Floquet Formalism

The current Floquet interface represents one periodic drive in a finite Fourier, or Sambe, space. This page fixes the notation used by the model-construction and truncation pages.

Periodic Hamiltonian

For a Hamiltonian with period \(T\),

\[ H(t+T)=H(t), \qquad \Omega=\frac{2\pi}{T}. \]

Write its Fourier components as

\[ H(t)=\sum_k H_k e^{ik\Omega t}. \]

The current initializer takes the positive angular frequency \(\Omega\) from [extra] Omega.

Floquet States

A Floquet state is written as

\[ |\Psi_\alpha(t)\rangle =e^{-i\epsilon_\alpha t}|u_\alpha(t)\rangle, \qquad |u_\alpha(t+T)\rangle=|u_\alpha(t)\rangle. \]

The periodic part obeys

\[ \left[H(t)-i\partial_t\right]|u_\alpha(t)\rangle =\epsilon_\alpha|u_\alpha(t)\rangle. \]

Expanding

\[ |u_\alpha(t)\rangle=\sum_m e^{im\Omega t}|u_{\alpha m}\rangle \]

gives the time-independent Sambe-space equation

\[ \sum_n\left[H_{m-n}+m\Omega\delta_{mn}\right] |u_{\alpha n}\rangle =\epsilon_\alpha|u_{\alpha m}\rangle. \]

The corresponding inner product averages over one period:

\[ \langle\!\langle u|v\rangle\!\rangle =\frac{1}{T}\int_0^T dt\,\langle u(t)|v(t)\rangle. \]

Quasienergy Replicas

Multiplying the periodic part by \(e^{i\ell\Omega t}\) produces another representation of the same time-dependent state:

\[ |u_{\alpha,\ell}(t)\rangle=e^{i\ell\Omega t}|u_\alpha(t)\rangle, \qquad \epsilon_{\alpha,\ell}=\epsilon_\alpha+\ell\Omega. \]

NRG Ljubljana retains the resulting extended-zone energies. The runtime uses a separate criterion to order states for truncation; it does not fold the stored energies into a selected quasienergy zone.

Finite Mode Window

The initializer restricts the Fourier index to

\[ m=-N_{\mathrm{cut}},\ldots,N_{\mathrm{cut}}, \]

so the Floquet factor has dimension \(2N_{\mathrm{cut}}+1\). The finite-space shift is the projected operator

\[ S_k=\sum_{\substack{m=-N_{\mathrm{cut}}\\ m+k\in[-N_{\mathrm{cut}},N_{\mathrm{cut}}]}}^{N_{\mathrm{cut}}} |m+k\rangle\langle m|. \]

Terms crossing either edge are absent. The two edges are not connected.

Notation In NRG Ljubljana

Symbol or name	Meaning
\(m\)	Integer Fourier-mode label.
(\hat M=\sum_m m	m\rangle\langle m
`m`	Serialized singlet operator containing \(\hat M\).
`m^2`	Optional observable containing \(\hat M^2\).
\(\mu_{Ii}\)	Expectation value of the recalculated \(\hat M\) in runtime state \((I,i)\).
`Ncut`	Non-negative mode cutoff supplied through `options=Ncut=...`.

After mode mixing, \(\mu_{Ii}\) need not be an integer even though the spectrum of the seed operator \(\hat M\) consists of integer mode labels.