Capítulo IV: Las TIC en el aprendizaje
5. Modelo de aprendizaje Significativo de Ausubel
We start by rewriting zero-energy wavefunction in Eq. 4.11 for a p-wave meta-topological wire as Ψ(N )=0 =hψ 1 . . . ψ4 i x=N d · T (−N d) · P · τN · P−1· c(0) (4.12)
where we substitute Nd= 4 which is the number of Hamiltonian eigenvectors that we can
construct our zero-energy wavefunction. We show that 4 amplitudes can be determined by imposing two boundary conditions on the wire and the zero-energy wavefunction decays along the wire as x → ∞. We now proceed to construct p-wave zero-energy wave- function in Eq. 4.12 using eigenvectors of the off-diagonalized Hamiltonian. We do this in order to account for the Majorana wavefunction which has a diverging and converg- ing block Hamiltonian eigenvectors in both upper and lower blocks. Up to this point, we do not consider the ordering of the Hamiltonian eigenvectors ψ
j during our calcula-
tions since the wavefunction decays along the wire and 4 constants can be determined by imposing boundary conditions. However; we specify the ordering of Hamiltonian eigen- vectors when using block Hamiltonian eigenvectors since this ordering directly affects the nature of the recursion matrix since we use Hamiltonian eigenvectors to construct it. So we put upper Hamiltonian block eigenvectors as ψ
1 and ψ2 and lower Hamiltonian
eigenvectors as ψ3 and ψ4. This choice puts the recursion matrix into a block form as R = R(u) 0 0 R(l)
where R(u)and R(l)are upper and lower recursion matrix blocks. The block recursion matrix also orders the diagonal eigenvalue matrix τ so that we can write it in terms of upper and lower block recursion matrix eigenvalues as τ(u)= Diaght(u)1 t(u)2
and τ(l) = Diaght(l)1 t(l)2 i
. The upper and lower recursion matrix eigenvalues are cal- culated as R(u)s(u)i = t(u)i s(u)i and R(l)s(l)i = t(l)i s(l)i | i = 1, 2. As we show in previous chapter, we should have a diverging and decaying block Hamiltonian solution in each block in order to have a Majorana mode. The asymptotic behavior of block eigenvectors are connected to recursion matrix eigenvalues as tN
i ∝ e2N d/A as ti is a recursion matrix
eigenvalue, N d is the extent of the wire which extends to infinity N d → ∞ [85] and A is a constant that is a function of the Lyapunov exponent and superconductor coherence length. The block diagonal eigenvalue moduli take the form |t(u)1 | > 1, |t(u)2 | < 1 and |t(l)1 | < 1, |t(l)2 | > 1. We thus derive the topological index
Q(p,MTI)D = 2 Y i=1 sgn |t(u)i | − 1 (4.13)
Our formula checks for the existence of Majorana mode by looking at the asymptotic na- ture of block solutions of the Hamiltonian and gives a topological index that is calculated using the block recursion matrix eigenvalues by yielding −1 when a topologicallay pro- tected Majorana mode is present.
In Fig. 4.2a; we plot the topological phase space of a p-wave topological supercon- ductor wire as a function of chemical potential µ and electrostatic potential V (x) = V0
that is constant along the wire. The blue area of the density plot show topologically non- trivial phases (µ > V0) and the red area signify topologically trivial phases (µ < V0) of
the topological superconductor wire. The green dashed line is the phase transition line plotted at µ = V0. In Fig. 4.2b; we calculate the topological phase space of a p-wave
meta-topological wire as a function of chemical potential µ and piecewise continuous electrostatic potential of amplitude V0 = max(VSL). Density plots are the results of nu-
merical calculations for Eq. 3.3 for V (x) = VSL(x) and white lines are calculated using
Eq. 4.13 for a piecewise continuous electrostatic potential with amplitude V0. The su-
perlattice potential VSL(x) changes the asymptotic behavior of the Lyapunov exponent in
Eq. 3.3 as well as the existence of the Majorana mode. We keep the dashed green topolog- ical phase transition line of a p-wave topological superconductor wire to show different phase spaces of both types of topological wires. The density plot and white lines that are calculated for the same potential with our two different formulations of the topological index fits very well so that we say that Eqs. 4.13 and 3.3 show significant agreement. We plot Figs. 4.2a and 4.2b for a single superlattice unit cell with an effective spin orbit
(a) (b)
Figure 4.2: Topological phase space of a p-wave topological superconductor wire as a func- tion of chemical potential µ and an applied constant electrostatic gate potential along wire is plotted in (a). Topological phase space of a p-wave meta-topological insulator as a function of chemical potential µ and electrostatic potential amplitude max(VSL) is plotted in (b). Density
plot is calculated using Eq. 4.13 where red areas signify topologically trivial configurations and blue areas signify topologically nontrivial configurations. White lines are calculated us- ing Eq. 3.3 where they converge to p-wave topological superconductor phase transition line µ = V0when the electrostatic potential is absent. Calculations are done for a p-wave topolog-
ical superconductor and meta-topological insulator wire with an effective spin-orbit coupling ∆eff = 0.03~2/2ma2unit cell length d = 2a. Phase transition boundary in the density is ex-
cellently covered by white lines so both formulations of the topological indices show strong agreement. The red specks below the p-wave topological superconductor phase transition line µ > V0show that a topologically nontrivial wire can be tuned into its trivial phase by the ap-
plication of an electrostatic potential. The topological order also flips from trivial to nontrivial for s-wave topological superconductor wires that are the tails of blue spikes that are above the green dashed topological superconductor wire phase transition line µ < V0.
coupling strength ∆eff = 0.03~2/2ma2 where the length of the superlattice unit cell is
d = 2a. The effect of the electrostatic superlattice on topological phase space can be seen in Fig. 4.2b as it leads to topologically nontrivial to trivial phase transitions that we see in red patches which are below the yellow topological superconductor wire phase transition line. We also see topologically trivial to nontrivial phase transitions as parts of spikes that are above the phase transition line. Our analytical formula fits the numerical solution well.
We continue to analyze s-wave meta-topological wires.