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A comprehensive investigation of the entanglement characteristics is carried out on tripartite spin-1/2 systems, examining prototypical tripartite states, the thermal Heisenberg model, and the transverse field Ising model. The entanglement is computed using the R\'enyi relative entropy. In the traditional R\'enyi relative entropy, the generalization parameter $\alpha$ can take values only in the range $0 \leq \alpha \leq 2$ due to the requirements of joint convexity of the measure. To use the R\'enyi relative entropy over a wider range of $\alpha$, we use the sandwiched form which is jointly convex in the regime $0.5 \leq \alpha \leq \infty$. In prototypical tripartite states, we find that GHZ states are monogamous, but surprisingly so are W states. On the other hand, star states exhibit polygamy, due to the higher level of purity of the bipartite subsystems. For spin models, we study the dependence of entanglement on various parameters such as temperature, spin-spin interaction, and anisotropy, and identify regions where entanglement is the largest. The R\'enyi parameter $\alpha$ scales the amount of entanglement in the system. The entanglement measure based on the traditional and the sandwiched R\'enyi relative entropies obey the Araki-Lieb-Thirring inequality. In the Heisenberg models, namely the XYZ, XXZ, and XY models, the system is always monogamous. However, in the transverse field Ising model, the state is initially polygamous and becomes monogamous with temperature and coupling., Comment: 24 Pages

Chiral edge states are the fingerprint of the bulk-edge correspondence in a Chern insulator. Co-propagating edge modes, known as antichiral edge states, have been predicted to occur in the so-called modified Haldane model describing a two-dimensional semi-metal with broken time reversal symmetry. These counterintuitive edge modes are argued to be immune to backscattering and extremely robust against disorder. Here, we investigate the robustness of the antichiral edge states in the presence of Anderson disorder. By analysing different localization parameters, we show that, contrary to the general belief, these edge modes are fragile against disorder, and can be easily localized. Our work provides insights to improve the transport efficiency in the burgeoning fields of antichiral topological photonics and acoustics., Comment: 9 pages, 6 figures

Graphene can be turned into a semimetal with broken time-reversal symmetry by adding a valley-dependent pseudo-scalar potential that shifts the Dirac point energies in opposite directions, as in the modified Haldane model. We consider a bilayer obtained by stacking two time-reversed copies of the modified Haldane model, where conduction and valence bands cross to give rise to a nodal line in each valleys. In the AB stacking, the interlayer hopping lifts the degeneracy of the nodal lines and induces a band repulsion, leading surprisingly to a chiral insulator with a Chern number $C=\pm2$. As a consequence a pair of chiral edge states appears at the boundaries of the ribbon bilayer geometry. In contrast, the AA stacking does not show nontrivial topological phases. We discuss possible experimental implementations of our results., Comment: version 2 with 13 pages, 6 figures -- five appendices

Several numerical studies have shown that the electronic properties of twisted bilayers of graphene (TBLG) and transition metal dichalcogenides (TMDs) are tunable by strain engineering of the stacking layers. In particular, the flatness of the low-energy moir\'e bands of the rigid and the relaxed TBLG was found to be, substantially, sensitive to the strain. However, to the best of our knowledge, there are no full analytical calculations of the effect of strain on such bands. We derive, based on the continuum model of moir\'e flat bands, the low-energy Hamiltonian of twisted homobilayers of graphene and TMDs under strain at small twist angles. We obtain the analytical expressions of the strain-renormalized Dirac velocities and explain the role of strain in the emergence of the flat bands. We discuss how strain could correct the twist angles and bring them closer to the magic angle $\theta_m\sim1.05^{\circ}$ of TBLG and how it may reduce the widths of the lowest-energy bands at charge neutrality of the twisted homobilayer of TMDs. The analytical results are compared with numerical and experimental findings and also with our numerical calculations based on the continuum model., Comment: 7 pages + supplemental material

We study the interplay between a uniaxial strain and the topology of the Haldane and the modified Haldane models which, respectively, exhibit chiral and antichiral edge modes. The latter were, recently, predicted by Colom\'es and Franz (Phys. Rev. Lett. {\bf 120}, 086603 (2018)) and expected to take place in the transition metal dichalcogenides. Using the continuum approximation and a tight-binding approach, we investigate the effect of the strain on the topological phases and the corresponding edge modes. We show that the strain could induce transitions between topological phases with opposite Chern numbers or tune a topological phase into a trivial one. As a consequence, the dispersions of the chiral and antichiral edge modes are found to be strain dependent. The strain may reverse the direction of propagation of these modes and eventually destroy them. This effect may be used for strain-tunable edge currents in topological insulators and two-dimensional transition metal dichalcogenides.