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Title:  Floquet theory of the XY spin chain
Author:  Sergio Enrique Tapias Arze
As Contributor:   Sergio Enrique Tapias Arze
Type: Ph.D.
Field: Physics
  • Condensed Matter Physics - Theory
Approach: Theoretical
Degree granting institution:  University of Amsterdam
Supervisor(s): Prof. Jean-S├ębastien Caux
Defense date:  2021-02-18


In this thesis we study the properties of the spin-1/2 XY chain under periodic quenches of a transverse magnetic field in the framework of Floquet theory. By exploiting the algebraic properties of the model, we are able to find an exact expression for the Floquet Hamiltonian valid for arbitrary choices of the system parameters, both for finite size systems and in the thermodynamic limit. Most notably, our analytical results are valid for any value of the driving frequency. This point is key, as it allows us to study the problem exactly even in the presence of resonances. We show that when the system is in resonance, the generated couplings in the Floquet Hamiltonian decay algebraically with distance, in contrast to the exponentially decaying couplings in the non-resonant regimes. We also study the out-of-equilibrium dynamics of the model by computing the time-dependent expectation values of local observables. These present an algebraically decaying envelope and reach a synchronised steady state in the infinite time limit, characterised by periodic fluctuations with the same frequency as the driving. The decaying part of these expectation values follows a different power law depending on whether or not a resonance is present. Furthermore, the synchronised steady state expectation values of local observables present non-analyticities at the transitions between regimes in the parameter space with different number of resonant states, leading us to propose the number of resonant modes as a criterion to determine the phase diagram of the model.

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