# Magnetoplasmons for the -T model with filled Landau levels

###### Abstract

The dynamical polarizability and the dispersion relation for magnetoplasmon modes for the -T model are calculated at zero temperature. In the absence of magnetic field, the low-energy spectrum consists of a pair of Dirac cones and a dispersionless (flat) band in the K and K valleys, i.e., two inequivalent Dirac points in the first Brillouin zone. However, the corresponding wave functions are valley-dependent. The Dirac-Weyl Hamiltonian for this structure with pseudospin is characterized by a parameter which is a measure of the coupling strength between an additional atom at the center of the honeycomb graphene lattice for the A and B atoms of graphene. We present results for a doped layer in the integer quantum-Hall regime for fixed and various magnetic fields, and chosen magnetic field and different in the random-phase approximation. We may assume that the electrons are in either the K or k valley. This is reasonable since the kinetic energy is degenerate in the two valleys and there is no scattering by the Coulomb interaction between valley states in our model. We investigate the Berry connection vector field, the quantum mechanical average of the position operator, for various Landau levels in the valence energy subband. These modes may be observed with the aid of inelastic light-scattering experiments.

###### pacs:

73.21.-b, 71.70.Ej, 73.20.Mf, 71.45.Gm, 71.10.Ca, 81.05.ue## I Introduction

In seminal work of Raoux, et al. Raoux (), it was demonstrated that Dirac cone structures 21 (); 15 (); 20 (); 16 (); 17 (); 18 (); 22 () with the same energy band structure in the absence of magnetic field show substantial differences in their orbital magnetic susceptibilities. These range from diamagnetism in graphene gr01 (); gr02 (); gr03 () to paramagnetism in the T or dice lattice Sutherland (); thesis2 (); ABpaper (). The dice lattice, a sketch of which is shown in Fig. 1, is defined by a Dirac-Weyl Hamiltonian similar to that for graphene, except that its pseudospin S = 1. The impact from Ref. Raoux () basically comes from its introduction of a lattice parameter which can be varied in a continuous way from the low-energy Dirac cone model to that for the dice lattice. A unique property of this model is that the Berry phase can be varied continuously from to by changing a parameter which represents the coupling strength between an additional atom at the center of the honeycomb graphene lattice and the A and B atoms of graphene, depicted in Fig. 1(a). Other properties of the -T model which have been investigated include the magneto-optical conductivity and the Hofstadter butterfly AIinsert (), Floquet topological phase transition IND1 (), the role of pseudospin polarization and transverse magnetic field on zitterbewegung IND2 (), its magnetotransport properties IND3 () as well as the Hall quantization and optical conductivity Illes (). Also,the electron states of the gapped âT lattice in the presence of an electrostatic field of a charged impurity were reported recently gapped (). We investigate the combined effect of varying and a perpendicular magnetic on the magnetoplasmon excitations of the -T model. One possible realization of this model was given as cold atoms in an optical lattice Raoux ()..

Diamagnetic materials are repelled by a magnetic field; an applied magnetic field creates an induced magnetic field in them in the opposite direction, causing a repulsive force. In contrast, paramagnetic and ferromagnetic materials are attracted by a magnetic field. Diamagnetism is a quantum mechanical effect that occurs in all materials; when it is the only contribution to the magnetism, the material is called diamagnetic. In paramagnetic and ferromagnetic substances the weak diamagnetic force is overcome by the attractive force of magnetic dipoles in the material. Hence, our investigation of the collective magnetoplasmons in the -T model should be of interest to experimentalists.

As is generally done for monolayer graphene, several authors have made a low-energy expansion of the band structure around the Dirac points and of the hexagonal Brillouin zone. In this notation, is the atom-atom lattice parameter. In this approximation, we investigate orbital susceptibility Raoux (); Raoux2 (), the fequency-dependent polarizability, impurity shielding, and plasmons Malcolm (); hu (); at1 (); malcolmMain (), Klein tunneling Klein (), and the magnetotransport properties Biswas (), for the pseudospin-1 dice lattice. In the low-energy regime, the energy subbands are given by , where is the Fermi velocity, for the valence and conduction bands and a third flat band with zero energy, independent of the wave vector , as is represented in Fig. 1(b). An interesting feature which the -T model exhibits is that a continuously variable Berry phase does not change the energy band spectrum but some key physical properties are strongly affected. However, this behavior is not maintained when there is a symmetry-breaking external field. Iurov, et al. Dressed () investigated interacting Floquet states due to off-resonant coupling of Dirac spin-1 electrons in the -T model from external radiation having various polarizations. In particular, these authors demonstrated that when the parameter is varied the electronic properties of the - model (consisting of a flat band and two cones) could be modified depending on the polarization of the external irradiation. Furthermore, under elliptically-polarized light the low-energy band structure depends on the valley index.

It would be of interest to consider superfluidity and Bose-Einstein condensation for dilute two-component dipolar excitons in -T. But, since this material is intrinsically gapless, we must find a way to open up a gap in order to separate the electrons in the conduction band from the holes in the valence band. This may be achieved by applying a perpendicular magnetic field Berman (). For this, one requires the electron-hole wave function, as it was done for graphene Fertig (), for which the electron is confined to one layer and the hole in the other layer with a dielectric material between them. This two-body problem could be treated in terms of a two-dimensional harmonic oscillator approximation and by employing either the Coulomb potential or taking appropriate screening effects into account using the Keldysh potential. Consequently, a natural first step is to completely understand the eigenstate properties of electrons and holes in this spin-1 material in a magnetic field applied to a monolayer and their resulting collective magnetoplasmon properties so that these results could be applied to a double layer with weakly interacting Bose gas of the dipolar excitons at low densities. There, one may assume that exciton-exciton dipole-dipole repulsion exists between excitons only for separations which exceed distances between the exciton and the classical turning point. The distance between two excitons cannot be less than this distance.

The rest of this paper is organized as follows. Section II is devoted to a description of the low-energy Hamiltonian of the -T model under a perpendicular magnetic field. There, we also determine the energy eigenstates in the two valleys which are then employed in Sec. III for calculating the form factors appearing in the polarizability. A thorough examination of the static polarization function at T=0 K is conducted. The effect on those results due to finite frequency are also discussed in Sec. V. We present our numerical results for magnetoplasmons corresponding to various coupling strengths and filling factors in Sec. V. In Sec. VI,we present our formalism for calculating the Berry connection vector field for each energy band as the quantum mechanical average of the position operator and give numerical results for each of its two components. Section VII is devoted to concluding remarks.

## Ii Low-energy -T Hamiltonian under a perpendicular magnetic field

### ii.1 Wave functions of the -T lattice in the K valley

In the absence of an applied magnetic field, and with nearest-neighbor hopping in a single layer, the kinetic energy part of the Hamiltonian for the -T model is obtained by including hopping contributions around the rim of the hexagon in a honeycomb lattice as well as from the hub atom to the rim. These contributing terms are described by the tight-binding Hamiltonian

(1) |

with denoting nearest-neighbor lattice sites, for electron intrinsic spin, is a hopping parameter and . The annihilation operators on the rim are and the hub is .

For a chosen spin state, the degrees of freedom for the (A,B) sublattices and pseudospin () lead to four-component wave functions, which are written in the basis (). In momentum space, we have from Eq. ( 1) the kinetic energy for an electron in the absence of an applied magnetic field is given by a matrix

(2) |

with submatrices for an electrons (e) and holes (h)

(3) |

with denoting the Fermi velocity and where and stands for the valley index at the and points.

We adopt the Hamiltonian describing the effects of a magnetic field perpendicular to the plane of the lattice as derived in Ref. thesis (). We will work in the Landau gauge, where the vector potential is chosen so that and is the magnetic field. Using that Hamiltonian, we will calculate the wave functions and Landau levels for the lattice. In the Landau gauge for the vector potential and using the usual Peierls substitution , where is the momentum eigenvalue in the absence of magnetic field and is the momentum operator, we have

(4) |

where with denoting the Fermi velocity and we introduced the destruction operator and the creation operator analogous to the harmonic oscillator. We note that when , the Hamiltonian submatrix consisting of the first two rows and columns is exactly used in Roldan (); Berman () for monolayer graphene. This important observation will come to have interesting consequences in this paper. The corresponding wave function of this spin-1 Hamiltonian is now written as

(5) |

where the graphene wave functions are

(6) |

expressed in terms of the Hermite polynomial , with , is an integer, is a normalization length, is a normalization area and is the magnetic length. For convenience, we suppress the dependence of and on the variables and . The quantities and are now determined by substituting this form of the wave function into the eigenvalue equation , we obtain

(7) |

which are satisfied when , then, . Therefore,

(8) |

(9) |

(10) |

These three equations for , and have the solutions when . Hence,

(11) |

Therefore, the eigenfunction is

(12) |

for . We note that when , the first two rows of Eq. (12) give exactly the spin- wave function for graphene. Furthermore, when , the solution of Eq. (7) yields for the eigenfunction of the lowest state

(13) |

which is not the same as that for graphene due to the appearance of the extra state in the third row Roldan ().

Turning now to the derivation of the eigenstates for the flat band, , we have

(14) |

which demands that a possible solution is so that we obtain

(15) |

From these results, we obtain the normalized wave function for the flat band as

(16) |

for . We may also have as solutions of Eq. (14), with , thereby yielding for the lowest flat band state

(17) |

### ii.2 Wave functions of the -T lattice in the K valley

The low-energy magnetic Hamiltonian in the valley takes the form

(18) |

whose eigenfunctions we express as

(19) |

so that from the eigenvalue equation we have the simultaneous equations

(20) |

From these equations, it follows that possible solutions are , and in the second line of Eq. (20) is replaced by . Consequently, the Landau levels are given by

(21) |

and the corresponding normalized eigenfunctions are

(22) |

for . Setting in Eq. (22), the matrix element in the third row vanishes but the resulting eigenvector has three rows and does not coincide with the pseudospin- wave function for graphene which has two rows only. Additionally, when , the solution of Eq. (20) yields for the eigenfunction of the lowest state

(23) |

where the row with ket makes the difference with graphene.

By employing a similar procedure to the one we followed above, we obtain the normalized wave function for the flat band in the valley as

(24) |

again for . We also have for the flat band the wave function

(25) |

## Iii Polarizability for the -T model

A central quantity in our investigation is the frequency () and wave vector () dependent longitudinal polarization function which is generally given by

(26) |

where is the Fermi-Dirac distribution function and is a form factor which we now discuss.

There are two distinct cases which we now address for the form factor corresponding to the allowed transitions from an occupied to an unoccupied state..

Case (i)

The first corresponds to transitions between states within the conduction band (one below and the other above the Fermi level so that ) or from the valence to the conduction band so that . For these, the form factor is given compactly by

(27) | |||||

In our notation,

(28) |

(29) |

where is a Laguerre polynomial. It is worthy noting that when we set , in Eq. (27), the form factor takes the form as that obtained in Roldan () for doped monolayer graphene at T=0 in a perpendicular magnetic field.

Case (ii)

It is now the turn for us to calculate the form factor for transitions from the (flat) band to the conduction band. The wave function for this degenerate level is given in Eq. (16) and that for the conduction band by Eq. (12). Therefore, the form factor for transitions from one of the discrete states with to the conduction band is given by

(30) |

Clearly, this form factor vanishes when clearly confirming that there is no contribution from the level states to the polarization for this case . This result is expected since when , the -T model yields the low-energy spin- graphene lattice.

## Iv Dependence of polarizability: on magnetic field and coupling parameter

Figure 2 shows the static polarizability as a function of the in-plane wave vector (in units of the inverse magnetic length ) for the -T lattice when the Landau level filling factor is . When , the total polarization differs slightly from that for graphene. The reason for this is that although this corresponds to the case when there is no hopping between the C atom located at the hub in Fig. 1 and the atoms on the rim, their eigenstates are different. For example, the lowest state eigenfunction in Eq. (13) evidently differs from its corresponding graphene counterpart. Also, the number of peaks in the polarization corresponds to the Landau level filling factor. The flat band makes no contribution to the polarizability when . The variation in the plots presented in Figs. 2(a) through 2(d) arises only from chosen values of the coupling parameter. In all panels of Fig. 2, the intraband contribution (blue curve) exceeds that arising from the interband transitions (green and red curves) at long wavelengths. However, as the wave number is increased, the interband contribution to the total polarizability dominates. In Fig. 3, we present the static polarization function of the -T lattice as a function of the transferred wave vector for and chosen values of the coupling parameter . The interband and intraband contributions for this higher filling factor in Figs. 3(a) through 3(d) are similar in nature to those when but its value is enhanced, which in turn has an effect on static impurity shielding and Friedel oscillations. Additionally, the total polarizability for each chosen filling factor in Figs. 2 and 3 tends to the same asymptotic limit at large wave vector.

Figure 4 presents a comparison of the static polarizability for the -T model for various values of the hopping parameter and filling factor . Interestingly, the polarizability when does not coincide with that for graphene. The reason for this is that although the flat band does not contribute when , since the form factor is zero, the eigenstates in Eq. (22) differ from the graphene wave function Roldan (). This result demonstrates one of the underlying reasons for the difference in the magnetic behaviors of graphene (diamagnetic) versus the -T model (paramagnetic) Raoux ().

In Figs. 5(a) and 5(b), we present our results showing a comparison between the dynamical (blue and green curves) with the static (red curve) polarization functions for and when . In both panels, the figures display plots for three chosen frequencies. In the long wavelength limit, the static polarizability is positive in contrast to negative value for finite frequency in the wave vector regime below the first peak. The range of wave number over which the polarizability is negative is expanded as the frequency is increased. In all these plots, we observe two peaks at small along with another one appearing at larger value of the wave vector. This rounded bump becomes less noticeable for smaller values of which means for the case of the dice lattice the third bump is still present although weak. Close examination of the plots reveals that all the peaks shift towards larger wave vector with increased frequency and the magnitude of the polarizability is increased significantly. A corresponding comparison of the magnitude of the polarizability for different shows the polarization is decreased as is increased to .

## V Magnetoplasmon Dispersion relation

Making use of the expression for polarization function in Eq. (26), we have numerically calculated the plasmon mode dispersion relation for magnetoplasmons for the model in the presence of a uniform perpendicular magentic field for various values of the coupling parameter of . These correspond to the resonances of the polarizability for interacting electrons which, in the random-phase approximation (RPA), is

(31) |

where is the Coulomb potential. In Figs. 6 and 7, we compare the dispersions of the -T lattice for various couplings represented by the choice for . Density plots are presented for when as well as and (dice lattice). We can see several bright branches originating at finite wave vector and the lower branches are generally more dispersive than those at higher frequency. Each magnetoplasmon branch is polarization shifted from a multiple of the cyclotron frequency . In the lower frequency regime, these magnetoplasmons are of high intensity for longer wavelength but eventually fade away when the wave vector is increased due to Landau damping by single-particle excitations. Each branch bifurcates into less bright branches at larger wave vector and this division point shifts to larger wave vector in the higher frequency region.

We also note that the frequency of the high-intensity portion of the low-energy magnetoplasmon branches increases monotonically and is then flattened for larger values of the wave vector where these lines are almost dispersionless. Another distinct feature seen in Figs. 6 and 7 is the minimum wave vector of the bright region for the magnetoplasmons. This critical wave vector is shifted to larger values for the higher branches in both Figs. 6 and 7. This shift is relatively larger for the smaller values of as can be seen in Figs. 6(a) and 6(b) as well as 7(a) and 7(b). The brightness of the magnetoplasmons decreases drastically as the coupling parameter is decreased from to for both and . Additionally, the overall intensity of the magnetoplasmons is significantly reduced for small .

## Vi Berry connection in a perpendicular magnetic field

As the first step, we calculate the Berry connection vector field , defined as

(32) | |||

In contrast to the energy dispersion, the wave function,depends on the wave vector compoenents . Additionally, for convenience, the conduction and valence bands with , may be writtem succinctly as

(33) |

where is the valley index. Also, for the flat band , the corresponding wavefunction is given compactly as

(34) |

where, we recall that is the -th state wave function of a one-dimensional simple harmonic oscillator, which depends on and . Since the Berry connection vector field of each band is the quantum mechanical average of the position operator , we present it as

(35) |

Now from Eq. (32) could be evaluated in a straightforward fashion for each energy band and valley .

The Berry phase is defined as a geometrical phase difference, which a purely quantum system receives over a complete cycle of adiabatic, or isoenergetic evolution, i.e.,

(36) |

where represents an arbitrary closed path within a lattice plane. Since the energy eigenstates (Landau levels) do not depend on the wave vector components, we can choose any closed path, as long as the state number is fixed.

Our numerical results for the Berry connection vector field, obtained from Eq. 32, are expressed as two components and in and , are presented in Figs. 8 and 9. The unit of length is , which corresponds to the electron density in graphene. The magnetic length is of . Our results show that both and show a nearly monotonic dependence on the coordinate , it periodically depends on , and is slightly modified for different .

## Vii Concluding Remarks

In summary, we have calculated the polarization function involving both analytical and numerical procedures and applied these results to a determination of the magnetoplasmon dispersion relation for the -T lattice in the presence of perpendicular magnetic field. Our numerical results are presented for the Landau levels at coupling strengths, expressed in terms of the parameter between the hub atom and the carbon atoms on the honeycomb lattice. In terms of Feynman diagrams, the RPA utilizes the polarizability in the form of a particle-hole bubble so that mathematically, this is given by Eq. (26). Our numerical results for the zero temperature static polarizability at large wave vector show an interesting difference between the cases when the parameter and the result for graphene which, as we emphasized, is not the same as that obtained when . The reason being that for any allowed value of , the ground eigenfunction for graphene is different from that for -T so that transitions from the zeroth energy level dominate the polarizability causing it to decrease as the transferred momentum is increased for all filling factors.

In all figures for the static polarizability and its real part at finite frequency, we have separated the contribution from interband transition (valence and flat bands to the conduction band), and the intraband (from below to above the Fermi level within the conduction band). The plots show that the combined intraband and interband contribution coming from the valence to conduction band is very much similar to the case in graphene for all values of . However, the extra contribution which appears due to the presence of the flat band when could be significant and dominates other terms.

All of these sum up to give the total polarizability we presented where there is a peak initially corresponding to the peaks of graphene-like contribution, depending on , followed by a linearly increasing portion. The slope of this linear part is decreased when the value of is decreased. A significant difference can be seen in Figs. 2(a) and 3(a), where , meaning that even in the absence of the flat band, there is still no coincidence of the total polarizability with that for graphene. Clearly, the intraband transitions dominate at low momentum transfer whereas the interband dictates the behavior when is large. Also, we have presented results of our calculations of the magnetoplasmon dispersion relation and showed their energy-momentum distributions.

The results we have obtained for the single-particle states in the presence of a perpendicular were employed in a calculation of the Berry connection vector field. We presented numerical results for the components of this vector field for several Landau levels in the valence band. The results for our single-particle states in the presence of a perpendicular magnetic field may also be used to calculate the electron-hole wave function for this spin-1 system in a perpendicular magnetic field, along the lines presented by Iyengar, et al. Fertig () for graphene, In particular, one may derive the excitonic wave function for a double layer structure with the electrons and holes in separate layers having a dielectric material between them. Consequently, one may investigate the conditions for the occurrence of Bose-Einstein condensation and superfluidity of indirect magnetoexcitons for a pair of quasi-two-dimensional spatially separated -T layers. The collective excitations, the spectrum of sound velocity in a dilute gas of excitons and the effective magnetic mass of magnetoexcitons could be obtained in the integer quantum Hall regime for strong magnetic fields. The superfluid density and the temperature of the Kosterlitz-Thouless phase transition may also be probed as functions of the excitonic density, magnetic field and the interlayer separation.

The -T lattice may be used in electronic applications such as scattering control in valleytronics since the wave function depends on the parameter . Its unique -dependent wave function may also be employed in coherent electro-optics, and its edge-current in a nano-ribbon to control pseudospin-atom interaction. Additionally, optical applications may arise from band gap engineering to separate the flat band from the conduction (or valence) band. This could be achieved by controlled light polarization for a topological transition.

We note that another structure which has recently been receiving a considerable amount of attention for its topological insulator properties and which also possesses a flat dispersionless band is the kagome lattice PRA (); PRB80 (); PRL114 (); NJP14 (); PRL15 (). Two of its bands touch each other at two inequivalent Dirac points located at the corners of the hexagonal Brillouin zone whereas two bands touch at the center of the Brillouin zone. The lowest band is completely full at filling and the dispersion relation for the electronic excitations from the lowest energy is similar to those for graphene. In Ref. PRB80 (), an insulator was produced by opening a gap at the Dirac point. This was achieved by either applying a spin-independent lattice dimerization, which breaks the inversion symmetry of the lattice, or through a spin-orbit interaction-induced hopping between next nearest-neighbors, which breaks the SU(2) spin symmetry.

###### Acknowledgements.

G. G.. would like to acknowledge the support from the Air Force Research Laboratory (AFRL) through Grant #12530960. D.H. would like to acknowledge the support from the Air Force Office of Scientific Research (AFOSR). D.H is also supported by the DoD Lab-University Collaborative Initiative (LUCI) program.## References

- (1) A. Raoux, M. Morigi, J.-N. Fuchs, F. Piéchon, and G. Montambaux, Phys. Rev. Lett. 112, 026402 (2014).
- (2) D. Abergel and T. Chakraborty, Applied Physics Letters 95, 062107 (2009).
- (3) O. Kibis, Phys. Rev. B81, 165433 (2010).
- (4) BalÃ¡zs Dóra, J. Cayssol, F. Simon, and R. Moessner, Phys. Rev. Lett. 108, 056602 (2012).
- (5) O. Kibis, K. Dini, I. Iorsh, and I. Shelykh, Phys. Rev. B95, 125401 (2017).
- (6) A. Iurov, L. Zhemchuzhna, G. Gumbs, and D. Huang, Journal of Applied Physics 122, 124301 (2017).
- (7) O. Kibis, K. Dini, I. Iorsh, and I. Shelykh, Semiconductors 52, 523 (2018).
- (8) H. L. Calvo, J. S. Luna, V. Dal Lago, and L. E. F. Torres, Phys. Rev. B98, 035423 (2018).
- (9) K. Novoselov, A. K. Geim, S. Morozov, D. Jiang, M. Katsnelson, I. Grigorieva, S. Dubonos, and A. Firsov, Nature 438, 197 (2005).
- (10) A. C. Ferrari, J. Meyer, V. Scardaci, C. Casiraghi, M. Lazzeri, F. Mauri, S. Piscanec, D. Jiang, K. Novoselov, S. Roth, et al., Phys. Rev. Lett. 97, 187401 (2006).
- (11) R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. Peres, and A. K. Geim, Science 320, 1308 (2008).
- (12) B. Sutherland, Phys. Rev. B 34, 5208 (1986).
- (13) E. Illes, J. P. Carbotte, and E. J. Nicol Phys. Rev. B 92, 245410 (2015).
- (14) E. V. Gorbar, V. P. Gusynin, and D. O. Oriekhov, Phys. Rev. B 99, 155124 (2019).
- (15) S. K. Firoz Islam and Paramita Dutta, Phys. Rev. B 96, 045418 (2017).
- (16) E. Illes and E. J. Nicol, Phys. Rev. B 94, 125435 (2016).
- (17) Bashab Dey and Tarun Kanti Ghosh, arXiv: 1901.10778.
- (18) Tutul Biswas and Tarun Kanti Ghosh, Journal of Physics: Condensed Matter 30, 075301 (2018).
- (19) Tutul Biswas and Tarun Kanti Ghosh, Journal of Physics: Condensed Matter 28, 495302 (2016) [arXiv: 1605.06680].
- (20) E. Illes, J. P. Carbotte, and E. J. Nicol, Phys. Rev. B 92, 245410 (2015).
- (21) F. Piéchon, J.-N. Fuchs, A. Raoux, and G Montambaux, Journal of Physics: Conference Series 603, 012001 (2015).
- (22) D. Malcolm and E. J. Nicol, Phys. Rev. B 93, 165433 (2016).
- (23) Bashab Dey and Tarun Kanti Ghosh, Phys. Rev. B 98, 075422 (2018).
- (24) M. Rizzi, V. Cataudella, and R. Fazio, Phys. Rev. B73, 144511 (2006).
- (25) Julien Vidal, Rémy Mosseri, and Benoit DouÃ§ot, Phys. Rev. Lett. 81, 5888 (1998).
- (26) E. Illes and E. J. Nicol Phys. Rev. B95,, 235432 (2017).
- (27) Tutul Biswas and Tarun Kanti Ghosh, J. Phys.: Condens. Matter 28, 495302 (2016)
- (28) Andrii Iurov, Godfrey Gumbs, and Danhong Huang, : arXiv:1806.09172 [cond-mat.mes-hall].
- (29) Oleg L. Berman, Yurii E. Lozovik, and Godfrey Gumbs, Phys. Rev. B 77, 155433 (2008).
- (30) A. Iyengar, Jianhui Wang, H. A. Fertig, and L. Brey, Phys. Rev. B 75, 125430 (2007).
- (31) Emilia Illes, “Properties of the Alpha-T Model” A thesis presented to The University of Guelph (2017).
- (32) R. Roldan, J.-N. Fuchs, and M. O. Goerbig, Phys. Rev. B80, 085408 (2009).
- (33) Godfrey Gumbs, Antonios Balassis, Dipendra Dahal, and M Lawrence Glasser, The European Physical Journal B 89234 (2016).
- (34) Mark J. Ablowitz and Justin T. Cole, Phys. Rev. A 99, 033821 (2019).
- (35) H.-M. Guo and M. Franz, Phys. Rev. B 80, 113102 (2009).
- (36) S. Mukherjee, A. Spracklen, D. Choudhury, N. Goldman, P. Ãhberg, E. Andersson, and R. R. Thomson, Phys. Rev. Lett. 114, 245504 (2015).
- (37) D. GuzmÃ¡n-Silva, C. MejÃa-CortÃ©s, M. A. Bandres, M. Rechtsman, S. Weimann, S. Nolte, M. Segev, A. Szameit, and R. A. Vicencio, New J. Phys. 16, 063061 (2014).
- (38) R. A. Vicencio, C. Cantillano, L. Morales-Inostroza, B. Real, C. MejÃa-CortÃ©s, S.Weimann, A. Szameit, andM. I.Molina, Phys. Rev. Lett. 114, 245503 (2015).