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Periodic Dynamics at Low Frequency of a Local Perturbation in the Isotropic XY Model

Livia Corsi, Giuseppe Genovese

Published Aug 23, 2019• 0 citations
OpenAlex

Abstract

We continue our study on the isotropic XY chain with a time-periodic transverse magnetic field acting on a single site. Under a suitable technical assumption on the external forcing, we show the state to be asymptotically a periodic function synchronised with the forcing for any frequency. For small frequencies our result is valid if the amplitude of the transverse field is small.

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We continue our study on the isotropic XY chain with a time-periodic transverse magnetic field acting on a single site. Under a suitable technical assumption on the external forcing, we show the state to be asymptotically a periodic function synchronised with the forcing for any frequency. For small frequencies our result is valid if the amplitude of the transverse field is small.
We study the isotropic XY quantum spin chain with a time-periodic transverse magnetic field acting on a single site. The asymptotic dynamics is described by a highly resonant Floquet–Schrödinger equation, for which we show the existence of a periodic solution if the forcing frequency is away from a discrete set of resonances. This in turn implies the state of the quantum spin chain to be asymptotically a periodic function synchronised with the forcing, also at arbitrarily low non-resonant frequencies. The behaviour at the resonances remains a challenging open problem.
Abstract We study the isotropic XY quantum spin chain with a time-periodic transverse magnetic field acting on a single site. The asymptotic dynamics is described by a highly resonant Floquet–Schrödinger equation, for which we show the existence of a periodic solution if the forcing frequency is away from a discrete set of resonances. This in turn implies the state of the quantum spin chain to be asymptotically a periodic function synchronised with the forcing, also at arbitrarily low non-resonant frequencies. The behaviour at the resonances remains a challenging open problem.
Using continued fractions we study the ground state properties of the spin-½ Ising chain in a transverse field with periodically varying interaction strengths and external fields. We consider in detail the chain having a period-2 modulation of the interaction strengths and compare the results obtained with those corresponding to the spin-½ isotropic XY chain in a transverse field. In contrast to the behaviour of the transverse XY chain, the transverse Ising chain does not exhibit a step-like magnetization versus field dependence caused by the alternation of bonds. Its susceptibility exhibits a logarithmic singularity at a field determined by interaction strengths, and it is stable with respect to spin-Peierls dimerization.
We consider the quantum XY spin chain with quasi-periodic two-valued exchange couplings and a uniform transverse field. The equivalence of the system with a free fermionic model permits a detailed analysis of its thermodynamics. The zero-temperature magnetization is a Cantor function of the applied field. The zero-field specific heat and susceptibility have a power-law behaviour at low temperature, modulated by periodic amplitudes. The critical exponents interpolate continuously between those of the periodic and of the random model.