Topological unification of time-reversal and particle-hole symmetries in non-Hermitian physics

Topological phases are enriched in non-equilibrium open systems effectively described by non-Hermitian Hamiltonians. While several properties unique to non-Hermitian topological systems were uncovered, the fundamental role of symmetry in non-Hermitian physics has yet to be fully understood, and it has remained unclear how symmetry protects non-Hermitian topological phases. Here we show that two fundamental anti-unitary symmetries, time-reversal and particle-hole symmetries, are topologically equivalent in the complex energy plane and hence unified in non-Hermitian physics. A striking consequence of this symmetry unification is the emergence of unique non-equilibrium topological phases that have no counterparts in Hermitian systems. We illustrate this by presenting a non-Hermitian counterpart of the Majorana chain in an insulator with time-reversal symmetry and that of the quantum spin Hall insulator in a superconductor with particle-hole symmetry. Our work establishes a fundamental symmetry principle in non-Hermitian physics and paves the way towards a unified framework for non-equilibrium topological phases.


SUPPLEMENTARY NOTE 1: PHYSICAL MEANING OF A COMPLEX GAP
The presence of a complex gap has a significant influence on the non-equilibrium wave dynamics. To understand this, we consider a three-level system shown in Supplementary Figure 1 (a) as an example. Here the ground state |g⟩ is resonantly coupled to the two excited states |e 1 ⟩ and |e 2 ⟩ that have the same energy levels but the different decay rates (linewidths) γ 1 and γ 2 (γ 1 < γ 2 ), and the driving strength (Rabi frequency) Ω between |g⟩ and |e 1 ⟩ is equal to that between |g⟩ and |e 2 ⟩. In the framework of the non-Hermitian band theory, this situation corresponds to the presence of two modes having the same real parts but the different imaginary parts of eigenenergies at a particular wavenumber k. Then when the wave function of the system is denoted as |ψ⟩, the wave dynamics (in the rotating frame) is governed by where the basis is chosen as (|e 1 ⟩ , |g⟩ , |e 2 ⟩). Here |ψ⟩ represents an electric-field envelope in classical photonics and a quantum wave function in quantum physics (the Planck constant ℏ is set to unity). Now let us consider driving the system at the Rabi frequency Ω. When the state is initially prepared to be |g⟩, the intensity or the population for the excited state |e i ⟩ (i = 1, 2) at time t is obtained as The difference of the decay rates (imaginary parts of the eigenenergies) significantly affects the behavior of the wave dynamics, although the two modes have the same energy levels (real parts of the eigenenergies). In fact, the excitation to the mode with the larger decay rate is suppressed [ Supplementary Figure 1 (b)]. We remark that such suppression of the wave dynamics due to the presence of dissipation is known as the quantum Zeno effect [1,2]. Although termed as the quantum Zeno effect due to some historical reasons, this phenomenon occurs in any linear dynamics of probability amplitudes and especially in classical wave dynamics. In fact, an early experimental demonstration was performed for classical light [3]. These dissipative effects are also relevant in many-body systems [4,5]. In particular, even a purely imaginary band gap can play a role similar to that of a real band gap and confine the dynamics of a wave packet within the lower band, provided that the driving strength is weak enough compared with the band gap [6].
Supplementary Figure 1. Physical meaning of a complex gap. (a) Energy-level diagram of a three-level system. The ground state |g⟩ is resonantly coupled to the two excited states |e1⟩ and |e2⟩ that have the same energy levels but the different decay rates (linewidths) γ1 and γ2 (γ1 < γ2), and the driving strength (Rabi frequency) is equal to Ω for both excited states. (b) Evolution of the intensity or the population for the excited states |e1⟩ and |e2⟩ (Ω = 1.0, γ1 = 0.5, γ2 = 5.0). The excitation to the mode |e2⟩ is suppressed due to the larger decay rate (γ1 < γ2).

SUPPLEMENTARY NOTE 3: NON-HERMITIAN MAJORANA CHAIN
Certain symmetry-protected topological phases survive even in the presence of non-Hermiticity. Such phases include superconducting wires that respect particle-hole symmetry with C 2 = +1 (1D class D). Let us express the Bogoliubov-de Gennes (BdG) Hamiltonian as H (k) = h 0 (k) I + ⃗ h (k) · ⃗ σ with the identity matrix I and Pauli matrices ⃗ σ := (σ x , σ y , σ z ). Then the presence of particle-hole symmetry C : Therefore, h z becomes real at the particle-hole-symmetric momentum k 0 ∈ {0, π}, and hence the topological invariant ν D can be defined by as long as h z (0) and h z (π) are nonzero. The presence of particle-hole symmetry allows one to introduce the topological invariant ν D as in the Hermitian topological superconductors [11], although h 0,x,y (k 0 ) can be nonzero (pure imaginary) in contrast to Hermitian systems. Moreover, when we introduce Majorana operatorsâ k andb k in momentum space asâ Thus, the non-Hermitian part of the momentum-space Hamiltonian at the particle-hole-symmetric momentum k 0 in the Majorana basis is obtained as X (k 0 ) = ih z (k 0 ) σ y , and the topological invariant defined above is also expressed as (−1) The above discussion applies to a non-Hermitian Majorana chain with asymmetric hopping: whereĉ j (ĉ † j ) annihilates (creates) a fermion on site j; t L > 0 (t R > 0) is the hopping amplitude from right to left (from left to right), ∆ ∈ C is the p-wave pairing gap, and µ ∈ R is the chemical potential. Here non-Hermiticity arises from the asymmetric hopping t L ̸ = t R [8,12,13]. The BdG Hamiltonian is given by Hence the energy dispersion is obtained as and the complex bands are separated from each other by the gap of magnitude min {2 |µ + (t L + t R )| , 2 |µ − (t L + t R )|} as schematically illustrated in Supplementary Figure 2 (a,b). Moreover, the topological invariant is obtained as , and the non-trivial topology of the bulk (ν D = 1; |µ| < t L + t R ) is accompanied by the emergence of the Majorana edge states that are topologically protected with particle-hole symmetry as shown in Supplementary Figure 2

SUPPLEMENTARY NOTE 4: SYMMETRY AND COMPLEX-BAND STRUCTURE
We consider in detail a complex-band structure of a generic one-dimensional system with time-reversal symmetry (1D class AI). We here investigate a two-band system (E + (k) , E − (k)) for the sake of simplicity, but the discussion can be straightforwardly generalized to arbitrary 2n-band systems. In the presence of Hermiticity, the real bands individually respect time-reversal symmetry [ Supplementary Figure 3

(a)]:
where topological phases are absent. In the presence of strong non-Hermiticity, on the other hand, time-reversal symmetry is spontaneously broken and the two complex bands are paired via time-reversal symmetry [Supplementary Figure 3 (d, e)]: Remarkably, this band structure becomes gapless in the presence of Hermiticity since  (11) and Hence the complex gap closes at a time-reversal-invariant momentum k 0 for this Hamiltonian and gap closing associated with a topological phase transition should be accompanied between these phases [Supplementary Figure 3  the emergent non-Hermitian topological phases cannot be continuously deformed into any Hermitian phase that belongs to the same symmetry class. Importantly, there is a fundamental and non-trivial relationship between symmetry classes AI and D as a direct consequence of the topological unification of time-reversal and particle-hole symmetries. If we begin with a Hermitian Hamiltonian that belongs to 1D class D and possesses the topological phase There is a crucial distinction between this emergent non-Hermitian topological insulator and the non-Hermitian topological insulator protected by chiral symmetry including the Su-Schrieffer-Heeger (SSH) model. In fact, the topological phase in the non-Hermitian SSH model can be continuously deformed into the topological phase in the Hermitian SSH model (Supplementary Figure 4), whereas those in the topological insulator induced by non-Hermiticity cannot be continuously deformed into any Hermitian phase as discussed above. Here the crucial distinction is that chiral symmetry is unitary (unrelated to complex conjugation) in stark contrast to the anti-unitary time-reversal and particle-hole symmetries (see Table I in the main text).
When A describes time-reversal symmetry (φ = 0), this reduces to the conventional Kramers degeneracy; when A describes particle-hole symmetry (φ = π), this implies the degeneracy with pure imaginary eigenenergies.

SUPPLEMENTARY NOTE 6: NON-HERMITIAN QUANTUM SPIN HALL INSULATOR
A generic non-Hermitian Hamiltonian that forms four bands in momentum space can be expressed by the identity matrix I, five Dirac matrices Γ i , and their ten commutators Γ ij := [Γ i , Γ j ]/2i as in Eq. (7) Therefore, the presence of both time-reversal and inversion symmetries [ Hence In particular, we consider the non-Hermitian quantum spin Hall insulator described by Eq. (7) in the main text with (15) where t, m, λ, and γ are real. The eigenstates form four bands in momentum space and their energy dispersions are obtained as which leads to The complex bands are gapped with the same parameters used in Fig. 5 in the main text (Supplementary Figure 5). The Z 2 topological invariant is determined as

SUPPLEMENTARY NOTE 7: ROBUSTNESS AGAINST DISORDER
The topological phase in the non-Hermitian Majorana chain given by Supplementary Equation (7) is immune to disorder that respects particle-hole symmetry. To confirm this robustness, we consider the following disordered non-Hermitian Majorana chain:Ĥ where (t L ) j [(t R ) j ] is the disordered hopping amplitude from site j + 1 to j (from site j to j + 1), and µ j is the disordered chemical potential on site j. The Majorana edge states are protected to have zero energy even in the presence of disorder (Supplementary Figure 6); they are topologically protected with particle-hole symmetry. The topological phase in the non-Hermitian topological insulator given by Eq. (3) in the main text is also immune to disorder that respects time-reversal symmetry. To confirm this robustness, we consider the following disordered non-Hermitian chain: where t j and δ j are the disordered hopping amplitudes between sites j − 1 and j, and γ j is the disordered gain/loss on site j. The edge states are protected to have zero energy even in the presence of disorder (Supplementary Figure 7); they are topologically protected with time-reversal symmetry. Finally, the topological phase in the non-Hermitian quantum spin Hall insulator given by Eq. (7) and shown in Fig. 5 in the main text is immune to disorder that respects time-reversal symmetry. To confirm this robustness, we consider the following disordered non-Hermitian insulator with open boundaries in the x direction and periodic boundaries in the y direction: where⃗ c x, ky (⃗ c † x, ky ) annihilates (creates) a fermion on site x and with momentum k y that has four internal degrees of freedom; m x , λ x , and γ x denote the disordered parameters. The helical edge states are topologically protected with time-reversal symmetry even in the presence of disorder [ Supplementary Figure 8 (a,b)]. Moreover, we confirm their robustness also in the following disordered non-Hermitian insulator with open boundaries in both x and y directions: Here⃗ c x, y (⃗ c † x, y ) annihilates (creates) a fermion on site (x, y), and m x,y , λ x,y , and γ x,y denote the disordered parameters. The helical edge states are robust also in this case [ Supplementary Figure 8

SUPPLEMENTARY NOTE 8: DYNAMICS OF TOPOLOGICAL EDGE STATES IN NON-HERMITIAN SYSTEMS: EXPERIMENTAL SIGNATURE OF NON-HERMITIAN TOPOLOGICAL PHASES
A hallmark of the topological phases is the emergence of robust edge states that reflect the non-trivial topology of the bulk. Remarkably, non-Hermiticity can make these edge states amplified (lasing) [15][16][17][18] and anomalous [19]. The non-Hermitian topological insulators considered in this work also exhibit unique topologically protected edge states that have no Hermitian counterparts, which serves as an experimental signature of the non-Hermitian topological phases. In the following, we consider Supplementary Figure 9. Population dynamics in the non-Hermitian topological insulator protected by time-reversal symmetry (1D class AI). An initial state is prepared to be a localized wave function |ψ0⟩ ∝ ∑ L x=1 e −x |x⟩, and the evolutions of the population (normalized intensity) |⟨x|ψt⟩| 2 / ⟨ψt|ψt⟩ are shown for (a) |x⟩ = |a1⟩, (b) |x⟩ = |b1⟩, (c) |x⟩ = |a2⟩ and (d) |x⟩ = |b2⟩ [see Fig. 3 (a) in the main text for details]. The insulator with L = 50 sites is characterized by the parameters t = δ = 1.0 and γ = 0.2 for the topological phase (blue curves) and γ = 3.0 for the trivial phase (orange curves). In the topological phase, there emerges time-reversal-symmetry-protected topological edge states whose imaginary parts vanish. As a consequence, the population of the wave function in the topological phase is greater than that in the trivial phase. the dynamics of non-Hermitian topological systems. When an initial state is prepared to be where |φ n ⟩ (|χ n ⟩) is a right (left) eigenstate of the non-Hermitian Hamiltonian [20], the wave function evolves as Although the system eventually reaches a stationary state |φ n ⟩ with the largest imaginary part of E n , unique topological features appear in the non-Hermitian transient dynamics, which have been observed in recent experiments [15][16][17][18].
For the non-Hermitian topological insulator given by Eq. (3) in the main text, the imaginary parts of the eigenenergies of the topologically protected edge states vanish due to the presence of time-reversal symmetry. Consequently, these edge states are stable, despite instability of the bulk states. The presence of such stable edge states can be detected by focusing on the evolution of the particle population near the edges. In fact, when an initial state is prepared as a localized state at one edge, its population near the edge in the topological phase is greater than that in the trivial phase (Supplementary Figure 9).
For the non-Hermitian quantum spin Hall insulator given by Eq. (7) and shown in Fig. 5 in the main text, the helical edge states emerge corresponding to the non-trivial topology of the bulk. As a consequence, when an initial state is prepared as a localized state at one edge, the wave packet remains localized near the edge in the topological phase (Supplementary Figure 10). In the trivial phase, on the other hand, the wave packet quickly diffuses into the bulk due to the absence of the edge states. There are several significant distinctions between the Hermitian and non-Hermitian quantum spin Hall insulators. In contrast to the Hermitian one, the spin-up component is more localized than the spin-down component due to the presence of the imaginary gap between the helical edge states [see Fig. 4 (b) in the main text]. Moreover, the wave packet does not move along the edges since the real parts of the energy dispersions are flat for the helical edge states [see Fig. 4 (a) in the main text].

SUPPLEMENTARY NOTE 9: BULK-EDGE CORRESPONDENCE IN NON-HERMITIAN SYSTEMS
The bulk-edge correspondence in Hermitian topological systems is generally understood with a continuum model described by a Dirac Hamiltonian [21,22]. Although the topological classification is drastically altered by non-Hermiticity as demonstrated in the main text, the bulk-edge correspondence can be generalized to a non-Hermitian continuum model in the same manner as the Hermitian one. We here consider a one-dimensional Dirac Hamiltonian that respects chiral symmetry S := σ z (hence belongs to 1D class AIII): where k is a wavenumber (momentum), and g, m, δ ∈ R are real parameters that characterize the topological or trivial phase. The energy dispersion of this continuum model is obtained as and hence the complex bands are gapped for |m| > |g|. As with the Hermitian case, the topological invariant (winding number) can be defined for the gapped phases as Here the half-integer topological invariant W is common to the continuum models and should be complemented by the structure of wave functions away from the Dirac point [23,24].
To investigate the emergence of topologically protected boundary states, we consider a domain wall defined by with a step function θ (x). Then we solve the Schrödinger equation ) σ x + (m (x) + iδ (x)) σ y ] φ (x) = 0 (30) to find the topologically protected boundary states with zero energy. For x > 0, if we take an ansatz φ + (x) = (a b) T e −x/ξ+ , the above Schrödinger equation reduces to ( ξ −1 We notice that Re [ξ + ] > 0 is needed for the localization of the boundary states. Then, for m + > 0, we have g + + m + > 0 and hence Re [ ξ −1 + + g + + m + + iδ + ] > 0; for m + < 0, we have g + − m + > 0 and hence Re [ ξ −1 + + g + − m + − iδ + ] > 0. Therefore, we obtain For x < 0, on the other hand, we obtain Due to the boundary condition φ + (0) = φ − (0), the bound states emerge if and only if the signs of m + and m − are different, reflecting the different winding numbers in each region; the emergence of the topologically protected bound states corresponds to the non-trivial topology of the bulk. Notably, similar discussions for non-Hermitian continuum models in two dimensions are found in Supplementary References [25,26].