Dyson hierarchical long-ranged quantum spin-glass via real-space renormalization

Abstract

We consider the Dyson hierarchical version of the quantum spin-glass with random Gaussian couplings characterized by the power-law decaying variance and a uniform transverse field h. The ground state is studied via real-space renormalization to characterize the spin-glass-paramagnetic zero temperature quantum phase transition as a function of the control parameter h. In the spin-glass phase , the typical renormalized coupling grows with the length scale L as the power-law with the classical droplet exponent , where the stiffness modulus vanishes at criticality , whereas the typical renormalized transverse field decays exponentially in terms of the diverging correlation length . At the critical point , the typical renormalized coupling and the typical renormalized transverse field display the same power-law behavior L−z with a finite dynamical exponent z. The RG rules are applied numerically to chains containing L = 212 = 4096 spins in order to measure these critical exponents for various values of σ in the region .