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.