The t-J model: From light to heavy doping
  The regions of hole concentrations $0 \leq x \alt 0.3$ and temperatures $0.005|t| \leq T \leq 0.02|t|$ are studied in the t-J model of Cu-O planes of perovskite high-$T_c$ superconductors. For this purpose self-energy equations for hole and spin Green's functions are derived using Mori's projection operator technique and these equations are self-consistently solved. The calculated hole band transforms radically at $x\approx 0.08$. A narrow low-concentration band with minima near $(\pm\frac{\pi}{2},\pm\frac{\pi}{2})$ is converted to a band resembling the case of weak electron correlations, with the minimum at $(\pi,\pi)$ or $(0,0)$. The hole Fermi surface is respectively changed from small ellipses at $(\pm\frac{\pi}{2},\pm\frac{\pi}{2})$ to a large rhombus centered at $(\pi,\pi)$ or $(0,0)$. The decrease of the magnetic susceptibility at the antiferromagnetic wave vector and spin correlations with doping is determined by the growth of the frequency of spin excitations at this momentum. The shape of the frequency dependence of the susceptibility depends heavily on the hole damping and varies from a broad feature similar to that observed in La$_{2-x}$Sr$_x$CuO$_4$ to a pronounced maximum which resembles the normal-state resonance peak in YBa$_2$Cu$_3$O$_{7-y}$. 
A. Sherman