Quantum Critical Behavior of the Infinite-range Transverse Ising Spin Glass : An Exact Numerical Diagonalization Study

We report exact numerical diagonalization results of the infinite-range Ising spin glass in a transverse field $\Gamma$ at zero temperature. Eigenvalues and eigenvectors are determined for various strengths of $\Gamma$ and for system sizes $N \le 16$. We obtain the moments of the distribution of the spin-glass order parameter, the spin-glass susceptibility and the mass gap at different values of $\Gamma$. The disorder averaging is done typically over 1000 configurations. Our finite size scaling analysis indicates a spin glass transition at $\Gamma_c \simeq 1.5$. Our estimates for the exponents at the transition are in agreement with those known from other approaches. For the dynamic exponent, we get $z=2.1 \pm 0.1$ which is in contradiction with a recent estimate ($z=4$). Our cumulant analysis indicates the existence of a replica symmetric spin glass phase for $\Gamma < \Gamma_c$.