Adiabatic quantum dynamics of a random Ising chain across its quantum critical point

We present here our study of the adiabatic quantum dynamics of a random Ising chain across its quantum critical point. The model investigated is an Ising chain in a transverse field with disorder present both in the exchange coupling and in the transverse field. The transverse field term is proportional to a function Gamma(t) which, as in the Kibble-Zurek mechanism, is linearly reduced to zero in time with a rate tau(-1), Gamma(t)=-t/tau, starting at t=-infinity from the quantum disordered phase (Gamma=infinity) and ending at t=0 in the classical ferromagnetic phase (Gamma=0). We first analyze the distribution of the gaps,occurring at the critical point Gamma(c)=1, which are relevant for breaking the adiabaticity of the dynamics. We then present extensive numerical simulations for the residual energy E(res) and density of defects rho(k) at the end of the annealing, as a function of the annealing inverse rate tau. Both the average E(res)(tau) and rho(k)(tau) are found to behave logarithmically for large tau, but with different exponents, [E(res)(tau)/L](av)similar to 1/ln(zeta)(tau) with zeta approximate to 3.4, and [rho(k)(tau)](av)similar to 1/ln(2)(tau). We propose a mechanism for 1/ln(2)tau behavior of [rho(k)](av) based on the Landau-Zener tunneling theory and on a Fisher's-type real-space renormalization group analysis of the relevant gaps. The model proposed shows therefore a paradigmatic example of how an adiabatic quantum computation can become very slow when disorder is at play, even in the absence of any source of frustration.