# Floquet Modification of the Bandgaps and Energy Spectrum in Flat-Band Pseudospin-1 Dirac Materials
## 1 Introduction  
Floquet theory, which was originally developed to solve partial differential equations [1], represents a unique tool for describing the electronic properties and behavior of a large number of diverse quantum systems [2; 3; 4] in the presence of high-frequency periodic dressing fields [5]. It has also advanced into the so-called Floquet engineering, a technique for obtaining the electronic dispersions and, specifically, bandgaps that are required for technical applications. This has quickly become one of the most promising research directions [6; 7] in condensed matter physics [8; 9; 10] and photonics. Floquet engineering is primarily applied to two-dimensional (2D) materials or the surface states of three-dimensional lattices. Because of the recent experimental advances in laser and microwave optics, it has become possible to verify the theoretical predictions of condensed matter quantum optics in actual experiments and realize them in optoelectronic devices [11]. Floquet theory has been successfully applied to original [12; 13; 14; 15; 16; 17], gapped [13; 18] and Kekule-patterned graphene [19]. It has also entered into extensive studies on anisotropic [20] and tilted Dirac materials [21; 22; 23; 24; 25; 26; 27] as well as other structures [12; 28; 29; 30; 31; 32]. Special interest has been given to the Floquet control of electronic transport properties of irradiated materials [33; 34; 35] in addition to indirect exchange interaction [36; 37] and collective behaviors [38; 39]. Its interests further extend to the fields of excitonic [40] and dissipative [9]Floquet systems, pseudospin textures, and nanoribbons [41; 42; 43; 44; 45]. In particular, an optical-dressing field was utilized to create localized trapped states [46], similarly to those in fullerenes [47; 48; 49; 50]. In fact, the electron-photon interaction and its dressed states are only one part of an ongoing effort to investigate all electronic and collective properties in recently discovered Dirac materials. These efforts include establishing a Wentzel-Kramers-Brillouin (WKB) theory for taking into consideration semi-classical electronic states in \(\alpha\)-\(\mathcal{T}_{3}\), dice, and even Kekule-patterned graphene [51; 52] in a way that is very similar to previously reported successful works on genuine graphene materials [53; 54; 55]. Additionally, Floquet engineering is utilized to alter the Bloch-electron states in Dirac materials with a flat band, such as a dice lattice [7; 56; 57]. The \(\alpha\)-\(\mathcal{T}_{3}\) model, which represents interpolation between a graphene and a dice lattice, is one of the most favorable and well-studied models for materials acquiring a flat band. A strong and ongoing interest on this \(\alpha\)-\(\mathcal{T}_{3}\) model is attributed to its unusual low-energy electronic band structure involving a regular graphene Dirac cone as well as a flat band. This actually makes \(\alpha\)-\(\mathcal{T}_{3}\) materials candidates for replacing graphene in modern electronic devices. The atomic structure of the \(\alpha\)-\(\mathcal{T}_{3}\) materials consists of an additional atom at the center of a graphene-like honeycomb lattice (HCL). Therefore, the new hub-to-rim (and rim-to-hub) hopping coefficient can be related to an existing rim-to-rim hopping simply by a scaling parameter \(\alpha\), which turns out to be a key factor for characterizing various \(\alpha\)-\(\mathcal{T}_{3}\) materials. For example, the case with \(\alpha=1\) corresponds to a dice lattice, while \(\alpha\to 1\) resembles graphene with a fully decoupled set of hub atoms at the center of each hexagon. Importantly, the electronic properties of the \(\alpha\)-\(\mathcal{T}_{3}\) model can be described by a pseudospin-1 Dirac-Weyl Hamiltonian which directly depends on an \(\alpha\) parameter or, more specifically, on its phase \(\phi=\tan^{-1}\alpha\).