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Hopf Exceptional Points

Tsuneya Yoshida, Emil J. Bergholtz, Tomáš Bzdušek

TL;DR

This work identifies Hopf exceptional points (HEPs), a novel class of non-Hermitian band touchings protected by Hopf invariants, including higher-dimensional generalizations. It shows that HEP3 and symmetry-protected HEP5 carry a $\mathbb{Z}_2$ topology, effectively making them their own antiparticles, while HEP4 can carry a Hopf ($\mathbb{Z}$) charge, with higher-dimensional classifications arising from higher homotopy groups of spheres. The authors develop concrete toy models and calculational frameworks (involving resultant vectors, Hopf invariants, and $\mathbb{Z}_2$ invariants $\nu_{F}$) to demonstrate these topologies and discuss extensions to PT symmetry, pseudo-Hermiticity, CP and chiral symmetry. They further predict a rich landscape of multifold HEPs and finite-group topologies ($\mathbb{Z}_3$, $\mathbb{Z}_{12}$, $\mathbb{Z}_{24}$, etc.) beyond existing symmetry-classifications, suggesting new avenues for experimental realization in photonics, metamaterials, and open quantum systems. The results open up a broad program to map non-Hermitian topological phases in higher dimensions and to explore novel fusion rules and invariants arising from higher homotopy structures.

Abstract

Exceptional points at which eigenvalues and eigenvectors of non-Hermitian matrices coalesce are ubiquitous in the description of a wide range of platforms from photonic or mechanical metamaterials to open quantum systems. Here, we introduce a class of Hopf exceptional points (HEPs) that are protected by the Hopf invariants (including the higher-dimensional generalizations) and which exhibit phenomenology sharply distinct from conventional exceptional points. Saliently, owing to their $\mathbb{Z}_2$ topological invariant related to the Witten anomaly, three-fold HEPs and symmetry-protected five-fold HEPs act as their own ``antiparticles". Furthermore, based on higher homotopy groups of spheres, we predict the existence of multifold HEPs and symmetry-protected HEPs with non-Hermitian topology captured by a range of finite groups (such as $\mathbb{Z}_3$, $\mathbb{Z}_{12}$, or $\mathbb{Z}_{24}$) beyond the periodic table of Bernard-LeClair symmetry classes.

Hopf Exceptional Points

TL;DR

This work identifies Hopf exceptional points (HEPs), a novel class of non-Hermitian band touchings protected by Hopf invariants, including higher-dimensional generalizations. It shows that HEP3 and symmetry-protected HEP5 carry a topology, effectively making them their own antiparticles, while HEP4 can carry a Hopf () charge, with higher-dimensional classifications arising from higher homotopy groups of spheres. The authors develop concrete toy models and calculational frameworks (involving resultant vectors, Hopf invariants, and invariants ) to demonstrate these topologies and discuss extensions to PT symmetry, pseudo-Hermiticity, CP and chiral symmetry. They further predict a rich landscape of multifold HEPs and finite-group topologies (, , , etc.) beyond existing symmetry-classifications, suggesting new avenues for experimental realization in photonics, metamaterials, and open quantum systems. The results open up a broad program to map non-Hermitian topological phases in higher dimensions and to explore novel fusion rules and invariants arising from higher homotopy structures.

Abstract

Exceptional points at which eigenvalues and eigenvectors of non-Hermitian matrices coalesce are ubiquitous in the description of a wide range of platforms from photonic or mechanical metamaterials to open quantum systems. Here, we introduce a class of Hopf exceptional points (HEPs) that are protected by the Hopf invariants (including the higher-dimensional generalizations) and which exhibit phenomenology sharply distinct from conventional exceptional points. Saliently, owing to their topological invariant related to the Witten anomaly, three-fold HEPs and symmetry-protected five-fold HEPs act as their own ``antiparticles". Furthermore, based on higher homotopy groups of spheres, we predict the existence of multifold HEPs and symmetry-protected HEPs with non-Hermitian topology captured by a range of finite groups (such as , , or ) beyond the periodic table of Bernard-LeClair symmetry classes.

Paper Structure

This paper contains 19 sections, 33 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. HEP3s with Z2 topology
  3. Symmetry-protected HEP5s with Z2 topology
  4. Symmetry-protected HEP4s with Z topology
  5. General characteristics of HEPs
  6. Multiply-charged aspect of HEPs
  7. Additional bands
  8. Higher dimensions
  9. Conclusion and outlook
  10. Funding information
  11. Defining topology from resultants
  12. Resultant of a pair of polynomials
  13. Resultant winding number
  14. Berry connection and curvature of resultant Hamiltonians for Hopf topology
  15. Details of the presented Hamiltonians
  16. ...and 4 more sections

Figures (5)

  • Figure 1: Illustration of $\mathbb{Z}_2$ topology protecting a three-fold Hopf exceptional point (HEP3). The resultant vector $\bm{R}(\bm{k})$ defines a map from a 4-sphere in the momentum (or parameter) space to a 3-sphere in the space of the resultant vector [see \ref{['eq: def r_j', 'eq: Rvec Generic 3x3']}]. If the map is topologically nontrivial, an HEP3 with $\mathbb{Z}_2$ topology exists inside the 4-sphere.
  • Figure 2: Energy bands of Hamiltonian \ref{['eq: toy Z2EP3']} for $m_0\,{=}\,1.5$. The real (imaginary) part is represented as height (color). In panel (a), the complex conjugate of the upper band is omitted. Panels (a) and (b) are obtained for $k_5\,{=}\,\pi/3$, $\delta\,{=}\,0.5$, and $f(k_5)\,{=}\,1$. Here, $\Delta k_4$ is defined as $\Delta k_4\,{=}\,k_4-\pi/2$. Panel (c) [(d)] displays pair annihilation of HEP3s for $(k_2,k_3,k_4)\,{=}\,(0,0,\pi/2)$, and $f(k_5)\,{=}\,1$ [$f(k_5)\,{=}\,2\sin (k_5/2)$]. In these panels, numerically computed $\nu_\mathrm{F}$ for $k_5\,{=}\,-\pi/2$, $0$, and $\pi/2$ at $\delta\,{=}\,0.5$ is represented by numbers highlighted in yellow. We used a mesh of $40^4$ points to evaluate the integrals with momenta $k_{1,2,3}\in[-\pi,\pi]$ and $k_4 \in [0,2\pi]$.
  • Figure 3: Energy bands of Hamiltonian \ref{['eq: toy Z2 EP5']} for $f(k_5)\,{=}\,1$, $k_5\,{=}\,\pi/3$, $m_0\,{=}\,1.5$ and $\delta\,{=}\,0.5$. The real (imaginary) part is represented as height (color). Panel (a) [(b)] displays the data for $(k_3,k_4)\,{=}\,(0,\pi/2)$ [$(k_1,k_2)\,{=}\,(0,0)$]. In panel (a), the complex conjugates of the upper and the lower bands are omitted. In panel (b), $\Delta k_4$ is defined as $\Delta k_4\,{=}\,k_4-\pi/2$.
  • Figure 4: (a) and (b): Energy eigenvalues of Hamiltonian in \ref{['eq: toy Hopf EP4']} for $m_0\,{=}\,3$ and $\delta\,{=}\,0$. The real (imaginary) part is represented as height (color). In panel (b), complex conjugates of the upper and lower bands are omitted. (c) [(d)]: Lines in the momentum space $(k_1,k_2,k_3)$ for $m_0\,{=}\,3$, $\delta\,{=}\,0$ and $k_4\,{=}\,-\pi/2$ [$k_4\,{=}\,\pi/2$], where red and blue lines denote the momenta satisfying $\bm{R}\,{\propto}\,(0,0,1)^{\mathrm{T}}$ and $\bm{R}\,{\propto}\,(0,0,-1)^{\mathrm{T}}$, respectively. The linking of these lines determines the value of the Hopf invariant $\nu_{\mathrm{H}}$. (e): Momenta satisfying $\bm{R}(\bm{k})\,{=}\,\bm{0}$ for $k_1\,{=}\,0$ and $\delta\,{=}\,0$ resp. $\delta\,{=}\,0.2$ (orange manifolds). As $\delta$ is introduced, the symmetry-protected HEP4 inflates into a loop. The gray oval and the blue loop illustrate the $S^3$ resp. $S^2$, both extending in the fourth dimension $k_1$ (not shown), on which one computes the Hopf invariant $\nu_{\mathrm{H}}$ resp. the resultant winding number $W_{2}$ (see \ref{['appsec: ResW']}) Yoshida_EPn_PRR2024.
  • Figure 5: (a) [(b)]: Argument of $R_1+\mathrm{i} R_2$ [band structure] of Hamiltonian \ref{['eq: toy H fake EP3']}. In panel (b), the complex conjugate of the top band is omitted.