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Intrinsic non-Hermitian topological phases

Ken Shiozaki

TL;DR

The paper develops a unified $K$-theoretic framework to classify intrinsic non-Hermitian topological phases by comparing line-gap and point-gap classifications through natural homomorphisms from line-gap to point-gap phases. It formalizes a distinction between extrinsic phases (those reducible to Hermitian or anti-Hermitian line-gapped phases) and intrinsic phases (genuinely non-Hermitian) and provides explicit computations for all internal symmetry classes. By embedding the problem in twisted equivariant $K$-theory and analyzing degree shifts (AZ and AZ$^ ext{dag}$ classes), the authors produce complete classifications and detailed tables for 54 symmetry blocks, including real- and imaginary-line-gap cases and their interrelations via the maps $f_{ m r}$ and $f_{ m i}$. The results reveal that intrinsic NH topology encompasses phenomena beyond Hermitian counterparts (e.g., the non-Hermitian skin effect) and offer a systematic road map for identifying and engineering intrinsic phases in photonic, phononic, and electronic platforms. The work also points to future directions such as incorporating crystalline symmetries, interactions, and concrete experimental realizations.

Abstract

We study the interplay of non-Hermitian topological phases under point- and line-gap conditions. Using natural homomorphisms from line-gap to point-gap phases, we distinguish extrinsic phases, reducible to Hermitian or anti-Hermitian line-gapped phases, from intrinsic phases, which are genuinely non-Hermitian without Hermitian counterparts. Although classification tables for all symmetry classes were already presented in earlier work, the present paper develops a unified formulation and provides explicit computations for all internal symmetries.

Intrinsic non-Hermitian topological phases

TL;DR

The paper develops a unified -theoretic framework to classify intrinsic non-Hermitian topological phases by comparing line-gap and point-gap classifications through natural homomorphisms from line-gap to point-gap phases. It formalizes a distinction between extrinsic phases (those reducible to Hermitian or anti-Hermitian line-gapped phases) and intrinsic phases (genuinely non-Hermitian) and provides explicit computations for all internal symmetry classes. By embedding the problem in twisted equivariant -theory and analyzing degree shifts (AZ and AZ classes), the authors produce complete classifications and detailed tables for 54 symmetry blocks, including real- and imaginary-line-gap cases and their interrelations via the maps and . The results reveal that intrinsic NH topology encompasses phenomena beyond Hermitian counterparts (e.g., the non-Hermitian skin effect) and offer a systematic road map for identifying and engineering intrinsic phases in photonic, phononic, and electronic platforms. The work also points to future directions such as incorporating crystalline symmetries, interactions, and concrete experimental realizations.

Abstract

We study the interplay of non-Hermitian topological phases under point- and line-gap conditions. Using natural homomorphisms from line-gap to point-gap phases, we distinguish extrinsic phases, reducible to Hermitian or anti-Hermitian line-gapped phases, from intrinsic phases, which are genuinely non-Hermitian without Hermitian counterparts. Although classification tables for all symmetry classes were already presented in earlier work, the present paper develops a unified formulation and provides explicit computations for all internal symmetries.

Paper Structure

This paper contains 45 sections, 1 theorem, 144 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Some generalities
  3. Symmetry
  4. Hermitization
  5. Point-gap condition
  6. Singular-value gap
  7. Constraint of symmetry matrices
  8. Choice of symmetry matrices
  9. Line gaps
  10. Real line gap.
  11. Imaginary line gap.
  12. Intrinsic and extrinsic non-Hermitian topological phases
  13. Symmetry
  14. Point-gap and line-gap topological phases
  15. Intrinsic and extrinsic
  16. ...and 30 more sections

Key Result

Lemma 2.1

If a Hamiltonian $H_{\bm{k}}$ has a real line gap, then it can be continuously deformed, while preserving both the real line gap and the symmetry (eq_def_sym_2), into a Hermitian Hamiltonian $H'_{\bm{k}}$ with $H'_{\bm{k}}=H'_{\bm{k}}{}^\dag$. Similarly, if $H_{\bm{k}}$ has an imaginary line gap, th

Theorems & Definitions (7)

  • Lemma 2.1: (Anti-)Hermitization
  • Definition 3.1: Point-gap topological phase
  • Definition 3.2: Real-line-gap topological phase
  • Definition 3.3: Imaginary-line-gap topological phase
  • Definition 3.4: Homomorphisms from line-gap to point-gap phases
  • Definition 3.5: Extrinsic topological phases
  • Definition 3.6: Intrinsic topological phases