TY - JOUR

T1 - EFFECTIVE GAPS IN CONTINUOUS FLOQUET HAMILTONIANS

AU - Sagiv, Amir

AU - Weinstein, Michael I.

N1 - Publisher Copyright:
© 2022 Society for Industrial and Applied Mathematics.

PY - 2022

Y1 - 2022

N2 - We consider two-dimensional Schrodinger equations with honeycomb potentials and slow time-periodic forcing of the form iψt(t, x) “H\varepsilon (t)ψ “(H0 + 2iεA(εt) ¨ ∇) ψ, H0 “´\Delta +V (x). The unforced Hamiltonian, H0, is known to generically have Dirac (conical) points in its band spectrum. The evolution under H\varepsilon (t) of band-limited Dirac wave-packets (spectrally localized near the Dirac point) is well-approximated on large time scales (t À ε´2`) by an effective time-periodic Dirac equation with a gap in its quasi-energy spectrum. This quasi-energy gap is typical of many reduced models of time-periodic (Floquet) materials and plays a role in conclusions drawn about the full system: conduction vs. insulation, topological vs. nontopological bands. Much is unknown about the nature of the quasi-energy spectrum of original time-periodic Schr odinger equations, and it is believed that no such quasi-energy gap occurs. In this paper, we explain how to transfer quasi-energy gap information about the effective Dirac dynamics to conclusions about the full Schrodinger dynamics. We introduce the notion of an effective quasi-energy gap and establish its existence in the Schrodinger model. In the current setting, an effective quasi-energy gap is an interval of quasi-energies which does not support modes with large spectral projection onto band-limited Dirac wave-packets. The notion of effective quasi-energy gap is a physically relevant relaxation of the strict notion of quasi-energy spectral gap; if a system is tuned to drive or measure at momenta and energies near the Dirac point of H0, then the resulting modes in the effective quasi-energy gap will only be weakly excited and detected.

AB - We consider two-dimensional Schrodinger equations with honeycomb potentials and slow time-periodic forcing of the form iψt(t, x) “H\varepsilon (t)ψ “(H0 + 2iεA(εt) ¨ ∇) ψ, H0 “´\Delta +V (x). The unforced Hamiltonian, H0, is known to generically have Dirac (conical) points in its band spectrum. The evolution under H\varepsilon (t) of band-limited Dirac wave-packets (spectrally localized near the Dirac point) is well-approximated on large time scales (t À ε´2`) by an effective time-periodic Dirac equation with a gap in its quasi-energy spectrum. This quasi-energy gap is typical of many reduced models of time-periodic (Floquet) materials and plays a role in conclusions drawn about the full system: conduction vs. insulation, topological vs. nontopological bands. Much is unknown about the nature of the quasi-energy spectrum of original time-periodic Schr odinger equations, and it is believed that no such quasi-energy gap occurs. In this paper, we explain how to transfer quasi-energy gap information about the effective Dirac dynamics to conclusions about the full Schrodinger dynamics. We introduce the notion of an effective quasi-energy gap and establish its existence in the Schrodinger model. In the current setting, an effective quasi-energy gap is an interval of quasi-energies which does not support modes with large spectral projection onto band-limited Dirac wave-packets. The notion of effective quasi-energy gap is a physically relevant relaxation of the strict notion of quasi-energy spectral gap; if a system is tuned to drive or measure at momenta and energies near the Dirac point of H0, then the resulting modes in the effective quasi-energy gap will only be weakly excited and detected.

KW - Dirac equation

KW - Floquet

KW - Schrodinger equation

KW - graphene

KW - parametric forcing

KW - topological insulator

UR - http://www.scopus.com/inward/record.url?scp=85128361231&partnerID=8YFLogxK

U2 - 10.1137/21M1417363

DO - 10.1137/21M1417363

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AN - SCOPUS:85128361231

SN - 0036-1410

VL - 54

SP - 986

EP - 1021

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

IS - 1

ER -