Quantum Spin Glass on the Bethe Lattice

We present an exact analytic solution of the short-range interaction disordered quantum spherical model with symmetric binary $\pm J$ bond distribution on the Bethe lattice (with coordination number $z$). We find the zero temperature quantum spin glass phase transition for coordinations $z\ge 3$ but not for the one-dimensional system $(z=2\mathrm{})$. Quantum dynamics is examined via various correlation functions on the infinite tree which are evaluated in a closed form. The relationship between the results for a fully connected lattice $(z\to \infty )$ and the usual mean-field approach is established.