Accurate results from Hamiltonian strong-coupling expansions of the Ising, planar,O(3), andO(4)spin systems

Strong-coupling expansions of the quantum-Hamiltonian form of the Ising, $O\left(2\right)\mathrm{}$, $O\left(3\right)\mathrm{}$, and $O\left(4\right)\mathrm{}$ models in two dimensions are analyzed. The critical point, correlation-length exponent $\nu$, and magnetization index $\beta$ are found exactly for the Ising model. The susceptibility index $\gamma$ and the specific-heat index $\alpha$ are computed to within 1% of their known values. The phase transition of the $O\left(2\right)\mathrm{}$ model is found and the correlation length is predicted to diverge with an essential singularity $\xi \mathrm{}\left(T\right)\mathrm{}\sim \exp \left[b{(T-T_c)}^{-\sigma }\right]$ where $\sigma =0.7\mathrm{}\pm 0.1$. The index of the spin-spin correlation function $\eta$ is predicted to be 0.26±0.03 at the critical point in agreement with experiment. The non-Abelian spin systems are argued to exist only in a disordered phase. The asymptotic freedom of these models plays an important role in these arguments by ruling out the possibility of weak-coupling phase transitions.