Frequency-dependent Thermal Response of the Charge System and Restricted Sum Rules in La$_{2-x}$Sr$_{x}$CuO$_4$

The infrared optical conductivity $\sigma_1^{ab} (\omega,T)$ of the $ab$-planes of La$_{2-x}$Sr$_x$CuO$_{4}$ (LSCO) has been measured for different temperatures and for $0.03 < x <0.26$. The spectral weight of the carriers $W = \int_0^\Omega {\sigma_1^{ab} (\omega,T) d\omega}$ fulfills the same law $W = W_0 - B(\Omega) T^2$ as in a conventional metal like gold, for any $\Omega$ up to the plasma edge. However in LSCO the thermal response $B(\Omega)$ points toward correlation effects and, unlike in gold, is related to an energy scale $t_T << t_0 \sim W_0$. This experimental result is in agreement with Dynamical mean Field Theory calculations. The Ferrell-Glover-Tinkham sum rule is fulfilled in LSCO superconductors for $\Omega \agt$ 2000 cm$^{-1}$, an energy much higher than predicted for conventional superconductors.