Floquet

This is a work in progress!

This is a modern Fortran library, companion to Ref. [1], which implements the computational scheme described in that reference to calculate the effective Floquet Hamiltonian of real materials and tight-binding models subjected to time-periodic driving fields.

Specifying a crystal

The method described in Ref. [1] supposes a crystal represented by its "exact tight-binding parameters", $\hat{H}_0$ and $\hat{\xi}^j$. In practice, crystals are specified employing WannInt's crystal object. Notice that this way to represent a physical system can be used to represent real materials, models, and non-extended systems.

Specifying $E^j(t)$

Once provided a crystal, all dynamics are solely determined by $E^j(t)$. In Ref. [1] we consider the most general expression for a $T$-periodic $E^j(t)$ compatible with the Floquet conditions,

$$ E^j(t) = \sum{n=1}^{N} \hat{E}n^j\cos\left(n\omega t - \varphin^j\right)\textbf{e}j, $$

where $N$ is the number of harmonics, $\hat{E}n^j$ and $\varphin^j$ are the amplitude and phase corresponding to harmonic $n$ and component $j$, respectively and $\omega = 2\pi/T$.

API

The module Quasienergies of Floquet defines the object fortran type, extends(task_specifier) :: quasienergies_calc_tsk which is an extension of the SsTC sampling task. This object is used to calculate the quasienergy spectrum $\varepsilon_n(\textbf{k})$ in the BZ in eV [1].

The module Currents* of Floquet defines the object fortran type, extends(task_specifier) :: currents_calc_tsk which is an extension of the SsTC sampling task. This object is used to calculate the integrand of the Fourier transform of the current denisty $j^l(\lambda)$ generated by the driving field $\textbf{E}(t)$ in Angstroms, or the integrand of the Fourier series component of the current denisty $j^l_{\lambda}$ generated by the driving field $\textbf{E}(t)$ in eV times Angstroms [3].

The module Tdep_Currents* of Floquet defines the object fortran type, extends(task_specifier) :: tdep_currents_calc_tsk which is an extension of the SsTC sampling task. This object is used to calculate the integrand of the time dependent current denisty $j^l(t)$ generated by the driving field $\textbf{E}(t)$ in eV times Angstroms [3].

Build

An automated build is available for Fortran Package Manager users. This is the recommended way to build and use Floquet in your projects. You can add Floquet to your project dependencies by including

[dependencies] Floquet = { git="https://github.com/irukoa/Floquet.git" } in the fpm.toml file.

The repository can be downloaded by running git clone https://github.com/irukoa/Floquet.git in a directory of the users' choice.

Usage

The repository includes four examples. These programs replicate the data used to create some of the plots in Ref. [1].

1. Particle_in_a_Box. This example simulates the particle in a box and calculates the quasienergy spectrum for variable monochromatic driving field.
2. BC2N_Driving_Scan. This example simulates the material BC$_2$N and calculates the quasienergy spectrum for variable $E^x$ of a monochromatic, X-linear driving field with frequency $\hbar\omega = 0.5$ $\text{eV}$ in the BZ point $\textbf{k} = (1/2, 1/4, 0)$.
3. BC2N_Kpath. This example simulates the material BC$_2$N and calculates the quasienergy spectrum for $E^x = 5.2\times10^8$ $\text{V/m}$ of a monochromatic, X-linear driving field with frequency $\hbar\omega = 0.5$ $\text{eV}$ in the BZ path $\Gamma$ - $\text{X}$ - $\text{S}$ - $\text{Y}$ - $\Gamma$.
4. BC2N_Current*. This example calculates the Fourier transform of the current generated in the material BC$_2$N by a monochromatic, X-linear driving field with frequency $\hbar\omega = 2.5$ $\text{eV}$ and amplitude $E^x = 5.2\times10^8$ $\text{V/m}$.
5. BC2N_BPE*. This example calculates the Fourier component $j^l{\lambda=0}$, corresponding to the bulk photovoltaic effect (BPE), in the material BC$2$N by a monochromatic, X-linear driving field with frequency $\hbar\omega = 2.5$ $\text{eV}$ and amplitude $E^x = 5.2\times10^8$ $\text{V/m}$.
6. Tdep_Current*. This example calculates $j^l(t)$, for the time span of two periods, in the material BC$2$N by a monochromatic, X-linear driving field with frequency $\hbar\omega = 2.5$ $\text{eV}$ and amplitude $E^x = 5.2\times10^8$ $\text{V/m}$.

These can be run using

fpm run --example Particle_in_a_Box fpm run --example BC2N_Driving_Scan fpm run --example BC2N_Kpath fpm run --example BC2N_Current fpm run --example BC2N_BPE fpm run --example Tdep_Current

respectively in the repository directory.

[1] Á. R. Puente-Uriona, M. Modugno, I. Souza, and J. Ibañez-Azpiroz, «Computing Floquet quasienergies in finite and extended systems: Role of electromagnetic and quantum-geometric gauges», Phys. Rev. B, vol. 110, iss. 12, p. 125203, sep. 2024, DOI: 10.1103/PhysRevB.110.125203.

[2] Á. R. Puente-Uriona, M. Modugno, and G. Pettini, «Topological phase diagram of optimally shaken honeycomb lattices: A dual perspective from stroboscopic and nonstroboscopic Floquet Hamiltonians», Phys. Rev. Research, vol. 6, iss. 2, p. 023244, jun. 2024, DOI: 10.1103/PhysRevResearch.6.023244.

[3] A. R. Puente-Uriona et al. In preparation.

* The Currents module, and examples employing the source code therein are considered a work in progress related to the completion of Ref. [3], and thus not ready for production calculations. Use it at your own discretion in compliance with the license agreement.