Abstract
Topological phases are enriched in nonequilibrium open systems effectively described by nonHermitian Hamiltonians. While several properties unique to nonHermitian topological systems were uncovered, the fundamental role of symmetry in nonHermitian physics has yet to be fully understood, and it has remained unclear how symmetry protects nonHermitian topological phases. Here we show that two fundamental antiunitary symmetries, timereversal and particlehole symmetries, are topologically equivalent in the complex energy plane and hence unified in nonHermitian physics. A striking consequence of this symmetry unification is the emergence of unique nonequilibrium topological phases that have no counterparts in Hermitian systems. We illustrate this by presenting a nonHermitian counterpart of the Majorana chain in an insulator with timereversal symmetry and that of the quantum spin Hall insulator in a superconductor with particlehole symmetry. Our work establishes a fundamental symmetry principle in nonHermitian physics and paves the way towards a unified framework for nonequilibrium topological phases.
Introduction
It was Wigner who showed that all symmetries are either unitary or antiunitary and identified the fundamental role of timereversal symmetry in antiunitary operations^{1}. Timereversal symmetry is complemented by particlehole and chiral symmetries, culminating in the Altland–Zirnbauer (AZ) tenfold classification^{2}. The AZ classification plays a key role in characterizing the topological phases^{3,4,5} of condensed matter such as insulators^{6,7,8,9,10,11,12} and superconductors^{13,14,15,16}, as well as photonic systems^{17} and ultracold atoms^{18}, all of which are classified into the periodic table^{19,20,21,22}. Whereas the topological phase in the quantum Hall insulator is free from any symmetry constraint and breaks down in the presence of timereversal symmetry^{6,7}, certain topological phases are protected by symmetry; for example, the quantum spin Hall insulator is protected by timereversal symmetry^{8,9,10,11} and the Majorana chain is protected by particlehole symmetry^{14}.
Despite its enormous success, the existing framework for topological phases mainly concerns equilibrium closed systems. Meanwhile, there has been growing interest in nonequilibrium open topological systems, especially nonHermitian topological systems^{23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42}. In general, nonHermiticity arises from the presence of energy or particle exchanges with an environment^{43,44}, and a number of phenomena and functionalities unique to nonconservative systems have been theoretically predicted^{45,46,47,48,49,50,51,52,53,54,55,56} and experimentally observed^{57,58,59,60,61,62,63,64,65,66,67}. Here symmetry again plays a key role; for example, spectra of nonHermitian Hamiltonians can be entirely real in the presence of paritytime symmetry^{46}. Recently, a topological band theory for nonHermitian Hamiltonians was developed, and the topological phase in the quantum Hall insulator was shown to persist even in the presence of nonHermiticity^{31}. Moreover, topological lasers were proposed and realized on the basis of the interplay between nonHermiticity and topology^{39,41,42}. However, it is yet to be understood how symmetry constrains nonHermitian systems in general and how symmetry protects nonHermitian topological phases.
Here we point out that two fundamental antiunitary symmetries, timereversal symmetry and particlehole symmetry, are the two sides of the same symmetry in nonHermitian physics. In fact, once we lift the Hermiticity constraint on the Hamiltonian H, the Wigner theorem dictates that an antiunitary operator \(\mathcal{A}\) is only required to satisfy
This suggests that timereversal symmetry (φ = 0) and particlehole symmetry (φ = π) can be continuously transformed into each other in the complex energy plane. This topological unification leads to striking predictions about topological phenomena. In particular, properties intrinsic to topological insulators can appear also in the corresponding topological superconductors, and vice versa: a counterpart of the Majorana chain in a nonHermitian insulator with timereversal symmetry and that of the quantum spin Hall insulator in a nonHermitian superconductor with particlehole symmetry. We emphasize that such topological phases are absent in Hermitian systems; nonHermiticity alters the topological classification in a fundamental manner, and nonequilibrium topological phases unique to nonHermitian systems emerge as a result of the topological unification of timereversal and particlehole symmetries.
Results
Symmetries and complex spectra
To go beyond the Hermitian paradigm, it is necessary to revisit some fundamental concepts relevant to topology. We start by defining a gapped complex band. Let us consider a complexband structure \(\{ E_n({\boldsymbol{k}}) \in {\Bbb C}\}\), where k is a crystal wavevector in the Brillouin zone and n is a band index. Since a band gap should refer to an energy range in which no states exist, it is reasonable to define a band n to be gapped such that E_{m} (k) ≠ E_{n} (k) for all the band indices m ≠ n and wavevectors k (Fig. 1)^{31}, which is a natural generalization of the gapped band structure in the Hermitian band theory and explains the experimentally observed topological edge states in nonHermitian systems^{36,37,38,39,41,42}. Notably, the presence of a complex gap has a significant influence on the nonequilibrium wave dynamics (see Supplementary Note 1 for details). This definition of a complex gap is distinct from that adopted in ref. ^{68} and hence the corresponding topological classification is different.
We next consider the constraints on complex spectra imposed by antiunitary symmetry (Table 1). A Hamiltonian H has timereversal and particlehole symmetries if and only if there exist antiunitary operators \(\mathcal{T}\) and \(\mathcal{C}\) such that
and \({\cal T}\,z\,{\cal T}^{  1} = z^ \ast\), \({\cal C}\,z\,{\cal C}^{  1} = z^ \ast\) for all \(z \in {\Bbb C}\). For Hermitian Hamiltonians with entirely real spectra, timereversal symmetry places no constraints on the real spectra and particlehole symmetry renders the real spectra symmetric about zero energy. By contrast, for nonHermitian Hamiltonians, of which spectra are not restricted to be real, timereversal symmetry renders the spectra symmetric about the real axis^{46}, while particlehole symmetry makes the spectra symmetric about the imaginary axis^{26,33,52}; they are topologically equivalent in the complex energy plane (see Supplementary Note 2 for details). This crucial observation leads to the expectation that nonHermiticity topologically unifies symmetry classes (Fig. 2), as shown below. We note that the role of chiral symmetry is unchanged in nonHermitian physics, since it is defined to be unitary and does not involve complex conjugation.
Topological unification
Motivated by the topological equivalence of timereversal and particlehole symmetries in the complex energy plane, we consider a general antiunitary symmetry \(\mathcal{A}\) defined by Eq. (1). Here \(\mathcal{A}\) reduces to the operator that corresponds to timereversal (particlehole) symmetry for φ = 0 (φ = π). Remarkably, only timereversal and particlehole symmetries are allowed when H is Hermitian. To see this, we take Hermitian conjugation of Eq. (1) and use Hermiticity of H \(\left( {H^\dagger = H} \right)\) and the definition of antiunitary symmetry \(\left( {{\cal A}^\dagger = {\cal A}^{  1}} \right)\). We then obtain \({\cal A}H{\cal A}^{  1} = e^{  {\mathrm{i}}\varphi }H\), which leads to φ = 0, π. For nonHermitian H, on the other hand, there are no such constraints.
We study the continuous deformations between a system with timereversal symmetry and a system with particlehole symmetry in the presence of a complexenergy gap and an antiunitary symmetry \(\mathcal{A}\). Such deformations cannot be performed for Hermitian Hamiltonians since only the discrete values φ = 0, π are allowed due to Hermiticity; a topological phase with timereversal symmetry and that with particlehole symmetry are distinguished in Hermitian physics. Surprisingly, an arbitrary nonHermitian Hamiltonian H_{0} with timereversal symmetry can be continuously deformed into a Hamiltonian with particlehole symmetry because H_{φ}: = e^{−iφ/2}H_{0} preserves both complex gap and antiunitary symmetry \(\mathcal{A}\) for all φ, and H_{π} has particlehole symmetry. Therefore, a topological phase with timereversal symmetry and that with particlehole symmetry are unified into the same topological class in nonHermitian physics. The topological unification of antiunitary symmetries presents a general symmetry principle in nonHermitian physics that holds regardless of the definition of a complex gap^{68}.
Topological insulator induced by nonHermiticity
As a consequence of the topological unification of timereversal and particlehole symmetries, unique nonHermitian topological phases emerge that are absent in Hermitian systems. In particular, in accordance with the topological phase in the Majorana chain (1D class D)^{14}, nonHermiticity induces topological phases in onedimensional insulators that respect timereversal symmetry with \({\cal T}^2 = + 1\) (1D class AI). Examples include a onedimensional lattice with two sites per unit cell (Fig. 3a):
where \(\hat{a}_j\) \(\left( {\hat a_j^\dagger } \right)\) and \(\hat b_j\) \(\left( {\hat b_j^\dagger } \right)\) denote the annihilation (creation) operators on each sublattice site j, t > 0 and \(\delta \in {\Bbb C}\) are the asymmetrichopping amplitudes, and \(\gamma \in {\Bbb R}\) is the balanced gain and loss. Such gain and loss have been experimentally implemented in various systems^{36,37,38,39,40,41,42,57,58,59,60,61,62,63,64,65,66} and the asymmetric hopping in optical systems^{67}. The system respects timereversal symmetry \(\left( {\hat{\cal T}\hat H_{{\mathrm{NHTI}}}\hat{\cal T}^{  1} = \hat H} \right)\), where the timereversal operation is defined by \(\hat{\cal T} \hat a_j \hat{\cal T}^{  1} = \hat b_j\), \(\hat{\cal T}{\kern 1pt} \hat b_j{\kern 1pt} \hat{\cal T}^{  1} = \hat a_j\), and \(\hat{\cal T}\,z\,\hat{\cal T}^{  1} = z^ \ast\) for all \(z \in {\Bbb C}\). The eigenstates form two bands in momentum space, for which the Hamiltonian is determined as \(\vec h\left( k \right) \cdot \vec \sigma\) with h_{x} = −2i Im[δ]sin k, h_{y} = = 2i Re [δ] sin k, h_{z} = i (γ + 2t cos k), and Pauli matrices \(\vec \sigma : = (\sigma _x,\sigma _y,\sigma _z)\). The energy dispersion is obtained as \(E_ \pm \left( k \right) = \pm {\mathrm{i}}\sqrt {\left( {\gamma + 2t\,{\mathrm{cos}}\,k} \right)^2 + 4\left \delta \right^2{\mathrm{sin}}^2k}\), and hence the complex bands are separated from each other by the energy gap with magnitude min {2γ + 2t, 2γ  2t} (Fig. 3b).
In parallel with the Majorana chain^{14}, the topological invariant \({\nu}_{\mathrm{AI}}\) is defined by
As a hallmark of the nonHermitian topological phase, a pair of edge states with zero imaginary energy appears when the bulk has nontrivial topology (ν_{AI} = 1; Fig. 3c). Whereas the bulk states that belong to the band E_{+} (E_{−}) are amplified (attenuated) with time, the midgap edge states are topologically protected from such amplification and attenuation. In the case of t = δ, for instance, the topologically protected edge states are obtained as
which satisfy \(\left\ {[\hat H_{{\mathrm{NHTI}}},\hat \Psi _{{\mathrm{edge}}}]} \right\ = O{\kern 1pt} (e^{  L/\xi })\) with the localization length \(\xi :=  (\log \gamma/2t)^{1}\). These edge states are immune to disorder that respects timereversal symmetry (see Supplementary Note 7 for details), which is a signature of the topological phase. We emphasize that topological phases are absent in 1D class AI in the presence of Hermiticity^{3,4,5}; nonHermiticity induces the unique nonequilibrium topological phase as a result of the topological unification of timereversal and particlehole symmetries. Whereas the system is an insulator and does not support nonAbelian Majorana fermions, the sublattice degrees of freedom \(\hat a_j\) and \(\hat b_j\) play the roles of particles and holes in the Majorana chain; the Majorana edge states, which are equalsuperposition states of particles and holes, correspond to the equal superposition states of the two sublattices \(\hat a_j\) and \(\hat b_j\) in the nonHermitian topological insulator.
Emergent nonHermitian topological phases
The topological phases induced by nonHermiticity are not specific to the above model but general for all the nonHermitian systems with antiunitary symmetry. To see this, we examine the complexband structure of a generic twoband system (E_{+} (k), E_{−} (k)) in 1D class AI. In the presence of Hermiticity, the real bands individually respect timereversal symmetry: \(E_ \pm \left( k \right) = E_ \pm ^ \ast \left( {  k} \right)\) (Fig. 4a), where topological phases are absent^{3,5}. In the presence of strong nonHermiticity, on the other hand, timereversal symmetry is spontaneously broken and the complex bands are paired via timereversal symmetry: \(E_ + \left( k \right) = E_  ^ \ast \left( {  k} \right)\). Importantly, this system has the same band structure as the Hermitian topological superconductor protected by particlehole symmetry (1D class D) as a direct consequence of the topological unification of timereversal and particlehole symmetries; it exhibits both trivial (Fig. 4b) and topological (Fig. 4c) phases according to the \({\Bbb Z}_2\) topological invariant defined by Eq. (4). The latter band structure becomes gapless in the presence of Hermiticity due to E_{+} (k_{0}) = E_{−} (k_{0}) for a timereversalinvariant momentum k_{0} ∈ {0, π}.
Remarkably, the emergent nonHermitian topological phases cannot be continuously deformed into any Hermitian phase that belongs to the same symmetry class. In fact, there should exist a nonHermitian Hamiltonian that satisfies E_{+} (k) = E_{−} (−k) between the two types of band structures, and the complex gap closes at k = k_{0}. Thus complexgap closing associated with a topological phase transition should occur between these phases. We also emphasize that the above discussions are applicable to all the nonHermitian topological phases in any spatial dimension protected by antiunitary symmetry. Here the corresponding topological invariants are solely determined by the relationship between symmetry and the complexband structure as in the Hermitian case^{22}.
Quantum spin Hall insulator
Topological phases survive nonHermiticity also in two dimensions. In fact, the \({\Bbb Z}_2\) topological invariant \(\nu_{\mathrm{AII}}\) can be defined in nonHermitian twodimensional insulators that respect both timereversal and parity (inversion) symmetries just like the Hermitian ones^{11}:
where k_{0} ∈ {(0, 0), (0, π), (π, 0), (π, π)} denotes the timereversalinvariant and inversionsymmetric momenta in the Brillouin zone, and π_{n}(k_{0}) ∈ {±1} is the parity eigenvalue of the nth Kramers pair at k = k_{0}. In particular, for fourband insulators such as the Kane–Mele model^{8} and the Bernevig–Hughes–Zhang model^{9}, the 4 × 4 Hamiltonian in momentum space that satisfies \({\cal T}{\kern 1pt} {\cal H}\left( {\boldsymbol{k}} \right){\cal T}^{  1} = {\cal H}\left( {  {\boldsymbol{k}}} \right)\) and \({\cal P}{\cal H}\left( {\boldsymbol{k}} \right) {\cal P}^{  1} = {\cal H}\left( {  {\boldsymbol{k}}} \right)\) is expressed as
where the coefficients d_{i}’s and d_{ij}’s are real, Γ_{I}’s are \({\cal P}{\cal T}\)symmetric five Dirac matrices, and Γ_{ij}’s are their commutators Γ_{ij}: = [Γ_{i}, Γ_{j}]/2i. We notice that Hermiticity leads to d_{ij} = 0. Here only Γ_{1} and Γ_{ij} (2 ≤ i < j ≤ 5) are invariant under space inversion when Γ_{1} is chosen as \(\mathcal{P}\)^{11}. Moreover, when the parity and timereversal operators are given as \({\cal P} = \sigma _z\) and \({\cal T} = {\mathrm{i}}s_y{\cal K}\), the Dirac matrices can be expressed as \({\mathrm{\Gamma }}_1 = \sigma _z( = {\cal P})\), \({\mathrm{\Gamma }}_2 = \sigma _y\), \({\mathrm{\Gamma }}_3 = \sigma _xs_x\), \({\mathrm{\Gamma }}_4 = \sigma _xs_y\), and \({\mathrm{\Gamma }}_5 = \sigma _xs_z\) ^{11}. Here σ_{i}’s (s_{i}’s) denote the Pauli matrices that describe the sublattice (spin) degrees of freedom. Since the Hamiltonian at k = k_{0} is invariant under inversion \({\cal P}{\cal H}\left( {{\boldsymbol{k}}_0} \right){\cal P}^{  1} = {\cal H}\left( {{\boldsymbol{k}}_0} \right)\), it reduces to \({\cal H}_{{\mathrm{QSH}}}\left( {{\boldsymbol{k}}_0} \right) = d_{0}\left( {{\boldsymbol{k}}_0} \right)I + d_1\left( {{\boldsymbol{k}}_0} \right) {\mathcal{P}} + {\mathrm{i}}\mathop {\sum}\nolimits_{1 \le i < j \le 5} d_{ij}\left( {{\boldsymbol{k}}_0} \right){\mathrm{\Gamma }}_{ij}\); the parity of a Kramers pair at k = k_{0} corresponds to the sign of d_{1}(k_{0}), and the \({\Bbb Z}_2\) topological invariant defined by Eq. (6) is obtained as \(\left( {  1} \right)^{\nu _{{\mathrm{AII}}}} = \mathop {\prod}\nolimits_{{\boldsymbol{k}}_0} {\mathrm{sgn}}\left[ {d_1\left( {{\boldsymbol{k}}_0} \right)} \right]\) as long as complex bands are gapped and d_{1}(k_{0}) is nonzero.
This bulk \({\Bbb Z}_2\) topological invariant corresponds to the emergence of helical edge states (Fig. 5). In stark contrast to Hermitian systems^{8,9,10,11}, the helical edge states form not a Dirac point but a pair of exceptional points^{30,31,40,63,69,70} and have nonzero imaginary energies at the timereversalinvariant momenta. Nevertheless, they are immune to disorder due to the generalized Kramers theorem (see Supplementary Notes 5 and 7 for details), which states that all the real parts of energies should be degenerate in the presence of timereversal symmetry with \({\cal T}^2 =  1\); the degeneracies of the real parts of energies forbid the continuous annihilation of a pair of helical edge states. Notably, the helical edge states are lasing (see Supplementary Note 8 for details) like chiral edge states in a nonHermitian Chern insulator^{42}.
The topological unification of antiunitary symmetries indicates that nonHermitian systems that respect particlehole symmetry with \({\cal C}^2 =  1\) (2D class C) also exhibit the \({\Bbb Z}_2\) topological phase, in contrast to the \(2{\Bbb Z}\) topological phase in Hermitian physics^{3,4,5}. Here the spinup and spindown particles in insulators correspond to particles and holes in superconductors. This emergent \({\Bbb Z}_2\) topological phase is due to the presence of Kramers pairs of particles and holes with imaginary energies, which are forbidden in Hermitian systems where energies are confined to the real axis; nonHermiticity brings about topological phases unique to nonequilibrium open systems.
Discussion
NonHermiticity manifests itself in many disciplines of physics as gain and loss or asymmetric hopping^{43,44}. We have shown that such nonHermiticity unifies the two fundamental antiunitary symmetries and consequently topological classification, leading to the prediction of unique nonequilibrium topological phases that are absent at equilibrium. The unveiled topological unification of timereversal and particlehole symmetries provides a general symmetry principle in nonHermitian physics that also justifies a different type of topological classification^{68}. The modified topological classification implies that the symmetry unification can bring about physics unique to nonHermitian systems. It merits further study to explore such unusual properties and functionalities that result from our symmetry principle.
This work has explored topological phases characterized by wave functions in nonHermitian gapped systems, which is a nontrivial generalization of the Hermitian topological phases. By contrast, nonHermitian gapless systems possess an intrinsic topological structure, which accompanies exceptional points^{69,70}, and has no counterparts in Hermitian systems. This topology can be characterized by a complexenergy dispersion^{28,30,31} and is distinct from the topology defined by wave functions. A complete theory of nonHermitian topological systems should be formulated on the basis of these two types of topology in a unified manner, which awaits further theoretical development.
Data availability
The data that support the findings of this study are available from the corresponding author on reasonable request.
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Acknowledgements
K.K. thanks Aashish A. Clerk, Li Ge, Hosho Katsura, Masatoshi Sato, Ken Shiozaki, Zhong Wang, and Haruki Watanabe for helpful discussions. This work was supported by KAKENHI Grant No. JP18H01145 and a GrantinAid 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., S.H., and Y.A. were supported by the JSPS through Program for Leading Graduate Schools (ALPS). S.H. and Y.A. acknowledge the support from JSPS (KAKENHI Grant Nos. JP16J03619 and JP16J03613). Z.G. acknowledges the support from MEXT.
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K.K. conceived the main idea, and K.K. and S.H. proved it. K.K. investigated the illustrative examples and generalized them. M.U. supervised the work. K.K., S.H., Z.G., Y.A., and M.U. extensively analyzed and interpreted the results and contributed to the writing of the manuscript.
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Kawabata, K., Higashikawa, S., Gong, Z. et al. Topological unification of timereversal and particlehole symmetries in nonHermitian physics. Nat Commun 10, 297 (2019). https://doi.org/10.1038/s4146701808254y
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