Typical Macroscopic Long-Time Behavior for Random Hamiltonians

Abstract

We consider a closed macroscopic quantum system in a pure state ${\psi }_t$ evolving unitarily and take for granted that different macro states correspond to mutually orthogonal subspaces ${\mathcal{H}}_{\nu }$ (macro spaces) of Hilbert space, each of which has large dimension. We extend previous work on the question what the evolution of ${\psi }_t$ looks like macroscopically, specifically on how much of ${\psi }_t$ lies in each ${\mathcal{H}}_{\nu }$. Previous bounds concerned the \emph{absolute} error for typical ${\psi }_0$ and/or $t$ and are valid for arbitrary Hamiltonians $H$; now, we provide bounds on the \emph{relative} error, which means much tighter bounds, with probability close to 1 by modeling $H$ as a random matrix, more precisely as a random band matrix (i.e., where only entries near the main diagonal are significantly nonzero) in a basis aligned with the macro spaces. We exploit particularly that the eigenvectors of $H$ are delocalized in this basis. Our main mathematical results confirm the two phenomena of generalized normal typicality (a type of long-time behavior) and dynamical typicality (a type of similarity within the ensemble of ${\psi }_0$ from an initial macro space). They are based on an extension we prove of a no-gaps delocalization result for random matrices by Rudelson and Vershynin.

46 pages LaTeX, 2 figure files; v2 has tighter error bounds in Sec. 7 and minor improvements throughout the paper

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Cited by in corpus (4)

• Macroscopic Thermalization for Highly Degenerate Hamiltonians After Slight Perturbation
• Long-Time Behavior of Typical Pure States from Thermal Equilibrium Ensembles
• Typical Positivity of Nonequilibrium Entropy Production for Pure States
• Normal Typicality and Dynamical Typicality for a Random Block-Band Matrix Model