The Adiabatic Theorem and Linear Response Theory for Extended Quantum Systems

Bachmann, Sven; Roeck, Wojciech de und Fraas, Martin (2018): The Adiabatic Theorem and Linear Response Theory for Extended Quantum Systems. In: Communications in Mathematical Physics, Bd. 361, Nr. 3: S. 997-1027

Abstract

The adiabatic theorem refers to a setup where an evolution equation contains a time-dependent parameter whose change is very slow, measured by a vanishing parameter epsilon. Under suitable assumptions the solution of the time-inhomogenous equation stays close to an instantaneous fixpoint. In the present paper, we prove an adiabatic theorem with an error bound that is independent of the number of degrees of freedom. Our setup is that of quantum spin systems where the manifold of ground states is separated from the rest of the spectrum by a spectral gap. One important application is the proof of the validity of linear response theory for such extended, genuinely interacting systems. In general, this is a long-standing mathematical problem, which can be solved in the present particular case of a gapped system, relevant e.g. for the integer quantum Hall effect.

• Dokumententyp: Zeitschriftenartikel
• Fakultät: Mathematik, Informatik und Statistik > Mathematik
• Themengebiete: 500 Naturwissenschaften und Mathematik > 510 Mathematik
• ISSN: 0010-3616
• Sprache: Englisch