Random matrix theory and critical phenomena in quantum spin chains

We compute critical properties of a general class of quantum spin chains which are quadratic in the Fermi operators and can be solved exactly under certain symmetry constraints related to the classical compact groups $U(N)$, $O(N)$ and $Sp(2N)$. In particular we calculate critical exponents $s$, $ u$ and $z$, corresponding to the energy gap, correlation length and dynamic exponent respectively. We also compute the ground state correlators $leftlangle sigma^{x}_{i} sigma^{x}_{i+n} rightrangle_{g}$, $leftlangle sigma^{y}_{i} sigma^{y}_{i+n} rightrangle_{g}$ and $leftlangle prod^{n}_{i=1} sigma^{z}_{i} rightrangle_{g}$, all of which display quasi-long-range order with a critical exponent dependent upon system parameters. Our approach establishes universality of the exponents for the class of systems in question.