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

摘要

We consider the Dyson hierarchical version of the quantum spin-glass with random Gaussian couplings characterized by the power-law decaying variance <(J(2)(r))over bar> proportional to r(-2 sigma) 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 h < h(c), the typical renormalized coupling grows with the length scale L as the power-law J(L)(typ)(h) proportional to Upsilon(h) L-theta with the classical droplet exponent theta = 1 - sigma, where the stiffness modulus vanishes at criticality Upsilon(h) proportional to (hc - h)(mu), whereas the typical renormalized transverse field decays exponentially h(L)(typ) (h) proportional to e(-L/xi) in terms of the diverging correlation length xi proportional to (h(c) - h)(-v). At the critical point h = h(c), the typical renormalized coupling J(L)(typ) (h(c)) and the typical renormalized transverse field J(L)(typ)(h(c)) display the same power-law behavior L-z with a finite dynamical exponent z. The RG rules are applied numerically to chains containing L = 2(12) = 4096 spins in order to measure these critical exponents for various values of s in the region 1/2 < sigma < 1.