Both the physics and applications of fermionic symmetry-protected topological phases rely heavily on a principle known as bulk-boundary correspondence, which predicts the emergence of protected boundary-localized energy excitations (boundary states) if the bulk is topologically non-trivial. Current theoretical approaches formulate a bulk-boundary correspondence as an equality between a bulk and a boundary topological invariant, where the latter is a property of boundary states. However, such an equality does not offer insight into the stability or the sensitivity of the boundary states to external perturbations. To solve this problem, we adopt a technique known as the Wiener–Hopf factorization of matrix functions. Using this technique, we first provide an elementary proof of the equality of the bulk and the boundary invariants for one-dimensional systems with arbitrary boundary conditions in all Altland-Zirnbauer symmetry classes. This equality also applies to quasi-one-dimensional systems (e.g., junctions) formed by bulks belonging to the same symmetry class. We then show that only topologically non-trivial Hamiltonians can host stable zero-energy edge modes, where stability refers to continuous deformation of zero-energy excitations with external perturbations that preserve the symmetries of the class. By leveraging the Wiener–Hopf factorization, we establish bounds on the sensitivity of such stable zero-energy modes to external perturbations. Our results show that the Wiener–Hopf factorization is a natural tool to investigate bulk-boundary correspondence in quasi-one-dimensional fermionic symmetry-protected topological phases. Our results on the stability and sensitivity of zero-energy modes are especially valuable for applications, including Majorana-based topological quantum computing.
Both the physics and applications of fermionic symmetry-protected topological phases rely heavily on a principle known as bulk-boundary correspondence, which predicts the emergence of protected boundary-localized energy excitations (boundary states) if the bulk is topologically non-trivial. Current theoretical approaches formulate a bulk-boundary correspondence as an equality between a bulk and a boundary topological invariant, where the latter is a property of boundary states. However, such an equality does not offer insight into the stability or the sensitivity of the boundary states to external perturbations. To solve this problem, we adopt a technique known as the Wiener–Hopf factorization of matrix functions. Using this technique, we first provide an elementary proof of the equality of the bulk and the boundary invariants for one-dimensional systems with arbitrary boundary conditions in all Altland-Zirnbauer symmetry classes. This equality also applies to quasi-one-dimensional systems (e.g., junctions) formed by bulks belonging to the same symmetry class. We then show that only topologically non-trivial Hamiltonians can host stable zero-energy edge modes, where stability refers to continuous deformation of zero-energy excitations with external perturbations that preserve the symmetries of the class. By leveraging the Wiener–Hopf factorization, we establish bounds on the sensitivity of such stable zero-energy modes to external perturbations. Our results show that the Wiener–Hopf factorization is a natural tool to investigate bulk-boundary correspondence in quasi-one-dimensional fermionic symmetry-protected topological phases. Our results on the stability and sensitivity of zero-energy modes are especially valuable for applications, including Majorana-based topological quantum computing.
Majorana zero modes are well studied in the gapped phases of topological systems. We investigate Majorana zero modes at the topological quantum criticality in one dimensional topological superconducting model with longer range interaction. We identify stable localized Majorana zero modes appearing at criticality under certain conditions. Topological invariant number for these non-trivial criticalities is obtained from zeros of a complex function associated with the Hamiltonian. Behavior of parametric curve at criticalities validate the invariant obtained and account for the appearance of Majorana zero modes at criticality. Trivial and non-trivial topological nature of criticality due to the presence of multicritical point cause an unusual topological transition along the critical line. We observe and investigate this unique transition in terms of eigenvalue spectrum. Appearance of MZMs at criticality demands integer value of topological invariant number in order to validate the concept of bulk-boundary correspondence. Hence we propose a scheme to separate the invariant number into fractional and integer contribution to establish bulk-boundary correspondence at criticality.
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.
In contrast to eigenvalue-based approaches, we formulate the bulk-boundary correspondence for two-dimensional non-Hermitian quadratic lattice Hamiltonians in terms of Toeplitz operators and singular values, which correctly capture the stability, localization, and scaling of edge and corner modes. We show that singular values, rather than eigenvalues, provide the only stable foundation for topological protection in non-Hermitian systems because they remain robust under translational-symmetry-breaking perturbations that destabilize the eigenvalue spectrum, rendering it unsuitable for topological classification. Building on Toeplitz operator theory, we establish general results for non-Hermitian Hamiltonians defined on half and quarter planes, relating the topological indices of the associated Toeplitz operators to the number of finite-size singular values that are separated from the bulk singular-value spectrum and vanish in the thermodynamic limit. This yields a precise bulk-boundary correspondence for edge and corner modes, including higher-order topological phases, without requiring crystalline symmetries. We illustrate our general results with detailed examples exhibiting topologically protected families of edge states, coexisting edge and corner modes, and phases with both gapped bulk and edges supporting only stable corner modes. The latter is exemplified by a non-Hermitian generalization of the Benalcazar-Bernevig-Hughes model.
While tremendous research has revealed that symmetry enriches topological phases of matter, more general principles that protect topological phases have yet to be explored. In this Letter, we elucidate the roles of subspaces in free-fermionic topological phases. A subspace property for Hamiltonians enables us to define new topological invariants. It results in peculiar topological boundary phenomena, i.e., the emergence of an unpaired zero mode or zero-winding skin modes, characterizing subspace-protected topological phases. We establish and demonstrate the bulk-boundary correspondence in subspace-protected topological phases. We further discuss the interplay of the subspace property and internal symmetries. Toward application, we also propose possible platforms possessing the subspace property.
Symmetry-protected topological (SPT) phases in insulators and superconductors are known for their robust edge modes, linked to bulk invariants through the bulk-boundary correspondence. While this principle traditionally applies to gapped phases, recent advances have extended it to gapless systems, where topological edge-states persist even in the absence of a bulk gap. We extend this framework to periodically driven chains with chiral symmetry, revealing the existence of topological edge zero and $\ensuremath{\pi}$ modes despite the lack of bulk gaps in the quasienergy spectrum. By examining the half-period decomposition of chiral evolutions, we construct topological invariants that circumvent the need to define the Floquet Hamiltonian, making them more suitable for generalization to the gapless regime. We provide explicit examples, including generalizations of the Kitaev chain and related spin models, where localized $\ensuremath{\pi}$ modes emerge even when the bulk is gapless at the same quasienergy as the edge modes. We numerically study the effect of interactions, which give a finite lifetime to the edge modes in the thermodynamic limit with the decay rate consistent with Fermi's golden rule.
Symmetry-protected topological (SPT) phases are gapped phases of matter that cannot be deformed to a trivial phase without breaking the symmetry or closing the bulk gap.Here we introduce a notion of a topological obstruction that is not captured by bulk energy gap closings in periodic boundary conditions.More specifically, given a symmetric boundary termination we say two bulk Hamiltonians belong to distinct boundary obstructed topological phases (BOTPs) if they can be deformed to each other on a system with periodic boundaries, but cannot be deformed to each other in the open system without closing the gap at at least one high-symmetry surface.BOTPs are not topological phases of matter in the standard sense since they are adiabatically deformable to each other on a torus, but, similar to SPTs, they are associated with boundary signatures in open systems such as surface states or fractional corner charges.In contrast to SPTs, these boundary signatures are not anomalous and can be removed by symmetrically adding lower-dimensional SPTs on the boundary, but they are stable as long as the spectral gap at high-symmetry edges/surfaces remains open.We show that the double-mirror quadrupole model of [W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Science 357, 61 (2017)] is a prototypical example of such phases, and present a detailed analysis of several aspects of boundary obstructions in this model.In addition, we introduce several three-dimensional models having boundary obstructions, which are characterized either by surface states or fractional corner charges.Furthermore, we provide a complete characterization of boundary obstructed phases in terms of symmetry representations.Namely, two distinct BOTP phases correspond to equivalent band representations in the periodic system which become inequivalent upon restricting the symmetry group to that of the open system.This is used to shown that for a given open boundary, there is only one class of BOTPs which corresponds to a local representation of the symmetry of the open system and thus can be designated as the trivial phase.All other BOTP classes do not correspond to local representation of the open system and as a result necessarily exhibit a filling anomaly or gapless surface states.