Adiabatic quantum dynamics of a random Ising chain across its quantum critical point

Abstract

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 $E_{\mathrm{res}}$ 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 $E_{\mathrm{res}}(\tau )$ and ${\rho }_k(\tau )$ are found to behave logarithmically for large $\tau$, but with different exponents, $[E_{\mathrm{res}}(\tau )/L{]}_{\mathrm{av}}\sim 1/{\ln }^{\zeta }(\tau )$ with $\zeta \approx 3.4$, and $[{\rho }_k(\tau ){]}_{\mathrm{av}}\sim 1/{\ln }^2(\tau )$. We propose a mechanism for $1/{\ln }^2\tau$-behavior of $[{\rho }_k{]}_{\mathrm{av}}$ 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.

10 pages, 11 figures; v2: added references, published version

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Cited by in corpus (23)

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• Defect production in non-linear quench across a quantum critical point
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• Optimal non-linear passage through a quantum critical point
• Adiabatic dynamics in open quantum critical many-body systems
• Dynamical non-ergodic scaling in continuous finite-order quantum phase transitions
• Adiabatic quantum dynamics of the Lipkin-Meshkov-Glick model
• Quench dynamics and defect production in the Kitaev and extended Kitaev models
• Non-adiabacity and large flucutations in a many particle Landau Zener problem
• Quenching along a gapless line: A different exponent for defect density
• Defect production due to quenching through a multicritical point
• Theory of defect production in nonlinear quench across a quantum critical point
• Adiabatic quenches through an extended quantum critical region
• Time evolution of 1D gapless models from a domain-wall initial state: SLE continued?
• Defect generation in a spin-1/2 transverse XY chain under repeated quenching of the transverse field
• Effects of interference in the dynamics of spin-1/2 transverse XY Chain driven periodically through quantum critical points
• Phase Transition in Space: How Far Does a Symmetry Bend Before It Breaks?
• Cooling dynamics of pure and random Ising chains
• How to fix a broken symmetry: Quantum dynamics of symmetry restoration in a ferromagnetic Bose-Einstein condensate
• Adiabatic dynamics in a spin-1 chain with uniaxial single-spin anisotropy
• Statistical Mechanics of the Quantum K-Satisfiability problem