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=-∞ from the quantum disordered phase (Γ=∞) 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 Γc=1, which are relevant for breaking the adiabaticity of the dynamics. We then present extensive numerical simulations for the residual energy Eᵣₑₛ and density of defects ρₖ at the end of the annealing, as a function of the annealing inverse rate τ. Both the average Eᵣₑₛ(τ) and ρₖ(τ) are found to behave logarithmically for large τ, but with different exponents, [Eᵣₑₛ(τ)∕L]ₐᵥ~1∕ln^ζ(τ) with ζ≈3.4, and [ρₖ(τ)]ₐᵥ~1∕ln²(τ). We propose a mechanism for 1∕ln²τ behavior of [ρₖ]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.