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 $s$, $\nu$ and $z$, corresponding to the energy gap, correlation length and dynamic exponent respectively. We also compute the ground state correlators ${⟨{\sigma }_i^x{\sigma }_{i+n}^x⟩}_g$, $\left\langle \sigma^{y}{i} \sigma^{y}{i+n} \right\rangle_{g}$and$\left\langle \prod^{n}{i=1} \sigma^{z}{i} \right\rangle_{g}$, 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.