Spin and topological order in a periodically driven spin chain

Abstract

The periodically driven quantum Ising chain has recently attracted a large attention in the context of Floquet engineering. In addition to the common paramagnet and ferromagnet, this driven model can give rise to new topological phases. In this work we systematically explore its quantum phase diagram, by examining the properties of its Floquet ground state. We specifically focus on driving protocols with time-reversal invariant points, and demonstrate the existence of an infinite number of distinct phases. These phases are separated by second-order quantum phase transitions, accompanied by continuous changes of local and string order parameters, as well as sudden changes of a topological winding number and of the number of protected edge states. When one of these phase transitions is adiabatically crossed, the correlator associated to the order parameter is nonvanishing over a length scale which shows a Kibble-Zurek scaling. In some phases, the Floquet ground state spontaneously breaks the discrete time-translation symmetry of the Hamiltonian. Our findings provide a better understanding of topological phases in periodically driven clean integrable models.

Discrete time crystals in many-body quantum chaos

“…DTCs have also been independently proposed in periodically driven spin-1/2 chain systems by other groups [14][15][16][17], which have stimulated follow up studies [18][19][20][21][22][23] and several experimental realizations [24][25][26][27][28]. The original proposal in Ref.…”

Integrable Floquet dynamics

“…Indeed with exception of trivial linear systems, like harmonic oscillators, the Floquet theorem for classical systems does not exist. Nevertheless one can construct examples of Floquet integrable systems in the thermodynamic limit like various driven band models or spin chains, which can be mapped to free particle systems [2,[11][12][13]. The second approach based on the Berry-Tabor conjecture is also possible to extend to Flo-arXiv:1701.05276v4 [cond-mat.stat-mech]…”

Homogeneous Floquet time crystal protected by gauge invariance

“…where we define · · · as the average over the N real realizations of pseudo-disorder, w α = u 1, α · · · u L, α | v 1, α · · · v L, α are the singleparticle Floquet states (see for instance [58] for more details) and α runs over the 2(L − 2) values corresponding to the bulk single-particle Floquet states (the ones with α = ±π). For all the considered values of φ we can clearly see that IPR bulk does not scale with the system size and the same occurs for the error bars (evaluated as the r. m. s. fluctuation over the pseudo-disorder realizations).…”

“…Here it is enough to say that the dynamics is induced by an effective 2(L − 1) × 2(L − 1) timeperiodic single-particle Hamiltonian. This is important to mention because we can compute the 2(L − 1) singleparticle Floquet states and the 2(L − 1) single-particle quasi-energies α (see for instance [58]). These quantities will play an important role in what follows.…”