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 (t) which, as in the Kibble-Zurek mechanism, is linearly reduced to zero in time with a rate ^-1, (t) =-t∕, starting at t=- from the quantum disordered phase (=) and ending at t=0 in the classical ferromagnetic phase (=0). We first analyze the distribution of the gaps, occurring at the critical point ₂=1, which are relevant for breaking the adiabaticity of the dynamics. We then present extensive numerical simulations for the residual energy Eₑ₄ₒ and density of defects ₊ at the end of the annealing, as a function of the annealing inverse rate. Both the average Eₑ₄ₒ () and ₊ () are found to behave logarithmically for large, but with different exponents, Eₑ₄ₒ () ∕L₀ₕ1∕ln^ () with 3. 4, and ₊ () ₀ₕ1∕ln^2 (). We propose a mechanism for 1∕ln^2 behavior of ₊₀ₕ 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.