Scaling of thermal resistivity of $^4$He in restricted geometries

The thermal resistivity and its scaling function in quasi-2D $^4 $He systems are studied by Monte Carlo and spin-dynamics simulation of the classical 3D XY model on $L \times L \times H$ lattices with $L \gg H$. Open boundary conditions are applied along the $H$ direction and periodic boundary conditions along the $L$ directions. A hybrid Monte Carlo algorithm is adopted to efficiently deal with the critical slowing down \footnote{M. Krech and D. P. Landau, Phys. Rev. B {\bf 60}, 3375 (1999)}. Fourth-order Suzuki-Trotter decomposition of exponential operators is used to solve numerically the coupled equation of motion for each spin. The thermal conductivity is calculated by a dynamic current- current correlation function. Our results are consistent with a universal scaling function $F (X)=(L/ \xi_0)^{\pi/ \nu} ( \rho / \rho_0)$, $X=(L/ \xi_0)^{1/ \nu}t$ using known values of the critical exponents $\pi$ and $\nu$ $(\rho = \rho_0 t^{- \pi} $ is the thermal resistivity , and $\xi = \xi_0 t^{- \nu}$ is the correlation length). The thermal resistivity scaling function agrees well with the available experimental results \footnote{Experimental data provided by G. Ahlers, S. Jerebers, Y. Liu and F. C. Liu} for slabs using the temperature scale and thermal resistivity scale as free fitting parameters. \\ \\ $^*$Research supported by NASA