The intersection Dold-Thom theorem

Gajer, Paweł

doi:10.1016/0040-9383(95)00053-4

Cited by 10 publications

(4 citation statements)

References 7 publications

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“…In this context, we can define equivalence relations on ESs (or ELs in 2D). This follows and adapts the mathematical notions originated from Goresky and MacPherson's work and further developed by Gajer [50][51][52] . Using the above procedures, we obtain the quotient space of the S 1 in Fig.…”

Section: Resultsmentioning

confidence: 86%

“…It is important to note that antipodal points located in the regions where eigenenergies are gapped cannot be identified, because their eigenstates are reversely ordered by the eigenenergies. Such a refined topological discrimination of the strata of the origin, the intersecting lines f 2 ¼ Çf 3 and the plane is a distinguished feature of intersection homotopy methods [50][51][52] . The intersection homotopy method, which is a mathematical technique used to address hypersurface singularities, differs significantly from the conventional homotopy method that focuses on the topology of isolated singularities.…”

Section: Resultsmentioning

confidence: 99%

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Topological classification for intersection singularities of exceptional surfaces in pseudo-Hermitian systems

Jia,

Zhang,

et al. 2023

Commun Phys

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Non-Hermitian systems are known for their intriguing topological properties, which underpin various exotic physical phenomena. Exceptional points, in particular, play a pivotal role in fine-tuning these systems for optimal device functionality and material characteristics. These points can give rise to exceptional surfaces with embedded lower-dimensional non-isolated singularities. Here we introduce a topological classification for non-defective intersection lines of exceptional surfaces, where exceptional surfaces intersect transversally. We achieve this classification by constructing a quotient space of an order-parameter space under equivalence relations of eigenstates. We unveil that the fundamental group of these gapless structures is a non-Abelian group on three generators. This classification not only reveals a unique form of non-Hermitian gapless phases featuring a chain of non-defective intersection lines but also predicts the unexpected existence of topological edge states in one-dimensional lattice models protected by the intersection singularities. Our classification opens avenues for realizing robust topological phases.

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Section: Resultsmentioning

confidence: 86%

Section: Resultsmentioning

confidence: 99%

Topological classification for intersection singularities of exceptional surfaces in pseudo-Hermitian systems

Jia,

Zhang,

et al. 2023

Commun Phys

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“…Segal, too, gave a generalized model for the infinite symmetric product in [Seg74], viewing it as a labelled configuration space. In 1996, Gajer gave an intersection-homology variant of the Dold-Thom theorem [Gaj96]. An equivariant formulation of the theorem was given by dos Santos in [Dos03], which generalizes an equivariant integral-coefficient formulation given by Lima-Filho in [L-F97].…”

Section: Introductionmentioning

confidence: 99%

The Dold–Thom theorem via factorization homology

Bandklayder

2018

J. Homotopy Relat. Struct.

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“…Both loops inevitably cut through the ESs, as the EL3s and NIL are hypersurface singularities. Such an approach employs notions of intersection homotopy 31 , which is different from the conventional homotopy description with encircling loops on which all the Hamiltonians are gapped (see details in Section 4 of supplementary information). The two loops have the same staring point (SP, purple dots) so that direct comparison can be performed.…”

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confidence: 99%

Non-Hermitian swallowtail catastrophe revealing transitions among diverse topological singularities

Zhang

Wang

et al. 2023

Nat. Phys.

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Exceptional points are a unique feature in non-Hermitian systems, where eigenvalues and their corresponding eigenstates of a Hamiltonian coalesce 1-12 . A lot of intriguing physical phenomena arise from the topology of exceptional points, such as "bulk Fermi-arcs" 2,3 and braiding of eigenvalues 10 . Here we report that a more exotic and structurally richer degeneracy morphology, known as the swallowtail catastrophe in singularity theory 13 , can naturally exist in non-Hermitian systems with both parity-time and pseudo-Hermitian symmetries. The swallowtail exhibits the coexistence and intriguing interactions of degeneracy lines of three different types, including an isolated nodal line, a pair of exceptional lines of order three and a non-defective intersection line, with the12 1 2 1 13 3 2 3 23

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The intersection Dold-Thom theorem

Cited by 10 publications

References 7 publications

Topological classification for intersection singularities of exceptional surfaces in pseudo-Hermitian systems

Topological classification for intersection singularities of exceptional surfaces in pseudo-Hermitian systems

The Dold–Thom theorem via factorization homology

Non-Hermitian swallowtail catastrophe revealing transitions among diverse topological singularities

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