Explicit forms of zero modes in symmetric interacting Kitaev chain without and with dimerization

The fermionic and bosonic zero modes of the one-dimensional（1D） interacting Kitaev chain at the symmetric point are unveiled. The many-body structures of the Majorana zero modes in the topological region are given explicitly by carrying out a perturbation expansion up to infinite order. We also give the analytic expressions of the bosonic zero modes in the topologically trivial phase. Our results are generalized to the hybrid fermion system comprised of the interacting Kitaev model and the Su–Schrieffer–Heeger（SSH） model, in which we show that these two types of zero modes can coexist in a certain region of its phase diagram.

Effect of Interaction on the Majorana Zero Modes in the Kitaev Chain at Half Filling

The one-dimensional interacting Kitaev chain at half filling is studied. The symmetry of the Hamiltonian is examined by dual transformations, and various physical quantities as a function of the fermion-fermion interaction U are calculated systematically using the density matrix renormalization group method. A special value of interaction Up is revealed in the topological region of the phase diagram. We show that at Up the ground states are strictly two-fold degenerate even though the chain length is finite and the zero-energy peak due to the Majorana zero modes is maximally enhanced and exactly localized at the end sites. Here Up may be attractive or repulsive depending on other system parameters. We also give a qualitative understanding of the effect of interaction under the self-consistent mean field framework.

Topological phase diagrams and Majorana zero modes of the Kitaev ladder and tube

In this paper,we study two quasi-one-dimensional（1 D） Kitaev models with ladder-like and tube-like spatial structures,respectively.Our results provide the phase diagrams and explicit expressions of the Majorana zero modes.The topological phase diagrams are obtained by decomposing the topological invariants and the topological conditions for topologically nontrivial phases are given precisely.For systems which belongs to topological class BDI,we obtain the regions in the phase diagrams where the topological numbers show even-odd effect.For the Kitaev tube model a phase factor induced by the magnetic flux in the axial direction of the tube is introduced to alter the classification of the tube Hamiltonian from class BDI to D.The Kitaev tube of class D is characterized by the Z2 index when the number of chains is odd while 0,1,2 when the number of chains is even.The phase diagrams show periodic behaviors with respect to the magnetic flux.The bulk-boundary correspondence is demonstrated by the observations that the topological conditions for the bulk topological invariant to take nontrivial values are precisely those for the existence of the Majorana zero modes.

Integrated single-cell RNA sequencing analysis reveals distinct cellular and transcriptional modules associated with survival in lung cancer

Lung adenocarcinoma(LUAD)and squamous carcinoma(LUSC)are two major subtypes of non-small cell lung cancer with distinct pathologic features and treatment paradigms.The heterogeneity can be attributed to genetic,transcriptional,and epigenetic parameters.Here,we established a multi-omics atlas,integrating 52 single-cell RNA sequencing and 2342 public bulk RNA sequencing.We investigated their differences in genetic amplification,cellular compositions,and expression modules.We revealed that LUAD and LUSC contained amplifications occurring selectively in subclusters of AT2 and basal cells,and had distinct cellular composition modules associated with poor survival of lung cancer.Malignant and stage-specific gene analyses further uncovered critical transcription factors and genes in tumor progression.Moreover,we identified subclusters with proliferating and differentiating properties in AT2 and basal cells.Overexpression assays of ten genes,including sub-cluster markers AQP5 and KPNA2,further indicated their functional roles,providing potential targets for early diagnosis and treatment in lung cancer.