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Tutorial 7 — Floquet Dynamics

Notebook: extra/tutorial/Tutorial7_Floquet.ipynb

Learn how to model periodically driven systems using the LatticeQM.Floquet module and interpret quasienergy spectra.

Learning goals

  • Build Floquet Hamiltonians from static tight-binding models and driving protocols.
  • Truncate harmonic spaces judiciously to capture resonant processes.
  • Compute quasienergy spectra and track avoided crossings.
  • Evaluate time-averaged observables and micromotion.

Prerequisites

  • Tutorials 1–2 for standard Hamiltonian workflows.
  • Basic understanding of Floquet theory.

Workflow outline

  1. Base model — Start with a lattice and tight-binding Hamiltonian (graphene or a custom model).
  2. Drive specification — Define periodic fields via helper constructors in Floquet (e.g. circularly polarised light, pulsed fields).
  3. Floquet construction — Use Floquet.makefloquet(hops; harmonics, ω) or equivalent functions demonstrated in the notebook.
  4. Diagonalisation — Solve for quasienergies with Floquet.getspectrum or the standard Spectrum.getbands applied to the extended Hamiltonian.
  5. Observables — Analyse mode occupations, compute time-averaged currents, and export plots.

Live example

Floquet.periodicDrive(5.0)
Angular frequency of the periodic drive: 5.0
Non-zero harmonic modes: Integer[]
Fourier components of the driving operator: AbstractMatrix[]

figdir = joinpath(pwd(), "figures")
mkpath(figdir)
nothing
bands = getbands(HF, ks)
Floquet.keepfirstFBZ!(bands, HF)
first(bands.bands, 3)
3-element Vector{Float64}:
 -3.000636914187236
 -3.000636914187227
  2.9993630858127576
p = plot(bands; ylabel="ε/t", marker=:none, size=(380, 240), title="Floquet quasienergies (first zone)")
savefig(p, joinpath(figdir, "floquet_bands.svg"))
nothing

Chern numbers for driven graphene (toy amplitude)

N = size(HF(ks.points[:, 1]), 1) ÷ 2
chern_vals = Spectrum.getcherns(HF, 16, 16, N-1:N+2)
chern_vals
4-element Vector{Float64}:
  4.1283097856330353e-10
 -4.128308541372083e-10
  8.490031804568065e-8
 -8.490031803813127e-8

Two-band toy model

static_cherns = Spectrum.getcherns(HB, 12, 12)
static_cherns
2-element Vector{Float64}:
 -0.9999999999999999
  1.0
p = plot(bands_static; size=(360, 240), title="Static two-band model")
savefig(p, joinpath(pwd(), "figures", "floquet_static.svg"))
nothing

Driven two-band model

ω = 12.0
D = 0.8
M = 3
drive2 = Floquet.periodicDrive(ω, [[D 0; 0 -D], [D 0; 0 -D]], [1, -1])
HF2 = Floquet.FloquetOperator(HB, drive2, M)
bands_driven = Spectrum.getbands(HF2, path2)
Floquet.keepfirstFBZ!(bands_driven, HF2)
p = plot(bands_driven; size=(360, 240), title="Driven bands", ylims=(-ω/2, ω/2))
savefig(p, joinpath(pwd(), "figures", "floquet_driven.svg"))
nothing

chern_drive = Spectrum.getcherns(HF2, 12, 12, [2M + 1])
wind_drive = Spectrum.getwindnum(HF2, 12, 12, 2M + 1)
(chern_drive, wind_drive)
([1.6653345369377348e-16], -0.9999999999999996)

Validation checklist

  • Confirm quasienergy zones repeat modulo the drive frequency.
  • Compare against static limits (ω → ∞) to ensure convergence.
  • Save plots (typically under output/floquet/) and verify they match the notebook reference.

Common pitfalls

  • Insufficient harmonic truncation can create spurious gaps. Increase M until bands and observables stabilise.
  • Beware energy folding: always compare spectra modulo ω and consider calling Floquet.keepfirstFBZ! before plotting.
  • For large systems, prefer sparse types and limit the number of requested bands to keep memory in check.

Suggested extensions

  • Couple Floquet results to linear-response calculations to estimate pump-probe signatures.
  • Explore different truncation strategies and document runtime vs. accuracy.
  • Benchmark parallel execution where available and capture typical runtimes for your hardware and parameter choices.