Simultaneous observation of small- and large-energy-transfer electron-electron scattering in three dimensional indium oxide thick films

In three dimensional (3D) disordered metals, the electron-phonon (\emph{e}-ph) scattering is the sole significant inelastic process. Thus the theoretical predication concerning the electron-electron (\emph{e}-\emph{e}) scattering rate $1/\tau_\varphi$ as a function of temperature $T$ in 3D disordered metal has not been fully tested thus far, though it was proposed 40 years ago [A. Schmid, Z. Phys. \textbf{271}, 251 (1974)]. We report here the simultaneous observation of small- and large-energy-transfer \emph{e}-\emph{e} scattering in 3D indium oxide thick films. In temperature region of $T\gtrsim100$\,K, the temperature dependence of resistivities curves of the films obey Bloch-Gr\"{u}neisen law, indicating the films possess degenerate semiconductor characteristics in electrical transport property. In the low temperature regime, $1/\tau_\varphi$ as a function of $T$ for each film can not be ascribed to \emph{e}-ph scattering. To quantitatively describe the temperature behavior of $1/\tau_\varphi$, both the 3D small- and large-energy-transfer \emph{e}-\emph{e} scattering processes should be considered (The small- and large-energy-transfer \emph{e}-\emph{e} scattering rates are proportional to $T^{3/2}$ and $T^2$, respectively). In addition, the experimental prefactors of $T^{3/2}$ and $T^{2}$ are proportional to $k_F^{-5/2}\ell^{-3/2}$ and $E_F^{-1}$ ($k_F$ is the Fermi wave number, $\ell$ is the electron elastic mean free path, and $E_F$ is the Fermi energy), respectively, which are completely consistent with the theoretical predications. Our experimental results fully demonstrate the validity of theoretical predications concerning both small- and large-energy-transfer \emph{e}-\emph{e} scattering rates.Comment: 5 pages and 4 figure

The tensor renormalization group study of the general spin-S Blume-Capel model

We focus on the special situation of $D=2J$ of the general spin-S Blume-Capel model on the square lattice. Under the infinitesimal external magnetic field, the phase transition behaviors due to the thermal fluctuations are discussed by the newly developed tensor renormalization group method. For the case of the integer spin-S, the system will undergo $S$ first-order phase transitions with the successive symmetry breaking with the magnetization $M=S,S-1,...0$. For the half-integer spin-S, there are similar $S-1/2$ first order phase transition with $M=S,S-1,...1/2$ stepwise structure, in addition, there is a continuous phase transition due to the spin-flip $Z_2$ symmetry breaking. In the low temperature regions, all first-order phase transitions are accompanied by the successive disappearance of the optional spin-component pairs($s,-s$), furthermore, the critical temperature for the nth first-order phase transition is the same, independent of the value of the spin-S. In the absence of the magnetic field, the visualization parameter characterizing the intrinsic degeneracy of the different phases clearly demonstrates the phase transition process.Comment: 6 pages, 7 figure