Classification of Exceptional Points and Non-Hermitian Topological   Semimetals

Kohei Kawabata; Takumi Bessho; Masatoshi Sato

arXiv:1902.08479·cond-mat.mes-hall·August 13, 2019

Classification of Exceptional Points and Non-Hermitian Topological Semimetals

Kohei Kawabata, Takumi Bessho, Masatoshi Sato

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TL;DR

This paper provides a comprehensive classification of exceptional points in non-Hermitian systems, revealing new topological structures and predicting novel non-Hermitian semimetals and nodal superconductors.

Contribution

It introduces a complete classification scheme based on energy gaps and symmetries, uncovering unique non-Hermitian topological phases with no Hermitian counterparts.

Findings

01

Classification of exceptional points based on complex-energy gaps and symmetries

02

Prediction of new non-Hermitian semimetals and nodal superconductors

03

Construction of a topological dumbbell of exceptional points in 3D

Abstract

Exceptional points are universal level degeneracies induced by non-Hermiticity. Whereas past decades witnessed their new physics, the unified understanding has yet to be obtained. Here we present the complete classification of generic topologically stable exceptional points according to two types of complex-energy gaps and fundamental symmetries of charge conjugation, parity, and time reversal. This classification reveals unique non-Hermitian gapless structures with no Hermitian analogs and systematically predicts unknown non-Hermitian semimetals and nodal superconductors; a topological dumbbell of exceptional points in three dimensions is constructed as an illustration. Our work paves the way toward richer phenomena and functionalities of exceptional points and non-Hermitian topological semimetals.

Tables15

Table 1. Table 1: Classification table of topologically stable exceptional points at generic momentum points. The codimension p 𝑝 p is defined as p := d − d EP assign 𝑝 𝑑 subscript 𝑑 EP p:=d-d_{\rm EP} with the spatial dimension d 𝑑 d and the dimension d EP subscript 𝑑 EP d_{\rm EP} of the gapless region; exceptional points, lines, and surfaces are described by d EP = 0 , 1 , 2 subscript 𝑑 EP 0 1 2 d_{\rm EP}=0,1,2 , respectively. Complex-energy gaps have two distinct types, a point (P) or line (L) gap, and the subscript of L specifies a line gap for the real or imaginary part of the complex spectrum. The sign of 𝒫 𝒯 𝒫 𝒯 {\cal PT} ( 𝒞 𝒫 𝒞 𝒫 {\cal CP} ) symmetry means ( 𝒫 𝒯 ) ( 𝒫 𝒯 ) ∗ 𝒫 𝒯 superscript 𝒫 𝒯 ({\cal PT})\,({\cal PT})^{*} [ ( 𝒞 𝒫 ) ( 𝒞 𝒫 ) ∗ 𝒞 𝒫 superscript 𝒞 𝒫 ({\cal CP})\,({\cal CP})^{*} ].

Symmetry	Gap	Classifying space	$p = 0$	$p = 1$	$p = 2$	$p = 3$	$p = 4$	$p = 5$	$p = 6$	$p = 7$
No	P	$𝒞_{p}$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$
No	L	$𝒞_{p + 1}$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$
Chiral	P	$𝒞_{p + 1}$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$
	$L_{r}$	$𝒞_{p}$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$
	$L_{i}$	$𝒞_{p + 1} \times 𝒞_{p + 1}$	$0$	$ℤ \oplus ℤ$	$0$	$ℤ \oplus ℤ$	$0$	$ℤ \oplus ℤ$	$0$	$ℤ \oplus ℤ$
Sublattice	P	$𝒞_{p} \times 𝒞_{p}$	$ℤ \oplus ℤ$	$0$	$ℤ \oplus ℤ$	$0$	$ℤ \oplus ℤ$	$0$	$ℤ \oplus ℤ$	$0$
Sublattice	L	$𝒞_{p}$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$
$𝒫 𝒯$ , $+ 1$	P	$ℛ_{p}$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$
	$L_{r}$	$ℛ_{p + 7}$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$
	$L_{i}$	$ℛ_{p + 1}$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$
$𝒫 𝒯$ , $- 1$	P	$ℛ_{p + 4}$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$
	$L_{r}$	$ℛ_{p + 3}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$
	$L_{i}$	$ℛ_{p + 5}$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$
$𝒞 𝒫$ , $+ 1$	P	$ℛ_{p + 2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$
$𝒞 𝒫$ , $+ 1$	L	$ℛ_{p + 1}$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$
$𝒞 𝒫$ , $- 1$	P	$ℛ_{p + 6}$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$
$𝒞 𝒫$ , $- 1$	L	$ℛ_{p + 5}$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$

Table 2. Table S1: 𝒫 𝒫 {\cal P} AZ and 𝒫 AZ † 𝒫 superscript AZ † {\cal P}\text{AZ}^{{\dagger}} symmetry classes for non-Hermitian Hamiltonians. 𝒫 𝒯 + 𝒫 subscript 𝒯 \mathcal{P}\mathcal{T}_{+} symmetry, 𝒞 − 𝒫 subscript 𝒞 𝒫 \mathcal{C}_{-}\mathcal{P} symmetry, and chiral symmetry are respectively defined by ( 𝒫 𝒯 + ) H ∗ ( 𝒌 ) ( 𝒫 𝒯 + ) − 1 = H ( 𝒌 ) 𝒫 subscript 𝒯 superscript 𝐻 𝒌 superscript 𝒫 subscript 𝒯 1 𝐻 𝒌 ({\cal P}{\cal T}_{+})\,H^{*}\left({\bm{k}}\right)({\cal P}{\cal T}_{+})^{-1}=H\left({\bm{k}}\right) with ( 𝒫 𝒯 + ) ( 𝒫 𝒯 + ) ∗ = ± 1 𝒫 subscript 𝒯 superscript 𝒫 subscript 𝒯 plus-or-minus 1 ({\cal P}{\cal T}_{+})({\cal P}{\cal T}_{+})^{*}=\pm 1 , ( 𝒞 − 𝒫 ) H T ( 𝒌 ) ( 𝒞 − 𝒫 ) − 1 = − H ( 𝒌 ) subscript 𝒞 𝒫 superscript 𝐻 𝑇 𝒌 superscript subscript 𝒞 𝒫 1 𝐻 𝒌 ({\cal C}_{-}{\cal P})\,H^{T}\left({\bm{k}}\right)({\cal C}_{-}{\cal P})^{-1}=-H\left({\bm{k}}\right) with ( 𝒞 − 𝒫 ) ( 𝒞 − 𝒫 ) ∗ = ± 1 subscript 𝒞 𝒫 superscript subscript 𝒞 𝒫 plus-or-minus 1 ({\cal C}_{-}{\cal P})({\cal C}_{-}{\cal P})^{*}=\pm 1 , and Γ H † ( 𝒌 ) Γ − 1 = − H ( 𝒌 ) Γ superscript 𝐻 † 𝒌 superscript Γ 1 𝐻 𝒌 \Gamma\,H^{{\dagger}}\left({\bm{k}}\right)\Gamma^{-1}=-H\left({\bm{k}}\right) with Γ 2 = 1 superscript Γ 2 1 \Gamma^{2}=1 , which constitute 𝒫 𝒫 {\cal P} AZ symmetry class. 𝒞 + 𝒫 subscript 𝒞 𝒫 \mathcal{C}_{+}\mathcal{P} symmetry and 𝒫 𝒯 − 𝒫 subscript 𝒯 \mathcal{P}\mathcal{T}_{-} symmetry are respectively defined by ( 𝒞 + 𝒫 ) H T ( 𝒌 ) ( 𝒞 + 𝒫 ) − 1 = H ( 𝒌 ) subscript 𝒞 𝒫 superscript 𝐻 𝑇 𝒌 superscript subscript 𝒞 𝒫 1 𝐻 𝒌 ({\cal C}_{+}{\cal P})\,H^{T}\left({\bm{k}}\right)({\cal C}_{+}{\cal P})^{-1}=H\left({\bm{k}}\right) with ( 𝒞 + 𝒫 ) ( 𝒞 + 𝒫 ) ∗ = ± 1 subscript 𝒞 𝒫 superscript subscript 𝒞 𝒫 plus-or-minus 1 ({\cal C}_{+}{\cal P})({\cal C}_{+}{\cal P})^{*}=\pm 1 and ( 𝒫 𝒯 − ) H ∗ ( 𝒌 ) ( 𝒫 𝒯 − ) − 1 = − H ( 𝒌 ) 𝒫 subscript 𝒯 superscript 𝐻 𝒌 superscript 𝒫 subscript 𝒯 1 𝐻 𝒌 ({\cal P}{\cal T}_{-})\,H^{*}\left({\bm{k}}\right)({\cal P}{\cal T}_{-})^{-1}=-H\left({\bm{k}}\right) with ( 𝒫 𝒯 − ) ( 𝒫 𝒯 − ) ∗ = ± 1 𝒫 subscript 𝒯 superscript 𝒫 subscript 𝒯 plus-or-minus 1 ({\cal P}{\cal T}_{-})({\cal P}{\cal T}_{-})^{*}=\pm 1 , which constitute 𝒫 AZ † 𝒫 superscript AZ † {\cal P}\text{AZ}^{{\dagger}} symmetry class.

Symmetry class		$𝒫 𝒯_{+}$	$𝒞_{-} 𝒫$	$𝒞_{+} 𝒫$	$𝒫 𝒯_{-}$	$Γ$
Complex AZ	A	$0$	$0$	$0$	$0$	$0$
Complex AZ	AIII	$0$	$0$	$0$	$0$	$1$
Real $𝒫$ AZ	$𝒫$ AI	$+ 1$	$0$	$0$	$0$	$0$
	$𝒫$ BDI	$+ 1$	$+ 1$	$0$	$0$	$1$
	$𝒫$ D	$0$	$+ 1$	$0$	$0$	$0$
	$𝒫$ DIII	$- 1$	$+ 1$	$0$	$0$	$1$
	$𝒫$ AII	$- 1$	$0$	$0$	$0$	$0$
	$𝒫$ CII	$- 1$	$- 1$	$0$	$0$	$1$
	$𝒫$ C	$0$	$- 1$	$0$	$0$	$0$
	$𝒫$ CI	$+ 1$	$- 1$	$0$	$0$	$1$
Real $𝒫 {AZ}^{†}$	$𝒫 {AI}^{†}$	$0$	$0$	$+ 1$	$0$	$0$
	$𝒫 {BDI}^{†}$	$0$	$0$	$+ 1$	$+ 1$	$1$
	$𝒫 D^{†}$	$0$	$0$	$0$	$+ 1$	$0$
	$𝒫 {DIII}^{†}$	$0$	$0$	$- 1$	$+ 1$	$1$
	$𝒫 {AII}^{†}$	$0$	$0$	$- 1$	$0$	$0$
	$𝒫 {CII}^{†}$	$0$	$0$	$- 1$	$- 1$	$1$
	$𝒫 C^{†}$	$0$	$0$	$0$	$- 1$	$0$
	$𝒫 {CI}^{†}$	$0$	$0$	$+ 1$	$- 1$	$1$

Table 3. Table S2: Topological classification table of exceptional points in 𝒫 𝒫 {\cal P} AZ symmetry class. Exceptional points are classified according to 𝒫 𝒫 {\cal P} AZ symmetry class, the codimension p 𝑝 p , and the definition of a complex-energy point (P) or line (L) gap. The codimension p 𝑝 p is defined as p := d − d EP assign 𝑝 𝑑 subscript 𝑑 EP p:=d-d_{\rm EP} with the spatial dimension d 𝑑 d and the dimension d EP subscript 𝑑 EP d_{\rm EP} of the exceptional point. The subscript of L specifies the line gap for the real or imaginary part of the complex spectrum.

$𝒫 AZ$ class	Gap	Classifying space	$p = 0$	$p = 1$	$p = 2$	$p = 3$	$p = 4$	$p = 5$	$p = 6$	$p = 7$
A	P	$𝒞_{p}$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$
A	L	$𝒞_{p + 1}$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$
AIII	P	$𝒞_{p + 1}$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$
	$L_{r}$	$𝒞_{p}$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$
	$L_{i}$	$𝒞_{p + 1} \times 𝒞_{p + 1}$	$0$	$ℤ \oplus ℤ$	$0$	$ℤ \oplus ℤ$	$0$	$ℤ \oplus ℤ$	$0$	$ℤ \oplus ℤ$
$𝒫$ AI	P	$ℛ_{p}$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$
	$L_{r}$	$ℛ_{p + 7}$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$
	$L_{i}$	$ℛ_{p + 1}$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$
$𝒫$ BDI	P	$ℛ_{p + 1}$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$
	$L_{r}$	$ℛ_{p}$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$
	$L_{i}$	$ℛ_{p + 1} \times ℛ_{p + 1}$	$ℤ_{2} \oplus ℤ_{2}$	$ℤ_{2} \oplus ℤ_{2}$	$0$	$2 ℤ \oplus 2 ℤ$	$0$	$0$	$0$	$ℤ \oplus ℤ$
$𝒫$ D	P	$ℛ_{p + 2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$
$𝒫$ D	L	$ℛ_{p + 1}$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$
$𝒫$ DIII	P	$ℛ_{p + 3}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$
	$L_{r}$	$ℛ_{p + 2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$
	$L_{i}$	$𝒞_{p + 1}$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$
$𝒫$ AII	P	$ℛ_{p + 4}$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$
	$L_{r}$	$ℛ_{p + 3}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$
	$L_{i}$	$ℛ_{p + 5}$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$
$𝒫$ CII	P	$ℛ_{p + 5}$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$
	$L_{r}$	$ℛ_{p + 4}$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$
	$L_{i}$	$ℛ_{p + 5} \times ℛ_{p + 5}$	$0$	$0$	$0$	$ℤ \oplus ℤ$	$ℤ_{2} \oplus ℤ_{2}$	$ℤ_{2} \oplus ℤ_{2}$	$0$	$2 ℤ \oplus 2 ℤ$
$𝒫$ C	P	$ℛ_{p + 6}$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$
$𝒫$ C	L	$ℛ_{p + 5}$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$
$𝒫$ CI	P	$ℛ_{p + 7}$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$
	$L_{r}$	$ℛ_{p + 6}$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$
	$L_{i}$	$𝒞_{p + 1}$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$

Table 4. Table S3: Topological classification table of exceptional points in real 𝒫 AZ † 𝒫 superscript AZ † {\cal P}\text{AZ}^{{\dagger}} symmetry class. Exceptional points are classified according to 𝒫 AZ † 𝒫 superscript AZ † {\cal P}\text{AZ}^{{\dagger}} symmetry class, the codimension p 𝑝 p , and the definition of a complex-energy point (P) or line (L) gap. The codimension p 𝑝 p is defined as p := d − d EP assign 𝑝 𝑑 subscript 𝑑 EP p:=d-d_{\rm EP} with the spatial dimension d 𝑑 d and the dimension d EP subscript 𝑑 EP d_{\rm EP} of the exceptional point. The subscript of L specifies the line gap for the real or imaginary part of the complex spectrum.

$𝒫 {AZ}^{†}$ class	Gap	Classifying space	$p = 0$	$p = 1$	$p = 2$	$p = 3$	$p = 4$	$p = 5$	$p = 6$	$p = 7$
$𝒫 {AI}^{†}$	P	$ℛ_{p + 6}$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$
$𝒫 {AI}^{†}$	L	$ℛ_{p + 7}$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$
$𝒫 {BDI}^{†}$	P	$ℛ_{p + 7}$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$
	$L_{r}$	$ℛ_{p}$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$
	$L_{i}$	$ℛ_{p + 7} \times ℛ_{p + 7}$	$0$	$ℤ \oplus ℤ$	$ℤ_{2} \oplus ℤ_{2}$	$ℤ_{2} \oplus ℤ_{2}$	$0$	$2 ℤ \oplus 2 ℤ$	$0$	$0$
$𝒫 D^{†}$	P	$ℛ_{p}$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$
	$L_{r}$	$ℛ_{p + 1}$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$
	$L_{i}$	$ℛ_{p + 7}$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$
$𝒫 {DIII}^{†}$	P	$ℛ_{p + 1}$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$
	$L_{r}$	$ℛ_{p + 2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$
	$L_{i}$	$𝒞_{p + 1}$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$
$𝒫 {AII}^{†}$	P	$ℛ_{p + 2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$
$𝒫 {AII}^{†}$	L	$ℛ_{p + 3}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$
$𝒫 {CII}^{†}$	P	$ℛ_{p + 3}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$
	$L_{r}$	$ℛ_{p + 4}$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$
	$L_{i}$	$ℛ_{p + 3} \times ℛ_{p + 3}$	$0$	$2 ℤ \oplus 2 ℤ$	$0$	$0$	$0$	$ℤ \oplus ℤ$	$ℤ_{2} \oplus ℤ_{2}$	$ℤ_{2} \oplus ℤ_{2}$
$𝒫 C^{†}$	P	$ℛ_{p + 4}$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$
	$L_{r}$	$ℛ_{p + 5}$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$
	$L_{i}$	$ℛ_{p + 3}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$
$𝒫 {CI}^{†}$	P	$ℛ_{p + 5}$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$
	$L_{r}$	$ℛ_{p + 6}$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$
	$L_{i}$	$𝒞_{p + 1}$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$

Table 5. Table S4: Topological classification table of exceptional points in complex AZ symmetry class with sublattice symmetry (SLS) 𝒮 𝒮 {\cal S} . Exceptional points are classified according to the symmetry class, the codimension p 𝑝 p , and the definition of a complex-energy point (P) or line (L) gap. The subscript of 𝒮 𝒮 {\cal S} specifies the commutation ( + + ) or anticommutation ( − - ) relation to chiral symmetry. The codimension p 𝑝 p is defined as p := d − d EP assign 𝑝 𝑑 subscript 𝑑 EP p:=d-d_{\rm EP} with the spatial dimension d 𝑑 d and the dimension d EP subscript 𝑑 EP d_{\rm EP} of the exceptional point. The subscript of L specifies the line gap for the real or imaginary part of the complex spectrum.

SLS	AZ class	Gap	Classifying space	$p = 0$	$p = 1$	$p = 2$	$p = 3$	$p = 4$	$p = 5$	$p = 6$	$p = 7$
$𝒮_{+}$	AIII	P	$𝒞_{p}$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$
		$L_{r}$	$𝒞_{p} \times 𝒞_{p}$	$ℤ \oplus ℤ$	$0$	$ℤ \oplus ℤ$	$0$	$ℤ \oplus ℤ$	$0$	$ℤ \oplus ℤ$	$0$
		$L_{i}$	$𝒞_{p} \times 𝒞_{p}$	$ℤ \oplus ℤ$	$0$	$ℤ \oplus ℤ$	$0$	$ℤ \oplus ℤ$	$0$	$ℤ \oplus ℤ$	$0$
$𝒮$	A	P	$𝒞_{p} \times 𝒞_{p}$	$ℤ \oplus ℤ$	$0$	$ℤ \oplus ℤ$	$0$	$ℤ \oplus ℤ$	$0$	$ℤ \oplus ℤ$	$0$
$𝒮$	A	L	$𝒞_{p}$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$
$𝒮_{-}$	AIII	P	$𝒞_{p + 1} \times 𝒞_{p + 1}$	$0$	$ℤ \oplus ℤ$	$0$	$ℤ \oplus ℤ$	$0$	$ℤ \oplus ℤ$	$0$	$ℤ \oplus ℤ$
		$L_{r}$	$𝒞_{p + 1}$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$
		$L_{i}$	$𝒞_{p + 1}$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$

Table 6. Table S5: Topological classification table of exceptional points in real 𝒫 𝒫 {\cal P} AZ symmetry class with sublattice symmetry (SLS) 𝒮 𝒮 {\cal S} . Exceptional points are classified according to the symmetry class, the codimension p 𝑝 p , and the definition of a complex-energy point (P) or line (L) gap. The subscript of 𝒮 𝒮 {\cal S} specifies the commutation ( + + ) or anticommutation ( − - ) relation to 𝒫 𝒯 + 𝒫 subscript 𝒯 \mathcal{P}\mathcal{T}_{+} symmetry and/or 𝒞 − 𝒫 subscript 𝒞 𝒫 \mathcal{C}_{-}\mathcal{P} symmetry; for the symmetry classes that involve both 𝒫 𝒯 + 𝒫 subscript 𝒯 \mathcal{P}\mathcal{T}_{+} and 𝒞 − 𝒫 subscript 𝒞 𝒫 \mathcal{C}_{-}\mathcal{P} symmetries ( 𝒫 𝒫 {\cal P} BDI, 𝒫 𝒫 {\cal P} DIII, 𝒫 𝒫 {\cal P} CII, and 𝒫 𝒫 {\cal P} CI), the first and second subscripts specify the relation to 𝒫 𝒯 + 𝒫 subscript 𝒯 \mathcal{P}\mathcal{T}_{+} and 𝒞 − 𝒫 subscript 𝒞 𝒫 \mathcal{C}_{-}\mathcal{P} symmetries, respectively. The codimension p 𝑝 p is defined as p := d − d EP assign 𝑝 𝑑 subscript 𝑑 EP p:=d-d_{\rm EP} with the spatial dimension d 𝑑 d and the dimension d EP subscript 𝑑 EP d_{\rm EP} of the exceptional point. The subscript of L specifies the line gap for the real or imaginary part of the complex spectrum.

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SLS	$𝒫$ AZ class	Gap	Classifying space	$p = 0$	$p = 1$	$p = 2$	$p = 3$	$p = 4$	$p = 5$	$p = 6$	$p = 7$
$𝒮_{+ +}$	$𝒫$ BDI	P	$ℛ_{p}$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$
		$L_{r}$	$ℛ_{p} \times ℛ_{p}$	$ℤ \oplus ℤ$	$ℤ_{2} \oplus ℤ_{2}$	$ℤ_{2} \oplus ℤ_{2}$	$0$	$2 ℤ \oplus 2 ℤ$	$0$	$0$	$0$
		$L_{i}$	$ℛ_{p} \times ℛ_{p}$	$ℤ \oplus ℤ$	$ℤ_{2} \oplus ℤ_{2}$	$ℤ_{2} \oplus ℤ_{2}$	$0$	$2 ℤ \oplus 2 ℤ$	$0$	$0$	$0$
$𝒮_{- -}$	$𝒫$ DIII	P	$ℛ_{p + 2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$
		$L_{r}$	$ℛ_{p + 2} \times ℛ_{p + 2}$	$ℤ_{2} \oplus ℤ_{2}$	$0$	$2 ℤ \oplus 2 ℤ$	$0$	$0$	$0$	$ℤ \oplus ℤ$	$ℤ_{2} \oplus ℤ_{2}$
		$L_{i}$	$𝒞_{p}$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$
$𝒮_{+ +}$	$𝒫$ CII	P	$ℛ_{p + 4}$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$
		$L_{r}$	$ℛ_{p + 4} \times ℛ_{p + 4}$	$2 ℤ \oplus 2 ℤ$	$0$	$0$	$0$	$ℤ \oplus ℤ$	$ℤ_{2} \oplus ℤ_{2}$	$ℤ_{2} \oplus ℤ_{2}$	$0$
		$L_{i}$	$ℛ_{p + 4} \times ℛ_{p + 4}$	$2 ℤ \oplus 2 ℤ$	$0$	$0$	$0$	$ℤ \oplus ℤ$	$ℤ_{2} \oplus ℤ_{2}$	$ℤ_{2} \oplus ℤ_{2}$	$0$
$𝒮_{- -}$	$𝒫$ CI	P	$ℛ_{p + 6}$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$
		$L_{r}$	$ℛ_{p + 6} \times ℛ_{p + 6}$	$0$	$0$	$ℤ \oplus ℤ$	$ℤ_{2} \oplus ℤ_{2}$	$ℤ_{2} \oplus ℤ_{2}$	$0$	$2 ℤ \oplus 2 ℤ$	$0$
		$L_{i}$	$𝒞_{p}$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$
$𝒮_{-}$	$𝒫$ AI	P	$𝒞_{p}$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$
		$L_{r}$	$ℛ_{p + 6}$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$
		$L_{i}$	$ℛ_{p + 2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$
$𝒮_{- +}$	$𝒫$ BDI	P	$𝒞_{p + 1}$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$
		$L_{r}$	$ℛ_{p + 7}$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$
		$L_{i}$	$ℛ_{p + 1}$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$
$𝒮_{+}$	$𝒫$ D	P	$𝒞_{p}$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$
$𝒮_{+}$	$𝒫$ D	L	$ℛ_{p}$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$
$𝒮_{- +}$	$𝒫$ DIII	P	$𝒞_{p + 1}$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$
		$L_{r}$	$ℛ_{p + 1}$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$
		$L_{i}$	$ℛ_{p + 7}$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$
$𝒮_{-}$	$𝒫$ AII	P	$𝒞_{p}$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$
		$L_{r}$	$ℛ_{p + 2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$
		$L_{i}$	$ℛ_{p + 6}$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$
$𝒮_{- +}$	$𝒫$ CII	P	$𝒞_{p + 1}$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$
		$L_{r}$	$ℛ_{p + 3}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$
		$L_{i}$	$ℛ_{p + 5}$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$
$𝒮_{+}$	$𝒫$ C	P	$𝒞_{p}$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$
$𝒮_{+}$	$𝒫$ C	L	$ℛ_{p + 4}$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$
$𝒮_{- +}$	$𝒫$ CI	P	$𝒞_{p + 1}$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$
		$L_{r}$	$ℛ_{p + 5}$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$
		$L_{i}$	$ℛ_{p + 3}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$

TABLE S5 — continued
$𝒮_{- -}$	$𝒫$ BDI	P	$ℛ_{p + 2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$
		$L_{r}$	$𝒞_{p}$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$
		$L_{i}$	$ℛ_{p + 2} \times ℛ_{p + 2}$	$ℤ_{2} \oplus ℤ_{2}$	$0$	$2 ℤ \oplus 2 ℤ$	$0$	$0$	$0$	$ℤ \oplus ℤ$	$ℤ_{2} \oplus ℤ_{2}$
$𝒮_{+ +}$	$𝒫$ DIII	P	$ℛ_{p + 4}$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$
		$L_{r}$	$𝒞_{p}$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$
		$L_{i}$	$𝒞_{p}$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$
$𝒮_{- -}$	$𝒫$ CII	P	$ℛ_{p + 6}$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$
		$L_{r}$	$𝒞_{p}$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$
		$L_{i}$	$ℛ_{p + 6} \times ℛ_{p + 6}$	$0$	$0$	$ℤ \oplus ℤ$	$ℤ_{2} \oplus ℤ_{2}$	$ℤ_{2} \oplus ℤ_{2}$	$0$	$2 ℤ \oplus 2 ℤ$	$0$
$𝒮_{+ +}$	$𝒫$ CI	P	$ℛ_{p}$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$
		$L_{r}$	$𝒞_{p}$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$
		$L_{i}$	$𝒞_{p}$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$
$𝒮_{+}$	$𝒫$ AI	P	$ℛ_{p} \times ℛ_{p}$	$ℤ \oplus ℤ$	$ℤ_{2} \oplus ℤ_{2}$	$ℤ_{2} \oplus ℤ_{2}$	$0$	$2 ℤ \oplus 2 ℤ$	$0$	$0$	$0$
		$L_{r}$	$ℛ_{p}$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$
		$L_{i}$	$ℛ_{p}$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$
$𝒮_{+ -}$	$𝒫$ BDI	P	$ℛ_{p + 1} \times ℛ_{p + 1}$	$ℤ_{2} \oplus ℤ_{2}$	$ℤ_{2} \oplus ℤ_{2}$	$0$	$2 ℤ \oplus 2 ℤ$	$0$	$0$	$0$	$ℤ \oplus ℤ$
		$L_{r}$	$ℛ_{p + 1}$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$
		$L_{i}$	$ℛ_{p + 1}$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$
$𝒮_{-}$	$𝒫$ D	P	$ℛ_{p + 2} \times ℛ_{p + 2}$	$ℤ_{2} \oplus ℤ_{2}$	$0$	$2 ℤ \oplus 2 ℤ$	$0$	$0$	$0$	$ℤ \oplus ℤ$	$ℤ_{2} \oplus ℤ_{2}$
$𝒮_{-}$	$𝒫$ D	L	$ℛ_{p + 2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$
$𝒮_{+ -}$	$𝒫$ DIII	P	$ℛ_{p + 3} \times ℛ_{p + 3}$	$0$	$2 ℤ \oplus 2 ℤ$	$0$	$0$	$0$	$ℤ \oplus ℤ$	$ℤ_{2} \oplus ℤ_{2}$	$ℤ_{2} \oplus ℤ_{2}$
		$L_{r}$	$ℛ_{p + 3}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$
		$L_{i}$	$ℛ_{p + 3}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$
$𝒮_{+}$	$𝒫$ AII	P	$ℛ_{p + 4} \times ℛ_{p + 4}$	$2 ℤ \oplus 2 ℤ$	$0$	$0$	$0$	$ℤ \oplus ℤ$	$ℤ_{2} \oplus ℤ_{2}$	$ℤ_{2} \oplus ℤ_{2}$	$0$
		$L_{r}$	$ℛ_{p + 4}$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$
		$L_{i}$	$ℛ_{p + 4}$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$
$𝒮_{+ -}$	$𝒫$ CII	P	$ℛ_{p + 5} \times ℛ_{p + 5}$	$0$	$0$	$0$	$ℤ \oplus ℤ$	$ℤ_{2} \oplus ℤ_{2}$	$ℤ_{2} \oplus ℤ_{2}$	$0$	$2 ℤ \oplus 2 ℤ$
		$L_{r}$	$ℛ_{p + 5}$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$
		$L_{i}$	$ℛ_{p + 5}$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$
$𝒮_{-}$	$𝒫$ C	P	$ℛ_{p + 6} \times ℛ_{p + 6}$	$0$	$0$	$ℤ \oplus ℤ$	$ℤ_{2} \oplus ℤ_{2}$	$ℤ_{2} \oplus ℤ_{2}$	$0$	$2 ℤ \oplus 2 ℤ$	$0$
$𝒮_{-}$	$𝒫$ C	L	$ℛ_{p + 6}$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$
$𝒮_{+ -}$	$𝒫$ CI	P	$ℛ_{p + 7} \times ℛ_{p + 7}$	$0$	$ℤ \oplus ℤ$	$ℤ_{2} \oplus ℤ_{2}$	$ℤ_{2} \oplus ℤ_{2}$	$0$	$2 ℤ \oplus 2 ℤ$	$0$	$0$
		$L_{r}$	$ℛ_{p + 7}$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$
		$L_{i}$	$ℛ_{p + 7}$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$

Table 8. Table S6: Topological classification table of exceptional points in complex AZ symmetry class with pseudo-Hermiticity (pH) η 𝜂 \eta . Exceptional points are classified according to the symmetry class, the codimension p 𝑝 p , and the definition of a complex-energy point (P) or line (L) gap. The subscript of η 𝜂 \eta specifies the commutation ( + + ) or anticommutation ( − - ) relation to chiral symmetry. The codimension p 𝑝 p is defined as p := d − d EP assign 𝑝 𝑑 subscript 𝑑 EP p:=d-d_{\rm EP} with the spatial dimension d 𝑑 d and the dimension d EP subscript 𝑑 EP d_{\rm EP} of the exceptional point. The subscript of L specifies the line gap for the real or imaginary part of the complex spectrum.

pH	AZ class	Gap	Classifying space	$p = 0$	$p = 1$	$p = 2$	$p = 3$	$p = 4$	$p = 5$	$p = 6$	$p = 7$
$η$	A	P	$𝒞_{p + 1}$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$
		$L_{r}$	$𝒞_{p + 1} \times 𝒞_{p + 1}$	$0$	$ℤ \oplus ℤ$	$0$	$ℤ \oplus ℤ$	$0$	$ℤ \oplus ℤ$	$0$	$ℤ \oplus ℤ$
		$L_{i}$	$𝒞_{p}$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$
$η_{+}$	AIII	P	$𝒞_{p}$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$
		$L_{r}$	$𝒞_{p} \times 𝒞_{p}$	$ℤ \oplus ℤ$	$0$	$ℤ \oplus ℤ$	$0$	$ℤ \oplus ℤ$	$0$	$ℤ \oplus ℤ$	$0$
		$L_{i}$	$𝒞_{p} \times 𝒞_{p}$	$ℤ \oplus ℤ$	$0$	$ℤ \oplus ℤ$	$0$	$ℤ \oplus ℤ$	$0$	$ℤ \oplus ℤ$	$0$
$η_{-}$	AIII	P	$𝒞_{p + 1} \times 𝒞_{p + 1}$	$0$	$ℤ \oplus ℤ$	$0$	$ℤ \oplus ℤ$	$0$	$ℤ \oplus ℤ$	$0$	$ℤ \oplus ℤ$
		$L_{r}$	$𝒞_{p + 1}$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$
		$L_{i}$	$𝒞_{p + 1}$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$

Table 9. Table S7: Topological classification table of exceptional points in real 𝒫 𝒫 {\cal P} AZ symmetry class with pseudo-Hermiticity (pH) η 𝜂 \eta . Exceptional points are classified according to the symmetry class, the codimension p 𝑝 p , and the definition of a complex-energy point (P) or line (L) gap. The subscript of η 𝜂 \eta specifies the commutation ( + + ) or anticommutation ( − - ) relation to 𝒫 𝒯 + 𝒫 subscript 𝒯 \mathcal{P}\mathcal{T}_{+} symmetry and/or 𝒞 − 𝒫 subscript 𝒞 𝒫 \mathcal{C}_{-}\mathcal{P} symmetry; for the symmetry classes that involve both 𝒫 𝒯 + 𝒫 subscript 𝒯 \mathcal{P}\mathcal{T}_{+} and 𝒞 − 𝒫 subscript 𝒞 𝒫 \mathcal{C}_{-}\mathcal{P} symmetries ( 𝒫 𝒫 {\cal P} BDI, 𝒫 𝒫 {\cal P} DIII, 𝒫 𝒫 {\cal P} CII, and 𝒫 𝒫 {\cal P} CI), the first and second subscripts specify the relation to 𝒫 𝒯 + 𝒫 subscript 𝒯 \mathcal{P}\mathcal{T}_{+} and 𝒞 − 𝒫 subscript 𝒞 𝒫 \mathcal{C}_{-}\mathcal{P} symmetries, respectively. The codimension p 𝑝 p is defined as p := d − d EP assign 𝑝 𝑑 subscript 𝑑 EP p:=d-d_{\rm EP} with the spatial dimension d 𝑑 d and the dimension d EP subscript 𝑑 EP d_{\rm EP} of the exceptional point. The subscript of L specifies the line gap for the real or imaginary part of the complex spectrum.

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pH	$𝒫$ AZ class	Gap	Classifying space	$p = 0$	$p = 1$	$p = 2$	$p = 3$	$p = 4$	$p = 5$	$p = 6$	$p = 7$
$η_{+}$	$𝒫$ AI	P	$ℛ_{p + 7}$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$
		$L_{r}$	$ℛ_{p + 7} \times ℛ_{p + 7}$	$0$	$ℤ \oplus ℤ$	$ℤ_{2} \oplus ℤ_{2}$	$ℤ_{2} \oplus ℤ_{2}$	$0$	$2 ℤ \oplus 2 ℤ$	$0$	$0$
		$L_{i}$	$ℛ_{p}$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$
$η_{+ +}$	$𝒫$ BDI	P	$ℛ_{p}$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$
		$L_{r}$	$ℛ_{p} \times ℛ_{p}$	$ℤ \oplus ℤ$	$ℤ_{2} \oplus ℤ_{2}$	$ℤ_{2} \oplus ℤ_{2}$	$0$	$2 ℤ \oplus 2 ℤ$	$0$	$0$	$0$
		$L_{i}$	$ℛ_{p} \times ℛ_{p}$	$ℤ \oplus ℤ$	$ℤ_{2} \oplus ℤ_{2}$	$ℤ_{2} \oplus ℤ_{2}$	$0$	$2 ℤ \oplus 2 ℤ$	$0$	$0$	$0$
$η_{+}$	$𝒫$ D	P	$ℛ_{p + 1}$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$
		$L_{r}$	$ℛ_{p + 1} \times ℛ_{p + 1}$	$ℤ_{2} \oplus ℤ_{2}$	$ℤ_{2} \oplus ℤ_{2}$	$0$	$2 ℤ \oplus 2 ℤ$	$0$	$0$	$0$	$ℤ \oplus ℤ$
		$L_{i}$	$ℛ_{p}$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$
$η_{+ +}$	$𝒫$ DIII	P	$ℛ_{p + 2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$
		$L_{r}$	$ℛ_{p + 2} \times ℛ_{p + 2}$	$ℤ_{2} \oplus ℤ_{2}$	$0$	$2 ℤ \oplus 2 ℤ$	$0$	$0$	$0$	$ℤ \oplus ℤ$	$ℤ_{2} \oplus ℤ_{2}$
		$L_{i}$	$𝒞_{p}$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$
$η_{+}$	$𝒫$ AII	P	$ℛ_{p + 3}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$
		$L_{r}$	$ℛ_{p + 3} \times ℛ_{p + 3}$	$0$	$2 ℤ \oplus 2 ℤ$	$0$	$0$	$0$	$ℤ \oplus ℤ$	$ℤ_{2} \oplus ℤ_{2}$	$ℤ_{2} \oplus ℤ_{2}$
		$L_{i}$	$ℛ_{p + 4}$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$
$η_{+ +}$	$𝒫$ CII	P	$ℛ_{p + 4}$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$
		$L_{r}$	$ℛ_{p + 4} \times ℛ_{p + 4}$	$2 ℤ \oplus 2 ℤ$	$0$	$0$	$0$	$ℤ \oplus ℤ$	$ℤ_{2} \oplus ℤ_{2}$	$ℤ_{2} \oplus ℤ_{2}$	$0$
		$L_{i}$	$ℛ_{p + 4} \times ℛ_{p + 4}$	$2 ℤ \oplus 2 ℤ$	$0$	$0$	$0$	$ℤ \oplus ℤ$	$ℤ_{2} \oplus ℤ_{2}$	$ℤ_{2} \oplus ℤ_{2}$	$0$
$η_{+}$	$𝒫$ C	P	$ℛ_{p + 5}$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$
		$L_{r}$	$ℛ_{p + 5} \times ℛ_{p + 5}$	$0$	$0$	$0$	$ℤ \oplus ℤ$	$ℤ_{2} \oplus ℤ_{2}$	$ℤ_{2} \oplus ℤ_{2}$	$0$	$2 ℤ \oplus 2 ℤ$
		$L_{i}$	$ℛ_{p + 4}$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$
$η_{+ +}$	$𝒫$ CI	P	$ℛ_{p + 6}$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$
		$L_{r}$	$ℛ_{p + 6} \times ℛ_{p + 6}$	$0$	$0$	$ℤ \oplus ℤ$	$ℤ_{2} \oplus ℤ_{2}$	$ℤ_{2} \oplus ℤ_{2}$	$0$	$2 ℤ \oplus 2 ℤ$	$0$
		$L_{i}$	$𝒞_{p}$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$

TABLE S7 — continued
$η_{+ -}$	$𝒫$ BDI	P	$𝒞_{p + 1}$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$
		$L_{r}$	$ℛ_{p + 7}$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$
		$L_{i}$	$ℛ_{p + 1}$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$
$η_{- +}$	$𝒫$ DIII	P	$𝒞_{p + 1}$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$
		$L_{r}$	$ℛ_{p + 1}$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$
		$L_{i}$	$ℛ_{p + 7}$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$
$η_{+ -}$	$𝒫$ CII	P	$𝒞_{p + 1}$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$
		$L_{r}$	$ℛ_{p + 3}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$
		$L_{i}$	$ℛ_{p + 5}$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$
$η_{- +}$	$𝒫$ CI	P	$𝒞_{p + 1}$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$
		$L_{r}$	$ℛ_{p + 5}$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$
		$L_{i}$	$ℛ_{p + 3}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$
$η_{-}$	$𝒫$ AI	P	$ℛ_{p + 1}$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$
		$L_{r}$	$𝒞_{p + 1}$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$
		$L_{i}$	$ℛ_{p + 2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$
$η_{- -}$	$𝒫$ BDI	P	$ℛ_{p + 2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$
		$L_{r}$	$𝒞_{p}$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$
		$L_{i}$	$ℛ_{p + 2} \times ℛ_{p + 2}$	$ℤ_{2} \oplus ℤ_{2}$	$0$	$2 ℤ \oplus 2 ℤ$	$0$	$0$	$0$	$ℤ \oplus ℤ$	$ℤ_{2} \oplus ℤ_{2}$
$η_{-}$	$𝒫$ D	P	$ℛ_{p + 3}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$
		$L_{r}$	$𝒞_{p + 1}$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$
		$L_{i}$	$ℛ_{p + 2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$
$η_{- -}$	$𝒫$ DIII	P	$ℛ_{p + 4}$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$
		$L_{r}$	$𝒞_{p}$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$
		$L_{i}$	$𝒞_{p}$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$
$η_{-}$	$𝒫$ AII	P	$ℛ_{p + 5}$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$
		$L_{r}$	$𝒞_{p + 1}$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$
		$L_{i}$	$ℛ_{p + 6}$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$
$η_{- -}$	$𝒫$ CII	P	$ℛ_{p + 6}$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$
		$L_{r}$	$𝒞_{p}$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$
		$L_{i}$	$ℛ_{p + 6} \times ℛ_{p + 6}$	$0$	$0$	$ℤ \oplus ℤ$	$ℤ_{2} \oplus ℤ_{2}$	$ℤ_{2} \oplus ℤ_{2}$	$0$	$2 ℤ \oplus 2 ℤ$	$0$
$η_{-}$	$𝒫$ C	P	$ℛ_{p + 7}$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$
		$L_{r}$	$𝒞_{p + 1}$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$
		$L_{i}$	$ℛ_{p + 6}$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$
$η_{- -}$	$𝒫$ CI	P	$ℛ_{p}$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$
		$L_{r}$	$𝒞_{p}$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$
		$L_{i}$	$𝒞_{p}$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$	$ℤ$	$0$
$η_{- +}$	$𝒫$ BDI	P	$ℛ_{p + 1} \times ℛ_{p + 1}$	$ℤ_{2} \oplus ℤ_{2}$	$ℤ_{2} \oplus ℤ_{2}$	$0$	$2 ℤ \oplus 2 ℤ$	$0$	$0$	$0$	$ℤ \oplus ℤ$
		$L_{r}$	$ℛ_{p + 1}$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$
		$L_{i}$	$ℛ_{p + 1}$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$
$η_{+ -}$	$𝒫$ DIII	P	$ℛ_{p + 3} \times ℛ_{p + 3}$	$0$	$2 ℤ \oplus 2 ℤ$	$0$	$0$	$0$	$ℤ \oplus ℤ$	$ℤ_{2} \oplus ℤ_{2}$	$ℤ_{2} \oplus ℤ_{2}$
		$L_{r}$	$ℛ_{p + 3}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$
		$L_{i}$	$ℛ_{p + 3}$	$0$	2 $ℤ$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$
$η_{- +}$	$𝒫$ CII	P	$ℛ_{p + 5} \times ℛ_{p + 5}$	$0$	$0$	$0$	$ℤ \oplus ℤ$	$ℤ_{2} \oplus ℤ_{2}$	$ℤ_{2} \oplus ℤ_{2}$	$0$	$2 ℤ \oplus 2 ℤ$
		$L_{r}$	$ℛ_{p + 5}$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$
		$L_{i}$	$ℛ_{p + 5}$	$0$	$0$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$
$η_{+ -}$	$𝒫$ CI	P	$ℛ_{p + 7} \times ℛ_{p + 7}$	$0$	$ℤ \oplus ℤ$	$ℤ_{2} \oplus ℤ_{2}$	$ℤ_{2} \oplus ℤ_{2}$	$0$	$2 ℤ \oplus 2 ℤ$	$0$	$0$
		$L_{r}$	$ℛ_{p + 7}$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$
		$L_{i}$	$ℛ_{p + 7}$	$0$	$ℤ$	$ℤ_{2}$	$ℤ_{2}$	$0$	2 $ℤ$	$0$	$0$

Table 11. Table S8: Classifying spaces and their homotopy groups.

	Classifying space	$π_{0} (⋆)$
$𝒞_{0}$	$[U (m + n) / U (m) \times U (n)] \times ℤ$	$ℤ$
$𝒞_{1}$	$U (n)$	$0$
$ℛ_{0}$	$[O (m + n) / O (m) \times O (n)] \times ℤ$	$ℤ$
$ℛ_{1}$	$O (n)$	$ℤ_{2}$
$ℛ_{2}$	$O (2 n) / U (n)$	$ℤ_{2}$
$ℛ_{3}$	$U (2 n) / Sp (n)$	$0$
$ℛ_{4}$	$[Sp (m + n) / Sp (m) \times Sp (n)] \times ℤ$	$ℤ$
$ℛ_{5}$	$Sp (n)$	$0$
$ℛ_{6}$	$Sp (n) / U (n)$	$0$
$ℛ_{7}$	$U (n) / O (n)$	$0$

Table 12. Table S9: Clifford algebra extensions and classifying spaces. 𝒫 𝒯 + 𝒫 subscript 𝒯 \mathcal{P}\mathcal{T}_{+} ( 𝖯𝖳 + subscript 𝖯𝖳 {\sf P}{\sf T}_{+} ) and 𝒞 − 𝒫 subscript 𝒞 𝒫 \mathcal{C}_{-}\mathcal{P} ( 𝖢 − 𝖯 subscript 𝖢 𝖯 {\sf C}_{-}{\sf P} ) symmetries are respectively denoted as 𝒫 𝒯 𝒫 𝒯 \mathcal{PT} ( 𝖯𝖳 𝖯𝖳 {\sf PT} ) and 𝒞 𝒫 𝒞 𝒫 \mathcal{CP} ( 𝖢𝖯 𝖢𝖯 {\sf CP} ) symmetries for simplicity.

continued on next page
Symmetry class	Gap	Generator	Extension	Classifying space
A	P	${γ_{1}, \dots, γ_{p}, Σ}$	$C l_{p} \to C l_{p + 1}$	$𝒞_{p}$
A	L	${γ_{1}, \dots, γ_{p}}$	$C l_{p - 1} \to C l_{p}$	$𝒞_{p + 1}$
AIII	P	${γ_{1}, \dots, γ_{p}, Σ, Γ}$	$C l_{p + 1} \to C l_{p + 2}$	$𝒞_{p + 1}$
	$L_{r}$	${γ_{1}, \dots, γ_{p}, Γ}$	$C l_{p} \to C l_{p + 1}$	$𝒞_{p}$
	$L_{i}$	${γ_{1}, \dots, γ_{p}} \otimes {Γ}$	$C l_{p - 1} \times C l_{p - 1} \to C l_{p} \times C l_{p}$	$𝒞_{p + 1} \times 𝒞_{p + 1}$
$𝒫$ AI	P	${J γ_{1}, \dots, J γ_{p}, J Σ; 𝖯𝖳 𝒦, J 𝖯𝖳 𝒦}$	$C l_{p, 2} \to C l_{p + 1, 2}$	$ℛ_{p}$
	$L_{r}$	${J γ_{1}, \dots, J γ_{p}; 𝒫 𝒯 𝒦, J 𝒫 𝒯 𝒦}$	$C l_{p - 1, 2} \to C l_{p, 2}$	$ℛ_{p + 7}$
	$L_{i}$	${γ_{1}, \dots, γ_{p}, 𝒫 𝒯 𝒦, J 𝒫 𝒯 𝒦}$	$C l_{0, p + 1} \to C l_{0, p + 2}$	$ℛ_{p + 1}$
$𝒫$ BDI	P	${J (𝖢𝖯 𝒦) (𝖯𝖳 𝒦); γ_{1}, \dots, γ_{p}, Σ, 𝖢𝖯 𝒦, J 𝖢𝖯 𝒦}$	$C l_{1, p + 2} \to C l_{1, p + 3}$	$ℛ_{p + 1}$
	$L_{r}$	${J (𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦); γ_{1}, \dots, γ_{p}, 𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦}$	$C l_{1, p + 1} \to C l_{1, p + 2}$	$ℛ_{p}$
	$L_{i}$	${γ_{1}, \dots, γ_{p}, 𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦} \otimes {(𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦)}$	$C l_{0, p + 1} \times C l_{0, p + 1} \to C l_{0, p + 2} \times C l_{0, p + 2}$	$ℛ_{p + 1} \times ℛ_{p + 1}$
$𝒫$ D	P	${γ_{1}, \dots, γ_{p}, Σ, 𝖢𝖯 𝒦, J 𝖢𝖯 𝒦}$	$C l_{0, p + 2} \to C l_{0, p + 3}$	$ℛ_{p + 2}$
$𝒫$ D	L	${γ_{1}, \dots, γ_{p}, 𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦}$	$C l_{0, p + 1} \to C l_{0, p + 2}$	$ℛ_{p + 1}$
$𝒫$ DIII	P	${γ_{1}, \dots, γ_{p}, Σ, 𝖢𝖯 𝒦, J 𝖢𝖯 𝒦, J (𝖢𝖯 𝒦) (𝖯𝖳 𝒦)}$	$C l_{0, p + 3} \to C l_{0, p + 4}$	$ℛ_{p + 3}$
	$L_{r}$	${γ_{1}, \dots, γ_{p}, 𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦, J (𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦)}$	$C l_{0, p + 2} \to C l_{0, p + 3}$	$ℛ_{p + 2}$
	$L_{i}$	${γ_{1}, \dots, γ_{p}, 𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦} \otimes {(𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦)}$	$C l_{p + 1} \to C l_{p + 2}$	$𝒞_{p + 1}$
$𝒫$ AII	P	${J γ_{1}, \dots, J γ_{p}, J Σ, 𝖯𝖳 𝒦, J 𝖯𝖳 𝒦}$	$C l_{p + 2, 0} \to C l_{p + 3, 0}$	$ℛ_{p + 4}$
	$L_{r}$	${J γ_{1}, \dots, J γ_{p}, 𝒫 𝒯 𝒦, J 𝒫 𝒯 𝒦}$	$C l_{p + 1, 0} \to C l_{p + 2, 0}$	$ℛ_{p + 3}$
	$L_{i}$	${𝒫 𝒯 𝒦, J 𝒫 𝒯 𝒦; γ_{1}, \dots, γ_{p}}$	$C l_{2, p - 1} \to C l_{2, p}$	$ℛ_{p + 5}$
$𝒫$ CII	P	${𝖢𝖯 𝒦, J 𝖢𝖯 𝒦, J (𝖢𝖯 𝒦) (𝖯𝖳 𝒦); γ_{1}, \dots, γ_{p}, Σ}$	$C l_{3, p} \to C l_{3, p + 1}$	$ℛ_{p + 5}$
	$L_{r}$	${𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦, J (𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦); γ_{1}, \dots, γ_{p}}$	$C l_{3, p - 1} \to C l_{3, p}$	$ℛ_{p + 4}$
	$L_{i}$	${𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦; γ_{1}, \dots, γ_{p}} \otimes {(𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦)}$	$C l_{2, p - 1} \times C l_{2, p - 1} \to C l_{2, p} \times C l_{2, p}$	$ℛ_{p + 5} \times ℛ_{p + 5}$
$𝒫$ C	P	${𝖢𝖯 𝒦, J 𝖢𝖯 𝒦; γ_{1}, \dots, γ_{p}, Σ}$	$C l_{2, p} \to C l_{2, p + 1}$	$ℛ_{p + 6}$
$𝒫$ C	L	${𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦; γ_{1}, \dots, γ_{p}}$	$C l_{2, p - 1} \to C l_{2, p}$	$ℛ_{p + 5}$
$𝒫$ CI	P	${𝖢𝖯 𝒦, J 𝖢𝖯 𝒦; γ_{1}, \dots, γ_{p}, Σ, J (𝖢𝖯 𝒦) (𝖯𝖳 𝒦)}$	$C l_{2, p + 1} \to C l_{2, p + 2}$	$ℛ_{p + 7}$
	$L_{r}$	${𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦; γ_{1}, \dots, γ_{p}, J (𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦)}$	$C l_{2, p} \to C l_{2, p + 1}$	$ℛ_{p + 6}$
	$L_{i}$	${𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦; γ_{1}, \dots, γ_{p}} \otimes {(𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦)}$	$C l_{p + 1} \to C l_{p + 2}$	$𝒞_{p + 1}$

TABLE S9 — continued
$𝒫 {AI}^{†}$	P	${J γ_{1}, \dots, J γ_{p}; Σ, 𝖢_{+} 𝖯 𝒦, J 𝖢_{+} 𝖯 𝒦}$	$C l_{p - 1, 3} \to C l_{p, 3}$	$ℛ_{p + 6}$
$𝒫 {AI}^{†}$	L	${J γ_{1}, \dots, J γ_{p}; 𝒞_{+} 𝒫 𝒦, J 𝒞_{+} 𝒫 𝒦}$	$C l_{p - 1, 2} \to C l_{p, 2}$	$ℛ_{p + 7}$
$𝒫 {BDI}^{†}$	P	${Σ (𝖢_{+} 𝖯 𝒦), J Σ (𝖢_{+} 𝖯 𝒦); γ_{1}, \dots, γ_{p}, Σ, ({𝖯𝖳}_{-} 𝒦) (𝖢_{+} 𝖯 𝒦)}$	$C l_{2, p + 1} \to C l_{2, p + 2}$	$ℛ_{p + 7}$
	$L_{r}$	${J (𝒫 𝒯_{-} 𝒦) (𝒞_{+} 𝒫 𝒦); γ_{1}, \dots, γ_{p}, 𝒫 𝒯_{-} 𝒦, J 𝒫 𝒯_{-} 𝒦}$	$C l_{1, p + 1} \to C l_{1, p + 2}$	$ℛ_{p}$
	$L_{i}$	${J γ_{1}, \dots, J γ_{p}; 𝒞_{+} 𝒫 𝒦, J 𝒞_{+} 𝒫 𝒦} \otimes {(𝒫 𝒯_{-} 𝒦) (𝒞_{+} 𝒫 𝒦)}$	$C l_{p - 1, 2} \times C l_{p - 1, 2} \to C l_{p, 2} \times C l_{p, 2}$	$ℛ_{p + 7} \times ℛ_{p + 7}$
$𝒫 D^{†}$	P	${J Σ; γ_{1}, \dots, γ_{p}, {𝖯𝖳}_{-} 𝒦, J {𝖯𝖳}_{-} 𝒦}$	$C l_{1, p + 1} \to C l_{1, p + 2}$	$ℛ_{p}$
	$L_{r}$	${γ_{1}, \dots, γ_{p}, 𝒫 𝒯_{-} 𝒦, J 𝒫 𝒯_{-} 𝒦}$	$C l_{0, p + 1} \to C l_{0, p + 2}$	$ℛ_{p + 1}$
	$L_{i}$	${J γ_{1}, \dots, J γ_{p}; 𝒫 𝒯_{-} 𝒦, J 𝒫 𝒯_{-} 𝒦}$	$C l_{p - 1, 2} \to C l_{p, 2}$	$ℛ_{p + 7}$
$𝒫 {DIII}^{†}$	P	${({𝖯𝖳}_{-} 𝒦) (𝖢_{+} 𝖯 𝒦); γ_{1}, \dots, γ_{p}, Σ, Σ (𝖢_{+} 𝖯 𝒦), J Σ (𝖢_{+} 𝖯 𝒦)}$	$C l_{1, p + 2} \to C l_{1, p + 3}$	$ℛ_{p + 1}$
	$L_{r}$	${γ_{1}, \dots, γ_{p}, 𝒫 𝒯_{-} 𝒦, J 𝒫 𝒯_{-} 𝒦, J (𝒫 𝒯_{-} 𝒦) (𝒞_{+} 𝒫 𝒦)}$	$C l_{0, p + 2} \to C l_{0, p + 3}$	$ℛ_{p + 2}$
	$L_{i}$	${J γ_{1}, \dots, J γ_{p}, 𝒞_{+} 𝒫 𝒦, J 𝒞_{+} 𝒫 𝒦} \otimes {(𝒫 𝒯_{-} 𝒦) (𝒞_{+} 𝒫 𝒦)}$	$C l_{p + 1} \to C l_{p + 2}$	$𝒞_{p + 1}$
$𝒫 {AII}^{†}$	P	${J γ_{1}, \dots, J γ_{p}, 𝖢_{+} 𝖯 𝒦, J 𝖢_{+} 𝖯 𝒦; Σ}$	$C l_{p + 1, 1} \to C l_{p + 2, 1}$	$ℛ_{p + 2}$
$𝒫 {AII}^{†}$	L	${J γ_{1}, \dots, J γ_{p}, 𝒞_{+} 𝒫 𝒦, J 𝒞_{+} 𝒫 𝒦}$	$C l_{p + 1, 0} \to C l_{p + 2, 0}$	$ℛ_{p + 3}$
$𝒫 {CII}^{†}$	P	${γ_{1}, \dots, γ_{p}, Σ, Σ (𝖢_{+} 𝖯 𝒦), J Σ (𝖢_{+} 𝖯 𝒦), ({𝖯𝖳}_{-} 𝒦) (𝖢_{+} 𝖯 𝒦)}$	$C l_{0, p + 3} \to C l_{0, p + 4}$	$ℛ_{p + 3}$
	$L_{r}$	${𝒫 𝒯_{-} 𝒦, J 𝒫 𝒯_{-} 𝒦, J (𝒫 𝒯_{-} 𝒦) (𝒞_{+} 𝒫 𝒦); γ_{1}, \dots, γ_{p}}$	$C l_{3, p - 1} \to C l_{3, p}$	$ℛ_{p + 4}$
	$L_{i}$	${J γ_{1}, \dots, J γ_{p}, 𝒞_{+} 𝒫 𝒦, J 𝒞_{+} 𝒫 𝒦} \otimes {(𝒫 𝒯_{-} 𝒦) (𝒞_{+} 𝒫 𝒦)}$	$C l_{p + 1, 0} \times C l_{p + 1, 0} \to C l_{p + 2, 0} \times C l_{p + 2, 0}$	$ℛ_{p + 3} \times ℛ_{p + 3}$
$𝒫 C^{†}$	P	${J Σ, {𝖯𝖳}_{-} 𝒦, J {𝖯𝖳}_{-} 𝒦; γ_{1}, \dots, γ_{p}}$	$C l_{3, p - 1} \to C l_{3, p}$	$ℛ_{p + 4}$
	$L_{r}$	${𝒫 𝒯_{-} 𝒦, J 𝒫 𝒯_{-} 𝒦; γ_{1}, \dots, γ_{p}}$	$C l_{2, p - 1} \to C l_{2, p}$	$ℛ_{p + 5}$
	$L_{i}$	${J γ_{1}, \dots, J γ_{p}, 𝒫 𝒯_{-} 𝒦, J 𝒫 𝒯_{-} 𝒦}$	$C l_{p + 1, 0} \to C l_{p + 2, 0}$	$ℛ_{p + 4}$
$𝒫 {CI}^{†}$	P	${Σ (𝖢_{+} 𝖯 𝒦), J Σ (𝖢_{+} 𝖯 𝒦), ({𝖯𝖳}_{-} 𝒦) (𝖢_{+} 𝖯 𝒦); γ_{1}, \dots, γ_{p}, Σ}$	$C l_{3, p} \to C l_{3, p + 1}$	$ℛ_{p + 5}$
	$L_{r}$	${𝒫 𝒯_{-} 𝒦, J 𝒫 𝒯_{-} 𝒦; γ_{1}, \dots, γ_{p}, J (𝒫 𝒯_{-} 𝒦) (𝒞_{+} 𝒫 𝒦)}$	$C l_{2, p} \to C l_{2, p + 1}$	$ℛ_{p + 6}$
	$L_{i}$	${J γ_{1}, \dots, J γ_{p}; 𝒞_{+} 𝒫 𝒦, J 𝒞_{+} 𝒫 𝒦} \otimes {({𝖯𝖳}_{-} 𝒦) (𝒞_{+} 𝒫 𝒦)}$	$C l_{p + 1} \to C l_{p + 2}$	$𝒞_{p + 1}$
AIII + $𝒮_{+}$	P	${γ_{1}, \dots, γ_{p}, Σ, Γ, Σ Γ 𝖲}$	$C l_{p + 2} \to C l_{p + 3}$	$𝒞_{p}$
	$L_{r}$	${γ_{1}, \dots, γ_{p}, Γ} \otimes {Γ 𝒮}$	$C l_{p} \times C l_{p} \to C l_{p + 1} \times C l_{p + 1}$	$𝒞_{p} \times 𝒞_{p}$
	$L_{i}$	${γ_{1}, \dots, γ_{p}, 𝒮} \otimes {Γ}$	$C l_{p} \times C l_{p} \to C l_{p + 1} \times C l_{p + 1}$	$𝒞_{p} \times 𝒞_{p}$
A + $𝒮$	P	${γ_{1}, \dots, γ_{p}, Σ} \otimes {𝖲}$	$C l_{p} \times C l_{p} \to C l_{p + 1} \times C l_{p + 1}$	$𝒞_{p} \times 𝒞_{p}$
A + $𝒮$	L	${γ_{1}, \dots, γ_{p}, 𝒮}$	$C l_{p} \to C l_{p + 1}$	$𝒞_{p}$
AIII + $𝒮_{-}$	P	${γ_{1}, \dots, γ_{p}, Σ, Γ} \otimes {Σ 𝖲}$	$C l_{p + 1} \times C l_{p + 1} \to C l_{p + 2} \times C l_{p + 2}$	$𝒞_{p + 1} \times 𝒞_{p + 1}$
	$L_{r}$	${γ_{1}, \dots, γ_{p}, Γ, 𝒮}$	$C l_{p + 1} \to C l_{p + 2}$	$𝒞_{p + 1}$
	$L_{i}$	${γ_{1}, \dots, γ_{p}, 𝒮, Γ 𝒮}$	$C l_{p + 1} \to C l_{p + 2}$	$𝒞_{p + 1}$
$𝒫$ BDI + $𝒮_{+ +}$	P	${J (𝖢𝖯 𝒦) (𝖯𝖳 𝒦), (𝖢𝖯 𝒦) (𝖯𝖳 𝒦) Σ 𝖲; γ_{1}, \dots, γ_{p}, Σ, 𝖢𝖯 𝒦, J 𝖢𝖯 𝒦}$	$C l_{2, p + 2} \to C l_{2, p + 3}$	$ℛ_{p}$
	$L_{r}$	${J (𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦); γ_{1}, \dots, γ_{p}, 𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦} \otimes {(𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦) 𝒮}$	$C l_{1, p + 1} \times C l_{1, p + 1} \to C l_{1, p + 2} \times C l_{1, p + 2}$	$ℛ_{p} \times ℛ_{p}$
	$L_{i}$	${J 𝒮; γ_{1}, \dots, γ_{p}, 𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦} \otimes {(𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦)}$	$C l_{1, p + 1} \times C l_{1, p + 1} \to C l_{1, p + 2} \times C l_{1, p + 2}$	$ℛ_{p} \times ℛ_{p}$
$𝒫$ DIII + $𝒮_{- -}$	P	${J (𝖢𝖯 𝒦) (𝖯𝖳 𝒦) Σ 𝖲; γ_{1}, \dots, γ_{p}, Σ, 𝖢𝖯 𝒦, J 𝖢𝖯 𝒦, J (𝖢𝖯 𝒦) (𝖯𝖳 𝒦)}$	$C l_{1, p + 3} \to C l_{1, p + 4}$	$ℛ_{p + 2}$
	$L_{r}$	${γ_{1}, \dots, γ_{p}, 𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦, J (𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦)} \otimes {(𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦) 𝒮}$	$C l_{0, p + 2} \times C l_{0, p + 2} \to C l_{0, p + 3} \times C l_{0, p + 3}$	$ℛ_{p + 2} \times ℛ_{p + 2}$
	$L_{i}$	${γ_{1}, \dots, γ_{p}, 𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦, 𝒮} \otimes {(𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦)}$	$C l_{p + 2} \to C l_{p + 3}$	$𝒞_{p}$
$𝒫$ CII + $𝒮_{+ +}$	P	${𝖢𝖯 𝒦, J 𝖢𝖯 𝒦, J (𝖢𝖯 𝒦) (𝖯𝖳 𝒦), (𝖢𝖯 𝒦) (𝖯𝖳 𝒦) Σ 𝖲; γ_{1}, \dots, γ_{p}, Σ}$	$C l_{4, p} \to C l_{4, p + 1}$	$ℛ_{p + 4}$
	$L_{r}$	${𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦, J (𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦); γ_{1}, \dots, γ_{p}} \otimes {(𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦) 𝒮}$	$C l_{3, p - 1} \times C l_{3, p - 1} \to C l_{3, p} \times C l_{3, p}$	$ℛ_{p + 4} \times ℛ_{p + 4}$
	$L_{i}$	${𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦, J 𝒮; γ_{1}, \dots, γ_{p}} \otimes {(𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦)}$	$C l_{3, p - 1} \times C l_{3, p - 1} \to C l_{3, p} \times C l_{3, p}$	$ℛ_{p + 4} \times ℛ_{p + 4}$
$𝒫$ CI + $𝒮_{- -}$	P	${𝖢𝖯 𝒦, J 𝖢𝖯 𝒦, J (𝖢𝖯 𝒦) (𝖯𝖳 𝒦) Σ 𝖲; γ_{1}, \dots, γ_{p}, Σ, J (𝖢𝖯 𝒦) (𝖯𝖳 𝒦)}$	$C l_{3, p + 1} \to C l_{3, p + 1}$	$ℛ_{p + 6}$
	$L_{r}$	${𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦; γ_{1}, \dots, γ_{p}, J (𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦)} \otimes {(𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦) 𝒮}$	$C l_{2, p} \times C l_{2, p} \to C l_{2, p + 1} \times C l_{2, p + 1}$	$ℛ_{p + 6} \times ℛ_{p + 6}$
	$L_{i}$	${𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦; γ_{1}, \dots, γ_{p}, 𝒮} \otimes {(𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦)}$	$C l_{p + 2} \to C l_{p + 3}$	$𝒞_{p}$
$𝒫$ AI + $𝒮_{-}$	P	${J γ_{1}, \dots, J γ_{p}, J Σ; 𝖯𝖳 𝒦, J 𝖯𝖳 𝒦} \otimes {J Σ 𝖲}$	$C l_{p + 2} \to C l_{p + 3}$	$𝒞_{p}$
	$L_{r}$	${J γ_{1}, \dots, J γ_{p}; 𝒫 𝒯 𝒦, J 𝒫 𝒯 𝒦, 𝒮}$	$C l_{p - 1, 3} \to C l_{p, 3}$	$ℛ_{p + 6}$
	$L_{i}$	${γ_{1}, \dots, γ_{p}, 𝒫 𝒯 𝒦, J 𝒫 𝒯 𝒦, 𝒮}$	$C l_{0, p + 2} \to C l_{0, p + 3}$	$ℛ_{p + 2}$
$𝒫$ BDI + $𝒮_{- +}$	P	${J (𝖢𝖯 𝒦) (𝖯𝖳 𝒦); γ_{1}, \dots, γ_{p}, Σ, 𝖢𝖯 𝒦, J 𝖢𝖯 𝒦} \otimes {J Σ 𝖲}$	$C l_{p + 3} \to C l_{p + 4}$	$𝒞_{p + 1}$
	$L_{r}$	${J 𝒮, J (𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦); γ_{1}, \dots, γ_{p}, 𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦}$	$C l_{2, p + 1} \to C l_{2, p + 2}$	$ℛ_{p + 7}$
	$L_{i}$	${J 𝒮; γ_{1}, \dots, γ_{p}, 𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦, J (𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦) 𝒮}$	$C l_{1, p + 2} \to C l_{1, p + 3}$	$ℛ_{p + 1}$
$𝒫$ D + $𝒮_{+}$	P	${γ_{1}, \dots, γ_{p}, Σ, 𝖢𝖯 𝒦, J 𝖢𝖯 𝒦} \otimes {J Σ 𝖲}$	$C l_{p + 2} \to C l_{p + 3}$	$𝒞_{p}$
$𝒫$ D + $𝒮_{+}$	L	${J 𝒮; γ_{1}, \dots, γ_{p}, 𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦}$	$C l_{1, p + 1} \to C l_{1, p + 2}$	$ℛ_{p}$
$𝒫$ DIII + $𝒮_{- +}$	P	${γ_{1}, \dots, γ_{p}, Σ, 𝖢𝖯 𝒦, J 𝖢𝖯 𝒦, J (𝖢𝖯 𝒦) (𝖯𝖳 𝒦)} \otimes {J Σ 𝖲}$	$C l_{p + 3} \to C l_{p + 4}$	$𝒞_{p + 1}$
	$L_{r}$	${J 𝒮; γ_{1}, \dots, γ_{p}, 𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦, J (𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦)}$	$C l_{1, p + 2} \to C l_{1, p + 3}$	$ℛ_{p + 1}$
	$L_{i}$	${J 𝒮, J (𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦) 𝒮; γ_{1}, \dots, γ_{p}, 𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦}$	$C l_{2, p + 1} \to C l_{2, p + 2}$	$ℛ_{p + 7}$
$𝒫$ AII + $𝒮_{-}$	P	${J γ_{1}, \dots, J γ_{p}, J Σ, 𝖯𝖳 𝒦, J 𝖯𝖳 𝒦} \otimes {J Σ 𝖲}$	$C l_{p + 2} \to C l_{p + 3}$	$𝒞_{p}$
	$L_{r}$	${J γ_{1}, \dots, J γ_{p}, 𝒫 𝒯 𝒦, J 𝒫 𝒯 𝒦; 𝒮}$	$C l_{p + 1, 1} \to C l_{p + 2, 1}$	$ℛ_{p + 2}$
	$L_{i}$	${𝒫 𝒯 𝒦, J 𝒫 𝒯 𝒦; γ_{1}, \dots, γ_{p}, 𝒮}$	$C l_{2, p} \to C l_{2, p + 1}$	$ℛ_{p + 6}$
$𝒫$ CII + $𝒮_{- +}$	P	${𝖢𝖯 𝒦, J 𝖢𝖯 𝒦, J (𝖢𝖯 𝒦) (𝖯𝖳 𝒦); γ_{1}, \dots, γ_{p}, Σ} \otimes {J Σ 𝖲}$	$C l_{p + 3} \to C l_{p + 4}$	$𝒞_{p + 1}$
	$L_{r}$	${𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦, J (𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦), J 𝒮; γ_{1}, \dots, γ_{p}}$	$C l_{4, p - 1} \to C l_{4, p}$	$ℛ_{p + 3}$
	$L_{i}$	${𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦, J 𝒮; γ_{1}, \dots, γ_{p}, J (𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦) 𝒮}$	$C l_{3, p} \to C l_{3, p + 1}$	$ℛ_{p + 5}$
$𝒫$ C + $𝒮_{+}$	P	${𝖢𝖯 𝒦, J 𝖢𝖯 𝒦; γ_{1}, \dots, γ_{p}, Σ} \otimes {J Σ 𝖲}$	$C l_{p + 2} \to C l_{p + 3}$	$𝒞_{p}$
$𝒫$ C + $𝒮_{+}$	L	${𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦, J 𝒮; γ_{1}, \dots, γ_{p}}$	$C l_{3, p - 1} \to C l_{3, p}$	$ℛ_{p + 4}$
$𝒫$ CI + $𝒮_{- +}$	P	${𝖢𝖯 𝒦, J 𝖢𝖯 𝒦; γ_{1}, \dots, γ_{p}, Σ, J (𝖢𝖯 𝒦) (𝖯𝖳 𝒦)} \otimes {J Σ 𝖲}$	$C l_{p + 3} \to C l_{p + 4}$	$𝒞_{p + 1}$
	$L_{r}$	${𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦, J 𝒮; γ_{1}, \dots, γ_{p}, J (𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦)}$	$C l_{3, p} \to C l_{3, p + 1}$	$ℛ_{p + 5}$
	$L_{i}$	${𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦, J 𝒮, J (𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦) 𝒮; γ_{1}, \dots, γ_{p}}$	$C l_{4, p - 1} \to C l_{4, p}$	$ℛ_{p + 3}$
continued on next page

TABLE S9 — continued
$𝒫$ BDI + $𝒮_{- -}$	P	${J (𝖢𝖯 𝒦) (𝖯𝖳 𝒦); γ_{1}, \dots, γ_{p}, Σ, 𝖢𝖯 𝒦, J 𝖢𝖯 𝒦, J (𝖢𝖯 𝒦) (𝖯𝖳 𝒦) Σ 𝖲}$	$C l_{1, p + 3} \to C l_{1, p + 4}$	$ℛ_{p + 2}$
	$L_{r}$	${J (𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦); γ_{1}, \dots, γ_{p}, 𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦} \otimes {(𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦) 𝒮}$	$C l_{p + 2} \to C l_{p + 3}$	$𝒞_{p}$
	$L_{i}$	${γ_{1}, \dots, γ_{p}, 𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦, 𝒮} \otimes {(𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦)}$	$C l_{0, p + 2} \times C l_{0, p + 2} \to C l_{0, p + 3} \times C l_{0, p + 3}$	$ℛ_{p + 2} \times ℛ_{p + 2}$
$𝒫$ DIII + $𝒮_{+ +}$	P	${γ_{1}, \dots, γ_{p}, Σ, 𝖢𝖯 𝒦, J 𝖢𝖯 𝒦, J (𝖢𝖯 𝒦) (𝖯𝖳 𝒦), (𝖢𝖯 𝒦) (𝖯𝖳 𝒦) Σ 𝖲}$	$C l_{0, p + 4} \to C l_{0, p + 5}$	$ℛ_{p + 4}$
	$L_{r}$	${γ_{1}, \dots, γ_{p}, 𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦, J (𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦)} \otimes {(𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦) 𝒮}$	$C l_{p + 2} \to C l_{p + 3}$	$𝒞_{p}$
	$L_{i}$	${J 𝒮; γ_{1}, \dots, γ_{p}, 𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦} \otimes {(𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦)}$	$C l_{p + 2} \to C l_{p + 3}$	$𝒞_{p}$
$𝒫$ CII + $𝒮_{- -}$	P	${𝖢𝖯 𝒦, J 𝖢𝖯 𝒦, J (𝖢𝖯 𝒦) (𝖯𝖳 𝒦); γ_{1}, \dots, γ_{p}, Σ, J (𝖢𝖯 𝒦) (𝖯𝖳 𝒦) Σ 𝖲}$	$C l_{3, p + 1} \to C l_{3, p + 1}$	$ℛ_{p + 6}$
	$L_{r}$	${𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦, J (𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦); γ_{1}, \dots, γ_{p}} \otimes {(𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦) 𝒮}$	$C l_{p + 2} \to C l_{p + 3}$	$𝒞_{p}$
	$L_{i}$	${𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦; γ_{1}, \dots, γ_{p}, 𝒮} \otimes {(𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦)}$	$C l_{2, p} \times C l_{2, p} \to C l_{2, p + 1} \times C l_{2, p + 1}$	$ℛ_{p + 6} \times ℛ_{p + 6}$
$𝒫$ CI + $𝒮_{+ +}$	P	${𝖢𝖯 𝒦, J 𝖢𝖯 𝒦; γ_{1}, \dots, γ_{p}, Σ, J (𝖢𝖯 𝒦) (𝖯𝖳 𝒦), (𝖢𝖯 𝒦) (𝖯𝖳 𝒦) Σ 𝖲}$	$C l_{2, p + 2} \to C l_{2, p + 3}$	$ℛ_{p}$
	$L_{r}$	${𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦; γ_{1}, \dots, γ_{p}, J (𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦)} \otimes {(𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦) 𝒮}$	$C l_{p + 2} \to C l_{p + 3}$	$𝒞_{p}$
	$L_{i}$	${𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦, J 𝒮; γ_{1}, \dots, γ_{p}} \otimes {(𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦)}$	$C l_{p + 2} \to C l_{p + 3}$	$𝒞_{p}$
$𝒫$ AI + $𝒮_{+}$	P	${J γ_{1}, \dots, J γ_{p}, J Σ; 𝖯𝖳 𝒦, J 𝖯𝖳 𝒦} \otimes {Σ 𝖲}$	$C l_{p, 2} \times C l_{p, 2} \to C l_{p + 1, 2} \times C l_{p + 1, 2}$	$ℛ_{p} \times ℛ_{p}$
	$L_{r}$	${J γ_{1}, \dots, J γ_{p}, 𝒮; 𝒫 𝒯 𝒦, J 𝒫 𝒯 𝒦}$	$C l_{p, 2} \to C l_{p + 1, 2}$	$ℛ_{p}$
	$L_{i}$	${J 𝒮; γ_{1}, \dots, γ_{p}, 𝒫 𝒯 𝒦, J 𝒫 𝒯 𝒦}$	$C l_{1, p + 1} \to C l_{1, p + 2}$	$ℛ_{p}$
$𝒫$ BDI + $𝒮_{+ -}$	P	${J (𝖢𝖯 𝒦) (𝖯𝖳 𝒦); γ_{1}, \dots, γ_{p}, Σ, 𝖢𝖯 𝒦, J 𝖢𝖯 𝒦} \otimes {Σ 𝖲}$	$C l_{1, p + 2} \times C l_{1, p + 2} \to C l_{1, p + 3} \times C l_{1, p + 3}$	$ℛ_{p + 1} \times ℛ_{p + 1}$
	$L_{r}$	${J (𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦); γ_{1}, \dots, γ_{p}, 𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦, 𝒮}$	$C l_{1, p + 2} \to C l_{1, p + 3}$	$ℛ_{p + 1}$
	$L_{i}$	${(𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦) 𝒮; γ_{1}, \dots, γ_{p}, 𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦, 𝒮}$	$C l_{1, p + 2} \to C l_{1, p + 3}$	$ℛ_{p + 1}$
$𝒫$ D + $𝒮_{-}$	P	${γ_{1}, \dots, γ_{p}, Σ, 𝖢𝖯 𝒦, J 𝖢𝖯 𝒦} \otimes {Σ 𝖲}$	$C l_{0, p + 2} \times C l_{0, p + 2} \to C l_{0, p + 3} \times C l_{0, p + 3}$	$ℛ_{p + 2} \times ℛ_{p + 2}$
$𝒫$ D + $𝒮_{-}$	L	${γ_{1}, \dots, γ_{p}, 𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦, 𝒮}$	$C l_{0, p + 2} \to C l_{0, p + 3}$	$ℛ_{p + 2}$
$𝒫$ DIII + $𝒮_{+ -}$	P	${γ_{1}, \dots, γ_{p}, Σ, 𝖢𝖯 𝒦, J 𝖢𝖯 𝒦, J (𝖢𝖯 𝒦) (𝖯𝖳 𝒦)} \otimes {Σ 𝖲}$	$C l_{0, p + 3} \times C l_{0, p + 3} \to C l_{0, p + 4} \times C l_{0, p + 4}$	$ℛ_{p + 3} \times ℛ_{p + 3}$
	$L_{r}$	${γ_{1}, \dots, γ_{p}, 𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦, J (𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦), 𝒮}$	$C l_{0, p + 3} \to C l_{0, p + 4}$	$ℛ_{p + 3}$
	$L_{i}$	${γ_{1}, \dots, γ_{p}, 𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦, 𝒮, (𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦) 𝒮}$	$C l_{0, p + 3} \to C l_{0, p + 4}$	$ℛ_{p + 3}$
$𝒫$ AII + $𝒮_{+}$	P	${J γ_{1}, \dots, J γ_{p}, J Σ, 𝖯𝖳 𝒦, J 𝖯𝖳 𝒦} \otimes {Σ 𝖲}$	$C l_{p + 2, 0} \times C l_{p + 2, 0} \to C l_{p + 3, 0} \times C l_{p + 3, 0}$	$ℛ_{p + 4} \times ℛ_{p + 4}$
	$L_{r}$	${J γ_{1}, \dots, J γ_{p}, 𝒫 𝒯 𝒦, J 𝒫 𝒯 𝒦, 𝒮}$	$C l_{p + 2, 0} \to C l_{p + 3, 0}$	$ℛ_{p + 4}$
	$L_{i}$	${𝒫 𝒯 𝒦, J 𝒫 𝒯 𝒦, J 𝒮; γ_{1}, \dots, γ_{p}}$	$C l_{3, p - 1} \to C l_{3, p}$	$ℛ_{p + 4}$
$𝒫$ CII + $𝒮_{+ -}$	P	${𝖢𝖯 𝒦, J 𝖢𝖯 𝒦, J (𝖢𝖯 𝒦) (𝖯𝖳 𝒦); γ_{1}, \dots, γ_{p}, Σ} \otimes {Σ 𝖲}$	$C l_{3, p} \times C l_{3, p} \to C l_{3, p + 1} \times C l_{3, p + 1}$	$ℛ_{p + 5} \times ℛ_{p + 5}$
	$L_{r}$	${𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦, J (𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦); γ_{1}, \dots, γ_{p}, 𝒮}$	$C l_{3, p} \to C l_{3, p + 1}$	$ℛ_{p + 5}$
	$L_{i}$	${𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦, (𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦) 𝒮; γ_{1}, \dots, γ_{p}, 𝒮}$	$C l_{3, p} \to C l_{3, p + 1}$	$ℛ_{p + 5}$
$𝒫$ C + $𝒮_{-}$	P	${𝖢𝖯 𝒦, J 𝖢𝖯 𝒦; γ_{1}, \dots, γ_{p}, Σ} \otimes {Σ 𝖲}$	$C l_{2, p} \times C l_{2, p} \to C l_{2, p + 1} \times C l_{2, p + 1}$	$ℛ_{p + 6} \times ℛ_{p + 6}$
$𝒫$ C + $𝒮_{-}$	L	${𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦; γ_{1}, \dots, γ_{p}, 𝒮}$	$C l_{2, p} \to C l_{2, p + 1}$	$ℛ_{p + 6}$
$𝒫$ CI + $𝒮_{+ -}$	P	${𝖢𝖯 𝒦, J 𝖢𝖯 𝒦; γ_{1}, \dots, γ_{p}, Σ, J (𝖢𝖯 𝒦) (𝖯𝖳 𝒦)} \otimes {Σ 𝖲}$	$C l_{2, p + 1} \times C l_{2, p + 1} \to C l_{2, p + 2} \times C l_{2, p + 2}$	$ℛ_{p + 7} \times ℛ_{p + 7}$
	$L_{r}$	${𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦; γ_{1}, \dots, γ_{p}, J (𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦), 𝒮}$	$C l_{2, p + 1} \to C l_{2, p + 2}$	$ℛ_{p + 7}$
	$L_{i}$	${𝒞 𝒫 𝒦, J 𝒞 𝒫 𝒦; γ_{1}, \dots, γ_{p}, 𝒮, (𝒞 𝒫 𝒦) (𝒫 𝒯 𝒦) 𝒮}$	$C l_{2, p + 1} \to C l_{2, p + 2}$	$ℛ_{p + 7}$

Table 15. Table S10: Classification of topologically stable exceptional points in the previous works Berry ( 2004 ); Esaki et al. ( 2011 ); Zhen et al. ( 2015 ); González and Molina ( 2016 ); Leykam et al. ( 2017 ); Xu et al. ( 2017 ); González and Molina ( 2017 ); Chernodub ( 2017 ); Zyuzin and Zyuzin ( 2018 ); Cerjan et al. ( 2018 ); Zhou et al. ( 2018 ); Molina and González ( 2018 ); Shen et al. ( 2018 ); Shen and Fu ( 2018 ); Carlström and Bergholtz ( 2018 ); Moors et al. ( 2019 ); Okugawa and Yokoyama ( 2019 ); Budich et al. ( 2019 ); Yang and Hu ( 2019 ); Zhou et al. ( 2019 ); Wang et al. ( 2019 ); Yoshida et al. ( 2019 ); Carlström et al. ( 2019 ); Zyuzin and Simon ( 2019 ); Papaj et al. ( 2019 ); Kozii and Fu ; Cerjan et al. ; Luo et al. ; Lee et al. ; Chernodub and Cortijo ; Bergholtz and Budich . The codimension p 𝑝 p is defined as p := d − d EP assign 𝑝 𝑑 subscript 𝑑 EP p:=d-d_{\rm EP} with the spatial dimension d 𝑑 d and the dimension d EP subscript 𝑑 EP d_{\rm EP} of the gapless region; exceptional points, lines, and surfaces are described by d EP = 0 , 1 , 2 subscript 𝑑 EP 0 1 2 d_{\rm EP}=0,1,2 , respectively. Complex-energy gaps have two distinct types, a point (P) or line (L) gap, and the subscript of L specifies a line gap for the real or imaginary part of the complex spectrum. The sign of 𝒫 𝒯 𝒫 𝒯 {\cal PT} symmetry means ( 𝒫 𝒯 ) ( 𝒫 𝒯 ) ∗ = + 1 𝒫 𝒯 superscript 𝒫 𝒯 1 ({\cal PT})\,({\cal PT})^{*}=+1 .

Symmetry	Gap	$p = 1$	$p = 2$	$p = 3$
No	P	$0$	$ℤ$ Berry (2004); Xu et al. (2017); González and Molina (2016, 2017); Zyuzin and Zyuzin (2018); Cerjan et al. (2018); Zhou et al. (2018); Molina and González (2018); Shen et al. (2018); Shen and Fu (2018); Carlström and Bergholtz (2018); Yang and Hu (2019); Carlström et al. (2019); Zyuzin and Simon (2019); Papaj et al. (2019); Kozii and Fu ; Cerjan et al. ; Luo et al. ; Lee et al. ; Chernodub and Cortijo ; Bergholtz and Budich	$0$
No	L	$ℤ$	$0$	$ℤ$ Xu et al. (2017); Cerjan et al. (2018, ); Chernodub and Cortijo
Chiral	P	$ℤ$ Esaki et al. (2011); Budich et al. (2019); Yoshida et al. (2019)	$0$	$ℤ$ [Eq. (8)]
	$L_{r}$	$0$	$ℤ$	$0$
	$L_{i}$	$ℤ$ $\oplus$ $ℤ$	$0$	$ℤ$ $\oplus$ $ℤ$
Sublattice	P	$0$	$ℤ$ $\oplus$ $ℤ$ Leykam et al. (2017); Wang et al. (2019); Moors et al. (2019)	$0$
Sublattice	L	$0$	$ℤ$	$0$
$𝒫 𝒯$ , $+ 1$	P	$ℤ_{2}$ Zhen et al. (2015); Chernodub (2017); Okugawa and Yokoyama (2019); Zhou et al. (2019)	$ℤ_{2}$	$0$
	$L_{r}$	$ℤ$ Chernodub (2017)	$ℤ_{2}$	$ℤ_{2}$
	$L_{i}$	$ℤ_{2}$	$0$	$2$ $ℤ$

Equations105

W := \oint_{S^{1}} \frac{d k}{2 π i} \cdot \nabla_{k} lo g det [H (k) - E (k_{EP})] .

W := \oint_{S^{1}} \frac{d k}{2 π i} \cdot \nabla_{k} lo g det [H (k) - E (k_{EP})] .

(P T) H^{*} (k) (P T)^{- 1}

(P T) H^{*} (k) (P T)^{- 1}

(C P) H^{T} (k) (C P)^{- 1}

Γ H^{†} (k) Γ^{- 1} = - H (k), Γ^{2} = 1,

Γ H^{†} (k) Γ^{- 1} = - H (k), Γ^{2} = 1,

S H (k) S^{- 1} = - H (k), S^{2} = 1,

S H (k) S^{- 1} = - H (k), S^{2} = 1,

H (k) = i = 1 \sum p v_{i} δ k_{i} γ_{i} + P J P^{- 1}, p := d - d_{EP},

H (k) = i = 1 \sum p v_{i} δ k_{i} γ_{i} + P J P^{- 1}, p := d - d_{EP},

E (k) = \pm k_{x}^{2} + k_{w}^{2} + (k_{y}^{2} + k_{z}^{2} \pm i γ)^{2},

E (k) = \pm k_{x}^{2} + k_{w}^{2} + (k_{y}^{2} + k_{z}^{2} \pm i γ)^{2},

H (k) = k_{x} σ_{x} τ_{x} + k_{y} σ_{x} τ_{y} + k_{z} σ_{x} τ_{z} + i γ σ_{z} τ_{x},

H (k) = k_{x} σ_{x} τ_{x} + k_{y} σ_{x} τ_{y} + k_{z} σ_{x} τ_{z} + i γ σ_{z} τ_{x},

(C_{+} P) H^{T} (k) (C_{+} P)^{- 1} = H (k), (C_{+} P) (C_{+} P)^{*} = \pm 1,

(C_{+} P) H^{T} (k) (C_{+} P)^{- 1} = H (k), (C_{+} P) (C_{+} P)^{*} = \pm 1,

(P T_{-}) H^{*} (k) (P T_{-})^{- 1} = - H (k), (P T_{-}) (P T_{-})^{*} = \pm 1,

(P T_{-}) H^{*} (k) (P T_{-})^{- 1} = - H (k), (P T_{-}) (P T_{-})^{*} = \pm 1,

γ_{i} := (0 (γ_{i}^{P})^{†} γ_{i}^{P} 0), γ_{i}^{†} = γ_{i}

γ_{i} := (0 (γ_{i}^{P})^{†} γ_{i}^{P} 0), γ_{i}^{†} = γ_{i}

{γ_{i}, γ_{j}} = 2 δ_{ij} \Leftrightarrow γ_{i}^{P} (γ_{j}^{P})^{†} + γ_{j}^{P} (γ_{i}^{P})^{†} = 2 δ_{ij} .

{γ_{i}, γ_{j}} = 2 δ_{ij} \Leftrightarrow γ_{i}^{P} (γ_{j}^{P})^{†} + γ_{j}^{P} (γ_{i}^{P})^{†} = 2 δ_{ij} .

(P T_{+}) γ_{i}^{*} (P T_{+})^{- 1}

(P T_{+}) γ_{i}^{*} (P T_{+})^{- 1}

(C_{+} P) γ_{i}^{*} (C_{+} P)^{- 1}

(C_{-} P) γ_{i}^{*} (C_{-} P)^{- 1}

(P T_{-}) γ_{i}^{*} (P T_{-})^{- 1}

Γ γ_{i} Γ^{- 1}

S γ_{i} S^{- 1}

{\sf\Sigma}\,\gamma_{i}{\sf\Sigma}^{-1}=-\gamma_{i},\quad{\sf\Sigma}:=\left(\begin{array}[]{@{\,}cc@{\,}}1&0\\ 0&-1\\ \end{array}\right).

{\sf\Sigma}\,\gamma_{i}{\sf\Sigma}^{-1}=-\gamma_{i},\quad{\sf\Sigma}:=\left(\begin{array}[]{@{\,}cc@{\,}}1&0\\ 0&-1\\ \end{array}\right).

C l_{p} \to C l_{p + 1} \Leftrightarrow C_{p}, C l_{p, q} \to C l_{p, q + 1} \Leftrightarrow R_{q - p}, C l_{p, q} \to C l_{p + 1, q} \Leftrightarrow R_{p + 2 - q},

C l_{p} \to C l_{p + 1} \Leftrightarrow C_{p}, C l_{p, q} \to C l_{p, q + 1} \Leftrightarrow R_{q - p}, C l_{p, q} \to C l_{p + 1, q} \Leftrightarrow R_{p + 2 - q},

Γ H^{†} (k) Γ^{- 1} = - H (k) .

Γ H^{†} (k) Γ^{- 1} = - H (k) .

{γ_{1}, \dots, γ_{p}, Σ, Γ}

{γ_{1}, \dots, γ_{p}, Σ, Γ}

{γ_{1}, \dots, γ_{p}, Γ}

{γ_{1}, \dots, γ_{p}, Γ}

Γ γ_{i} Γ^{- 1} = γ_{i} .

Γ γ_{i} Γ^{- 1} = γ_{i} .

{γ_{1}, \dots, γ_{p}}

{γ_{1}, \dots, γ_{p}}

(P T) H^{*} (k) (P T)^{- 1} = H (k), (P T) (P T)^{*} = + 1;

(P T) H^{*} (k) (P T)^{- 1} = H (k), (P T) (P T)^{*} = + 1;

(C P) H^{T} (k) (C P)^{- 1} = - H (k), (C P) (C P)^{*} = - 1.

[PT K, CP K] = [PT K, Σ] = {CP K, Σ} = 0,

[PT K, CP K] = [PT K, Σ] = {CP K, Σ} = 0,

{CP K, J CP K; γ_{1}, \dots, γ_{p}, Σ, J (CP K) (PT K)}

{CP K, J CP K; γ_{1}, \dots, γ_{p}, Σ, J (CP K) (PT K)}

{C P K, J C P K; γ_{1}, \dots, γ_{p}, J (C P K) (P T K)}

{C P K, J C P K; γ_{1}, \dots, γ_{p}, J (C P K) (P T K)}

(P T) γ_{i}^{*} (P T)^{- 1} = - γ_{i}, (C P) γ_{i}^{*} (C P)^{- 1} = - γ_{i} .

(P T) γ_{i}^{*} (P T)^{- 1} = - γ_{i}, (C P) γ_{i}^{*} (C P)^{- 1} = - γ_{i} .

{C P K, J C P K; γ_{1}, \dots, γ_{p}}

{C P K, J C P K; γ_{1}, \dots, γ_{p}}

P T ∣ ψ_{1} (k) ⟩^{*} = ∣ ψ_{1} (k) ⟩, P T ∣ ψ_{2} (k) ⟩^{*} = ∣ ψ_{2} (k) ⟩

P T ∣ ψ_{1} (k) ⟩^{*} = ∣ ψ_{1} (k) ⟩, P T ∣ ψ_{2} (k) ⟩^{*} = ∣ ψ_{2} (k) ⟩

P T ∣ ψ_{1} (k) ⟩^{*} = ∣ ψ_{2} (k) ⟩ .

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Full text

Classification of Exceptional Points and Non-Hermitian Topological Semimetals

Kohei Kawabata

[email protected]

Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

Takumi Bessho

[email protected]

Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan

Masatoshi Sato

[email protected]

Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan

Abstract

Exceptional points are universal level degeneracies induced by non-Hermiticity. Whereas past decades witnessed their new physics, the unified understanding has yet to be obtained. Here we present the complete classification of generic topologically stable exceptional points according to two types of complex-energy gaps and fundamental symmetries of charge conjugation, parity, and time reversal. This classification reveals unique non-Hermitian gapless structures with no Hermitian analogs and systematically predicts unknown non-Hermitian semimetals and nodal superconductors; a topological dumbbell of exceptional points in three dimensions is constructed as an illustration. Our work paves the way toward richer phenomena and functionalities of exceptional points and non-Hermitian topological semimetals.

Topology plays a pivotal role in the understanding of phases of matter Hasan and Kane (2010); *Zhang-review; *Schnyder-Ryu-review. In gapless systems such as semimetals and nodal superconductors, topology guarantees stable degeneracies accompanying distinctive excitations Nielsen and Ninomiya (1983); Murakami (2007); Wan et al. (2011); Burkov and Balents (2011); *BHB-11; *HB-12; *MB-12; Young et al. (2012); Armitage et al. (2018). A prime example is the Weyl semimetal in three dimensions, where each gapless point is topologically protected by the Chern number defined on the enclosing surface. Symmetry further brings about diverse types of topological semimetals. Their unified understanding is developed as the classification theory according to fundamental symmetries such as $\mathcal{PT}$ and $\mathcal{CP}$ symmetries Hořava (2005); Matsuura et al. (2013); Zhao and Wang (2013); *Zhao-14; Morimoto and Furusaki (2014); Kobayashi et al. (2014); Shiozaki and Sato (2014); Chiu and Schnyder (2014); Zhao et al. (2016); *Zhao-17.

Recently, the interplay between topology and non-Hermiticity has attracted widespread interest in a non-Hermitian extension of topological insulators Rudner and Levitov (2009); *Zeuner-15; Hu and Hughes (2011); Esaki et al. (2011); *Sato-12; Poli et al. (2015); Malzard et al. (2015); Lee (2016); Weimann et al. (2017); Leykam et al. (2017); Kim et al. ; *Xiao-17; St-Jean et al. (2017); Bahari et al. (2017); Harari et al. (2018); *Bandres-18; Shen et al. (2018); Kunst et al. (2018); Kawabata et al. (2018a); Yao and Wang (2018); *YSW-18; Lieu (2018); Gong et al. (2018); Philip et al. (2018); *Hirsbrunner-19; Kawabata et al. (2018b); Takata and Notomi (2018); McDonald et al. (2018); Kawabata et al. (2019); Bliokh et al. (2019); Liu et al. (2019); Edvardsson et al. (2019); Lee and Thomale (2019); Herviou et al. (2019); Kunst and Dwivedi ; Kawabata et al. ; Zhou and Lee ; Zeng et al. ; Zirnstein et al. ; Rui et al. ; Borgnia et al. ; Yokomizo and Murakami ; McClarty and Rau ; Longhi and semimetals Zhen et al. (2015); González and Molina (2016); *Gonzalez-17; *Molina-18; Xu et al. (2017); Chernodub (2017); *Chernodub-19; Zyuzin and Zyuzin (2018); *Moors-19; *Zyuzin-19; Cerjan et al. (2018); Zhou et al. (2018); Carlström and Bergholtz (2018); *Carlstrom-19; Okugawa and Yokoyama (2019); Budich et al. (2019); Yang and Hu (2019); Zhou et al. (2019); Wang et al. (2019); Yoshida et al. (2019); Kozii and Fu ; *Papaj-19; *Shen-Fu-18; Cerjan et al. ; Luo et al. ; Lee et al. ; Bergholtz and Budich . Non-Hermiticity ubiquitously appears, for instance, in nonequilibrium open systems Konotop et al. (2016); *Feng-review; *Christodoulides-review; *Alu-review and correlated electron systems Kozii and Fu ; *Papaj-19; *Shen-Fu-18, leading to unusual properties with no Hermitian counterparts. One of their salient characteristics is the emergence of exceptional points Kato (1966); Berry (2004); Heiss (2012), i.e., universal non-Hermitian level degeneracies at which eigenstates coalesce with and linearly depend on each other def . The past decade has witnessed a plethora of rich phenomena and functionalities induced by exceptional points Konotop et al. (2016); *Feng-review; *Christodoulides-review; *Alu-review, including unidirectional invisibility Lin et al. (2011); Regensburger et al. (2012); Feng et al. (2013); Peng et al. (2014), chiral transport Dembowski et al. (2001); Gao et al. (2015); Doppler et al. (2016); Xu et al. (2016); Yoon et al. (2018), enhanced sensitivity Wiersig (2014); Liu et al. (2016); Hodaei et al. (2017); Chen et al. (2017); Lau and Clerk (2018); Zhang et al. , and unusual quantum criticality Lee and Chan (2014); Kawabata et al. (2017); Nakagawa et al. (2018); Li et al. (2019); Dóra et al. (2019); Wu et al. (2019); Xiao et al. .

Such exceptional points also alter the nodal structures of topological semimetals in a fundamental manner Zhen et al. (2015); González and Molina (2016); *Gonzalez-17; *Molina-18; Xu et al. (2017); Chernodub (2017); *Chernodub-19; Zyuzin and Zyuzin (2018); *Moors-19; *Zyuzin-19; Cerjan et al. (2018); Zhou et al. (2018); Carlström and Bergholtz (2018); *Carlstrom-19; Okugawa and Yokoyama (2019); Budich et al. (2019); Yang and Hu (2019); Zhou et al. (2019); Wang et al. (2019); Yoshida et al. (2019); Kozii and Fu ; *Papaj-19; *Shen-Fu-18; Cerjan et al. ; Luo et al. ; Lee et al. ; Bergholtz and Budich . Notably, non-Hermiticity deforms a Weyl point and spawns a ring of exceptional points Berry (2004). This Weyl exceptional ring is characterized by two topological charges Xu et al. (2017), a Chern number and a quantized Berry phase, and such a multiple topological structure has no analogs to Weyl and Dirac points in Hermitian systems Nielsen and Ninomiya (1983); Murakami (2007); Wan et al. (2011); Burkov and Balents (2011); *BHB-11; *HB-12; *MB-12; Young et al. (2012); Hořava (2005); Matsuura et al. (2013); Zhao and Wang (2013); *Zhao-14; Morimoto and Furusaki (2014); Kobayashi et al. (2014); Shiozaki and Sato (2014); Chiu and Schnyder (2014); Zhao et al. (2016); *Zhao-17; Armitage et al. (2018). Its experimental observation has also been reported in an optical waveguide array Cerjan et al. . Moreover, pseudo-Hermiticity Budich et al. (2019), $\mathcal{PT}$ symmetry Okugawa and Yokoyama (2019); Zhou et al. (2019), and chiral symmetry Yoshida et al. (2019) enable an exceptional ring (surface) in two (three) dimensions. A symmetry-protected exceptional ring has been experimentally observed in a two-dimensional photonic crystal slab with $\mathcal{PT}$ symmetry Zhen et al. (2015). Whereas these unconventional nodal structures imply a fundamental change in the existing classification for Hermitian topological semimetals, its non-Hermitian counterpart has yet to be established. The non-Hermitian topological classification is crucial not only because it provides a general theoretical framework but also because it predicts novel non-Hermitian topological materials and serves as a benchmark for experiments.

This Letter presents the general theory that completely classifies topologically stable exceptional points at generic momentum points in non-Hermitian topological semimetals according to fundamental 38-fold symmetry and two types of complex-energy gaps (Table 1 sup ). Our classification is based on the observation that non-Hermiticity enables a unique gapless structure with no Hermitian analogs: only one type of the complex-energy gaps is open around an exceptional point [Fig. 1 (d)], which is sharply contrasted with conventional Weyl and Dirac points in Hermitian systems around which both types are open [Fig. 1 (c)]. We also elucidate that exceptional points generally possess multiple topological structures due to the two types of complex-energy gaps. Our theory provides the unified understanding of non-Hermitian topological semimetals studied in previous works Zhen et al. (2015); González and Molina (2016); *Gonzalez-17; *Molina-18; Xu et al. (2017); Chernodub (2017); *Chernodub-19; Zyuzin and Zyuzin (2018); *Moors-19; *Zyuzin-19; Cerjan et al. (2018); Zhou et al. (2018); Carlström and Bergholtz (2018); *Carlstrom-19; Okugawa and Yokoyama (2019); Budich et al. (2019); Yang and Hu (2019); Zhou et al. (2019); Wang et al. (2019); Yoshida et al. (2019); Kozii and Fu ; *Papaj-19; *Shen-Fu-18; Cerjan et al. ; Luo et al. ; Lee et al. ; Bergholtz and Budich . Furthermore, it systematically predicts novel non-Hermitian topological materials unnoticed in the literature; we construct as an illustration a topological dumbbell of exceptional points [Fig. 2 (b)], i.e., a three-dimensional variant of the bulk Fermi arcs Zhou et al. (2018); Kozii and Fu ; *Papaj-19; *Shen-Fu-18.

Non-Hermitian gapless structures. —

A unique characteristic of non-Hermitian systems is complex-valued eigenenergy, from which two different energy gaps are defined, a point gap [Fig. 1 (a)] and a line gap [Fig. 1 (b)] Kawabata et al. . In the presence of a point (line) gap, complex-energy bands do not cross a reference point (line) in the complex-energy plane. A point gap is physically relevant to the localization transition in one-dimensional non-Hermitian systems Hatano and Nelson (1996); *Hatano-97; *Hatano-98; Gong et al. (2018); Longhi , whereas topologically protected edge states Esaki et al. (2011); *Sato-12; Poli et al. (2015); Weimann et al. (2017); Kim et al. ; *Xiao-17; St-Jean et al. (2017); Bahari et al. (2017); Harari et al. (2018); *Bandres-18; Shen et al. (2018) are understood by a line gap for the real part of the complex spectrum. A line gap for the imaginary part also has a significant influence on the nonequilibrium dynamics Kawabata et al. (2019). If symmetry exists, both complex-energy gaps should be invariant under the symmetry. Without loss of generality, the reference point is supposed to be placed on the reference line. Then, a line gap is always closed when a point gap is closed. However, the converse is not necessarily true; a point gap can be open even when a line gap is closed.

This nature of complex-energy gaps enables two distinct types of non-Hermitian gapless structures. We encircle a gapless region (point, line, surface, and so on) in momentum space by a $(p-1)$ -dimensional sphere $S^{p-1}$ ( $p\geq 1$ ), where $S^{0}$ , $S^{1}$ , and $S^{2}$ denote a pair of points, a circle, and a surface, respectively. The system has a complex-energy gap on $S^{p-1}$ , but two different situations may happen: (i) both point and line gaps are open [Fig. 1 (c)] or (ii) only a point gap is open [Fig. 1 (d)]. In the former case, the non-Hermitian Hamiltonian on $S^{p-1}$ can be continuously deformed into a Hermitian (or an anti-Hermitian) one as rigorously proven in Ref. Kawabata et al. . Thus, in a manner analogous to the Chern number for the conventional Weyl point, the gapless region hosts a topological charge essentially identical to that in the Hermitian case. In the latter case, by contrast, we need to assign a different topological charge to the gapless region on the basis of the point gap on $S^{p-1}$ . It should be noted that the latter is intrinsic to non-Hermitian systems and impossible in Hermitian ones, since there is no distinction between point and line gaps for Hermitian Hamiltonians.

We find that exceptional points realize the latter unique gapless structure as shown in Fig. 1 (d) and are characterized by topological charges for point gaps. A distinctive property of an exceptional point is swapping of eigenenergies and eigenstates upon its encirclement Heiss (2012); Konotop et al. (2016); *Feng-review; *Christodoulides-review; *Alu-review; Dembowski et al. (2001); Gao et al. (2015); Doppler et al. (2016); Xu et al. (2016); Zhou et al. (2018); Yoon et al. (2018). For illustration, let us consider a two-dimensional system with no symmetry that has an exceptional point ${\bm{k}}={\bm{k}}_{\rm EP}$ at which two complex bands $E_{\pm}\left({\bm{k}}\right)$ coalesce. A representative model is given as $H\left({\bm{k}}\right)=k_{x}\sigma_{x}+\left(k_{y}+\text{i}\gamma\right)\sigma_{y}$ with Pauli matrices $\sigma_{x,y}$ and the degree of non-Hermiticity $\gamma$ . The eigenenergies are $E_{\pm}\left({\bm{k}}\right)=\pm\sqrt{{\bm{k}}^{2}-\gamma^{2}+2\text{i}\gamma k_{y}}$ with the square root singularity around the exceptional points ${\bm{k}}_{\rm EP}=\left(\pm\gamma,0\right)$ . As a direct result of this singularity, a branch cut and a self-intersecting Riemann surface appear in the complex-energy plane and thus $E_{+}\left({\bm{k}}\right)$ and $E_{-}\left({\bm{k}}\right)$ are swapped when we go around a loop $S^{1}$ in momentum space that encircles one of the exceptional points. Importantly, a point gap for the reference point $E\left({\bm{k}}_{\rm EP}\right)$ is open but a line gap is closed on $S^{1}$ . As a result, the determinant of $H\left(\bm{k}\right)-E\left({\bm{k}}_{\rm EP}\right)$ does not vanish on $S^{1}$ , which enables us to define the winding number for a point gap as

[TABLE]

An exceptional point with $W\neq 0$ is topologically stable. For the above representative model, the exceptional points ${\bm{k}}_{\rm EP}=\left(\pm\gamma,0\right)$ are characterized by $W=\pm 1$ , respectively. We note that a similar topological invariant $\nu$ called vorticity is introduced in Refs. Leykam et al. (2017); Shen et al. (2018): $\nu:=-\left(2\pi\right)^{-1}\oint_{S^{1}}d{\bm{k}}\cdot\nabla_{\bm{k}}\,\mathrm{arg}\left[E_{+}\left({\bm{k}}\right)-E_{-}\left({\bm{k}}\right)\right]$ . Whereas both invariants are equivalent to each other for the above simple two-band model ( $\nu=-W/2$ ), only $W$ is straightforwardly applicable to general models at which more than two complex bands coalesce.

It is shown that exceptional points that accompany spontaneous ${\cal PT}$ -symmetry breaking Heiss (2012); Konotop et al. (2016); *Feng-review; *Christodoulides-review; *Alu-review; Bender and Boettcher (1998); *Bender-02; *Bender-review also possess a similar gapless structure (see Sec. SV in Ref. sup ): only a point gap is open and a line gap is closed around them.

Symmetry. —

To classify topologically stable exceptional points at generic momentum points for both complex-energy gaps in a general manner, we consider fundamental symmetries that keep all momenta invariant. The prime examples are ${\cal PT}$ and ${\cal CP}$ symmetries defined by

[TABLE]

where ${\cal PT}$ and ${\cal CP}$ are unitary operators. Here ${\cal PT}$ symmetry emerges in open systems with balanced gain and loss Konotop et al. (2016); *Feng-review; *Christodoulides-review; *Alu-review, while ${\cal CP}$ symmetry is respected in non-Hermitian superconductors and superfluids with inversion symmetry PHS . Whereas both space inversion ${\cal P}$ and time reversal ${\cal T}$ (charge conjugation ${\cal C}$ ) may be broken individually, the combined symmetry ${\cal PT}$ ( ${\cal CP}$ ) is required for the topological stability away from high-symmetry points.

Another relevant symmetry is chiral symmetry, which is a combination of ${\cal PT}$ and ${\cal CP}$ symmetries and hence is defined by

[TABLE]

with a unitary operator $\Gamma$ . It is notable that chiral symmetry is equivalent to pseudo-Hermiticity, defined by $\eta H^{{\dagger}}\left(\bm{k}\right)\eta^{-1}=H\left(\bm{k}\right)$ with a unitary and Hermitian operator $\eta$ Mostafazadeh (2002a); *Mostafazadeh-02-2; *Mostafazadeh-02-3. In fact, when $H\left({\bm{k}}\right)$ respects chiral symmetry, $\text{i}H\left({\bm{k}}\right)$ respects pseudo-Hermiticity Kawabata et al. (2019). Moreover, chiral symmetry is distinct from sublattice symmetry due to $H\left({\bm{k}}\right)\neq H^{{\dagger}}\left({\bm{k}}\right)$ Kawabata et al. , defined by

[TABLE]

with a unitary operator ${\cal S}$ . The above symmetries constitute the fundamental 38-fold symmetry class for the non-Hermitian gapless phases sup in a similar manner to the gapped ones in Ref. Kawabata et al. , which updates Bernard-LeClair symmetry Ber ; Esaki et al. (2011); *Sato-12; Lieu (2018); Budich et al. (2019); Zhou and Lee ; 38 (38).

Topological classification. —

Complex-energy gaps are closed at an exceptional point. Near such a gapless point, $H\left({\bm{k}}\right)$ is expressed as Blo

[TABLE]

where $v_{i}$ ’s are complex constants, $\delta k_{i}$ ’s are momentum deviations from the exceptional point, and $\gamma_{i}$ ’s are non-Hermitian Dirac matrices subject to certain symmetries. The gapless point $\delta{\bm{k}}=0$ in general takes a Jordan canonical form $PJP^{-1}$ with a Jordan matrix $J$ and an invertible matrix $P$ , which is defective unless $J$ is diagonal Kato (1966). In addition, $p$ is the codimension of the exceptional point with the spatial dimension $d$ and the dimension $d_{\rm EP}$ of the gapless region; exceptional points, lines, and surfaces are described by $d_{\rm EP}=0,1,2$ , respectively. Whereas a complex-energy gap is closed at the gapless point $\delta{\bm{k}}=0$ , it may be open, and a topological invariant can be defined on a ( $p-1$ )-dimensional surface $S^{p-1}$ that encloses the gapless point, in a similar manner to Fermi surfaces in Hermitian systems Morimoto and Furusaki (2014); Kobayashi et al. (2014); Zhao et al. (2016); Karoubi (2008); Chiu et al. (2016). Thus, topology of the exceptional point is determined by classifying the gapped topological phases on $S^{p-1}$ .

As discussed above, only a point gap is open for a sufficiently small $S^{p-1}$ , but a line gap can also be open for a larger $S^{p^{\prime}-1}$ [ $p^{\prime}$ can be different from $p$ ; for instance, $p=2$ and $p^{\prime}=3$ in Fig. 2 (a)]. Depending on types of the complex-energy gaps, an exceptional point may support multiple topological invariants. Taking into account both possibilities, we classify topologically stable exceptional points for each type of complex-energy gaps and all of the 38 symmetry classes sup . Our results are summarized in Table 1 and Tables S2-S7 in Supplemental Material sup . These periodic tables specify exceptional points and non-Hermitian topological semimetals in a general manner and describe their unconventional nodal structures. In fact, they corroborate previous works Zhen et al. (2015); González and Molina (2016); *Gonzalez-17; *Molina-18; Xu et al. (2017); Chernodub (2017); *Chernodub-19; Zyuzin and Zyuzin (2018); *Moors-19; *Zyuzin-19; Cerjan et al. (2018); Zhou et al. (2018); Carlström and Bergholtz (2018); *Carlstrom-19; Okugawa and Yokoyama (2019); Budich et al. (2019); Yang and Hu (2019); Zhou et al. (2019); Wang et al. (2019); Yoshida et al. (2019); Kozii and Fu ; *Papaj-19; *Shen-Fu-18; Cerjan et al. ; Luo et al. ; Lee et al. ; Bergholtz and Budich ; sup . For example, stable exceptional points in two dimensions Shen et al. (2018); Zhou et al. (2018); Kozii and Fu ; *Papaj-19; *Shen-Fu-18; Bergholtz and Budich are explained by the $\mathbb{Z}$ index in Table 1 with no symmetry, $p=2$ , and point (P) gap; symmetry-protected exceptional rings in two dimensions Zhen et al. (2015); Okugawa and Yokoyama (2019); Budich et al. (2019); Yoshida et al. (2019); Zhou et al. (2019) are explained by the $\mathbb{Z}$ or $\mathbb{Z}_{2}$ index in Table 1 with chiral or ${\cal PT}$ symmetry, $p=1$ , and point (P) gap. These unique nodal structures, as well as the consequent physical phenomena, are topologically protected against symmetry-preserving perturbations.

Multiple topological structure. —

A remarkable feature of Table 1 is the multiple topological indices for each symmetry class due to the two types of complex-energy gaps. As a result, exceptional points can be characterized by a couple of independent topological charges. A prime example is the Weyl exceptional ring in three-dimensional systems with no symmetry [Fig. 2 (a)] Berry (2004); Xu et al. (2017). For the Hermitian case, the topological stability of a Weyl point in three dimensions is ensured by the Chern number defined on an enclosing surface Nielsen and Ninomiya (1983); Armitage et al. (2018). In the presence of non-Hermiticity, such a Weyl point morphs into an exceptional ring. A representative model is given as $H\left({\bm{k}}\right)=k_{x}\sigma_{x}+k_{y}\sigma_{y}+\left(k_{z}+\text{i}\gamma\right)\sigma_{z}$ with a Weyl exceptional ring at $k_{x}^{2}+k_{y}^{2}=\gamma^{2}$ , $k_{z}=0$ Berry (2004); Xu et al. (2017). On a sphere [ $S^{2}$ in Fig. 2 (a)] that encloses the ring, a line gap is open and the Chern number remains to be well defined unless it annihilates with another ring, which corresponds to the $\mathbb{Z}$ index for a line gap with $p=3$ (see Table 1 with no symmetry). Moreover, on $S^{1}$ across the ring, a point gap is open and the winding number in Eq. (1) is well defined, which corresponds to the $\mathbb{Z}$ index for a point gap with $p=2$ . Notably, the two topological charges are independent of each other and individually ensure the topological stability of the Weyl exceptional ring.

Such a multiple topological structure is a general hallmark of non-Hermitian topological semimetals. For instance, according to our classification tables, symmetry-protected exceptional rings in two dimensions Zhen et al. (2015); Okugawa and Yokoyama (2019); Budich et al. (2019); Zhou et al. (2019); Yoshida et al. (2019) may host different topological charges for each $p\leq 2$ and each complex-energy gap. Whereas the unique nodal structures were discussed in Refs. Zhen et al. (2015); Okugawa and Yokoyama (2019); Budich et al. (2019); Yoshida et al. (2019); Zhou et al. (2019), their multiple topology has not been revealed until this Letter.

Exceptional point in three dimensions. —

Our classification also predicts unknown non-Hermitian topological semimetals and nodal superconductors. As an illustrative example, we consider an exceptional point in three dimensions ( $p=3$ ) protected by chiral symmetry. Whereas exceptional rings Berry (2004); Xu et al. (2017) and surfaces Okugawa and Yokoyama (2019); Zhou et al. (2019) were discussed before, such an exceptional point in three dimensions has not hitherto been known. A representative model is systematically constructed in the following manner. We begin with a Hermitian gapless system in four dimensions $H\left({\bm{k}}\right)=k_{x}\sigma_{x}\tau_{x}+k_{y}\sigma_{x}\tau_{y}+k_{z}\sigma_{x}\tau_{z}+k_{w}\sigma_{y}$ with Pauli matrices $\sigma_{i}$ ’s and $\tau_{i}$ ’s. Corresponding to the $\mathbb{Z}$ index for a real line gap with $p=4$ (see Table 1 with chiral symmetry), it possesses a topologically stable gapless point at ${\bm{k}}=0$ , around which the three-dimensional winding number sup ; Schnyder et al. (2008); *Ryu-10; Qi et al. (2010); Kawakami et al. (2018) is defined due to chiral symmetry in Eq. (4) with $\Gamma=\sigma_{z}$ . Now we add a non-Hermitian perturbation $\text{i}\gamma\sigma_{z}\tau_{x}$ to this Hermitian model. As with the Weyl exceptional ring, this non-Hermiticity spawns an exceptional ring. In fact, the complex spectrum is obtained as

[TABLE]

and an exceptional ring appears at $k_{x}^{2}+k_{w}^{2}=\gamma^{2}$ , $k_{y}=k_{z}=0$ . Finally, we take $k_{w}=0$ and regard this four-dimensional model as a three-dimensional one,

[TABLE]

which has a pair of exceptional points at ${\bm{k}}_{\rm EP}=\left(\pm\gamma,0,0\right)$ . As illustrated in Fig. 2 (b), these exceptional points are connected by a Fermi arc with $\mathrm{Re}\,E=0$ , forming a topologically stable dumbbell configuration. The topological stability of the exceptional points is ensured by the $\mathbb{Z}$ index in Table 1 with chiral symmetry, $p=3$ , and point gap, which is given as the Chern number $\pm 1$ of the Hermitian matrix $\text{i}H\left({\bm{k}}\right)\Gamma$ defined on a surface that encloses each of the exceptional points sup .

We stress that topologically stable gapless points are absent in Hermitian three-dimensional systems with chiral symmetry. These gapless points are thus unique to non-Hermitian systems. Furthermore, the above recipe is widely applicable to different symmetry classes and spatial dimensions and unknown non-Hermitian topological semimetals can be systematically predicted according to our classification.

Discussion. —

The emergence of exceptional points is one of the most striking and universal characteristics in non-Hermitian physics Berry (2004); Heiss (2012); Konotop et al. (2016); *Feng-review; *Christodoulides-review; *Alu-review. In this Letter, we have formulated the general classification theory on non-Hermitian topology of exceptional points. Our theory is widely applicable to non-Hermitian topological semimetals, providing the unified understanding of their unique nodal structures and predicting novel non-Hermitian topological materials. Since the consequent non-Hermitian topological phenomena and functionalities have yet to be explored, it merits further research to investigate such richer physics on the basis of our theory.

We are grateful to Ken Shiozaki and Masahito Ueda for the valuable discussions. K.K. also thanks Emil J. Bergholtz, Ryo Okugawa, and Tsuneya Yoshida for the helpful discussions. This work was supported by a Grant-in-Aid for Scientific Research on Innovative Areas “Topological Materials Science” (KAKENHI Grant No. JP15H05855) from the Japan Society for the Promotion of Science (JSPS). K.K. was supported by the JSPS through the Program for Leading Graduate Schools (ALPS) and KAKENHI Grant No. JP19J21927, and in part through the domestic junior researcher exchange program of “Topological Materials Science.” M.S. was supported by KAKENHI Grant No. JP17H02922 from the JSPS.

K.K. and T.B. contributed equally to this work.

I SI. Symmetry

To investigate topologically stable exceptional points that appear at generic momentum points, we consider the following 38-fold symmetry Kawabata et al. that keeps all momenta invariant. First of all, ${\cal PT}$ symmetry in Eq. (2), ${\cal CP}$ symmetry in Eq. (3), and chiral symmetry in Eq. (4) constitute the 10-fold symmetry class, which is denoted as parity-equipped Altland-Zirnbauer ( $\mathcal{P}$ AZ) symmetry class (Table S1). In stark contrast to Hermitian systems, a Hermitian-conjugate counterpart of $\mathcal{P}$ AZ symmetry can be introduced independently. To distinguish between them, we denote ${\cal PT}$ ( ${\cal CP}$ ) symmetry in the main text as $\mathcal{P}\mathcal{T}_{+}$ ( $\mathcal{C}_{-}\mathcal{P}$ ) symmetry in Table S1. The Hermitian-conjugate counterpart of $\mathcal{P}\mathcal{T}_{+}$ symmetry is defined with transposition instead of complex conjugation by

[TABLE]

with a unitary operator $\mathcal{C}_{+}\mathcal{P}$ , and the Hermitian-conjugate counterpart of $\mathcal{C}_{-}\mathcal{P}$ symmetry is defined with complex conjugation instead of transposition by

[TABLE]

with a unitary operator $\mathcal{P}\mathcal{T}_{-}$ . In our notation, $\mathcal{C}_{\pm}$ and $\mathcal{T}_{\pm}$ imply transposition and complex conjugation, respectively. Importantly, $\mathcal{P}\mathcal{T}_{+}$ and $\mathcal{C}_{-}\mathcal{P}$ symmetries respectively coincide with $\mathcal{C}_{+}\mathcal{P}$ and $\mathcal{P}\mathcal{T}_{-}$ symmetries for Hermitian Hamiltonians [ $H\left({\bm{k}}\right)=H^{{\dagger}}\left({\bm{k}}\right)$ ], but they do not in the presence of non-Hermiticity since complex conjugation and transposition are not equivalent by definition [ $H^{*}\left({\bm{k}}\right)\neq H^{T}\left({\bm{k}}\right)$ ]. Here $\mathcal{C}_{+}\mathcal{P}$ and $\mathcal{P}\mathcal{T}_{-}$ symmetries in addition to chiral symmetry also constitute the 10-fold symmetry class, which is denoted as $\mathcal{P}\text{AZ}^{{\dagger}}$ symmetry class (Table S1). Here we abbreviate the Hermitian-conjugate counterpart of $\mathcal{P}$ AZ symmetry class as $\mathcal{P}\text{AZ}^{{\dagger}}$ symmetry class (similar abbreviations are used below). It should be noted that all the symmetry classes in Table S1 are not independent. Complex AZ class and its Hermitian conjugate (i.e., complex $\text{AZ}^{{\dagger}}$ class) are equivalent to each other. Class ${\cal P}$ AI ( ${\cal P}$ AII) is equivalent to class ${\cal P}\text{D}^{{\dagger}}$ ( ${\cal P}\text{C}^{{\dagger}}$ ). In fact, when a non-Hermitian Hamiltonian $H\left({\bm{k}}\right)$ belongs to class ${\cal P}$ AI ( ${\cal P}$ AII), another non-Hermitian Hamiltonian $\text{i}H\left({\bm{k}}\right)$ belongs to class ${\cal P}\text{D}^{{\dagger}}$ ( ${\cal P}\text{C}^{{\dagger}}$ ) Kawabata et al. (2019). In this manner, we totally have 16 independent symmetry classes in Table S1.

Since sublattice symmetry in Eq. (5) is distinct from chiral symmetry in Eq. (4) for non-Hermitian Hamiltonians, it can be considered as an additional symmetry to ${\cal P}$ AZ symmetry. Specifying the commutation or anticommutation relation between $\mathcal{S}$ and the other symmetry operators, we have 3 (19) additional symmetry classes for the complex AZ (real ${\cal P}$ AZ) class with sublattice symmetry Kawabata et al. . We can also add sublattice symmetry to ${\cal P}\text{AZ}^{{\dagger}}$ symmetry class, but each real ${\cal P}\text{AZ}^{{\dagger}}$ class with sublattice symmetry is equivalent to the corresponding real ${\cal P}$ AZ class with sublattice symmetry. Whereas pseudo-Hermiticity can also be considered as an additional symmetry to ${\cal P}$ AZ symmetry, each ${\cal P}$ AZ or ${\cal P}\text{AZ}^{{\dagger}}$ symmetry class with pseudo-Hermiticity is equivalent to the corresponding ${\cal P}$ AZ or ${\cal P}\text{AZ}^{{\dagger}}$ symmetry class, or that with sublattice symmetry. In total, we have $16+3+19=38$ symmetry classes.

II SII. Classification tables

In Tables S2-S7, we provide topological classification of exceptional points at generic momentum points for all the 38 symmetry classes and two types of complex-energy gaps. A part of these tables is shown in Table 1 in the main text. Detailed derivations of these classification tables are provided in Sec. SIII. We note that each symmetry class in Tables S6 and S7 is equivalent to the corresponding symmetry class in Tables S2-S5 Kawabata et al. . Moreover, the following seven pairs of the symmetry classes in Tables S2-S5 are equivalent to each other, respectively: ( ${\cal P}$ AI & ${\cal P}\text{D}^{{\dagger}}$ ), ( ${\cal P}$ AII & ${\cal P}\text{C}^{{\dagger}}$ ), ( ${\cal P}$ AI + ${\cal S}_{-}$ & ${\cal P}$ AII + ${\cal S}_{-}$ ), ( ${\cal P}$ BDI + ${\cal S}_{-+}$ & ${\cal P}$ DIII + ${\cal S}_{-+}$ ), ( ${\cal P}$ BDI + ${\cal S}_{--}$ & ${\cal P}$ DIII + ${\cal S}_{--}$ ), ( ${\cal P}$ CII + ${\cal S}_{-+}$ & ${\cal P}$ CI + ${\cal S}_{-+}$ ), and ( ${\cal P}$ CII + ${\cal S}_{--}$ & ${\cal P}$ CII + ${\cal S}_{-+}$ ). Here ${\cal P}$ AI + ${\cal S}_{-}$ denotes class ${\cal P}$ AI with additional sublattice symmetry ${\cal S}_{-}$ that anticommutes with $\mathcal{P}\mathcal{T}_{+}$ symmetry, and so on.

III SIII. Derivations of topological classification

Topology of exceptional points is determined in a similar manner to that of Fermi surfaces in Hermitian systems Morimoto and Furusaki (2014); Kobayashi et al. (2014); Zhao et al. (2016). In fact, whereas a complex-energy gap is closed at the gapless point, it may be open and a topological invariant can be assigned on a $\left(p-1\right)$ -dimensional surface $S^{p-1}$ that encloses the gapless point. On this enclosing surface $S^{p-1}$ , one of the kinetic terms in Eq. (6) in the main text can be regarded as a mass term and the topology of the exceptional point is determined by classifying this mass term, which reduces to the extension of Clifford algebra Chiu et al. (2016); Karoubi (2008) as with the gapped case.

As discussed in the main text, non-Hermitian topology crucially depends on the types of complex-energy gaps Kawabata et al. . In particular, a proper representation of non-Hermitian Dirac matrices depends on the types of complex-energy gaps. We first consider the case where the Hamiltonian has a point gap on $S^{p-1}$ . Without loss of generality, we take the reference point of the gap as $E=0$ . In the presence of the point gap [i.e., $\forall\,{\bm{k}}~{}~{}E\left({\bm{k}}\right)\neq 0$ ], a generic non-Hermitian Hamiltonian can be flattened into a unitary matrix Kawabata et al. . As a result, a set of non-Hermitian Dirac matrices $\{\gamma_{i}^{\rm P}\}$ for the point gap does not necessarily follow Clifford algebra. Instead, a set of Hermitian matrices $\{\gamma_{i}\}$ defined by

[TABLE]

follows Clifford algebra

[TABLE]

Accordingly, symmetries impose the following constraints on the extended Hermitian Dirac matrices:

[TABLE]

Moreover, $\gamma_{i}$ respects additional chiral (sublattice) symmetry by construction:

[TABLE]

The Dirac matrices $\gamma_{i}$ ’s and the above symmetry operators form Clifford algebra, and the topology of the “point-gapless” point is determined by the extension of Clifford algebra Chiu et al. (2016).

On the other hand, a non-Hermitian Hamiltonian with a line gap can be flattened into a Hermitian (an anti-Hermitian) matrix in the presence of a real (imaginary) gap [i.e., $\forall\,{\bm{k}}~{}~{}{\rm Re}\,E\left({\bm{k}}\right)\neq 0$ ( ${\rm Im}\,E\left({\bm{k}}\right)\neq 0$ )] Kawabata et al. . Without loss of generality, the reference line of the gap is taken to be ${\rm Re}\,E=0$ ( ${\rm Im}\,E=0$ ). As a result, a set of Dirac matrices $\{\gamma_{i}^{\rm L}\}$ follows Clifford algebra. As with the case of the point gap, the Dirac matrices $\gamma_{i}^{\rm L}$ ’s and the symmetry operators form Clifford algebra and the topology of the “line-gapless” point is determined by its extension. We note that the following formulas are useful for the extension of Clifford algebra Karoubi (2008):

[TABLE]

where $Cl_{n}$ and $Cl_{m,n}$ are complex and real Clifford algebras, respectively, and ${\cal C}_{n}$ and ${\cal R}_{n}$ are the corresponding classifying spaces, which have Bott periodicity, i.e., ${\cal C}_{n+2}={\cal C}_{n}$ , ${\cal R}_{n+8}={\cal R}_{n}$ . The classifying spaces and their homotopy groups are summarized in Table S8.

In Table S9, we explicitly provide the extensions of Clifford algebras and the corresponding classifying spaces for all the 38 symmetry classes and two types of complex-energy gaps, which leads to classification tables S2-S7 including Table 1 in the main text. We respectively denote $\mathcal{P}\mathcal{T}_{+}$ ( ${\sf P}{\sf T}_{+}$ ) and $\mathcal{C}_{-}\mathcal{P}$ ( ${\sf C}_{-}{\sf P}$ ) symmetries as $\mathcal{PT}$ ( ${\sf PT}$ ) and $\mathcal{CP}$ ( ${\sf CP}$ ) symmetries for simplicity in Table S9.

Below we explicitly provide a couple of examples of the classification procedure. First, let us consider class AIII. A non-Hermitian Hamiltonian in class AIII respects chiral symmetry [Eq. (4) in the main text]:

[TABLE]

In the presence of a point gap, the extended Hermitian Dirac matrices $\gamma_{i}$ ’s defined as Eq. (S3) respect chiral symmetries defined by Eqs. (S19) and (S23). Since ${\sf\Sigma}$ and ${\sf\Gamma}$ anticommute with each other, a set of operators

[TABLE]

forms complex Clifford algebra $Cl_{p+2}$ . Hence, when one of the Dirac matrices is regarded as a mass term, we have an extension of Clifford algebra $Cl_{p+1}\rightarrow Cl_{p+2}$ , which leads to the classifying space ${\cal C}_{p+1}$ . The topology of point-gapless points with codimension $p$ in class AIII is determined by $\pi_{0}\left({\cal C}_{p+1}\right)$ [see Table S8 for $\pi_{0}\left({\cal C}_{p+1}\right)$ ]. By contrast, different classifying spaces characterize exceptional points when a line gap is relevant. In the presence of a real gap, $\gamma_{i}^{\rm L}=\gamma_{i}$ respects chiral symmetry, and hence a set of operators

[TABLE]

forms complex Clifford algebra $Cl_{p+1}$ , which leads to the classifying space ${\cal C}_{p}$ with its topology $\pi_{0}\left({\cal C}_{p}\right)$ . On the other hand, in the presence of an imaginary gap, $\gamma_{i}^{\rm L}=\text{i}\gamma_{i}$ respects chiral symmetry, i.e.,

[TABLE]

Hence, a set of operators

[TABLE]

forms complex Clifford algebra $Cl_{p}$ and all of them commute with $\Gamma$ , which leads to the classifying space ${\cal C}_{p-1}\times{\cal C}_{p-1}$ with its topology $\pi_{0}\left({\cal C}_{p-1}\right)\oplus\pi_{0}\left({\cal C}_{p-1}\right)$ .

Let us also consider class ${\cal P}$ CI as another example. A non-Hermitian Hamiltonian in class ${\cal P}$ CI respects ${\cal PT}$ symmetry and ${\cal CP}$ symmetry [Eqs. (2) and (3) in the main text]:

[TABLE]

In the presence of a point gap, the extended Hermitian Dirac matrices $\gamma_{i}$ ’s defined as Eq. (S3) respect Eqs. (S7), (S13), and (S23). Since the symmetry operators satisfy

[TABLE]

a set of operators

[TABLE]

forms real Clifford algebra $Cl_{2,p+2}$ with the imaginary unit $J$ and complex conjugation ${\cal K}$ . Hence, when one of the Dirac matrices is regarded as a mass term, we have an extension of Clifford algebra $Cl_{2,p+1}\rightarrow Cl_{2,p+2}$ , which leads to the classifying space ${\cal R}_{p+7}$ . The topology of point-gapless points with codimension $p$ in class ${\cal P}$ CI is determined by $\pi_{0}\left({\cal R}_{p+7}\right)$ . By contrast, different classifying spaces characterize exceptional points when a line gap is relevant. In the presence of a real gap, $\gamma_{i}^{\rm L}=\gamma_{i}$ respects ${\cal PT}$ and ${\cal CP}$ symmetries, and hence a set of operators

[TABLE]

forms real Clifford algebra $Cl_{2,p+1}$ , which leads to the classifying space ${\cal R}_{p+6}$ with its topology $\pi_{0}\left({\cal R}_{p+6}\right)$ . On the other hand, in the presence of an imaginary gap, $\gamma_{i}^{\rm L}=\text{i}\gamma_{i}$ respects ${\cal PT}$ and ${\cal CP}$ symmetries, i.e.,

[TABLE]

Hence, a set of operators

[TABLE]

forms real Clifford algebra $Cl_{2,p}$ and all of them commute with $({\cal CPK})({\cal PTK})$ . Noting $[({\cal CPK})({\cal PTK})]^{2}=-1$ , we have the classifying space ${\cal C}_{p+1}$ with its topology $\pi_{0}\left({\cal C}_{p+1}\right)$ . The extensions of Clifford algebra and the corresponding classifying spaces for all the other symmetry classes are similarly determined as Table S9.

IV SIV. Classification of previous works

The previous works Berry (2004); Esaki et al. (2011); Zhen et al. (2015); González and Molina (2016); Leykam et al. (2017); Xu et al. (2017); González and Molina (2017); Chernodub (2017); Zyuzin and Zyuzin (2018); Cerjan et al. (2018); Zhou et al. (2018); Molina and González (2018); Shen et al. (2018); Shen and Fu (2018); Carlström and Bergholtz (2018); Moors et al. (2019); Okugawa and Yokoyama (2019); Budich et al. (2019); Yang and Hu (2019); Zhou et al. (2019); Wang et al. (2019); Yoshida et al. (2019); Carlström et al. (2019); Zyuzin and Simon (2019); Papaj et al. (2019); Kozii and Fu ; Cerjan et al. ; Luo et al. ; Lee et al. ; Chernodub and Cortijo ; Bergholtz and Budich on exceptional points and non-Hermitian topological semimetals are categorized into our classification (Table S10). For example, topologically stable exceptional points in two dimensions Berry (2004); Zhou et al. (2018); Shen et al. (2018); Shen and Fu (2018); Papaj et al. (2019); Kozii and Fu ; Luo et al. ; Bergholtz and Budich and exceptional rings in three dimensions Berry (2004); Xu et al. (2017); González and Molina (2016, 2017); Zyuzin and Zyuzin (2018); Cerjan et al. (2018); Molina and González (2018); Carlström and Bergholtz (2018); Yang and Hu (2019); Carlström et al. (2019); Zyuzin and Simon (2019); Cerjan et al. ; Luo et al. ; Lee et al. ; Chernodub and Cortijo correspond to the $\mathbb{Z}$ index for no symmetry, $p=2$ , and point gap. Topologically stable exceptional rings in two-dimensional non-Hermitian systems with chiral symmetry Esaki et al. (2011); Budich et al. (2019); Yoshida et al. (2019) ( ${\cal PT}$ symmetry Zhen et al. (2015); Okugawa and Yokoyama (2019); Zhou et al. (2019)) correspond to the $\mathbb{Z}$ ( $\mathbb{Z}_{2}$ ) index for chiral ( ${\cal PT}$ ) symmetry, $p=1$ , and point gap. We note that an exceptional ring in two dimensions protected by pseudo-Hermiticity Budich et al. (2019) (the Hermitian conjugate of $\mathcal{C}\mathcal{P}$ symmetry, or $\mathcal{P}\mathcal{T}_{-}$ symmetry Okugawa and Yokoyama (2019)) is equivalent to that protected by chiral symmetry ( $\mathcal{P}\mathcal{T}$ symmetry).

V SV. Exceptional points and parity-time-symmetry breaking

As discussed in the main text, a distinctive property of an exceptional point is swapping of eigenenergies and eigenstates upon its encircling [Fig. S1 (a)]. Such an exceptional point indeed has the unique gapless structure with no Hermitian analogs around which a point gap is open but a line gap is closed [Fig. 1 (d) in the main text]. Another typical exceptional points appear upon ${\cal PT}$ -symmetry breaking Heiss (2012); Konotop et al. (2016); *Feng-review; *Christodoulides-review; *Alu-review; Bender and Boettcher (1998); *Bender-02; *Bender-review [see Eq. (2) in the main text for the definition of ${\cal PT}$ symmetry]. $\mathcal{PT}$ -symmetric systems have entirely real spectra when $\mathcal{PT}$ symmetry is unbroken, whereas some eigenenergies form complex-conjugate pairs when it is spontaneously broken; between the two phases an exceptional point appears. We here consider a ${\cal PT}$ -transition point ${\bm{k}}={\bm{k}}_{\rm EP}$ at which two complex bands coalesce with each other and denote their right and left eigenstates as $\ket{\psi_{1,2}\left({\bm{k}}\right)}$ and $\langle\!\langle\psi_{1,2}\left({\bm{k}}\right)|$ , normalized by $\langle\!\langle\psi_{m}\left({\bm{k}}\right)|\psi_{n}\left({\bm{k}}\right)\rangle=\delta_{mn}$ for ${\bm{k}}\neq{\bm{k}}_{\rm EP}$ . When $\mathcal{PT}$ symmetry is unbroken and the spectrum is real, we have Bender and Boettcher (1998); *Bender-02; *Bender-review

[TABLE]

with a proper choice of the gauges. On the other hand, when $\mathcal{PT}$ symmetry is spontaneously broken and the two eigenstates form a complex-conjugate pair, we have Bender and Boettcher (1998); *Bender-02; *Bender-review

[TABLE]

Since both equalities are compatible at the ${\cal PT}$ -transition point ${\bm{k}}={\bm{k}}_{\rm EP}$ , the two bands coalesce, i.e.,

[TABLE]

and thus form an exceptional point.

Remarkably, this $\mathcal{PT}$ -transition-induced exceptional point can also have the unique gapless structure. For simplicity, we consider a ${\cal PT}$ -symmetric system with $({\cal PT})({\cal PT})^{*}=+1$ in one dimension. A representative model is given as

[TABLE]

which has exceptional points $k_{\rm EP}=\pm\gamma$ . In fact, a point gap is open except for $k=k_{\rm EP}$ , whereas a line gap for the imaginary (real) part is closed in the ${\cal PT}$ -unbroken (-broken) phase [Fig. S1 (b)]. Due to the point gap, the $\mathbb{Z}_{2}$ topological charge $\nu\in\{0,1\}$ can be defined by Gong et al. (2018); Okugawa and Yokoyama (2019)

[TABLE]

The change in $\nu$ signals ${\cal PT}$ -symmetry breaking that accompanies an exceptional point. We note that this expression is applicable even if the degeneracy is not defective. Another expression for this $\mathbb{Z}_{2}$ topological charge is given by

[TABLE]

where $w$ is a real matrix that satisfies $w_{mn}\left(k\right):=\langle\!\langle\psi_{m}\left(k\right)|\,{\cal PT}\,|\psi_{n}\left(k\right)\rangle^{*}$ and $w^{2}=ww^{*}=1$ . Since this charge takes [math] ( $1$ ) in the ${\cal PT}$ -unbroken (broken) phase, its change signals ${\cal PT}$ -symmetry breaking that accompanies an exceptional point.

VI SVI. Recipe for exceptional points

Based on our classification, we can systematically obtain non-Hermitian gapless systems having exceptional points:

In the classification table, choose a symmetry class that has nontrivial topology for a line gap in $p$ codimension. 2. 2.

Consider a $p$ -dimensional Hermitian (or anti-Hermitian) gapless Dirac model $H\left(\bm{k}\right)=\sum_{i=1}^{p}v_{i}k_{i}\gamma_{i}$ that respects symmetry of the class and in which the matrix size $n$ of $\gamma_{i}$ ’s is minimal. The Dirac point at ${\bm{k}}=0$ is topologically stable due to the nontrivial topology for a line gap in $p$ codimension. Here note that topological charges for a line gap reduce to those of Hermitian systems. Since the energy spectrum is entirely real (or pure imaginary) and a line gap is closed at this Dirac point, it is located at zero energy (i.e., a reference point for a point gap in the complex-energy plane). Moreover, it is maximally degenerate ( $n$ -fold degenerate). 3. 3.

Add a non-Hermitian perturbation that preserves the symmetry and is independent of ${\bm{k}}$ . The non-Hermiticity makes the spectrum complex-valued, and the Dirac point spreads in both momentum space and complex-energy plane. Because of the nontrivial topology for a line gap, there remains a gapless region with respect to a line gap, i.e., a finite segment of a reference line for the line gap. Then, if this gapless segment contains the reference point for a point gap, an exceptional point can appear at this reference point. Indeed, the line gap is closed but the point gap is open around the reference point, which is a characteristic of exceptional points. 4. 4.

The obtained exceptional point has $p$ or less codimension, and its topological stability is ensured by the nontrivial topology for a point gap in the corresponding codimension. Reading off the possible stable structures (point, line, surface, and so on) and the topological invariants for a point gap from the classification table, we can confirm that it is indeed an exceptional point. 5. 5.

Very frequently, the symmetry-preserving non-Hermitian perturbation can be chosen so that the reference point keeps the $n$ -fold degeneracy. In this case, we can easily prove that this reference point is defective (nondiagonalizable). In fact, since the perturbation is ${\bm{k}}$ -independent, the perturbed energy spectrum passes the reference point at ${\bm{k}}={\bm{k}}_{\rm EP}\neq 0$ . Then because $H\left({\bm{k}}_{\rm EP}\right)$ is not proportional to the identity matrix, the $n$ -fold degeneracy of $H\left({\bm{k}}_{\rm EP}\right)$ implies that $H\left({\bm{k}}_{\rm EP}\right)$ has a nontrivial Jordan form, i.e., $H\left({\bm{k}}_{\rm EP}\right)$ is defective.

A Weyl exceptional ring Berry (2004); Xu et al. (2017) and a topologically stable exceptional point in two dimensions Zhou et al. (2018); Shen et al. (2018) are examples that can be constructed with this recipe. In the absence of symmetry (i.e., class A), there appear a line-gapless point that is characterized by a $\mathbb{Z}$ topological charge (Chern number) in three dimensions ( $p=3$ ), as well as a point-gapless point that is characterized by a $\mathbb{Z}$ topological charge (winding number) in two dimensions ( $p=2$ ). Based on this fact, we begin with a Hermitian Weyl semimetal

[TABLE]

which has a robust degenerate point at ${\bm{k}}=0$ . We next add a non-Hermitian perturbation to it:

[TABLE]

The Weyl point morphs into a ring of exceptional points at $k_{x}^{2}+k_{y}^{2}=\gamma^{2}$ , $k_{z}=0$ Berry (2004); Xu et al. (2017). After taking $k_{y}=0$ and regarding this non-Hermitian three-dimensional model as a two-dimensional one, we have

[TABLE]

where a pair of exceptional points appears at $\left(k_{x},k_{z}\right)=\left(\pm\gamma,0\right)$ in two dimensions Zhou et al. (2018); Shen et al. (2018).

Another example is a chiral-symmetry-protected exceptional point in three dimensions discussed in the main text. In the presence of chiral symmetry (i.e., class AIII), there appear a line-gapless point that is characterized by a $\mathbb{Z}$ topological charge (winding number) in four dimensions ( $p=4$ ), as well as a point-gapless point that is characterized by a $\mathbb{Z}$ topological charge (Chern number) in three dimensions ( $p=3$ ). Based on this fact, we begin with a Hermitian gapless Dirac model with chiral symmetry in four dimensions:

[TABLE]

which respects chiral symmetry with $\Gamma=\sigma_{z}$ [i.e., $\sigma_{z}H^{{\dagger}}\left({\bm{k}}\right)\sigma_{z}^{-1}=-H\left({\bm{k}}\right)$ ]. Due to chiral symmetry, this model possesses a robust gapless point at ${\bm{k}}=0$ , which is topologically protected by the three-dimensional winding number $W=1$ defined as Schnyder et al. (2008); *Ryu-10; Qi et al. (2010); Kawakami et al. (2018)

[TABLE]

where $S^{3}$ is a hypersphere surrounding the gapless point ${\bm{k}}=0$ and $\epsilon^{\alpha\beta\gamma}$ is the Levi-Civita symbol. Moreover, this gapless point is four-fold degenerate. Now we add a non-Hermitian perturbation to it:

[TABLE]

Noting $\{\sigma_{x}\tau_{x},\sigma_{z}\tau_{x}\}=\{\sigma_{y},\sigma_{z}\tau_{x}\}=0$ and $[\sigma_{x}\tau_{y},\sigma_{z}\tau_{x}]=[\sigma_{x}\tau_{z},\sigma_{z}\tau_{x}]=0$ , we have

[TABLE]

leading to the complex spectrum

[TABLE]

The original four-fold degenerate point thus morphs into an exceptional ring at $k_{x}^{2}+k_{w}^{2}=\gamma^{2}$ , $k_{y}=k_{z}=0$ . After taking $k_{w}=0$ and regarding this non-Hermitian four-dimensional model as a three-dimensional one, we have

[TABLE]

where a pair of exceptional points appears at $\left(k_{x},k_{y},k_{z}\right)=\left(\pm\gamma,0,0\right)$ in three dimensions. In fact, the Hamiltonian is defective (nondiagonalizable) at these points:

[TABLE]

whose Jordan matrix $J$ is given as

[TABLE]

VII SVII. Topological charge of a chiral-symmetry-protected gapless point

In the presence of chiral symmetry defined by Eq. (4) in the main text, i.e.,

[TABLE]

with a unitary matrix $\Gamma$ , the topological charge of a non-Hermitian gapless point with odd $p\geq 1$ is given as the Chern number of the Hermitian matrix $\text{i}H\left({\bm{k}}\right)\Gamma$ that is defined on a surface $S^{p-1}$ enclosing the gapless point Kawabata et al. ; Yoshida et al. (2019). To see this fact, let us consider the following Hermitian matrix:

[TABLE]

Here the presence of a point gap for $H\left({\bm{k}}\right)$ is equivalent to the presence of an energy gap for ${\sf H}\left({\bm{k}}\right)$ Gong et al. (2018); Kawabata et al. . Due to Eq. (4) in the main text, ${\sf H}\left({\bm{k}}\right)$ also respects chiral symmetry:

[TABLE]

Moreover, ${\sf H}\left({\bm{k}}\right)$ respects intrinsic chiral symmetry by construction:

[TABLE]

From the two chiral symmetries, we have the following commutation relation

[TABLE]

As a result, two Hermitian matrices ${\sf H}\left({\bm{k}}\right)$ and $\text{i}{\sf\Gamma\Sigma}$ can be simultaneously diagonalized with a unitary matrix ${\sf U}$ as

[TABLE]

Therefore, the topology of non-Hermitian $H\left({\bm{k}}\right)$ is captured by Hermitian $\text{i}H\left({\bm{k}}\right)\Gamma$ with no symmetry, and the Chern number can be defined for $\text{i}H\left({\bm{k}}\right)\Gamma$ on the surface $S^{p-1}$ with odd $p\geq 1$ .

For the non-Hermitian model in Eq. (8) in the main text, we have

[TABLE]

where a $SU\left(2\right)$ rotation $U:=e^{-\text{i}\left(\pi/4\right)\sigma_{x}}$ changes $\sigma_{y}$ to $\sigma_{z}$ (i.e., $U\sigma_{y}U^{{\dagger}}=\sigma_{z}$ ). Since this expression is a direct sum of the two Weyl semimetals whose gapless points are located at ${\bm{k}}={\bm{k}}_{\rm EP}=\left(\pm\gamma,0,0\right)$ , we have the Chern number $\pm 1$ for each exceptional point ${\bm{k}}_{\rm EP}$ .

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