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 Γ(t) which, as in the Kibble-Zurek mechanism, is linearly reduced to zero in time with a rate τ^-1, Γ(t)=-t/τ, starting at t=-infty from the quantum disordered phase $Γ=infty$ and ending at t=0 in the classical ferromagnetic phase $Γ=0$. We first analyze the distribution of the gaps - occurring at the critical point Γ_c=1 - which are relevant for breaking the adiabaticity of the dynamics. We then present extensive numerical simulations for the residual energy E_{rm res} and density of defects ρ_k at the end of the annealing, as a function of the annealing inverse rate τ. %for different lenghts of the chain. Both the average E_{rm res}(τ) and ρ_k(τ) are found to behave logarithmically for large τ, but with different exponents, \[E_{rm res}(τ)/L\]_{rm av}sim 1/ln^ζ(τ) with ζapprox 3.4, and \[ρ_k(τ)\]_{rm av}sim 1/ln²(τ). We propose a mechanism for 1/ln²τ-behavior of \[ρ_k\]_{rm 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 absence of any source of frustration.