Physics – Condensed Matter – Other Condensed Matter
Scientific paper
2007-06-13
Phys. Rev. B 76, 144427 (2007)
Physics
Condensed Matter
Other Condensed Matter
10 pages, 11 figures; v2: added references, published version
Scientific paper
10.1103/PhysRevB.76.144427
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.
Caneva Tommaso
Fazio Rosario
Santoro Giuseppe E.
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