Effective gaps for time-periodic Hamiltonians modeling Floquet materials

Abstract: Featured Article: "Effective gaps in continuous Floquet Hamiltonians", SIAM J Math Analysis 54 #1 2022.

Featured Article Authors: Amir Sagiv and Michael Weinstein

Floquet media are a type of material, in which time-periodic forcing is applied to alter the material’s energy transport properties. Examples in classical and quantum physics include (i) applying a time-oscillating electric field to a graphene sheet to influence quantum electronic transport, and (ii) replacing a periodic array of standard optical waveguides by an array of helically coiled waveguides to influence the flow of light propagation in the array. The ability to modify energy transport via time modulation is recognized as having great potential for applications to information transfer, information processing, and other applications.

We discuss Floquet materials governed by a class of parametrically forced Schrödinger equations, which arises in the above examples: i ψt = Hε (t) ψ, Hε(t) = H0 + ε W ( ε t , -i ∇ ). (1)

The operator H0 = - Δ + V ( x ) models a graphene-like static material; the potential V has the symmetries of a honeycomb tiling of the plane. The family of self-adjoint operators T ↦ W ( T, -i ∇ ) is assumed to be Tper – periodic.

While the energy transport of the unforced system ( W ≡ 0 ) is governed by the band structure (Floquet-Bloch spectral theory) of H0 , that of the forced system is characterized by the monodromy operator, the unitary mapping ψ0 ↦ ψ ( ε-1 Tper ) , and its associated quasi-energy spectrum. Gaps in the band spectrum and quasi-energy spectrum, respectively, of static and Floquet materials, play an important role in their transport properties.

The evolution under Hε (t) of band-limited Dirac wave-packets (a natural model of physical excitations) is well-approximated on very large time scales by an effective time-periodic Dirac equation, which has a gap in its quasi-energy spectrum. But little is known about the nature of the quasi-energy spectrum of (1), and it is believed that no such quasi-energy gap occurs.

We explain how to transfer quasi-energy gap information about the effective Dirac dynamics to conclusions about the full Schrödinger dynamics (1). In particular, we show that (1) has an effective quasi-energy gap, an interval of quasi-energies whose corresponding modes have only a very small spectral projection onto the subspace of 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. Physically, 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.

This lecture focuses on joint work with Amir Sagiv (Columbia University) appearing in the article: Effective gaps in continuous Floquet Hamiltonians, SIMA 54 #1 2022.

analysis of PDEs

Audience: researchers in the topic

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