We calculate the gauge-invariant cumulants (and moments) associated with the Zak phase in the Rice-Mele model. We reconstruct the underlying probability distribution by maximizing the information entropy and applying the moments as constraints. When the Wannier functions are localized within one unit cell, the probability distribution so obtained corresponds to that of the Wannier function. We show that in the fully dimerized limit the magnitudes of the moments are all equal. In this limit, if the on-site interaction is decreased towards zero, the distribution shifts towards the midpoint of the unit cell, but the overall shape of the distribution remains the same. Away from this limit, if alternate hoppings are finite and the on-site interaction is decreased, the distribution also shifts towards the midpoint of the unit cell, but it does this by changing shape, by becoming asymmetric around the maximum, and by shifting. We also follow the probability distribution of the polarization in cycles around the topologically nontrivial point of the model. The distribution moves across to the next unit cell, its shape distorting considerably in the process. If the radius of the cycle is large, the shift of the distribution is accompanied by large variations in the maximum.
We study an extended Aubry-André-Harper model with simultaneous modulation of hopping, on-site potential, and p-wave superconducting pairing. For the case of commensurate modulation of β = 1/2 it is shown that the model hosts four different types of topological states: adiabatic cycles can be defined which pump particles, two types of Majorana fermions, or Cooper pairs. In the incommensurate case we calculate the phase diagram of the model in several regions. We characterize the phases by calculating the mean inverse participation ratio and perform multi-fractal analysis. In addition, we characterize whether the phases found are topologically trivial or not. We find an interesting critical extended phase when incommensurate hopping modulation is present. The rise between the inverse participation ratio in regions separating localized and extended states is gradual, rather than sharp. When, in addition, the on-site potential modulation is incommensurate, we find several sharp rises and falls in the inverse participation ratio. In these two cases all different phases exhibit topological edge states. For the commensurate case we calculate the evolution of the Hofstadter butterfly and the band Chern numbers upon variation of the pairing parameter for zero and finite on-site potential. For zero on-site potential the butterflies are triangular-like near zero pairing, when gap-closure occurs, they are square-like, and hexagonal-like for larger pairing, but with the Chern numbers switched compared to the triangular case. For the finite case gaps at quarter and three-quarters filling close and lead to a switch in Chern numbers.
We construct a topological ladder model, one-dimensional, following the steps which lead to the Kane-Mele model in two dimensions. Starting with a Creutz ladder we modify it so that the gap closure points can occur at either k = π/2 or −π/2. We then couple two such models, one for each spin channel, in such a way that time-reversal invariance is restored. We also add a Rashba spin-orbit coupling term. The model falls in the CII symmetry class. We derive the relevant 2Z topological index, calculate the phase diagram and demonstrate the existence of edge states. We also give the thermodynamic derivation of the quantum spin Hall conductance (Středa-Widom). Approximate implementation of this result indicates that this quantity is sensitive to the topological behavior of the model.
EuMg2Bi2 has been investigated to understand the electronic and magnetic behaviors as an antiferromagnetic (AFM) topological semimetal candidate. High-quality single crystals of EuMg2Bi2 were grown via a Bi flux and, subsequently, characterized to be consistent with the previously reported bulk magnetic and resistivity properties. A ferromagnetic interaction is indicated by the positive Curie–Weiss temperature obtained through fitting the bulk magnetic susceptibility data. The bulk resistivity measurements reveal an interesting electronic behavior that is potentially influenced by a competing antiferromagnetic and ferromagnetic interaction in and out of the ab plane. From the resulting refinement of the neutron diffraction data, EuMg2Bi2 was found to exhibit an A-type magnetic structure with Eu2+ moments ferromagnetically aligned in the plane and antiferromagnetically stacked between neighbor ferromagnetic Eu layers. The power law fitting magnetic ordering parameter below TN ∼ 8 K agrees with the 2D Heisenberg model, indicating a weak interlayer antiferromagnetic interaction. Considering the magnetic structure determined by neutron diffraction, the surface state calculation suggests that EuMg2Bi2 is an AFM topological insulator candidate. Linearly dispersed Dirac surface states were also observed in our angle-resolved photoemission spectroscopy measurements, consistent with the calculation.
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