Magnetic impurity coupled to a strongly correlated electron system in two dimensions

The dynamics of an impurity spin that couples to a two-dimensional system of strongly correlated electrons is studied. They give rise to a fluctuating exchange field acting on the impurity spin. We assume that the fluctuations can be treated as a Gauss-Markoff process. First the case is considered where the strongly correlated electrons are described by a Heisenberg Hamiltonian. The stochastic equation is solved for the spectral function of the impurity-spin correlation function for both cases, i.e., slow and fast fluctuation of the exchange field. The solution is generalized to the quantum mechanical case by relating it to the retarded Green's function of the impurity spin from which the susceptibility and specific-heat contribution of the impurity can be calculated. It is found that the susceptibility is of the Curie form while the specific heat shows a Schottky anomaly due to the slow relaxation of the exchange field at low temperatures. The theory is generalized to a Heisenberg antiferromagnet doped with holes (or doubly occupied sites) having lost the Néel order at $T=0\mathrm{}$ by simply assuming that the relaxation time of the exchange field remains finite even at $T=0\mathrm{}$. This situation resembles ${\mathrm{Nd}}_{2-x}{\mathrm{Ce}}_x\mathrm{Cu}{\mathrm{O}}_4$. However, instead of heavy-fermion behavior we find a marginal Fermi-liquid behavior. The specific heat has a term linear in $T$ in addition to the Schottky anomaly but the susceptibility is of the form $\ln T$ and not Pauli-like. This may be due to the neglect of the feedback effect of the impurity spin on the strongly correlated electron system.