Extended exceptional points in projected non-Hermitian systems

Exceptional points are interesting physical phenomena in non-Hermitian physics at which the eigenvalues are degenerate and the eigenvectors coalesce. In this paper, we find that in projected non-Hermitian two-level systems (sub-systems under projecting partial Hilbert space) the singularities of exceptional points (EPs) is due to basis defectiveness rather than energy degeneracy or state coalescence. This leads to the discovery of extended exceptional points (EEPs). For EEPs, more subtle structures (e.g. the so-called Bloch peach), additional classification, and ‘hidden’ quantum phase transitions are explored. By using the topologically protected sub-space from two edge states in the non-Hermitian Su–Schrieffer–Heeger model as an example, we illustrate the physical properties of different types of EEPs.

On the other hand, in some quantum many-body models, due to special conditions of symmetry/topology, there may exist protected sub-systems.For example, for topological insulators, there exist topologically protected edge states with gapless energy spectra (or zero modes for one dimensional cases) [58,59]; For the many-body systems with spontaneously symmetry breaking there exist symmetry-protected degenerate ground states; for the topological orders with long range entanglement, there exist topologically protected degenerate ground states (on a torus) that make up topological qubits andmay be possible to be applied to incorporate intrinsic fault tolerance into a quantum computer [60][61][62][63].For these topologically/symmetry protected sub-systems in different quantum many-body models, the quantum properties will be changed under non-Hermitian perturbations.In this paper, we investigate the influence of non-Hermitian perturbations on the quantum properties of quantum many-body systems through the topologically/symmetry protected sub-system.We find that basis defectiveness plays a key role in EPs and that in projected non-Hermitian systems there may exist EPs without eigenvalue degeneracy, EPs without the coalescence of different eigenvectors, or those without both features.This leads to the discovery of extended EPs based on the basis defectiveness.
The remainder of the paper is organized as follows.In section 2, we review the theory of singularity for usual EPs in a simple two-level systems and show the reason why eigenstates coalesce at an EP.In section 3, we discuss a general theory for projected non-Hermitian sub-spaces that are protected by topologically/symmetry in different quantum many-body models and show how to derive the effective Hamiltonian and the corresponding (initial) basis.In section 4, we develop a theory of singularity for the extended EPs and show their complete classification.In section 5, an example of a one-dimensional (1D) non-Hermitian topological insulator-nonreciprocal Su-Schrieffer-Heeger (SSH) model is focused on.According to the global phase diagram of the topologically protected sub-systems from the two edge states, we show the occurrence of different types of extended EPs.Finally, conclusions are given in section 6.
2. Singularity at an EP in a traditional non-Hermitian two-level PT system 2.1.EP in a traditional non-Hermitian two-level PT system To learn the nature of the singularity at an EP, we study a traditional two-level PT system described by the following Hamiltonian: For this two-level non-Hermitian system, {|ψ R 0 ⟩} = {|ψ 0 ⟩} = {|↑⟩ , |↓⟩} denotes an initial basis obeying orthogonal and normalization conditions, i.e. ⟨↑ | ↓⟩ = 0 and ⟨↑ | ↑⟩ = ⟨↓ | ↓⟩ = 1.At h x = h z , a typical spontaneous PT -symmetry breaking occurs: for the case of h x > h z , the energy levels |+⟩ and |−⟩ are x − h 2 z ; For the case h x < h z , these two energy levels are For the case of h x = h z , the system is at an EP with eigenstate coalescence and energy degeneracy.

Basis defectiveness with singular non-Hermitian similarity transformation
In this part, we solve the problem why eigenvectors coalesce at EPs.By performing a non-Hermitian similarity transformation S M = e −β M σy , we transform the original non-Hermitian Hamiltonian ĤNH into a Hermitian/anti-Hermitian one, where β M = 1 2 ln hx+hz hx−hz .The eigenvalue of Ĥ0 is the same as that of ĤNH .Under the non-Hermitian similarity transformation S M , the basis {|ψ R ⟩} of Ĥ0 becomes, In the following, we refer to {|ψ R ⟩} as the matrix basis.If we select the initial basis {|ψ R 0 ⟩} to be the eigenstates of σ y , i.e. where R } obeys the orthogonal condition R ⟨1|2⟩ R = 0 but does not obey the normalization conditions, i.e.R ⟨1|1⟩ R = 1 and R ⟨2|2⟩ R = e −2β M .Approaching the EP, the non-Hermitian similarity transformation becomes singular, i.e.
As a result, the matrix basis becomes defective, i.e.
Here, the base |2⟩ R disappears !To illustrate the defectiveness of the matrix basis, in figure 1(c), we plot

| ψR
and Now, near the EP, where β M → ∞, the state similarity obviously approaches 1, which is consistent with figure 1(b).Thus, one can see that the singularity of the EP arises from the defective matrix basis {|1⟩ R , 0} due to the singular non-Hermitian similarity transformation S M (σ y , β M → ∞).

Bloch peach for two-level states under the non-Hermitian similarity transformation
To further illustrate the singularity of EPs, we use a geometric approach to illustrate the deformation of the Bloch sphere under a singular non-Hermitian similarity transformation S M (σ y , β M → ∞).For the Hermitian case, one may use a point on the Bloch sphere with SU(2) rotation symmetry to represent an arbitrary quantum state of the two-level system, i.e.
Here, θ ∈ [0, π] and φ ∈ [0, 2π] are real numbers, and the radius r of the Bloch sphere can be obtained as r = ⟨ψ|ψ ⟩ = 1.For a two-level PT system, a quantum state under the non-Hermitian similarity transformation S M (σ y , β M ) becomes (with θ ∈ [0, π] and φ ∈ [0, 2π]), and the radius R of the Bloch sphere can be obtained as Therefore, the original Bloch sphere changes into a peach-like closed surface with residue U(1) symmetry along the y-axis (we call this the Bloch peach).With increasing β, one pole of the Bloch peach moves upward.At the EP, this pole touches the origin of the coordinate system.See the illustration of the Bloch peach in the limit of β M → ∞ in figure 1(d).

Projected non-Hermitian sub-systems
For a many-body system, due to special conditions of symmetry/topology, there may exist projected sub-systems, such as the subsystem formed by edge states in topological insulators and subsystem formed by topologically protected degenerate ground states in intrinsic topological orders.Through these projected sub-systems that are projected by symmetry/topology, one can study the quantum properties of many-body systems more concisely and intuitively.To completely characterize the many-body model and its projected sub-systems, we give their definitions, { ĤMB , B MB } and { ĤS , B S }, respectively.For the Hermitian case, both where P is a projective operator on the basis of the many-body system.It is obvious that K < N. When one adds a non-Hermitian perturbation iδ Ĥ on the many-body model, the total Hamiltonian becomes non-Hermitian, i.e.
Under the non-Hermitian perturbation, the basis has no changing, the many-body model is denoted by In general, for a non-Hermitian system, the biorthogonal set for the basis is defined by and and ⟨Ψ L j |Ψ R j ⟩ = 1 where j is state index.We then assume the non-Hermitian terms are perturbation and do not change the existence of the projected sub-system and { ĤNH−S , B S } = {P ĤMB P −1 , PB MB } is used to describe the projected non-Hermitian sub-system.Here, ĤNH−S = P ĤMB P −1 is the effective Hamiltonian of the protected sub-system.Under the projected operation P, the basis of the projected sub-system is obtained as The basis of projected sub-system B S = {|ψ R j ⟩} is always abnormal under the projection P of the non-Hermitian system.For example, it does't follow usual normalization ⟨ψ R j |ψ R j ⟩ ̸ = 1.In addition, we show the method to calculate ĤNH−S .Based on the basis B S = {|ψ R j ⟩}, an effective Hamiltonian of projected sub-system is derived as where In this paper, We consider the projected two-level sub-system of pseudo-Hermitian systems that can perform the similarity transformation [24].So, we only focus on the case of K = 2, the dimension of the sub-system is 2, shown as figure 2.

Extended EPs-universal feature and classification in projected non-Hermitian systems
We next develop the theory for the projected non-Hermitian systems with a singularity and introduce the concept of extended EPs.This phenomenon always occurs in subsystems of certain non-Hermitian models, for example, the defective edge states of a non-Hermitian topological insulator, the defective degenerate ground states in non-Hermitian systems with spontaneous symmetry breaking, or the topologically protected degenerate ground states in intrinsic topological orders.An arbitrarily non-Hermitian protected two-level sub-system can be described as { ĤNH , {|ψ R 0 ⟩}}.The Hamiltonian is where h 0 is a complex number, ⃗ h = (h x , h y , h z ) is a complex vector and ⃗ σ = (σ x , σy , σz ) is the vector of Pauli matrices.For Hermitian systems, the Hamiltonian itself can fully describe the energy level structure and wave function of the system, but in non-Hermitian systems, the combination of the Hamiltonian and the system basis vector can fully describe the energy level structure and corresponding wave function of the system, so we use terms like { ĤNH , {|ψ R 0 ⟩}} to describe the non-Hermitian two-level sub-system(the same goes for { ĤNH−MB , B MB } and { ĤNH−S , B S } mentioned earlier).

Three equivalent basis representations
For non-Hermitian two-level systems mentioned above, we found that there are three representations that can equivalently describe the system.

Initial basis representation
Shown as figure 2, the basis {|ψ R 0 ⟩ of the (projected) sub-system ĤNH is always abnormal under the projective operator P from the non-Hermitian system to its sub-system.At the same time, ĤNH is a non-Hermitian Hamiltonian.For this non-Hermitian system, we can just use the combination of the Hamiltonian ĤNH and the basis {|ψ R 0 ⟩} to directly describe this two-level system.We define { ĤNH , {|ψ R 0 ⟩}} as the initial basis representation.

Matrix basis representation
In this part, we transform the non-Hermitian Hamiltonian ĤNH into a Hermitian/anti-Hermitian one with the similarity transformation and derive the Matrix basis representation that can equivalently describe the non-Hermitian sub-system.To transform ĤNH into a Hermitian/anti-Hermitian one, Firstly, we have to rewrite where ⃗ σ Re and ⃗ σ Im (⃗ σ Re • ⃗ σ Re = 1 and ⃗ σ Im • ⃗ σ Im = 1) are the Pauli matrices corresponding to the real and imaginary parts of ⃗ h, respectively.And secondly, we divide i( Im , with ⃗ σ C Im ,⃗ σ Re = 0 and {⃗ σ A Im ,⃗ σ Re } = 0.Then, the non-Hermitian Hamiltonian becomes where Then, we can transform the original NH Hamiltonian ĤNH into a Hermitian/anti-Hermitian one with a non-Hermitian similarity transformation where with . As a result, the eigenvalue of Ĥ0 is the same as that of ĤNH , i.e.
Finally, one can see that under the non-Hermitian similarity transformation, the initial basis {|ψ R 0 ⟩} is correspondingly changed into the unique matrix basis {|ψ R ⟩}, i.e.
So, we can use the combination of the Hamiltonian Ĥ0 ( Ĥ0 is a Hermitian/anti-Hermitian Hamiltonian)and the basis {|ψ R ⟩ ({|ψ R 0 ⟩ is abnormal)to directly describe this two-level system equivalently as the initial basis representation.We define { Ĥ0 , {|ψ R ⟩}} as the initial basis representation.

Normal basis representation
In addition, to describe the same non-Hermitian two-level system, one can also use another representation, with NH is a non-Hermitian one, and the |ψ 0 ⟩ is a normal basis consisting of eigenstates of ⃗ σ B (or ⃗ σ B |ψ 0 ⟩ = ±|ψ 0 ⟩) based on the normal basis, so the two-level system can also equivalently describe by the normal basis representation { Ĥβ B NH , {|ψ 0 ⟩}}.Besides, we can associate the normal basis representation with the matrix basis representation through similarity transformation, To clearly show the relationship among the three representations for the Hamiltonians of the same non-Hermitian two-level system, { ĤNH , {|ψ R 0 ⟩}} under the initial basis, { Ĥ0 , {|ψ R ⟩}}} under the matrix basis, and { Ĥβ B NH , {|ψ 0 ⟩}} under the normal basis, we plot figure 2. S M , S B , and S M S B are different non-Hermitian similarity transformations relating these representations.

Definition of extended EPs
We point out that the universal feature of the projected non-Hermitian system with singularities is defectiveness of the matrix basis {|ψ R ⟩} as β M → ∞ and/or β B → ∞rather than energy degeneracy or state coalescence.Thus, we define extended EPs as follows: Definition-

extended EPs (EEPs): For an arbitrary projected two-level non-Hermitian systems system, EEPs exist if and only if the matrix basis
a normalization factor.As a result, at EEPs, the two energy levels may not necessarily be degenerate, i.e.E + = E − and E + ̸ = E − are both allowed; the two eigenstates ψR + and ψR − may not necessarily coalesce, i.e.Λ ≡ 1 and Λ ≡ 0 are both allowed.In addition, in the form of Jordan block matrix, at EEPs the algebra multiplier of a matrix may be same to its geometric multiplier.

Classification of extended EPs
By considering different behaviors of defective bases, we can classify EEPs for an arbitrary projected non-Hermitian two-level system.Depending on the behavior of the defective matrix basis

M-EEP B-EEP H-EEP Class
and IIB-EEPs.For IB-EEPs, the two eigenstates will never coalesce with each other Λ ≡ 0; for IIB-EEPs, the two eigenstates will always coalesce with each other Λ ≡ 1, shown as the table 1.
To show the physical properties of these two classes of B-EEPs, we perform a non-Hermitian similarity transformation S −1 B on the initial basis {|ψ R 0 ⟩} and obtain a representation under the normal basis, i.e.
Correspondingly, the original Hamiltonian ĤNH is transformed into There aretwo possibilities for Ĥβ B NH , which correspond to the two classes of B-EEPs: one possibility is that all elements of Ĥβ B NH are finite, in which case the eigenstates do not coalesce (or Λ ≡ 0), and the other is one or more elements diverge, i.e. ( Ĥβ B NH ) ij → ∞, in which case the eigenstates coalesce (or Λ ≡ 1).It is obvious that at IB-EEPs the algebra multiplier of the Hamiltonian ĤNH is equal to its geometric multiplier.Therefore, there exists a quantum phase transition between IB-EEPs (the region without eigenstate coalescence) and IIB-EEPs (the region with eigenstate coalescence).Let us give a simple explanation of this fact.For IB-EEPs, the Hamiltonian ĤNH commutes with ⃗ σ B , i.e. [ ĤNH ,⃗ σ B ] = 0.As a result, the Hamiltonian ĤNH must be written as λ⃗ σ B with λ ̸ = 0.Under the non-Hermitian similarity transformation S −1 B = e β B ⃗ σ B , we have Now, the basis of Here, the |ψ 0 ⟩ are eigenstates of ⃗ σ B , or ⃗ σ B |ψ 0 ⟩ = ±|ψ 0 ⟩.Because the eigenstates |ψ R ± ⟩ of ĤNH = λ⃗ σ B are also those of |ψ 0 ⟩, the state similarity of |ψ R ± ⟩ must be zero, i.e.
On the other hand, for IIB-EEPs, the Hamiltonian ĤNH does not commute with ⃗ σ B , i.e. [ ĤNH ,⃗ σ B ] ̸ = 0.The Hamiltonian ĤNH must be written as On the basis of {|ψ 0 ⟩}, the divergent term is proportional to ( 0 1 0 0 ) or ( 0 0 1 0 ).As a result, the Hamiltonian is dominated by this divergent term, and we can ignore other terms.In this case, the state similarity of |ψ R ± ⟩ must be 1, i.e.
Without sudden changing the energy levels and the defectiveness of matrix basis, this quantum phase transition is always 'hidden' .

The model
In this section, we take the 1D nonreciprocal SSH model as an example to illustrate the different types of EEPs for its topologically protected sub-space of two edge states.The Bloch Hamiltonian for the nonreciprocal SSH model under periodic boundary conditions (PBC) is given by where ; the τ i are the Pauli matrices acting on the (A or B) sublattice subspaces; t 1 and t 2 describe the intracell and intercell hopping strengths, respectively; γ describes unequal intracell hopping; and ε denotes the strength of an imaginary staggered potential on the two sublattices.t 1 , t 2 , γ, and ε are all real.In this paper, we set t 2 = 1.
Under a non-Hermitian similarity transformation S NHP , the physics properties of the 1D nonreciprocal SSH model under open boundary conditions (OBC) are characterized by ĤOBC (k) rather than ĤPBC (k) [13].
To characterize the topological properties of the non-Hermitian topological system, the non-Bloch topological invariant w of ĤOBC (k) is introduced, i.e.
where φ(k) = tan −1 ( hy / hx ) and hx = t1 + t2 cos k, hy = t2 sin k.In the region of | t1 | < | t2 | and w = 1, the system is a topological insulator (the gray region in figure 3(a)); in the region of | t1 | > | t2 | and w = 0, the system is a normal insulator (the white region in figure 3(a)).A quantum phase transition occurs at where the bulk energy gap under OBC is closed.

Two-level systems from two edge states in topological phase
In the topological phase with w = 1, there exist two edge states ψ R 1 and ψ R 2 .These two edge states make up a topologically protected subspace denoted by { ĤNH , {|ψ R 0 ⟩}}.Under the biorthogonal set, the initial basis The effective Hamiltonian ĤNH is written as where Through straightforward calculations [18], we can analytically obtain the effective Hamiltonian of the two edge states as where ∆ = t2 ( t1 t2 ) N .The two energy levels for the two eigenstates
, (1, 0) T and (0, 1) T represent sub-lattice degrees of freedom.The basis vector satisfies orthogonal normalization, As mentioned earlier, the relationship between the wave functions of the two systems before and after similarity transformation satisfies The basis of the ĤPBC (k) has the following orthogonality normalization, This indicates that the basis vectors of the non-Hermitian system satisfy orthogonality, but do not satisfy normalization relationships.{|b1 ⟩, | b2 ⟩} of the non-Hermitian are shown in figure 4(a) and (b) with the distribution of n.Here the basis b2 | b2 ⟩ is approximately equal to 0, this is consist with equations ( 46) and (47).And this also indicates that the basis vector of the system is defective.Then we discuss the different types of EEPs separately.

M-EEPs
Firstly, we consider the case of M-EEPs by setting ε to a purely real value and t 1 = 0. Now, the initial basis becomes normal, |ψ R 0 ⟩ = {|ψ 0 ⟩} (48) with S B (⃗ σ B , β B ) = 1.The Hamiltonian of the effective two-level model is reduced to where A spontaneous PT -symmetry-breaking transition occurs at |ε| = ∆ 0 .The energy levels become degenerate, shown as figure 5(a), and the non-Hermitian similarity transformation becomes singular, We have an M-EEP with the following defective matrix basis:  4, we can obtain that at M-EEPs two energy levels become degenerate and two energy states coalesce (E + ̸ = E − , Λ = 1), shown as the gray dotted line at γ = ±0.86.In addition, as shown in figure 3(b 1 ), the Bloch peach has a symmetric axis along the y-direction.
) is an imaginary wave vector that characterizes the non-Hermitian skin effect.In the thermodynamic limit N → ∞, we have a B-EEP with the following defective matrix basis: In figure 3(a), except for the black lines and red dots, B-EEPs exist throughout the whole topological insulator region.An interesting fact is that the energy levels are not degenerate, shown as figure 6(a), i.e.

H-EEPs
Thirdly, we consider the case of H-EEPs by setting ε to a purely real value, ̸ = 0, ∆ = |ε|.the one hand, the initial basis is defective, with β B = Nq 0 → ∞.In the thermodynamic limit N → ∞, the two-level system may be regarded as a B-EEP.On the other hand, for the case of ∆ ε = 1, the same system can be regarded as an M-EEP with another singular non-Hermitian similarity transformation S B (β M → ∞).Therefore, in thermodynamic limit N → ∞, we have an H-EEP with the following defective matrix basis: Shown the gray dotted line in figure 5 at t 1 = 1.03.We can see that the two energy levels become degenerate (E + = E − ) ,where the H-EEP occurs with Λ = 1.Now, the quantum state under the non-Hermitian similarity transformations S M (σ y , β M ) and S B (σ z , β B ) becomes As shown in figure 6(b 3 ), the Bloch peach, with a symmetric axis along the z-direction shrinks.In addition, for the non-Hermitian SSH model, the H-EEP is an unusual spontaneous PT -symmetry breaking accompanied by a transition from real to complex spectra.
Finally, we numerically calculated the similarity between two edge states(Λ = 1) in the entire topology region of the SSH model, as shown in figure 7.This is completely consistent with the analytical phase diagram in figure 6(a).

Conclusion and discussion
It is widely accepted that exceptional points (EPs) are unique and ubiquitous features of non-Hermitian systems, at which both eigenvalues and eigenvectors coalesce.In this paper, we illustrate that the phenomenon of EPs in projected non-Hermitian systems is much more interesting than expected.we found that the essential reason for the coalesce of eigenvectors at the EP point is the basis defectiveness.Based on defective basis vectors, we define the concept of Extended exceptional points (EEPs).For projected non-Hermitian two-level systems, there may exist three types of EEPs-M-EEPs (with a defective matrix basis and a normal initial basis), B-EEPs (with a normal matrix basis and a defective initial basis), and H-EEPs (with a defective matrix basis and a defective initial basis).In addition, there are two classes of B-EEPs-IB-EEPs , without eigenstate coalescence, and IIB-EEPs, with eigenstate coalescence.
By taking the topologically protected edge states in the non-Hermitian SSH model as an example, we demonstrate the phenomenon of basis defectiveness and explore the physical properties of EEPs.In the future, we will study higher-order EEPs in projected non-Hermitian systems and attempt to develop a complete theory of EEPs.
Then, based on { ĤNH , {|ψ R 0 ⟩}}, we study the EEPs for the two edge states in the 1D nonreciprocal SSH model.In topological phase with w = 1 (or | t1 | < | t2 |), we have EEPs for the two edge states except for the Hermitian/anti-Hermitian cases at γ = 0 or t 1 = 0.

Figure 5 .
Figure 5. (a) is the energy levels of two edge states; (b) is the state similarity of two edge states, Λ = |⟨ ψR + | ψR − ⟩|.(a) and (b) are all for the case of N = 50 , t2 = 1, ε = 0.001 and t1 = 0.The two energy levels become degenerate and the state similarity becomes 1 at the gray dotted line (γ = ±0.86),where the M-EEP occurs.

Figure 3 (
Figure 3(a) is an illustration of the global phase diagram for EEPs: the four red dots correspond to M-EEPs, the solid black lines correspond to H-EEPs, the yellow region corresponds to IB-EEPs and the gray region corresponds to IIB-EEPs.The Bloch peaches for different types of EEPs are illustrated in figure 3(b 1 ) (an example for M-EEPs), figure 3(b 2 ) (an example for B-EEPs), and figure 3(b 3 ) (an example for H-EEPs).Then we discuss the different types of EEPs separately.

Table 1 .
Three types of EEP definitions under normal basis representation.