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 $t=-\infty$ from the quantum disordered phase ($\Gamma=\infty$) 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 $E_{ \rm res}$ and density of defects $\rho_k$ at the end of the annealing, as a function of the annealing inverse rate $\tau$. %for different lenghts of the chain. Both the average $E_{ \rm res}(\tau)$ and $\rho_k(\tau)$ are found to behave logarithmically for large $\tau$, but with different exponents, $[E_{ \rm res}(\tau)/L]_{ \rm av}\sim 1/\ln^{ \zeta}(\tau)$ with $\zeta\approx 3.4$, and $[\rho_k(\tau)]_{ \rm av}\sim 1/\ln^{ 2}(\tau)$. We propose a mechanism for $1/\ln^2{ \tau}$-behavior of $[\rho_k]_{ \rm av}$ 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.
Categories: cond-mat.other quant-ph