Floquet Model Construction With nrginit

The Mathematica initializer builds a finite one-frequency Floquet factor and appends it to the model basis. The model owns the drive Hamiltonian and the mode operators exported to the runtime.

Configuration

A Floquet initialization supplies the mode cutoff, angular frequency, runtime switch, and mode operator:

[extra]
Omega=0.234

[param]
model=model.m
symtype=SPU1
floquet=true
options=Ncut=1
ops=m
diagratio=1

Ncut is a non-negative decimal integer. Omega must round to a finite, positive C++ double; a positive input that rounds to zero is rejected. The same param file is read by the initializer and the C++ runtime. Omega uses decimal-literal syntax and does not accept the initializer's !expression extension.

Floquet Basis

floquetbasis[Ncut] returns

transformtoFL[basis,{Ncut}] forms the tensor-product basis. Existing invariant labels are unchanged, and the Floquet index is appended as the final ket and bra coordinate. Floquet extension is the last built-in basis stage, so spin, orbital, and phonon factors precede it. A hook_basis implementation receives the completed non-Floquet basis; the initializer applies the Floquet factor after the hook, so hook-provided auxiliary coordinates also precede it. The hook must leave bvc as a canonical occupation-vector basis: occupation entries must remain valid, states must be nonzero, and auxiliary factors must form one consolidated ket without unresolved Null coordinates.

The initializer sets the global lowercase symbol ncut from the exact option token Ncut=.... A model that participates in this basis extension sets MAKEFLOQUET = 1.

Mode-Space Operators

The built-in constructors are:

These helpers resolve against the state on which they act and preserve all ket coordinates preceding the final Floquet coordinate. This makes the mode number, identity, and shifts independent of the number of earlier tensor factors.

For Fourier components \(H_k\), the finite Sambe Hamiltonian has the form

floquetcos contains the sum of the two shifts and no factor of \(1/2\). Thus a term \(A O\cos(\Omega t)\) contributes

floquetlift[op,k,Ncut] is the direct construction for a Fourier component. For a static operator that itself contains auxiliary ket/bra factors, use floquetlift[op,0,Ncut] to extend it over the Floquet identity. Fermionic-only static terms already preserve a trailing auxiliary ket during application.

Three-Mode Example

For Ncut=1, in the ordered basis \((|-1\rangle,|0\rangle,|1\rangle)\),

Its eigenvalues are

This matrix is also the analytic case used by the Floquet primitive regression test.

Custom Model File

A model can define the finite-space terms directly:

MAKEFLOQUET = 1;
snegrealconstants[g, Omega];

Hfl = g floquetlift[number[d[]], 1, ncut] +
      g floquetlift[number[d[]], -1, ncut] +
      Omega floquetnumber[ncut];
opm = floquetnumber[ncut];
opm2 = pow[opm, 2];

H = H0 + Hc + H1 + Hfl;

Here g is the coefficient of each of the two nearest-mode shifts. Static model terms act diagonally in the appended mode coordinate.

Mode Operator Export

The runtime truncation path reads exactly one even-parity singlet declaration named m. A modeloperators.m file can export the model-owned definitions as:

Module[{t = {}},
  t = Join[t, mtSingletOp["m", opm]];
  t = Join[t, mtSingletOp["m^2", opm2]];
  t
]

m^2 is an optional observable and is not used to form the truncation criterion. The generated m declaration contains one Hermitian diagonal block for every seed invariant sector. The runtime obtains the allowed mode interval from the spectrum of those seed blocks.

The repository example is in test/nrginit+nrgrun/floquet_spu1/, including its param, model.m, and modeloperators.m files.

Related Pages

• Floquet formalism
• Floquet truncation
• nrginit workflow
• Parameter reference