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Quantum-statistical theory for laser-tuned transport and optical conductivities of dressed electrons in α − T 3 materials

Andrii Iurov1,2∗, Liubov Zhemchuzhna, Dipendra Dahal, Godfrey Gumbs, and Danhong Huang Department of Physics and Computer Science, Medgar Evers College of City University of New York, Brooklyn, NY 11225, USA Center for High Technology Materials, University of New Mexico, 1313 Goddard SE, Albuquerque, New Mexico, 87106, USA Department of Physics and Astronomy, Hunter College of the City University of New York, 695 Park Avenue, New York, New York 10065, USA Donostia International Physics Center (DIPC), P de Manuel Lardizabal, 4, 20018 San Sebastian, Basque Country, Spain Air Force Research Laboratory, Space Vehicles Directorate, Kirtland Air Force Base, New Mexico 87117, USA (Dated: August 21, 2019)


I. INTRODUCTION
Currently, the α − T 3 model seems to present prospective opportunities for revolutionizing low-dimensional physics through novel two-dimensional (2D) materials. 1 Its atomic configuration consists of a graphene-type honeycomb lattice along with an additional site, i.e., a hub atom at the center of each hexagon. 2 An essential material parameter α = tan φ, which enters into its low-energy Dirac-Weyl pseudospin-1 Hamiltonian, represents the ratio between the rim-to-hub and rim-to-rim hopping coefficients. This parameter affects all fundamental electronic properties of the lattice. It is believed to vary from 0 to 1 for different types of α − T 3 materials, which may lead to some important technological applications in electronic devices. The case when α = 0 corresponds to graphene with a completely separated flat band, whereas when α = 1, we have the pseudospin-1 dice lattice, which has been fabricated and studied considerably. 3,4 Consequently, the α − T 3 model may be regarded as an interpolation between graphene and the dice lattice, or pseudospin-1 T 3 model. Its low-energy dispersion consists of a cone, which is similar to that for graphene, 5 and a flat band with zero-energy separating the valence from the conduction band for these pseudospin-1 materials. 6,7 In recent times, there have been numerous attempts leading to an actual experimental realization of the α−T 3 model. Its topological characteristics, comprising a Dirac cone with three bands located at a single point, were exhibited in the triplon band structure of SrCu 2 (BO 3 ) 2 , which is an example of a Mott-Hubbard insulator. 8 Dielectric photonic crystals with zero refractive index have Dirac cone dispersion at the center of the Brillouin zone in the presence of an accidental degeneracy. 9,10 Most importantly, there are various types of photonic Lieb lattices 11,12 consisting of a 2D array of optical waveguides. Such structure demonstrate a three-band structure, which includes a perfectly flat middle band.
Further to a relatively recent discovery of α − T 3 , there have been a few crucial publications devoted to investigating their magnetic, 1,13-15 optical, 16,17 many-body 7 and electron transport properties, 18,19 as well as generalized versions of the model. 20 A number of compelling properties of graphene and other low-dimensional materials 21 have also been investigated in α − T 3 , including Klein tunneling 22,23 and the Hofstadter butterfly. 24 On the other hand, all pseudospin-1 structures display some previously unobserved phenomena, originating from the presence of the flat band in its energy dispersion, 25 for example, very specific plasmons with branch "pinching" at the Fermi level. 7 We note that an exciting emergent technical application for condensed matter quantum optics, is Floquet engineering. This has wide-range optical tuning capability and control of the electronic behavior and transport in 2D materials by applying off-resonant periodic dressing field in the terahertz or microwave range. [26][27][28][29][30][31][32] Such external irradiation imposed on a 2D material leads to a dramatic change in most of its electronic properties due to creating so-called dressed states. This defines a single quantum object, consisting of an electron interacting with a photon. It is described by modified energy dispersion relations, which now depend on the intensity and polarization of the incoming radiation.
Our investigation of the electron dressed states is based on Floquet theory of quantum systems, driven by external periodic potentials. This results in various 1/( ω) series expansions, such as for Floquet-Magnus, Brillouin-Wigner and others, 33 playing an essential role in the investigation of the electron-light interaction in a variety of novel 2D materials [34][35][36][37] and optically induced topological surface states. 38 The modification of the single-particle dispersion relations as well as basic electronic properties strongly affects the many-body properties 21,39,40 in addition to the electronic transport, 41,42 specifically, the Boltzmann conductivity. 43 This is due mainly due to the opening of an energy bandgap 34 between the valence and conduction bands, and the corresponding change in the electronic wave functions. 44 For the case of linearly-polarized dressing field, we observe strong in-plane anisotropy, while initially anisotropic dispersion relations in phosphorene also undergo significant variations. 45 Once Floquet engineering is applied to α − T 3 materials, we will be impressed by the counter-intuitive findings, such as anisotropy, opening inequivalent energy band gaps within each band, breaking of the electron-hole symmetry and valley degeneracy, 16 significant variations of the electron wave function, their symmetries and the Berry phases. 46 Specifically, applying circularly-polarized light induces a topological phase transition from a gapless semimetal to a topological insulator with a non-zero Chern number. 17 This has some resemblance with a topological insulator, obtained from a periodic array of quantum rings under circularly-polarized light field. 47

II. ELECTRON DRESSED STATES IN α − T3 MODEL
In the presence of an external field, the single-electron states are obtained by making the following replacement for the wave vector components k x,y → k x,y − e/ A (C) x,y (t) in the low-energy Hamiltonian of the considered material. In this notation, A (C) (t) is the vector potential of the applied circularly polarized field. Therefore, the resulting Hamiltonian experiences an additional time-dependent term, corresponding to the electron-light interaction. In this paper, we deal with circular polarization of the dressing field so that A (C) (t) = E/ω {cos(ωt), sin(ωt)} T with the two equal and π/2 phase-shifted components. Here, E is the amplitude of the electric field of the imposed radiation and ω is its frequency in off-resonance regime.
We have used Floquet-Magnus perturbation expansion theory. This procedure is applicable to any periodically driven quantum structure, 33 to obtain valley-degenerate, isotropic and symmetric near the flat energy band structure. 46 This is given by ε d (k) = 0 and where c 0 = v F e E/ω is the interaction coefficient (energy), and λ 0 = c 0 /( ω) = v F eE/( ω 2 ) is a dimensionless electronlight coupling parameter. Since our consideration is limited within the off-resonance frequencies of the dressing field, in which the photon energy greatly exceeds any of the representative electron energies of the material, such as Fermi energy E F , λ 0 1 disregarding of the actual light intensity I = 0 c E 2 /2 10 W/cm 2 .
These dispersions reveal an energy band gap ε G = 2∆ 0 = λ 0 c 0 /2, which is twice as small as the corresponding gap in graphene with the same interaction coefficient c 0 34 .
The parameter ∆ (λ) = 2λ 0 c 0 /(4 + λ 2 0 ) is not equivalent to the actual energy bandgap 2∆ 0 (β = 1, λ 0 ) = λ 0 c 0 . For the flat band, we obtain where The obtained electronic states are specifically related to an irradiated dice material and not equivalent to those derived by adding a ∆ 0Σ
In this notation, α = e 2 /(4π 0 r ) is the inverse dielectric constant, determined by the substrate material with background dielectric constant r . The ynamical polarizability is also involved in determining impurity shielding. The screened potential for a dilute distribution of impurities embedded in a dice lattice has been investigated 7 as well as for a general type of α − T 3 . 54 Furthermore, the polarizability of any α − T 3 lattice at T=0 K may be substantially differs from that fot graphene as shown in in Fig. 1. This is due to the additional electron transitions from the flat band, as depicted in the inset in Fig. 1(b), especially for frequencies near the Fermi level.
The particle-hole continuum comprising the single-particle excitation regions are defined as the regions with finite imaginary part of the polarization function. Once a plasmon branch enters such a region, there is Landau damping resulting in the decay of a plasmon into single-particle excitations. Thus, we are interested in finding the regions of long-lived plasmons with Im Π (0) (q, ω) = 0, which are free from Landau damping.
In our calculations, we scan (q, ω) space where the plasmons could possibly exist. We find that for all α − T 3 materials, there is one triangle region below the Fermi level E F and another above the main diagonal ω = v F q, as shown in Fig. 2. Here, v F is the Fermi velocity. Even though there are additional areas free from Landau damping for q > 2k F , where k F is the Fermi wave vector, plasmon branches in free-standing 2D materials cannot be located there. Strictly speaking, the plasmon begins to decay once the branch enters the region above ω = E F , however the strength of this damping varies for φ and becomes infinitesimally small and eventually disappears for graphene with φ → 0.
Our results for the polarizability at T= 0 K for finite hopping parameter satisfying the condition 0 < α < 1 (or 0 < φ < π/4) are presented in Figs. 1(a) through 1(d). Its imaginary part shows a noticeable increase at low energies, compared with graphene in Fig. 1(a). At large wave vectors q ≥ k F , there is a single infinite pole −|q 2 − ω 2 | −1/2 , just it was previously demonstrated for graphene referring to Fig. 1(b). Similar behavior for the real part of the polarizability is noticed in Fig. 1(c), which also shows an additional peak, which affects the location of a plasmon mode dwhose frequency in the chosen range is determined by the plot in Fig. 1

().
Plasmon energy essentially determines at what extent each branch is affected by the electron transitions from and to the flat band. The presence of this additional band is most apparent next to the Fermi level E F , where various plasmon branches with different α in a dice lattice are pinched at a single point ω = E F = v F q. 7 We have noticed that for a intermediate case of φ = π/6 such plasmon branches still demonstrate two different peaks of their separation from the diagonal v F q below and above E F , but no actual pinching occurs -all the branches cross the ω = E F energy level at different wave vectors q (see Fig. 2 (c)).
An interesting result was extracted by analyzing the the degree to which a plasmon mode, corresponding to a chosen phase φ, is separated from the boundary of single-particle excitations v F q. For all α − T 3 materials, the plasmon modes are located below those in graphene (φ = 0, red line in Fig. 2(d)). For each chosen φ, there are a pair of plasmon branches corresponding to two clearly well-defined loss peaks depicting their "distance" from v F q, except for φ = 0. The undamped part of each branch when ω < E F spans a much larger (up to ×10) range of the wave vector q compared to the case of graphene. Except for their dispersion near E F , all the plasmon curves show quite similar behavior in the long wavelength limit q E F /( v F ), corresponding to the short-range electron transitions across the Fermi level, and for the frequencies much above the Fermi energy, related to the far-inter-cone transitions with large energy and momentum transfer.
In summary, we believe that the best opportunity for observing the φ-, or α-dependence of the plasmon dispersions and damping occurs when φ 0, i.e., in the vicinity of the graphene limit. This is so because the results for φ = π/6 and dice show only little difference as can be seen by comparing Figs. 2(a) and 2(b)). In accordance with the trigonometric expression for the main Hamiltonian terms of α − T 3 , qualitatively, such dependence could be presented as cos(2φ).
Next, we turn our attention to the dependence of the polarizability and plasmon dispersion of α − T 3 materials on circularly-polarized irradiation. Specifically, we focus on a dice lattice, which is the most diverse from graphene.In graphene,the effect of circularly-polarized is almost equivalent to opening a gap via adding aσ (2) z term to the initial Dirac Hamiltonian. 44 The most important difference for the dice lattice, and for all α > 0, is that its static polarizability Π (0) (q, ω = 0) is not constant for q < 2k F (see Fig. 3 (a)). At the same time, applying the irradiation leads to a certain reduction of the static polarization function.
The static screening factor |1/ (q, ω = 0| λ 0 )| 2 , shown in the inset (i1) of Fig. 3, is a fundamental component in evaluating the Boltzmann conductivity. It is found to be strongly affected by the irradiation, even if the coupling λ 0 is weak, its ratio between the non-irradiated material and with λ 0 = 0.5, is nearly twice. In the presence of finite electron-light coupling λ 0 , the peaks of the imaginary part of the olarizability are shifted towards higher frequencies, while its real part undergoes a completely opposite change (compare panels (b) and (c) of Fig. 3).  In general, we often observe that the increase of the phase φ from graphene to a dice lattice and the presence of a dressing field lead to opposite and conflicting effects. Specifically, in the static limit, finite dressing results in a drop of the polarization. Since the actual effects of external irradiation in off-resonance regime is significantly smaller than any change of α, the obtained results provide a unique possibility for fine-tuning of all crucial electronic properties, including the conductivity, in all α − T 3 materials.
For an irradiated material, very strong Landau damping is present above the main v F q, as we show at Fig. 4(b). This affects the frequency range of long-lived plasmon excitations compared with λ 0 = 0. Our results for the plasmon dispersion presented in Figs. 4(c) and 4(d), demonstrate a totally unexpected robust increase of their frequency due to the radiation. This reveals a fundamental difference with the plasmon dispersions in graphene, which have a lowered frequency The reason for this is attributed to the presence of a flat band, which substantially changes both real and imaginary parts of the polarization function, and the fact that the energy gap is now twice as small. We also cannot detect a finite separation between the intra-and inter-band parts of the singleparticle excitation regions since the plasmon mode is always damped for ω > E F / , corresponding to the electron transition, related to the flat band.

IV. TRANSPORT PROPERTIES
Our formalism for calculating the conductivity of interacting electrons is based on the polarization function, which we discussed above. Specifically, the optical conductivity σ O (λ 0 , ω) is directly proportional to its long wave limit as σ O (λ 0 , ω) = lim q→0 ie 2 ω/q 2 Π(q, ω | λ 0 ) . The real or absorptive part of σ O corresponds to the imaginary part of the polarizability.
Our results for the optical conductivity, presented in Fig. 5, show that its peak appearing just above the Fermi energy is estimated for graphene as 1 + 2∆ G 0 /( ω) 2 . This is higher than a flat constant portion, corresponding to a gapless Dirac cone, 55,56 which is now observed for much wider frequency ranges. Similarly, the imaginary part of σ O shows a steady difference due to the irradiation for all allowable frequencies. The λ 0 dependence, presented in Figs. 5(c) and 5(d) is decreased for the larger ω, similarly to the polarizability and plasmon dispersion.
Finally, we consider the Boltzmann conductivity of an irradiated dice lattice. We calculated the conductivity in the relaxation time approximation, while the scattering potential is equivalent to the Coulomb interaction with a charged impurity Q -U C (r) = eQ/(4π 0 r). For finite electron doping E F > 0, the inverse relaxation time is given by 57-60 where θ k,k is the angle between k and k , the complete wave function is Ψ(k | r) = Ψ d (λ 0 , k) exp [i k · r ] For isotropic dispersions and electronic states, corresponding to the circularly-polarized irradiation, the relaxation time depends only on k = |k|. In the absence of screening, the relaxation time could be easily obtained analytically as we have shown in our Supplementary materials.
For the isotropic case of circularly polarized irradiation, the only non-zero component of the electric current is directed along the applied field E e and is expressed as In this notation, the delta function δ[ε(k) d − E F ] corresponds to the zero-temperature limit of the negative d/dεderivative of the Fermi-Dirac distribution, and no integration is needed.
The relaxation time for a non-irradiated dice lattice is 3/4× of that in graphene due to the difference in wave function overlap factor, which in the former case is equal to (1+cos β k,k ) 2 /4. Once the circularly polarized irradiation is applied, the inverse relaxation time is modified to be 3π/4 + 3π/4 ξ − 37π/16 ξ 2 + · · · , where ξ = c 0 λ 0 /E F = eE/{( ω) 2 k F }. This strong increase of the first order of λ 0 c 0 /E F essentially determines the decrease in conductivity in the presence of light. Such an expansion could be obtained only for a pseudospin-1 wave function with three inequivalent components, but not for graphene or a 2D electron gas.
Another modification of the conductivity comes from the λ 0 -dependent variation of the electron group velocity v g (λ 0 , k) in Eq. (7), given as The two terms are related to the modification of the slope of the Dirac cone in the dispersion in Eq. (1) and to the opening of a band gap, and in fact have a conflicting effect on the conductivity. The ratio σ B (λ 0 )/σ B (λ 0 = 0) depends on the intensity of the irradiation. For normally used I = 10 W/cm 2 , c 0 /E F 9.75 there is always a decrease.
These results are quite different from graphene, in which the square of the group velocity is decreased as 1 − (∆ G 0 /E F ) 2 , and the relaxation time is also reduced according to 43 i.e., both contributions work together to lower the Boltzmann conductivity. The same goes for the screening, which is also reduced and must be taken into account for an accurate determination of σ B (λ).
Our numerical results for the inverse relaxation time and the conductivity when the screening factor 1/| (q, ω = 0)| 2 is taken into account are presented in Fig. 6. Even though the dependence on k and λ 0 qualitatively agrees with our analytical approximations, the actual results differ quite a lot, since the static dielectric constant is correlated with λ 0 substantially, as we showed in the inset of Fig. 3.

V. CONCLUDING REMARKS AND SUMMARY
In this paper, we have obtained and analyzed theoretical results for plasmon dispersion relations, their Landau damping, as well as the optical and Boltzmann conductivities in various α − T 3 lattices with 0 ≤ α ≤ 1. For the dice lattice, we analyzed the way in which all these properties are modified in the presence of circularly-polarized off-resonance dressing fields. As a consequence, we propose two different types of tuning of the collective and transport properties of the α − T 3 model. These are a strong dependence on the lattice parameter α and a small modification of the material parameters, related to both single electron states and collective behavior, by varying the intensity of the electron-light interaction with λ 0 << 1.
The φ-dependence is strongest in the graphene limit when φ 0, and is weak for materials whose hopping parameter is close to that for the dice lattice, i.e., satisfying φ ≈ π/4. We observe a significant increase in the plasmon damping above the Fermi level, as well as at lower energies of plasmon excitations, when α is increased from 0 to 1. The effect on the plasmon dispersion relation is particularly strong around the Fermi energy, for which the presence of the flat band in the band structure is decisive.
Once the external irradiation is applied, modification of the polarizability of a dice lattice is often opposite to that due to increasing φ. We see a substantial increase in a plasmon energy at a chosen wave vector, compared to a non-irradiated material. The Boltzmann conductivity, calculated in the relaxation time approximation, shows a strong decrease, once the dressing field is applied. All these results are expected to provide useful information and guidance when designing nano-electronic and nano-plasmonic devices which are based on innovative low-dimensional α − T 3 materials.

A. Single electronic states
The electronic states for irradiated dice lattice, pertaining to the valence and conduction bands γ = ±1, are obtained as where Parameter ∆ (λ) = 2λ 0 c 0 /(4 + λ 2 0 ) is not equivalent to the actual energy bandgap 2∆ 0 (β = 1, λ 0 ) = λ 0 c 0 . We rewrite it as where for τ = 1, corresponding to the K-valley, and for λ 0 1, we approximate as , c For the flat band, we obtain where The components of wave function are no longer equivalent, as expected for a finite energy gap ∆ D 0 = λ 0 c 0 /2. This wave function could be presented as where
For an irradiated λ 0 > 0 dice lattice with φ = π/4 we obtain which corresponds to the transitions from the flat band γ 1 = 0 to the conduction band with γ 2 = +1, (0 ↔ +1 and back, for the transitions between the valence γ 1 = −1 and conduction band with γ 2 = +1, (−1 ↔ +1), and finally, We do not consider the remaining possible transitions −1 ↔ −1 inside the valence band and between the flat and valences bands 0 ↔ −1, which are excluded at zero temperature for the electron doping E F > 0 since all such states are completely occupied n F (E < 0, In the absence of irradiation, the prefactors are provided in Table VI C.

C. Boltzmann transport
For a finite electron doping E F > 0, the inverse relaxation time is given as where θ k,k is the angle between k and k , the complete wave function is Ψ(k | r) = Ψ d (λ 0 , k) exp [i k · r ]. For isotropic dispersions and electronic states, corresponding to the circularly-polarized irradiation, the relaxation time depends only on k = |k|.
In our analytical evaluation, we are going to leave out the screening factor 1/| s (q)| 2 . Since we are interested in a ratio of the irradiated material to that with λ 0 , we only have an error due to the change of the dielectric function We begin with the scattering potential matrix element as W(k, k ) = d 2 r Ψ(k | r) U C (r) Ψ(k | r) .
The scattering potential is equivalent to the Coulomb interaction with a charged impurity Q: U C (r) = eQ/(4π 0 r) = α C /r. The invese relaxation time is given as where O(k, k ) is the overlap in the conduction band, defined in Eq. (19).
Indeed, (k − k ) · r = ∆k · r = ∆k x x + ∆k y y = ∆k x r cos θ + ∆k y r sin θ. Substituting ξ(θ) = ∆k x cos θ + ∆k y sin θ and making use of the following identity ∞ −∞ e iξ r dr = 2πδ(ξ) , we obtain Transition rate in the Born approximation Since the electron states, corresponding to k and k momenta have identical energy gaps, the condition ε(|k|) = ε(|k |) is equivalent to k = k and |k − k | = 2k sin β k,k 2 , which eliminates the radial component of k−integral. The inverse relaxation time where θ k,k is the angle between k and k .