Intrinsic non-Hermitian topological phases

1. Problem Statement

Non-Hermitian quantum systems exhibit unique phenomena absent in Hermitian systems, such as complex eigenvalues, exceptional points, and the non-Hermitian skin effect. While topological classifications for non-Hermitian systems have been proposed based on point-gap conditions (spectra avoiding a reference point in the complex plane) and line-gap conditions (spectra avoiding a line, typically the real or imaginary axis), a fundamental question remains: Which topological phases are genuinely intrinsic to non-Hermitian systems?

Specifically, the paper addresses the need to distinguish between:

• Extrinsic phases: Non-Hermitian phases that can be continuously deformed into Hermitian (real-line-gap) or anti-Hermitian (imaginary-line-gap) systems without closing the gap. These have Hermitian counterparts.
• Intrinsic phases: Phases that exist only under point-gap conditions and cannot be reduced to any line-gap (Hermitian or anti-Hermitian) phase. These represent genuinely new non-Hermitian topology.

2. Methodology

The author develops a unified mathematical framework based on K-theory and homomorphisms to systematically classify these phases.

A. Symmetry Framework

The paper defines symmetries for non-Hermitian Hamiltonians $H_k$ using three homomorphisms $\phi, c, \kappa: G \to \{ \pm 1 \}$ acting on a finite symmetry group $G$:

• $\phi$: Determines if the symmetry involves complex conjugation ($K$).
• $c$: Determines the sign of the symmetry action ($H \to \pm H$).
• $\kappa$: Distinguishes between unitary and anti-unitary actions on the Hamiltonian vs. its conjugate.
  This leads to 54 internal symmetry classes (derived from combinations of Time-Reversal, Particle-Hole, Chiral, Pseudo-Hermiticity, and Sublattice symmetries).

B. Hermitization and Gap Conditions

• Hermitization: A non-Hermitian Hamiltonian $H_k$ is mapped to a doubled Hermitian Hamiltonian $\tilde{H}_k = \begin{pmatrix} 0 & H_k \\ H_k^\dagger & 0 \end{pmatrix}$.
• Point Gap: Defined as $\det H_k \neq 0$. This is equivalent to $\tilde{H}_k$ having no zero eigenvalues.
• Line Gaps:
  • Real-line gap: Spectrum avoids the imaginary axis. Equivalent to $\tilde{H}_k$ having an additional chiral symmetry ($\sigma_y$).
  • Imaginary-line gap: Spectrum avoids the real axis. Equivalent to $\tilde{H}_k$ having an additional chiral symmetry ($\sigma_x$).

C. Homomorphisms and Classification Strategy

The core methodology involves defining natural homomorphisms from line-gap classification groups to point-gap classification groups:

1. $K_P$: Classification group for point-gap phases.
2. $K_{Lr}$: Classification group for real-line-gap (Hermitian-like) phases.
3. $K_{Li}$: Classification group for imaginary-line-gap (anti-Hermitian-like) phases.

The author constructs homomorphisms:
$$f_r: K_{Lr} \to K_P \quad \text{and} \quad f_i: K_{Li} \to K_P$$
These maps "forget" the additional chiral symmetry required for line gaps.

• Extrinsic Phases: The image of these maps ($\text{im } f_r + \text{im } f_i$).
• Intrinsic Phases: The quotient group $K_P / (\text{im } f_r + \text{im } f_i)$. These are the phases that cannot be reached from Hermitian or anti-Hermitian limits.

D. Explicit Computations

The paper computes these homomorphisms for all 54 internal symmetry classes across various dimensions ($d$) and defect codimensions ($D$).

• The problem is reduced to zero-dimensional (0D) calculations using suspension isomorphism.
• Representative Hamiltonians and symmetry operators are constructed for each class to determine the map structure (e.g., $\mathbb{Z} \to \mathbb{Z}$, $\mathbb{Z} \oplus \mathbb{Z} \to \mathbb{Z}$, etc.).
• Special attention is paid to cases where the relative orientation of $f_r$ and $f_i$ matters (e.g., when both map $\mathbb{Z} \to \mathbb{Z} \oplus \mathbb{Z}$), requiring analysis of winding numbers and Chern numbers in specific dimensions.

3. Key Contributions

1. Unified Formalism: Establishes a rigorous algebraic structure distinguishing intrinsic vs. extrinsic non-Hermitian topology using K-theory and homomorphisms.
2. Complete Classification: Provides explicit classification tables for all 38 independent non-Hermitian symmetry classes (accounting for equivalences under $H \to iH$).
3. Explicit Homomorphisms: Calculates the specific maps $f_r$ and $f_i$ for all 54 symmetry classes, detailing how line-gap invariants map to point-gap invariants (e.g., identifying cases where the map is surjective, zero, or a doubling map).
4. Identification of Intrinsic Phases: Systematically identifies which symmetry classes and dimensions host intrinsic non-Hermitian phases (those with non-trivial quotient groups).

4. Key Results

The results are summarized in Tables 11–16 of the paper. Key findings include:

• Intrinsic Phases Exist Widely: Intrinsic non-Hermitian phases appear in a broad range of symmetry classes and dimensions, not just in the absence of symmetry.
• Dimensional Dependence: The existence of intrinsic phases depends heavily on spatial dimension. For example, in certain classes, intrinsic phases vanish in even dimensions but appear in odd dimensions (and vice versa).
• Specific Examples:
  • The non-Hermitian skin effect is identified as a manifestation of intrinsic topology (specifically related to spectral winding numbers that vanish in Hermitian limits).
  • Classes like A + $\eta$ (Pseudo-Hermiticity) and AIII + S (Chiral + Sublattice) show rich structures where the point-gap classification ($\mathbb{Z} \oplus \mathbb{Z}$) is larger than the sum of its line-gap images, indicating robust intrinsic phases.
  • In some cases (e.g., Class AI), the map from real-line-gap to point-gap is surjective ($\mathbb{Z} \to \mathbb{Z}_2$), implying all point-gap phases are extrinsic in that specific context.
• Quotient Groups: The paper provides the explicit quotient groups $K_P / (\text{im } f_r + \text{im } f_i)$, which serve as the definitive classification of intrinsic phases.

5. Significance

• Theoretical Clarity: The work resolves the ambiguity in non-Hermitian topology by providing a clear criterion for "genuinely non-Hermitian" phenomena. It moves beyond ad-hoc examples to a systematic theory.
• Experimental Guidance: By identifying specific symmetry classes and dimensions that support intrinsic phases, the paper guides experimentalists in photonic, acoustic, and mechanical metamaterials on where to search for robust non-Hermitian effects (like the skin effect) that have no Hermitian analog.
• Foundation for Future Work: The framework sets the stage for extending these classifications to systems with crystalline symmetries (spatial symmetries) and interacting systems, which are the next frontiers in non-Hermitian physics.

In summary, Shiozaki's paper provides the "periodic table" for intrinsic non-Hermitian topological phases, rigorously separating the new physics of open systems from the familiar physics of Hermitian systems.