We investigate the topological properties of a generalized Dirac--Kronig--Penney model, a continuum one-dimensional model for a relativistic quantum chain. By tuning the coupling parameters this model can accommodate five Altland--Zirnbauer--Cartan symmetry classes, three of which (AIII, BDI and D) support non-trivial topological phases in dimension one. We characterize analytically the spectral properties of the Hamiltonian in terms of a spectral function, and numerically compute the Zak phase to probe the bulk topological content of the insulating phases. Our findings reveal that, while the Zak phase is quantized in classes AIII and BDI, it exhibits non-quantized values in class D, challenging its traditional role as a topological marker in continuum settings. We also discuss the bulk-boundary correspondence for a truncated version of the chain, analyzing how the emergence of edge states depends on both the truncation position and the boundary conditions. In classes AIII and BDI, we find that the Zak phase effectively detects edge states as a relative boundary topological index, although the correspondence is highly sensitive to the parameters characterizing the truncation.
We investigate the topological properties of a generalized Dirac--Kronig--Penney model, a continuum one-dimensional model for a relativistic quantum chain. By tuning the coupling parameters this model can accommodate five Altland--Zirnbauer--Cartan symmetry classes, three of which (AIII, BDI and D) support non-trivial topological phases in dimension one. We characterize analytically the spectral properties of the Hamiltonian in terms of a spectral function, and numerically compute the Zak phase to probe the bulk topological content of the insulating phases. Our findings reveal that, while the Zak phase is quantized in classes AIII and BDI, it exhibits non-quantized values in class D, challenging its traditional role as a topological marker in continuum settings. We also discuss the bulk-boundary correspondence for a truncated version of the chain, analyzing how the emergence of edge states depends on both the truncation position and the boundary conditions. In classes AIII and BDI, we find that the Zak phase effectively detects edge states as a relative boundary topological index, although the correspondence is highly sensitive to the parameters characterizing the truncation.
The bulk-boundary correspondence relates topologically protected edge modes to bulk topological invariants and is well understood for short-range free-fermion chains. Although case studies have considered long-range Hamiltonians whose couplings decay with a power-law exponent α, there has been no systematic study for a free-fermion symmetry class. We introduce a technique for solving gapped, translationally invariant models in the 1D BDI and AIII symmetry classes with α>1, linking together the quantized winding invariant, bulk topological string-order parameters, and a complete solution of the edge modes. The physics of these chains is elucidated by studying a complex function determined by the couplings of the Hamiltonian: in contrast to the short-range case where edge modes are associated to roots of this function, we find that they are now associated to singularities. A remarkable consequence is that the finite-size splitting of the edge modes depends on the topological winding number, which can be used as a probe of the latter. We furthermore generalize these results by (i) identifying a family of BDI chains with α<1 where our results still hold and (ii) showing that gapless symmetry-protected topological chains can have topological invariants and edge modes when α-1 exceeds the dynamical critical exponent.
To clarify the relationship between edge electronic states in open-boundary crystalline systems and their corresponding bulk electronic structure, A. Alase et al., [Phys. Rev. Lett. 117, 076804 (2016)] have recently generalized Bloch's theorem to lattice models with broken translational symmetry. Their formalism provides a bulk-boundary correspondence indicator, D, sensitive directly to the localization of edge states. We extend this formalism in two significant ways. First, we explicitly classify the edge-state basis provided by the theory according to their underlying protecting symmetries. Second, acknowledging that the true topological invariant is the Zak phase, inherently defined only for periodic boundary conditions, we introduce an analogous quantity ${\ensuremath{\gamma}}_{Z}$, suitable for open-boundary systems. We illustrate these developments on the example of the Kitaev wire model. We demonstrate that both indicators, D and our open-boundary analog, ${\ensuremath{\gamma}}_{Z}$, effectively capture topological localization: while D diverges within the topological phase, ${\ensuremath{\gamma}}_{Z}=\ensuremath{\pi}$. We further examine eigenvalues (z) of the lattice shift operator, showing that variations in these eigenvalues as a function of chemical potential provide additional signatures of topological phase transitions. Additionally, we study a periodic Kitaev wire containing a bond impurity, revealing that while the topological phase remains characterized by ${\ensuremath{\gamma}}_{Z}=\ensuremath{\pi}$, significant state localization near the impurity emerges only at critical values of the chemical potential.
In this paper, we study five-dimensional Dirac fermions of which the extra-dimension is compactified on quantum graphs. We find that there is a nontrivial correspondence between matrices specifying boundary conditions at the vertex of the quantum graphs and zero-dimensional Hamiltonians in gapped free-fermion systems. Based on the correspondence, we provide a complete topological classification of the boundary conditions in terms of noninteracting fermionic topological phases. The ten symmetry classes of topological phases are fully identified in the language of five-dimensional Dirac fermions, and topological numbers of the boundary conditions are given. In analogy with the bulk-boundary correspondence in noninteracting fermionic topological phases, the boundary condition topological numbers predict four-dimensional massless fermions localized at the vertex of the quantum graphs and thus govern the low energy physics in four dimensions.
We investigate the appearance of new types of insulators and superconductors in long-range (LR) fermionic quantum systems. These phases are not included in the famous 'ten-fold way classification' (TWC), valid in the short-range (SR) limit. This conclusion is obtained analysing at first specific one-dimensional models, in particular their phase diagrams and entanglement properties. The LR phases are signalled, for instance, by the violation of the area-law for the von Neumann entropy and by a corresponding peculiar entanglement spectrum (ES). Later on, the origin of the deviations from the TWC is investigated from a more general point of view and in any dimension, showing that it is related with the presence of divergences occurring in the spectrum, due to the LR couplings. A satisfying characterization for the LR phases can be achieved, at least for one-dimensional quantum systems, as well as the definition of a nontrivial topology for them, resulting in the presence of massive edge states, provided a careful evaluation of the LR contributions. Our results allows to infer, at least for one-dimensional models, the weakening of the bulk-boundary correspondence, due to the important correlations between bulk and edges, and consequently to clarify the nature of the massive edge states. The emergence of this peculiar edge structure is signalled again by the bulk ES. The stability of the LR phases against local disorder is also discussed, showing notably that this ingredient can even strengthen the effect of the LR couplings. Finally, we analyse the entanglement content of the paradigmatic LR Ising chain, inferring again important deviations from the SR regime, as well as the limitations of bulk-boundary (tensor-network based) approaches to classify LR spin models.
Topological insulators are noninteracting, gapped fermionic systems which have gapless boundary excitations. They are characterized by topological invariants, which can be written in many different ways, including in terms of Green's functions. Here we show that the existence of the edge states directly follows from the existence of the topological invariant written in terms of the Green's functions, for all ten classes of topological insulators in all spatial dimensions. We also show that the resulting edge states are characterized by their own topological invariant, whose value is equal to the topological invariant of the bulk insulator. This can be used to test whether a given model Hamiltonian can describe an edge of a topological insulator. Finally, we observe that the results discussed here apply equally well to interacting topological insulators, with certain modifications.
We discover novel topological effects in the one-dimensional Kitaev chain modified by long-range Hamiltonian deformations in the hopping and pairing terms. This class of models display symmetry-protected topological order measured by the Berry/Zak phase of the lower-band eigenvector and the winding number of the Hamiltonians. For exponentially decaying hopping amplitudes, the topological sector can be significantly augmented as the penetration length increases, something experimentally achievable. For power-law decaying superconducting pairings, the massless Majorana modes at the edges get paired together into a massive nonlocal Dirac fermion localized at both edges of the chain: a new topological quasiparticle that we call topological massive Dirac fermion. This topological phase has fractional topological numbers as a consequence of the long-range couplings. Possible applications to current experimental setups and topological quantum computation are also discussed.