Dynamics of a quasiparticle in the α -T 3 model: role of pseudospin polarization and transverse magnetic field on zitterbewegung

The quote only partly states this claim.

In the a≫1 limit for rim site C, v_y(t) is approximated by Eq. (19).

“In the a ≫ 1 limit, we obtain”

• Also stated in: § 2.5.1. z-polarization. · plain

  “v y (t) ≈ - v F 2 sin 2 ϕ cos(2ϕ)e -τ 2 sin(2aτ ) -2 1 + cos(2ϕ) e -τ 2 /4 sin(aτ ) .(19)”

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For finite α, choices (i) and (ii) differ twofold in ZB: amplitudes differ by α-dependent factors sin^2φ and cos^2φ, and the second square-bracket terms differ, possibly causing a phase difference.

“For a finite α, comparing equations ( 16) and ( 18), one may say that choices (i) and (ii) impose twofold differences in ZB.Firstly, the amplitudes differ by a α-dependent factor, namely, sin 2 ϕ and cos 2 ϕ.Secondly, the second terms in the square brackets in equations (16) and ( 18) are different which may bring a phase difference in the oscillations.”

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At α=0, ZB decays rapidly because of the e^{-τ^2} term, with characteristic decay time of order d/v_F.

“For α = 0 the ZB decays rapidly due to the presence of the term e -τ 2 in equation ( 17).Here, the characteristic time scale corresponding to the decay in ZB amplitude is of the order of d/v F .”

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Figure 4 shows that ZBs in position and velocity for different choices differ in phase and amplitude; the (0,1,0) ZB decays more rapidly than others because of Eq. (21); the position ZB for (1,0,0) is negative while others are positive.

“As illustrated in figure 4 the ZBs in position and velocity corresponding to different choices clearly differ in phase and amplitude.The ZB corresponding to the choice (c 1 , c 2 , c 3 ) = (0, 1, 0) decays more rapidly than the other choices due to the structure of the equation (21).Moreover, the ZB in position for (c 1 , c 2 , c 3 ) = (1, 0, 0) is negative while the ZBs for other choices are”

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At α=0 (φ=0), the second term in Eq. (16) does not contribute to v_y(t), so the velocity average has a single-frequency ZB from conduction-valence interference, as in graphene.

“When α = 0 i.e. ϕ = 0 the second term in the square bracket of equation ( 16) contributes nothing to v y (t) .In this case the velocity average exhibits single frequency ZB determined from the interference between CB and VB which basically reminiscences the case of graphene.”

• AssumptionRests on an assumption: ZB arises from interference between positive and negative energy solutions of the Dirac equation.

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For z-polarization with the initial wave packet only on rim site B (c1=1,c2=0,c3=0), v_x(t)=0 and v_y(t) is given by Eq. (16), which is a two-frequency ZB for finite α.

“First we consider c 1 = 1, c 2 = 0, and c 3 = 0 i.e. the initial wave packet was located in the rim site B only.From equations ( 13)-( 15), after a straightforward calculation, we find v x (t) = 0 whereas v y (t) takes the following form”

• Also stated in: § 2.5.1. z-polarization. · plain

  “Note that equation ( 16) is an example of two-frequency ZB for a finite α.”

The quote only partly states this claim.

The Hamiltonian's spectrum has three branches: two linearly dispersing conic bands ε_λk = λ v_F k with λ = ±1 and one dispersionless flat band.

“The energy spectrum corresponding to the Hamiltonian given in equation (1) consists of three branches.Out of them two are linearly dispersing ε λ k = λ v F k with λ = ±1 and k = |p|/ , known as conic band.”

• Also stated in: § 2.1. Lattice structure and low lying energy states · plain

  “The other energy branch is dispersionless and is known as flat band (FB).”

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The resulting ZB for rim site B is transient.

“As evident from figures 2 and 3, the resultant ZB is transient in character.”

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At α=1 (φ=π/4), Eq. (16) keeps only the second term; v_y(t) has a single-frequency ZB from coupling between the flat band and either conduction or valence band, and conduction-valence interference is completely prevented by the flat band.

“In the opposite limit i.e. for α = 1 we have ϕ = π/4.In this case, equation (16) retains only the second term in square bracket.”

• Also stated in: § 2.5.1. z-polarization. · plain

  “In other words, for α = 1, the interference between CB and VB is completely prevented by FB.”

• AssumptionRests on an assumption: ZB arises from interference between positive and negative energy solutions of the Dirac equation.

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For z-polarization with the initial wave packet only on rim site C (c1=0,c2=0,c3=1), v_x(t)=0 and v_y(t) is given by Eq. (18).

“Next, we consider another choice, namely, c 1 = 0, c 2 = 0, and c 3 = 1.In this case initially the electronic wave function was concentrated only in the other rim site C.Here we find v x (t) = 0 and”

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For x-polarization, v_x(t) shows a tiny oscillation while x(t) depends linearly on time.

“Although v x (t) shows a tiny oscillation as depicted in the insets of figure 5(a) but x(t) exhibits a linear dependence with time (figure 5(b)).”

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For 0<α<1 the ZB frequency is hard to specify; at α=0.4 the two frequencies produce a complicated pattern, and roughly one frequency dominates when α is below or above 0.4.

“It is difficult to comment on the frequency of ZB for 0 < α < 1. Due to the interplay of two frequencies, namely, 2Ω q and Ω q a complicated pattern is obtained for α = 0.4.It is worthy to notice that only one frequency either 2Ω q or Ω q roughly dominates when α is below or above the value α = 0.4.”

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For x-polarization with (c1,c2,c3)=(sinφ,0,-cosφ), both x(t) and v_x(t) are zero.

“When (c 1 , c 2 , c 3 ) = (sin ϕ, 0, -cos ϕ), both x(t) and v x (t) are zero.”

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For x-polarization, the y-components of position and velocity show ZB oscillation, but their amplitudes are much smaller than in the z-polarization case.

“On the other hand the y-component of position and velocity exhibit the expected ZB oscillation.However, their amplitudes are much smaller in compariso n with the case 1.”

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ZB behavior depends significantly on initial pseudospin polarization: rim-site initial wave packets give two frequencies for 0<α<1, while hub-site initial wave packet gives a single frequency for all α.

“Interestingly, we also note that the behavior of ZB is significantly dependent on the type of the initial pseudospin polarization of the wave packet.More specifically, when the initial wave packet was located entirely in any of the rim sites, as understood from equations ( 16) and ( 18), corresponding ZB consists of two frequencies for 0 < α < 1.But when the initial wave packet was in the hub site”

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At α=1, ZB decays more slowly than at α=0 because of the e^{-τ^2/4} term, with characteristic decay time 2d/v_F.

“When α = 1 ZB decays slowly in comparison to the α = 0.This behavior is justified by the presence of e -τ 2 /4 term in equation (17).Here the characteristic decay time scale is 2d/v F .”

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Equations (14) and (15) contain two frequencies, Ω_q and 2Ω_q; Ω_q comes from interference between the flat band and either valence or conduction band, while 2Ω_q comes from conduction-valence coupling.

“Note that in equations ( 14) and ( 15) there exist two frequencies namely Ω q and 2Ω q .The former is a result of interference between either FB and VB or CB and FB while the latter arises due to the coupling between CB and VB.”

• AssumptionRests on an assumption: ZB arises from interference between positive and negative energy solutions of the Dirac equation.

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In Eq. (16), the first square-bracket term arises from conduction-valence interference and the second from interference between the flat band and either conduction or valence band.

“The interference between CB and VB leads to the first term in the square bracket in equation ( 16) while the second term is appeared as a consequence of the interference between either CB and FB or VB and FB.”

• AssumptionRests on an assumption: ZB arises from interference between positive and negative energy solutions of the Dirac equation.

The quote only partly states this claim.

For x-polarization, unlike z-polarization, both x(t) and v_x(t) do not vanish for all choices.

“In contrast to case 1, here both x(t) and v x (t) do not vanish for all choices of (c 1 , c 2 , c 3 ).”

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Numerical evaluation for a=5 shows a crossover from 2Ω_q-frequency ZB to Ω_q-frequency ZB as α goes from 0 to 1; near α=1 the ZB frequency is half its α=0 value.

“Both figures clearly demonstrate a crossover from a ZB of frequency 2Ω q to a ZB of frequency Ω q with the smooth evolution of α from 0 to 1.In other words, as α approaches 1, the frequency of ZB gets reduced to half of the value corresponding to α = 0.”

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For z-polarization with the initial wave packet at hub site A (c1=0,c2=1,c3=0), v_x(t)=0 and v_y(t) is given by Eq. (20); ZB has only one frequency governed by the conduction-valence energy difference for all α, and the flat band contributes nothing.

“Finally, we consider c 1 = 0, c 2 = 1, and c 3 = 0.The physical meaning of this particular choice corresponds to the initial localization of the wave function at the hub site A. In this case, we obtain v x (t) = 0 and”

• Also stated in: § 2.5.1. z-polarization. · boosted

  “It is transparent from equation ( 20) that ZB consists of only one frequency governed by the energy difference between CB and VB for all values of α.For this specific case, the FB contributes nothing to ZB.”

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In the large wave-packet-width limit a≫1, v_y(t) is approximated by Eq. (17), whose decaying exponentials explain the transient ZB.

“To understand this transient behavior, it is possible to find an approximate analytical expression of equation ( 16) when the width of the wave packet is large enough i.e. for a 1.”

• Also stated in: § 2.5.1. z-polarization. · boosted

  “Note that the presence of decaying exponential terms in equation ( 17) clearly explains the transient behavior of ZB.”

Probably not a substantial claim: may not pass S3 (load-bearing). The quote only partly states this claim.

For the z-polarized wave packet, the ZB frequency components for α=0, 0.5, and 1 are listed in Table 1.

“The revealed frequencies corresponding to different values of α are given in table 1.”

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For y-polarization, x(t)=0 and v_x(t)=0 for the considered choices.

“Similar to choice 1, we obtain x(t) = 0 and v x (t) = 0.”

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For higher k0x, the positions of extrema are nearly independent of α; α changes only the ZB amplitude.

“Note that the locations of maxima and minima are almost insensitive to α.Only the amplitude of ZB changes with α.”

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For z-polarization with (c1,c2,c3)=(0,0,1), v_x(t)=0 and v_y(t) is given by Eq. (42).

“(iii) When (c 1 , c 2 , c 3 ) = (0, 0, 1), it is obtained that v x (t) = 0 and v y (t) = 2v F √ 3 ∞ n=0 1 2 n n! √ n + 1 Im sin ϕ p * n r n ξ 2n+1 + cos ϕ 2 (n + 1)(n + 2) h * n+1 p n ξ 2n+3 .(42)”

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For z-polarization with (c1,c2,c3)=(0,1,0), v_x(t)=0 and v_y(t) is given by Eq. (41).

“(ii) For (c 1 , c 2 , c 3 ) = (0, 1, 0) we find v x (t) = 0 and”

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For z-polarization (1,0,0), the products f*_{n+2}g_{n+1} and g*_{n+2}h_{n+1} are imaginary, so v_x(t)=0 and v_y(t) is given by Eq. (40).

“Note that the products f * n+2 g n+1 and g * n+2 h n+1 are purely imaginary.As a result, we readily obtain from equation (34) that v x (t) = 0 and”

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For higher k0x=6×10^8 m^-1, the ZB in position is transient, while the velocity ZB has a highly oscillatory pattern superimposed on transient behavior.

“In this case the ZB in position exhibits transient character.However, for ZB in velocity a highly oscillatory pattern is superimposed on the transient character.”

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For x-polarization, the expectation values of both x-velocity and x-position are nonzero.

“Interestingly we find that the x-component of the expectation values of velocity and position operators are non-zero as evident from figure 10.”

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For z-polarization (1,0,0) with k0x=2×10^8 m^-1, B=10 T, l0=8.11 nm, the ZBs in position and velocity show permanent oscillations whose patterns depend significantly on α.

“It is clear from figure 7, the ZBs appeared in both position and velocity undergo permanent oscillations and oscillatory patterns depend significantly on α.”

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For x-polarization, v_x(t) for (cosφ,1,sinφ) and (-cosφ,1,-sinφ) are not exactly mirror images; the amplitude is greater in the first case.

“However, the values of v x (t) corresponding to (c 1 , c 2 , c 3 ) = (cos ϕ, 1, sin ϕ) and (c 1 , c 2 , c 3 ) = (-cos ϕ, 1, -sin ϕ) are not exactly mirror images of each other.In fact, the amplitude of v x (t) in the first case is greater than that in the second case.”

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For the z-polarized packet (c1=1,c2=0,c3=0), the FFT of v_y(t) gives about eight frequencies at α=0, twelve at α=0.5, and nine at α=1.

“approximately, 8, 12, and 9 frequencies for α = 0, α = 0.5, and α = 1, respectively.”

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For x-polarization, y(t) and v_y(t) show regular oscillations for (sinφ,0,-cosφ), while irregularities appear for other choices.

“Both y(t) and v y (t) exhibit regular oscillations for (c 1 , c 2 , c 3 ) = (sin ϕ, 0, -cos ϕ).Irregularities in the oscillations appear for other choices of (c 1 , c 2 , c 3 ).”

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For x-polarization with (sinφ,0,-cosφ), v_x(t)=0 and x(t)=0.

“However, for (c 1 , c 2 , c 3 ) = (sin ϕ, 0, -cos ϕ), it is obtained that v x (t) = 0 and x(t) = 0.”

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For y-polarization, ZB appears only in the y-direction, and different choices of (c1,c2,c3) introduce a phase in the oscillation.

“For this particular case, ZB appears only in y-direction.Different possibilities of (c 1 , c 2 , c 3 ) introduce a phase in the oscillation.”

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For all z-polarization choices, v_x(t)=0 but v_y(t)≠0, implying ZB occurs perpendicular to the initial wave vector and pseudospin.

“For all the choices of (c 1 , c 2 , c 3 ), we obtain v x (t) = 0 but v y (t) = 0.This implies that ZB occurs in a direction perpend icular to the direction of initial wave vector and pseudo spin of the wave packet.”

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The α-T3 lattice is a honeycomb with an extra central site per hexagon, so each unit cell has three sites: two rim sites (B and C, coordination 3) and one hub site (A, coordination 6); nearest-neighbor pairs are rim-hub, with A-B hopping t and A-C hopping αt.

“As depicted in figure 1(a), α-T 3 model has honeycomb lattice structure with an additional site at the center of each hexagon.Each unit cell (shown by the dashed rhombus) contains three sites.”

• Also stated in: § 2.1. Lattice structure and low lying energy states · plain

  “The sites A and B are connected through hopping parameter t while hopping energy between A and C is αt.”

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Low-energy excitations near the Dirac point in one valley are described by a Hamiltonian with f_p = v_F(ζp_x - i p_y), valley index ζ = ±1, and α = tan φ.

“The low energy excitations near the Dirac point in a particular valley are described by the following Hamiltonian [46]”

• Also stated in: § 2.1. Lattice structure and low lying energy states · plain

  “Finally, following relation α = tan ϕ holds between α and ϕ.”

• cites [46] 46. Wang F and Ran Y 2011 Phys. Rev. B 84 241103Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

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For that choice, v_y(t)=0 at α=0, which contradicts established graphene results; the discrepancy is only apparent because the model has pseudospin-1 and graphene lacks c3, so graphene results cannot be correctly interpreted from Eq. (18).

“It is obvious from equation (18) that v y (t) = 0 when α = 0.This clearly contradicts the established results corresponding to graphene.However, the discrepancy arose here is apparent because we are dealing with pseudospin-1.For graphene the pseudospin component c 3 is completely absent.Hence, the results corresponding to graphene can not be correctly interpreted from equation (18).”

• AssumptionRests on an assumption: For graphene, the pseudospin component c3 is completely absent.

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One ZB frequency comes from conduction-valence band interference; the other comes from interference involving the flat band and either the conduction or valence band.

“One frequency is originating due to the interference between conduction and valence band whereas the other frequency is a result of interference between either conduction and flat band or flat and valence band.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The position expectation values in figures 7-9 are shifted from their true values by no more than a constant y0.

“The expectation values of position operator as illustrated in figures 7-9 differ from their actual values at most by a constant shift y 0 .”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

A narrow-gap semiconductor can host ZB and its length scale can be enhanced by up to five orders of magnitude relative to vacuum.

“However, a ray of hope was shown in 2005 when Zawadki [2] argued that a narrow gap semiconductor not only can host the intriguing phenomenon ZB but also the associated length scale can be enhanced up to five orders higher than that in vacuum.”

• cites [2] 2. Barut A O and Bracken A J 1981 Phys. Rev. D 23 2454Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim: may not pass S3 (load-bearing).

The α-T3 model smoothly changes from graphene at α=0 to dice or T3 lattice at α=1.

“Via the variation of a parameter α, the α-T 3 model reveals a smooth changeover from graphene (α = 0) to dice or T 3 lattice (α = 1).”

Probably not a substantial claim: may not pass S3 (load-bearing).

The T3 model has been predicted to exist in optical lattices and semiconductor structures.

“Moreover, the existence of the T 3 -model within the framework of optical lattice [44] and semiconductor structure [45] is also predicted recently.”

• cites [44] 44. Malcolm J D and Nicol E J 2016 Phys. Rev. B 93 165433Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [45] 45. Bercioux D, Urban D F, Grabert H and Häusler W 2009 Phys. Rev. A 80 063603Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim: may not pass S3 (load-bearing).

A Hg1-xCdxTe quantum well can be considered an α-T3 model with α=1/√3 at a particular doping.

“According to Malcolm and Nicol [47] a Hg 1-x Cd x Te quantum well can be considered as α-T 3 model with α = 1/ √ 3 at a particular doping.”

• cites [47] 47. Raoux A, Morigi M, Fuchs J-N, Piechon F and Montambaux G 2014 Phys. Rev. Lett. 112 026402Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

ZB is generally permanent, but for an electron described by a wave packet it becomes transient.

“In general ZB has permanent character i.e. oscillations do not die out in time.When an electron is illustrated by a wave packet the resulting ZB undergoes a transient nature according to Lock [31].”

• cites [31] 31. Cserti J and David G 2010 Phys. Rev. B 82 201405Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.