Adiabatic perturbation theory: from Landau-Zener problem to quenching through a quantum critical point

TL;DR

This paper develops and applies adiabatic perturbation theory to analyze slow parameter changes in quantum systems, deriving scaling laws for defect formation and energy, especially near quantum critical points, and connecting slow and sudden quench dynamics.

Contribution

It extends adiabatic perturbation theory to many-particle systems and quantum critical points, providing new scaling relations and insights into quench dynamics.

Findings

Derived asymptotics for transition probabilities in two-level systems.

Established scaling laws for defect density and energy near quantum critical points.

Connected adiabatic and sudden quench regimes in quantum phase transitions.

Abstract

We discuss the application of the adiabatic perturbation theory to analyze the dynamics in various systems in the limit of slow parametric changes of the Hamiltonian. We first consider a two-level system and give an elementary derivation of the asymptotics of the transition probability when the tuning parameter slowly changes in the finite range. Then we apply this perturbation theory to many-particle systems with low energy spectrum characterized by quasiparticle excitations. Within this approach we derive the scaling of various quantities such as the density of generated defects, entropy and energy. We discuss the applications of this approach to a specific situation where the system crosses a quantum critical point. We also show the connection between adiabatic and sudden quenches near a quantum phase transitions and discuss the effects of quasiparticle statistics on slow and sudden quenches at finite temperatures.