Magnetoplasmons for the α -T 3 model with filled Landau levels
Antonios Balassis, Dipendra Dahal, Godfrey Gumbs, Andrii Iurov, Danhong Huang, Oleksiy Roslyak
MMagnetoplasmons for the α -T model with filled Landau levels Antonios Balassis , Dipendra Dahal ,Godfrey Gumbs , , Andrii Iurov , and Danhong Huang , Department of Physics & Engineering Physics, Fordham University,441 East Fordham Road, Bronx, NY 10458 USA Department of Physics and Astronomy,Hunter College of the City University of New York,695 Park Avenue, New York, NY 10065, USA Donostia International Physics Center (DIPC),P de Manuel Lardizabal, 4, 20018 San Sebastian, Basque Country, Spain Center for High Technology Materials,University of New Mexico, 1313 Goddard SE,Albuquerque, New Mexico, 87106, USA Air Force Research Laboratory, Space Vehicles Directorate,Kirtland Air Force Base, New Mexico 87117, USA a r X i v : . [ c ond - m a t . m e s - h a ll ] M a y bstract The dynamical polarizability and the dispersion relation for magnetoplasmon modes for the α -T model are calculated at zero temperature. In the absence of magnetic field, the low-energyspectrum consists of a pair of Dirac cones and a dispersionless (flat) band in the K and K (cid:48) valleys,i.e., two inequivalent Dirac points in the first Brillouin zone. However, the corresponding wavefunctions are valley-dependent. The Dirac-Weyl Hamiltonian for this structure with pseudospin S = 1 is characterized by a parameter α which is a measure of the coupling strength betweenan additional atom at the center of the honeycomb graphene lattice for the A and B atoms ofgraphene. We present results for a doped layer in the integer quantum-Hall regime for fixed α and various magnetic fields, and chosen magnetic field and different α in the random-phaseapproximation. We may assume that the electrons are in either the K or K (cid:48) valley. This isreasonable since the kinetic energy is degenerate in the two valleys and there is no scattering bythe Coulomb interaction between valley states in our model. We investigate the Berry connectionvector field, the quantum mechanical average of the position operator, for various Landau levels inthe valence energy subband. These modes may be observed with the aid of inelastic light-scatteringexperiments. . INTRODUCTION In seminal work of Raoux, et al. [1], it was demonstrated that Dirac cone structures [2–8]with the same energy band structure in the absence of magnetic field show substantial differ-ences in their orbital magnetic susceptibilities. These range from diamagnetism in graphene[9–12] to paramagnetism in the T or dice lattice [13–15]. The dice lattice, a sketch of which isshown in Fig. 1, is defined by a Dirac-Weyl Hamiltonian similar to that for graphene, exceptthat its pseudospin S = 1. The impact from Ref. [1] basically comes from its introductionof a lattice parameter α which can be varied in a continuous way from the low-energy Diraccone model to that for the dice lattice. A unique property of this model is that the Berryphase can be varied continuously from 0 to π by changing a parameter α which representsthe coupling strength between an additional atom at the center of the honeycomb graphenelattice and the A and B atoms of graphene, depicted in Fig. 1(a). Other properties of the α -T model which have been investigated include the magneto-optical conductivity and theHofstadter butterfly [16], Floquet topological phase transition [17], the role of pseudospinpolarization and transverse magnetic field on zitterbewegung [18], its frequency-dependentmagneto-optical conductivity [19], its magnetotransport properties [20] as well as the Hallquantization and optical conductivity [14]. Also, the electron states of the gapped α -T lattice in the presence of an electrostatic field of a charged impurity were reported recently[21]. We investigate the combined effect of varying α and a perpendicular magnetic field onthe magnetoplasmon excitations of the α -T model. One possible realization of this modelwas given as cold atoms in an optical lattice[1]. Furthermore, there has been a proposal forits use as an optical lattice [22], and it has been mentioned as having potential applicationto topology-induced phase transitions [23].Diamagnetic materials are repelled by a magnetic field; an applied magnetic field createsan induced magnetic field in them in the opposite direction, causing a repulsive force. Incontrast, paramagnetic and ferromagnetic materials are attracted by a magnetic field. Dia-magnetism is a quantum mechanical effect that occurs in all materials; when it is the onlycontribution to the magnetism, the material is called diamagnetic. In paramagnetic andferromagnetic substances the weak diamagnetic force is overcome by the attractive force ofmagnetic dipoles in the material. Hence, our investigation of the collective magnetoplasmonsin the α -T model should be of interest to experimentalists.3s is generally done for monolayer graphene, several authors have made a low-energy ex-pansion of the band structure around the Dirac points K = (cid:16) π √ a , (cid:17) and K (cid:48) = (cid:16) − π √ a , (cid:17) of the hexagonal Brillouin zone. In this notation, a is the atom-atom lattice parameter.In this approximation, we investigate orbital susceptibility [1, 24], the fequency-dependentpolarizability, impurity shielding, and plasmons [25–28], Klein tunneling [29], and the mag-netotransport properties [30], for the pseudospin-1 dice lattice. In the low-energy regime,the energy subbands are given by (cid:15) ( k ) = ± (cid:126) v F | k | , where v F is the Fermi velocity, for thevalence and conduction bands and a third flat band with zero energy, independent of thewave vector k , as is represented in Fig. 1(b). An interesting feature which the α -T modelexhibits is that a continuously variable Berry phase does not change the energy band spec-trum but some key physical properties are strongly affected. However, this behavior is notmaintained when there is a symmetry-breaking external field. Iurov, et al. [31] investi-gated interacting Floquet states due to off-resonant coupling of Dirac spin-1 electrons inthe α -T model from external radiation having various polarizations. In particular, theseauthors demonstrated that when the parameter α is varied the electronic properties of the α - T model (consisting of a flat band and two cones) could be modified depending on thepolarization of the external irradiation. Furthermore, under elliptically-polarized light thelow-energy band structure depends on the valley index.It would be of interest to consider superfluidity and Bose-Einstein condensation for dilutetwo-component dipolar excitons in α -T . But, since this material is intrinsically gapless, wemust find a way to open up a gap in order to separate the electrons in the conduction bandfrom the holes in the valence band. This may be achieved by applying a perpendicularmagnetic field [32, 33]. For this, one requires the electron-hole wave function, as it was donefor graphene [34], for which the electron is confined to one layer and the hole in the otherlayer with a dielectric material between them. This two-body problem could be treated interms of a two-dimensional harmonic oscillator approximation and by employing either theCoulomb potential or taking appropriate screening effects into account using the Keldyshpotential. Consequently, a natural first step is to completely understand the eigenstateproperties of electrons and holes in this spin-1 material in a magnetic field applied to amonolayer and their resulting collective magnetoplasmon properties so that these resultscould be applied to a double layer with weakly interacting Bose gas of the dipolar excitonsat low densities. There, one may assume that exciton-exciton dipole-dipole repulsion exists4 )( ) a ( )( ) b A BC
FIG. 1: (a) Schematic of the α -T lattice showing an atom C at the hub of a honeycomb structurehaving types A and B atoms on sublattices. Panel (b) shows the low-energy band structure alonghigh symmetry directions in the tight-binding approximation. At low energy, there exists a pairof linear bands intersecting at the high symmetric K point because of the honeycomb symmetryalong with a flat dispersionless subband. between excitons only for separations which exceed distances between the exciton and theclassical 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 ofthe low-energy Hamiltonian of the α -T model under a perpendicular magnetic field. There,we also present 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 examinationof the static polarization function at T=0 K is conducted. The effect on those resultsdue to finite frequency are also discussed in Sec. V. We present our numerical results formagnetoplasmons 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 eachenergy band as the quantum mechanical average of the position operator and give numericalresults for each of its two components. Section VII is devoted to concluding remarks.5 I. LOW-ENERGY α -T HAMILTONIAN UNDER A PERPENDICULAR MAG-NETIC FIELDA. Wave functions of the α -T lattice in the K valley In the absence of an applied magnetic field, and with nearest-neighbor hopping in a singlelayer, the kinetic energy part of the Hamiltonian for the α -T model is obtained by includinghopping contributions around the rim of the hexagon in a honeycomb lattice as well as fromthe hub atom to the rim. These contributing terms are described by the tight-bindingHamiltonian H = − t (cid:88) ,σ ˆ a † i,σ ˆ b j,σ − αt (cid:88) ,σ ˆ a † i,σ ˆ c j,σ + H.c. (1)with < i, j > denoting nearest-neighbor lattice sites, σ = ↑ , ↓ for electron intrinsic spin, t isa hopping parameter and 0 ≤ α ≤
1. The annihilation operators on the rim are ˆ a, ˆ b and thehub is ˆ c .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 ( A ⇑ , B ⇑ , A ⇓ , B ⇓ ). In momentum space, we have from Eq. ( 1) the kinetic energy for an electron in theabsence of an applied magnetic field is given by a 6 × H s = H ( e ) s H ( h ) s (2)with 3 × H ( (cid:96) ) s = v F (cid:16) s ˆ p ( (cid:96) ) x + i ˆ p ( (cid:96) ) y (cid:17) cos( ϕ ) 0 (cid:16) s ˆ p ( (cid:96) ) x − i ˆ p ( (cid:96) ) y (cid:17) cos( ϕ ) 0 (cid:16) s ˆ p ( (cid:96) ) x + i ˆ p ( (cid:96) ) y (cid:17) sin( ϕ )0 (cid:16) s ˆ p ( (cid:96) ) x − i ˆ p ( (cid:96) ) y (cid:17) sin( ϕ ) 0 , (3)with v F denoting the Fermi velocity and where (cid:96) = e, h and s = ± K and K (cid:48) points.We adopt the Hamiltonian describing the effects of a magnetic field B z ˆ z perpendicularto the plane of the lattice as derived in Ref. [35]. We will work in the Landau gauge, where6he vector potential A is chosen so that A = − B z y ˆ x and ∇ × A = B z ˆ z is the magneticfield. Using that Hamiltonian, we will calculate the wave functions and Landau levels forthe lattice. In the Landau gauge for the vector potential A = − B z y ˆ x and using the usualPeierls substitution (cid:126) k → p → p + e A , where k is the momentum eigenvalue in the absenceof magnetic field and p is the momentum operator, we haveˆ H K = − ˆ H ∗ K (cid:48) = γ B φ ˆ a φ a + φ ˆ a φ ˆ a + , (4)where γ B ≡ v F √ eB z (cid:126) with v F denoting the Fermi velocity and we introduced the destruc-tion operator ˆ a = √ (cid:126) eB z (ˆ p x − eB z ˆ y − i ˆ p y ) and the creation operator ˆ a + = √ (cid:126) eB z (ˆ p x − eB z ˆ y + i ˆ p y ) analogous to the harmonic oscillator. We note that when φ = 0, the Hamiltonian sub-matrix consisting of the first two rows and columns is exactly used in [32, 36] for monolayergraphene. This important observation will come to have interesting consequences in thispaper. The corresponding wave function of this spin-1 Hamiltonian is now written as | ψ > = β | l >β | m >β | n > e ik y y L / y , (5)where the graphene wave functions areΨ( x ; n, k y ) ≡ | n > = 1 √ n n ! πl H exp (cid:40) − (cid:18) x − k y l H l H (cid:19) (cid:41) H n (cid:18) x − k y l H l H (cid:19) , (6)expressed in terms of the Hermite polynomial H n ( x ), k y = 2 πm y /L y with 0 ≤ m y ≤ A/ (2 πl H ), m y is an integer, L y is a normalization length, A is a normalization area and l H = (cid:112) (cid:126) /eB z is the magnetic length. For convenience, we suppress the dependence of | ψ >, | l >, | m > and | n > on the variables x and k y . The quantities β , β and β arenow determined by substituting this form of the wave function into the eigenvalue equationˆ H | ψ > = (cid:15) | ψ > , we obtain 7 B cos φ β √ m | m − > = (cid:15)β | l >γ B cos φ β √ l + 1 | l + 1 > + γ B sin φ β √ n | n − > = (cid:15)β | m >γ B sin φ β √ m + 1 | m + 1 > = (cid:15)β | n > (7)which are satisfied when l = n −
2, then, m = n −
1. Therefore, γ B cos φ β √ n − (cid:15)β (8) γ B cos φ β √ n − γ B sin φ β √ n = (cid:15)β (9)sin φ β √ n = (cid:15)β (10)These three equations for β , β and β have the solutions when (cid:15) = ± γ B (cid:112) n − φ .Hence, β = β cos φ (cid:112) n − φ √ n − β = β sin φ (cid:112) n − φ √ n . (11)Therefore, the eigenfunction is | ψ K ± ,n > = 1 √ √ n − √ n − φ cos φ | n − > ±| n − > √ n √ n − φ sin φ | n > e ik y y L / y , (12)for n ≥
2. We note that when φ = 0, the first two rows of Eq. (12) give exactly the spin- wave function for graphene. Furthermore, when n = 1, the solution of Eq. (7) yields for theeigenfunction of the lowest state | ψ K ± , > = 1 √ ± | > | > e ik y y L / y , (13)8hich is not the same as that for graphene due to the appearance of the | > extra state inthe third row [36]. This fundamental difference between the eigenstates of α − T model for φ = 0 and graphene is not restricted to the ground state but to all Landau levels.Turning now to the derivation of the eigenstates for the flat band, (cid:15) = 0, we have γ B ˆ a cos φβ | m > = 0 γ B cos φβ √ m | m − > = 0 γ B cos φβ √ l + 1 + γ B β sin φ √ n | n − > = 0 (14)which demands that a possible solution is l = n − β β = − sin φ cos φ √ n √ l + 1 . (15)From these results, we obtain the normalized wave function for the flat band as | ψ K ,n > = (cid:113) n sin φn − cos φ | n − > | n − > − (cid:113) ( n −
1) cos φn − cos φ | n > e ik y y L / y , (16)for n ≥
2. We may also have as solutions of Eq. (14), n = 0 with β = β = m = 0, therebyyielding for the lowest flat band state | ψ K ± , > = 1 √ | > e ik y y L / y . (17) B. Wave functions of the α -T lattice in the K (cid:48) valley The low-energy magnetic Hamiltonian in the K (cid:48) valley takes the formˆ H K (cid:48) = − γ B φ ˆ a † φ ˆ a φ ˆ a † φ ˆ a (18)whose eigenfunctions we express as 9 ψ > = β (cid:48) | l >β (cid:48) | m >β (cid:48) | n > e ik y y L / y (19)so that from the eigenvalue equation we have the simultaneous equations − γ B cos φβ (cid:48) √ m + 1 | m + 1 > = (cid:15)β (cid:48) | l > − γ B cos φβ (cid:48) √ l | l − > − γ B β (cid:48) sin φ √ n + 1 | n + 1 > = (cid:15)β (cid:48) | m > − γ B sin φ √ m | m − > β (cid:48) = (cid:15)β (cid:48) | n > . (20)From these equations, it follows that possible solutions are l = n , m = n − n in thesecond line of Eq. (20) is replaced by n −
2. Consequently, the Landau levels are given by (cid:15) n = ± γ B (cid:112) n − φ (21)and the corresponding normalized eigenfunctions are | ψ K (cid:48) ± ,n > = 1 √ − (cid:113) n cos φn − sin φ | n > ±| n − > − (cid:113) ( n −
1) sin φn − sin φ | n − > e ik y y L / y , (22)for n ≥
2. Setting φ = 0 in Eq. (22), the matrix element in the third row vanishes butthe resulting eigenvector has three rows and does not coincide with the pseudospin- wavefunction for graphene which has two rows only. Additionally, when n = 1, the solution ofEq. (20) yields for the eigenfunction of the lowest state | ψ K (cid:48) ± , > = 1 √ −| > ± | > e ik y y L / y , (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 normalizedwave function for the flat band in the K (cid:48) valley as10 ψ K (cid:48) ,n > = (cid:113) ( n −
1) sin φn − sin φ | n > | n − > − (cid:113) n cos φn − sin φ | n − > e ik y y L / y , (24)again for n ≥
2. We also have for the flat band the n = 0 wave function | ψ K (cid:48) , > = 1 √ | > e ik y y L / y . (25) III. POLARIZABILITY FOR THE α -T MODEL
A central quantity in our investigation is the frequency ( ω ) and wave vector ( q ) dependentlongitudinal polarization function which is generally given byΠ( q, ω ) = (cid:88) s,s (cid:48) (cid:88) n,n (cid:48) f ( (cid:15) s (cid:48) ,n (cid:48) ) − f ( (cid:15) s,n ) (cid:126) ω + (cid:15) s (cid:48) ,n (cid:48) − (cid:15) s,n + i + F n,n (cid:48) s,s (cid:48) ( q ) , (26)where f ( (cid:15) s,n ) is the Fermi-Dirac distribution function and F n,n (cid:48) s,s (cid:48) ( q ) is a form factor which wenow discuss.There are two distinct cases which we now address for the form factor corresponding tothe allowed transitions from an occupied to an unoccupied state.. Case (i)
The first corresponds to transitions between states within the conduction band (one belowand the other above the Fermi level so that ss (cid:48) = +1) or from the valence to the conductionband so that ss (cid:48) = −
1. For these, the form factor is given compactly by F n,n (cid:48) s,s (cid:48) ( q ) ≡ | < ψ Ksn | e i q · r | ψ Ks (cid:48) n (cid:48) > | = 14 (cid:12)(cid:12)(cid:12)(cid:12) (cid:16) C ( n ) cos φ < n − | s < n − | C ( n ) sin φ < n | (cid:17) e i q · r C ( n (cid:48) ) cos φ | n (cid:48) − >s (cid:48) | n (cid:48) − >C ( n (cid:48) ) sin φ | n (cid:48) > (cid:12)(cid:12)(cid:12)(cid:12) = 14 (cid:12)(cid:12) C ( n ) C ( n (cid:48) ) R n − ,n (cid:48) − cos φ + ss (cid:48) R n − ,n (cid:48) − + C ( n ) C ( n (cid:48) ) R n,n (cid:48) sin φ (cid:12)(cid:12) . (27)11n our notation, C ( n ) = √ n − (cid:112) n − φC ( n ) = √ n (cid:112) n − φ . (28) R n,n (cid:48) = (cid:10) n | e i q · r | n (cid:48) (cid:11) = (cid:115) n < n < !2 n > n > ! ( − ( n > − n < ) / e iq x k y l H ( q x l H ) n > − n < e − ( q x l H / L n > − n < n< (cid:18) q x l H (cid:19) , (29)where L mn ( x ) is a Laguerre polynomial. It is worthy noting that when we set φ = 0, in Eq.(27), the form factor takes the form as that obtained in [36] for doped monolayer grapheneat 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 (cid:15) = 0 (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 transitionsfrom one of the discrete states with (cid:15) = 0 to the conduction band is given by F n,n (cid:48) ,s (cid:48) ( q ) = 12 | C ( n (cid:48) ) C ( n ) R n − ,n (cid:48) − − C ( n ) C ( n (cid:48) ) R n,n (cid:48) | sin φ cos φ (30)Clearly, this form factor vanishes when φ = 0 clearly confirming that there is no contributionfrom the (cid:15) = 0 level states to the polarization for this case . This result is expected sincewhen φ = 0, the α -T model yields the low-energy spin-1 / IV. DEPENDENCE OF POLARIZABILITY: ON MAGNETIC FIELD AND COU-PLING PARAMETER
Figure 2 shows the static polarizability as a function of the in-plane wave vector q (in units of the inverse magnetic length 1 /l H ) for the α -T lattice when the Landau levelfilling factor is N F = 2. When α = 0, the total polarization differs slightly from that forgraphene. The reason for this is that although this corresponds to the case when there12s no hopping between the C atom located at the hub in Fig. 1 and the atoms on therim, their eigenstates are different. For example, the lowest state eigenfunction in Eq. (13)evidently differs from its corresponding graphene counterpart. Also, the number of peaksin the polarization corresponds to the Landau level filling factor. The flat band makes nocontribution to the polarizability when α = 0. The variation in the plots presented in Figs.2(a) through 2(d) arises only from chosen values of the coupling parameter. In all panelsof Fig. 2, the intraband contribution (blue curve) exceeds that arising from the interbandtransitions (green and red curves) at long wavelengths. However, as the wave number isincreased, the interband contribution to the total polarizability dominates. In Fig. 3, wepresent the static polarization function of the α -T lattice as a function of the transferredwave vector q for N F = 3 and chosen values of the coupling parameter α . The interband andintraband contributions for this higher filling factor in Figs. 3(a) through 3(d) are similarin nature to those when N F = 2 but its value is enhanced, which in turn has an effect onstatic impurity shielding and Friedel oscillations. Additionally, the total polarizability foreach chosen filling factor in Figs. 2 and 3 tends to the same asymptotic limit at large wavevector.Figure 4 presents a comparison of the static polarizability for the α -T model for variousvalues of the hopping parameter α and filling factor N F = 3. Interestingly, the polarizabilitywhen α = 0 does not coincide with that for graphene. The reason for this is that althoughthe flat band does not contribute when α = 0, since the form factor is zero, the eigenstatesin Eq. (22) differ from the graphene wave function [36]. This result demonstrates one of theunderlying reasons for the difference in the magnetic behaviors of graphene (diamagnetic)versus the α -T model (paramagnetic) [1].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 α = 0 . . N F = 3. In both panels, the figures display plots for three chosen frequencies. In thelong wavelength limit, the static polarizability is positive in contrast to negative value forfinite frequency in the wave vector regime below the first peak. The range of wave numberover which the polarizability is negative is expanded as the frequency is increased. In allthese plots, we observe two peaks at small q along with another one appearing at larger valueof the wave vector. This rounded bump becomes less noticeable for smaller values of α which13 BFBCBtotalgraphene ql H Π ( q , ) α = N F = ( a ) - ql H Π ( q , ) α = N F = ( b ) - ql H Π ( q , ) α = N F = ( c ) - ql H Π ( q , ) α = N F = ( d ) FIG. 2: (Color online) Static polarization function, in units of π (cid:126) v F l H , at T=0 K versus in-planewave vector q , in units of the inverse magnetic length 1 /l H , for the α -T lattice for Landau levelfilling factor N F = 2 and chosen (a) α = 0, (b) α = 0 .
25, (c) α = 0 . α = 1 .
0. Thefigures reveal the separate contributions to the polarizability due to transitions from the valenceband (VB), the conduction band (CB), the flat band (FB) as well as the total from all allowedtransitions. In (a), we also compare with the polarizability for graphene. means for the case of the dice lattice α = 1 . α shows thepolarization is decreased as α is increased to 1 . V. MAGNETOPLASMON DISPERSION RELATION
Making use of the expression for polarization function in Eq. (26), we have numericallycalculated the plasmon mode dispersion relation for magnetoplasmons for the α − T modelin the presence of a uniform perpendicular magentic field for various values of the couplingparameter of α . These correspond to the resonances of the polarizability for interactingelectrons which, in the random-phase approximation (RPA), isΠ RP A ( q, ω ) = Π( q, ω )1 − v ( q )Π( q, ω ) ≡ Π( q, ω ) (cid:15) ( q, ω ) , (31)14 BFBCBtotalgraphene ql H Π ( q , ) α = N F = ( a ) - ql H Π ( q , ) α = N F = ( b ) - ql H Π ( q , ) α = N F = ( c ) - ql H Π ( q , ) α = N F = ( d ) FIG. 3: (Color online) Static polarization function, in units of π (cid:126) v F l H , at T=0 K versus in-planewave vector q , in units of the inverse magnetic length 1 /l H , for the α -T lattice for Landau levelfilling factor N F = 3 and chosen (a) α = 0, (b) α = 0 .
25, (c) α = 0 . α = 1 .
0. The figuresshow the separate contributions, using different colors, to the polarizability due to transitions fromthe valence band (VB), the conduction band (CB), the flat band (FB) as well as the total from allallowed transitions. In (a), we also compare with the polarizability for graphene. where v ( q ) is the Coulomb potential. In Figs. 6 and 7, we compare the dispersions ofthe α -T lattice for various couplings represented by the choice for α . Density plots arepresented for N F = 2 , α = 0 as well as α = 0 . , . . ω c . In the lower frequencyregime, these magnetoplasmons are of high intensity for longer wavelength but eventuallyfade away when the wave vector is increased due to Landau damping by single-particleexcitations. Each branch bifurcates into less bright branches at larger wave vector and thisdivision 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 magneto-plasmon branches increases monotonically and is then flattened for larger values of the wavevector 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. Thiscritical wave vector is shifted to larger values for the higher branches in both Figs. 6 and15 ql H Π ( q , ) N F = α = α = α = α = FIG. 4: (Color online) Comparison of the static polarizability, in units of π (cid:126) v F l H , at T=0 K forthe α -T model, having chosen value of the hopping parameter α , for filling factor N F = 3. Forcomparison, we also present the static polarizability at T=0 K for for graphene with the samefilling factor. ω = ω = ω = - ql H Π ( q , ω ) α = N F = ( a ) - ql H Π ( q , ω ) α = N F = ( b ) FIG. 5: (Color online) The real part of the dynamic polarization function, in units of π (cid:126) v F l H ,at T=0 K as a function of wave vector q , measured in units of the inverse magnetic length 1 /l H for chosen frequency ω in units v F /l H . The filling factor is N F = 3 and coupling paramer is (a) α = 0 . α = 1 .
7. This shift is relatively larger for the smaller values of α as can be seen in Figs. 6(a) and6(b) as well as 7(a) and 7(b). The brightness of the magnetoplasmons decreases drasticallyas the coupling parameter is decreased from α = 1 to α = 0 for both N F = 2 and N F = 3.Additionally, the overall intensity of the magnetoplasmons is significantly reduced for small α . VI. BERRY CONNECTION IN A PERPENDICULAR MAGNETIC FIELD
As the first step, we calculate the Berry connection vector field A γτ,φ ( k , λ ), defined as16 IG. 6: (Color online) Density plots for the inverse dielectric function (cid:15) − ( q, ω ) yielding the mag-netoplasmon dispersion relation for filling factor N F = 2 and various chosen coupling constants α in the α -T lattice. A τs,φ ( k ) ≡ (cid:104) Ψ τs ( φ | k ) | i ∇ k | Ψ τs ( φ | k ) (cid:105) . (32)In contrast to the energy dispersion, the wave function, depends on the wave vector com-ponents k = ( k x , k y ). Additionally, for convenience, the conduction and valence bands with s = ±
1, may be written succinctly as | Ψ τs ( φ | k ) (cid:105) = 1 √ τ (cid:110) (2 n − τ −
1) cos φ n − [1+ τ cos(2 φ )] (cid:111) / | n − τ − (cid:105) s | n − (cid:105) τ (cid:110) (2 n + τ −
1) sin φ n − − τ cos(2 φ ) (cid:111) / | n + τ − (cid:105) e ik y y L / y , (33)where τ = ± s = 0, the corresponding wave-function is given compactly as | Ψ τs =0 ( φ | k ) (cid:105) = (cid:110) (2 n + τ −
1) sin φ n − − τ cos(2 φ ) (cid:111) / | n − τ − (cid:105) | n − (cid:105)≡ − (cid:110) (2 n − − τ ) cos φ n − − τ cos(2 φ ) (cid:111) / | n + τ − (cid:105) e ik y y L / y , (34)17 IG. 7: (Color online) Density plots for the inverse dielectric function (cid:15) − ( q, ω ) yielding the mag-netoplasmon dispersion relation for filling factor N F = 3 and various chosen coupling constants α in the α -T lattice. where, we recall that | n (cid:105) is the n -th state wave function of a one-dimensional simple harmonicoscillator, which depends on x and k y . Since the Berry connection vector field of each bandis the quantum mechanical average of the position operator r = i ∇ k , we present it as i ∇ k = x ˆ e x + i ∂∂k y ˆ e y . (35)Now A τs,φ ( k ) from Eq. (32) could be evaluated in a straightforward fashion for each energyband s = 0 , ± τ = ± ) d ( ) b ( ) f0.00.501.501.00 ( ) a ( ) c ( ) e FIG. 8: (Color online) The x − component C x ( n, φ, s, x, k y ) of the Berry connection field A τs,φ ( k )for the n -th Landau level of.the valence band. Panel (a) shows the x − coordinate dependence of C x for Landau levels n = 3, 4 and 5. Panel (b) shows a weak dependence of C x on the parameter φ for x = 0 . k − F , 0 . k − F and 0 . k − F , as labeled. Panel (c) and (d) describe the k y dependence of C x for chosen level index n or phase φ . The two lowest panels (e) and (f) are density plots whichshow the x − and k y − dependence of C x for φ = π/ n = 2 and 3, correspondingly. All theresults were obtained for K valley for which τ = 1. system receives over a complete cycle of adiabatic, or isoenergetic evolution, i.e., φ B ( γ, k | τ, φ ) = − (cid:73) C d k · (cid:104) Ψ τs ( φ | k ) | i ∇ k | Ψ τs ( φ | k ) (cid:105) , (36)where C represents an arbitrary closed path within a lattice plane. Since the energy eigen-states (Landau levels) (cid:15) n = sγ B (cid:112) n − φ do not depend on the wave vector compo-nents, we can choose any closed path, as long as the state number n is fixed.19 ) b ( ) c ( ) a ( ) d FIG. 9: (Color online) The y − component C y ( n, φ, s, x, k y ) of the Berry connection field A τs,φ ( k )for the n -th Landau level in the e valence band. Panel (a) showss the x − coordinate dependenceof C y for the levels n = 3, 4 and 5. In (b), we show a weak dependence of C y on φ for x = 0 . k − F ,0 . k − F and 0 . k − F , as indicated. The panels (c) and (d) describe the k y dependence of C y forvarious levels n and coupling parameter φ . All the results were obtained for the ( τ = 1) K valley. Our numerical results for the Berry connection vector field, obtained from Eq. 32, areexpressed as two components C x ( n, φ, s, x, k y ) and C y ( n, φ, s, x, k y ) in ˆ e x and ˆ e y , are pre-sented in Figs. 8 and 9. The unit of length is k − F = 7 . · − m , which corresponds to theelectron density n = 5 . · cm − in graphene. The magnetic length l H is 1 .
43 of k − F . Ourresults show that both C x and C y show a nearly monotonic dependence on the coordinate x , it periodically depends on k y , and is slightly modified for different φ . VII. CONCLUDING REMARKS
In summary, we have calculated the polarization function involving both analytical andnumerical procedures and applied these results to a determination of the magnetoplasmondispersion relation for the α -T lattice in the presence of perpendicular magnetic field. Ournumerical results are presented for the Landau levels at coupling strengths, expressed interms of the parameter α between the hub atom and the carbon atoms on the honeycomblattice. In terms of Feynman diagrams, the RPA utilizes the polarizability in the formof a particle-hole bubble so that mathematically, this is given by Eq. (26). Our numericalresults for the zero temperature static polarizability at large wave vector show an interesting20ifference between the cases when the parameter 0 ≤ α ≤ α = 0. The reason beingthat for any allowed value of α , the ground eigenfunction for graphene is different from thatfor α -T so that transitions from the zeroth energy level dominate the polarizability causingit 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 sep-arated the contribution from interband transition (valence and flat bands to the conductionband), 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 thevalence to conduction band is very much similar to the case in graphene for all values of α .However, the extra contribution which appears due to the presence of the flat band when α (cid:54) = 0 could be significant and dominates other terms.All of these sum up to give the total polarizability we presented where there is a peakinitially corresponding to the peaks of graphene-like contribution, depending on N F , followedby a linearly increasing portion. The slope of this linear part is decreased when the valueof α is decreased. A significant difference can be seen in Figs. 2(a) and 3(a), where α = 0,meaning that when the central atom is in inert state, there is still no coincidence of the totalpolarizability with that for graphene. Clearly, the intraband transitions dominate at lowmomentum transfer whereas the interband dictates the behavior when ql H is large. Also,we have presented results of our calculations of the magnetoplasmon dispersion relation andshowed their energy-momentum distributions.The results we have obtained for the single-particle states in the presence of a perpendic-ular were employed in a calculation of the Berry connection vector field. We presented nu-merical results for the components of this vector field for several Landau levels in the valenceband. The results for our single-particle states in the presence of a perpendicular magneticfield may also be used to calculate the electron-hole wave function for this spin-1 system in aperpendicular magnetic field, along the lines presented by Iyengar, et al. [34] for graphene,In particular, one may derive the excitonic wave function for a double layer structure withthe electrons and holes in separate layers having a dielectric material between them. Conse-quently, one may investigate the conditions for the occurrence of Bose-Einstein condensationand superfluidity of indirect magnetoexcitons for a pair of quasi-two-dimensional spatially21eparated α -T layers. The collective excitations, the spectrum of sound velocity in a dilutegas of excitons and the effective magnetic mass of magnetoexcitons could be obtained inthe integer quantum Hall regime for strong magnetic fields. The superfluid density and thetemperature of the Kosterlitz-Thouless phase transition may also be probed as functions ofthe excitonic density, magnetic field and the interlayer separation.The α -T lattice may be used in electronic applications such as scattering control invalleytronics since the wave function depends on the parameter α . Its unique α -dependentwave function may also be employed in coherent electro-optics, and its edge-current in anano-ribbon to control pseudospin-atom interaction. Additionally, optical applications mayarise 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 ofattention for its topological insulator properties and which also possesses a flat dispersionlessband is the kagome lattice [37–41]. Two of its bands touch each other at two inequivalentDirac points located at the corners of the hexagonal Brillouin zone whereas two bands touchat the center of the Brillouin zone. The lowest band is completely full at filling and thedispersion relation for the electronic excitations from the lowest energy is similar to thosefor graphene. In Ref. [38], an insulator was produced by opening a gap at the Dirac point.This was achieved by either applying a spin-independent lattice dimerization, which breaksthe inversion symmetry of the lattice, or through a spin-orbit interaction-induced hoppingbetween next nearest-neighbors, which breaks the SU(2) spin symmetry. Acknowledgments
G. G.. would like to acknowledge the support from the Air Force Research Laboratory(AFRL) through Contract
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