Essential implications of similarities in non-Hermitian systems

In recent years non-Hermitian (NH) Hamiltonians have attracted increasing attention, and one active branch of research focuses on the role of symmetries in NH systems.¹ A complete classification in terms of 38 symmetry classes was derived by Kawabata et al.,² and the topological features of these classes as well as the connection between some of them is studied in the literature.^2,3 Further, it is generally recognized that certain unitary and anti-unitary symmetries lower the codimension of exceptional points (EPs), which are degeneracies where the eigenvalues and the corresponding eigenvectors coalesce.^2,4–7 The generic appearance of EPns, where n is the order of the EP set by the number of coalescing eigenvectors, is determined by the codimension of the EP. As such, unitary and anti-unitary symmetries, which are local in parameter space—namely, parity-time $(PT)$ and anti-$PT$ symmetry, pseudo-Hermitian symmetry, as well as sublattice symmetry, chiral symmetry and parity-particle-hole $(CP)$ symmetry—inflict symmetries on the spectrum, and therefore reduce the codimension of the EPs.^8–12 

The (anti-)unitary symmetries can be grouped into three pairs based on the type of constraint they enforce on the set of eigenenergies.^8,16 In this work, we show that the spectral symmetries already come about in the presence of similarity relations and not just in the presence of more restrictive symmetries. These similarity relations, namely, pseudo-Hermiticity, pseudo-anti-Hermiticity and self skew-similarity, naturally pair the anti-unitary and unitary symmetries, cf. Fig. 1, and enforce the spectral symmetry. The symmetries appear as special cases of these three EP-inducing generalized similarities.

The relation of $PT$ symmetry and pseudo-Hermiticity, which denotes the similarity of H and its adjoint H^†, is well established. Quantum mechanics formulated on the basis of $PT$-symmetric operators was investigated by Bender and co-workers,^17–19 and has been related to pseudo-Hermitian operators. For diagonalizable $PT$-symmetric operators, Mostafazadeh proved pseudo-Hermiticity by explicitly showing the similarity between H and H^†.^20–23 Later this was extended to any finite $PT$-symmetric Hamiltonian by Zhang et al.²⁴ In Sec. II we summarize and expand upon their results by showing a further connection to Hermitian and pseudo-Hermitian symmetric matrices, where we note the subtle difference between pseudo-Hermiticity and pseudo-Hermitian symmetry. Pseudo-Hermiticity alone already enforces symmetries on the spectrum, and thus lowers the codimension of EPs, while $PT$ symmetry and pseudo-Hermitian symmetry constitute two special cases. Further we include Hermitian Hamiltonians as a special case of pseudo-Hermitian symmetric systems, which has additional spectral symmetry that naturally prevents EPs from emerging. We also comment on real Hamiltonians as a special case of $PT$-symmetry.

We find a similar structure for anti-$PT$-symmetric and chiral-symmetric systems. Both symmetries enforce pseudo-anti-Hermiticity on the systems, which we define in Sec. III. We prove that all anti-$PT$-symmetric and chiral-symmetric systems are pseudo-anti-Hermitian. We show the spectral constraint follows from pseudo-anti-Hermiticity, and relate anti-Hermiticity to pseudo-anti-Hermiticity. We compare pseudo-Hermiticity and pseudo-anti-Hermiticity, which have a resembling effect on the spectrum, and point out the similarities and differences between them.

In Sec. IV we follow the same approach for $CP$-symmetry and sublattice symmetry, where we find that in both cases the Hamiltonian exhibits self skew-similarity. This self skew-similarity is the origin of the spectral symmetry. We note that self skew-similarity behaves differently from pseudo-Hermiticity and pseudo-anti-Hermiticity, because it does not relate the Hamiltonian to its adjoint, but instead is a property of the Hamiltonian itself. This results in differences in the treatment of self skew-similarity.

We provide a conclusion in Sec. V.

The Proof of Theorem 1.2 can be found in Ref. 24. They show that the necessity follows from the definition of pseudo-Hermiticity and the similarity of every matrix to its transpose. The proof of sufficiency is shown explicitly. The similarity of H and H* results in real eigenvalues or pairs of complex conjugate eigenvalues. By ordering the Jordan canonical form of H in real Jordan blocks and block structures of complex conjugate Jordan blocks, Zhang et al. are able to construct the Hermitian similarity transformation η for any Hamiltonian that is similar to its complex conjugate. Theorem 2.1 and 2.2 are equivalent criteria for pseudo-Hermiticity.

For any n ≥ 3 to find unitary similarity it is necessary that the traces of the words for n = 3 have to be equal, while there are more non-redundant words for n > 3. However, for non-normal H, i.e., [H, H^†] ≠ 0, equality of the word traces of H and H* as well as of H and H^† does not follow from pseudo-Hermiticity. Thus for n ≥ 3 pseudo-Hermiticity is more general and not equivalent to $PT$-symmetry or pseudo-Hermitian symmetry.

We note that normality of H restores the equivalence of similarity and symmetry, which can be shown from the diagonalisability of H and the pseudo-Hermitian spectral properties. For any normal H pseudo-Hermiticity is equivalent to both $PT$-symmetry and pseudo-Hermitian symmetry. However, normality prohibits the emergence of EPs altogether, because the Hamiltonian must be diagonalizable in the whole parameter space. Therefore, we consider non-normal Hamiltonians in the following for which the pseudo-Hermiticity is more general then any of the two symmetries.

For $PT$ symmetry and pseudo-Hermitian symmetry this was shown in Ref. 8, but the symmetries are special cases of pseudo-Hermiticity according to Theorem 2.3. From Theorem 2.4 we know that pseudo-Hermiticity is more general than the two symmetries, and it already induces the EPs in lower dimension without the need of symmetry. Thus we have shown that not symmetry but similarity drives the emergence of exceptional points in lower dimensions. Further the spectral structure surrounding the similarity-induced EPs is fully determined by the similarity even in the presence of the more restrictive $PT$-symmetry or pseudo-Hermitian symmetry. This spectral structure is discussed in detail in previous papers on symmetry-induced EPs. Symmetry-protected EP2 rings were found,¹ and the rich spectral features surrounding symmetry-induced EP3s, EP4s and EP5s in two dimensions have also been analyzed.^16,34

In addition to $PT$ symmetry and pseudo-Hermitian symmetry, pseudo-Hermiticity encloses two more special cases, namely, Hermitian and real matrices. We note that a Hermitian matrix is a special case of a pseudo-Hermitian symmetric systems, and a real Hamiltonian a special case of $PT$-symmetric systems, with the symmetry generator being the identity operation in both cases. Our results concerning EPs are applicable to real matrices, while Hermiticity does not allow for EPs.

A proof of Corollary 3.2 by explicit construction of the similarity generator is carried out analogous to the construction of the generator of pseudo-Hermiticity in Ref. 24.

Due to the connection between the $PT$ and the pseudo-Hermitian symmetry of $H̃≔iH$ to the anti-$PT$ and the chiral symmetry of H, respectively, we can infer properties of pseudo-anti-Hermitian matrices from the theorems 2.3, 2.4 and 3.1.

While this Corollary follows from Theorem 2.5 and 3.1 we expand on this here in more detail to clarify the reasoning for the pseudo-anti-Hermitian system. We consider the complex conditions det[H] = 0 and tr[H^k] = 0 for 2 ≤ k < n. The spectral symmetry $ϵ=−ϵ*$⁠, which is a consequence of the pseudo-anti-Hermiticity, reduces the number of constraints, because the determinant of H and each of the traces of H^k is either real or purely imaginary. This reduces the codimension of the EP to n − 1. For the two symmetries enclosed by pseudo-anti-Hermiticity according to Corollary 3.3 this was shown in Ref. 8. However, from Corollary 3.4 it is clear that pseudo-anti-Hermiticity is more general than either anti-$PT$-symmetry or chiral symmetry. Pseudo-anti-Hermiticity already induced EPs by lowering their codimension without the need of symmetries of the Hamiltonian. Further, the spectral structure surrounding a pseudo-anti-Hermiticity-induced exceptional point is fully determined by the skew-similarity of the Hamiltonian. For EP3s in two dimensions the spectral structure is equivalent to the structures described in Ref. 16 for anti-$PT$-symmetry and chiral symmetry.

Besides anti-$PT$-symmetric and chiral-symmetric systems there are two notable special cases of pseudo-anti-Hermitian systems. The first case is anti-Hermiticity of H, meaning H = −H^†, which can be interpreted as chiral symmetry with the identity as generator. Because anti-Hermiticity implies normality no exceptional points can emerge in anti-Hermitian systems. The other case are imaginary matrices H = −H*, which are anti-$PT$-symmetric with the identity as generator. Our results are thus also applicable for imaginary matrices.

For the two symmetries that realize self skew-similar Hamiltonians this was shown in Ref. 8. From Theorem 3.3 it is clear that self skew-similarity is more general than either of the two symmetries. According to Theorem 3.4 self skew-similarity already induces EPns by lowering their codimension. Because the spectral symmetry is enforced by the similarity, symmetry of the Hamiltonian is not needed. Further the spectral structure accompanying the similarity-induced EPs is determined by the self skew-similarity, and also not affected by the additional constraints of $CP$-symmetry or sublattice symmetry.

A special case of self skew-similarity are anti-symmetric Hamiltonians H = −H^T, which are $CP$-symmetric with the identity as generator. All our results are applicable for anti-symmetric matrices.

It was previously shown that $PT$-symmetry entails pseudo-Hermiticity for finite dimensional systems. In this paper we show that this relation can be generalized to all unitary and anti-unitary symmetries, which lower the codimension of exceptional points. We prove that each of these symmetries is a special case of one of three generalized similarities, namely, pseudo-Hermiticity, pseudo-anti-Hermiticity, and self skew-similarity. Each similarity encompasses two symmetries, and in the case of pseudo-Hermiticity and pseudo-anti-Hermiticity for finite-dimensional systems of size n > 2 the similarities are more general than the respective symmetries. In the case of self skew-similar Hamiltonians this even holds for n ≥ 2.

Overall we find that the spectral features of non-Hermitian systems and the emergence of stable EPs is linked to the relevant similarity, and not the symmetry of the system contrary to previous assumptions. The similarities are far less restrictive compared to unitary or anti-unitary symmetries. As such, the presence of similarities may lead to the robustness of symmetry-stabilized non-Hermitian features to symmetry-breaking perturbations.

We are grateful to Julius Gohsrich and Alexander Felski for insightful discussions. We acknowledge funding from the Max Planck Society Lise Meitner Excellence Program 2.0. We also acknowledge funding from the European Union via the ERC Starting Grant “NTopQuant.” Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council (ERC). Neither the European Union nor the granting authority can be held responsible for them.

The authors have no conflicts to disclose.

Anton Montag: Conceptualization (lead); Investigation (lead); Methodology (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). Flore K. Kunst: Funding acquisition (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article.