signal showing the dependence of the two resonant modes on the temperature, the brightness of the colour marks the amplitude of the signal. Here for $D_c$ = 15 nm. (e,f) Dissipation of both resonant modes for both signals close to the the lambda point for heating up and cooldown characterized by the inverse quality factor. See the presence of the dissipation peaks at $\approx0.02$ K. Here for $D_c$ = 10 nm.
• Figure 2: (a,b) The confined $\rho_\mathrm{sc}$ (crosses) superfluid density, obtained using the Eq. \ref{['eq:conf_den']} against the shifted temperature $T_{\lambda} - T$ for both signals and all three resonators. The grey lines mark the $T_\mathrm{KT}$ calculated from the Eq.\ref{['eq:KT_static']}, from the left (20 nm) to the right (10 nm), the vertical pink dashed lines then correspond to the $T_\mathrm{KT}^\mathrm{corr}$ and horizontal to the universal jump in superfluid density at $T_\mathrm{KT}^\mathrm{corr}$. The black lines show the superfluid density renormalized by the dielectric constant $\epsilon$ obtained from the fit of dissipation. The insets then show both the $\rho_\mathrm{sb}$ and $\rho_\mathrm{sc}$ against reduced temperature $t = (T_{\lambda} - T)/T_{\lambda}$, the bulk density follows well the tabulated critical behaviour $\rho_\mathrm{sb} \propto t^{-0.67}$.
• Figure 3: (a) The inverse quality factor $Q^{-1}$ against the shifter temperature $T_{\lambda} - T$ for all three film thickness. $Q^{-1}$ characterizes the dissipation of the resonance mode during the KT-transition. The black lines are fits of the AHNS theory and the pink dashed lines correspond to the $T_\mathrm{KT}^\mathrm{corr}$. (b) The measured $\rho_\mathrm{sc}$ against $T_{\lambda} - T$ for the cross signal. The dashed lines are calculated confined densities using the tabulated bulk superfluid density and real part of the dielectric constant obtained from the fits in Fig. (a).
• Figure 4: Superfluid phase transition temperatures as a function of confinement. The static $T_\mathrm{KT}$\ref{['eq:KT_static']} (blue line) corrected for superfluid density suppression due to roton excitations (orange line) accounts for nearly all observed shift of the transition temperature with slab thickness, with no adjustable parameters. Historical data taken from Fig. 24 of the review gasparini_2008, which contains data from refs. gasparini_1998kimball_2001yu_1989ganshin_2006.
• Figure 5: (a) Scheme of the actual measurement circuit. (b) Example of the measured signal with a double lorentzian fit for two different temperatures. The measurement of the resonator with the 10 nm nanochannel.