Enhanced eigenvector sensitivity and algebraic classification of sublattice-symmetric exceptional points

Exceptional points (EPs) are degeneracy of non-Hermitian Hamiltonians, at which the eigenvalues, along with their eigenvectors, coalesce. Their orders are given by the Jordan decomposition. Here, we focus on higher-order EPs arising in fermionic systems with a sublattice symmetry, which restricts the eigenvalues of the Hamitlonian to appear in pairs of $\lbrace E, -E\rbrace $. Thus, a naive prediction might lead to only even-order EPs at zero energy. However, we show that odd-order EPs can exist and exhibit enhanced sensitivity in the behaviour of eigenvector-coalescence in their neighbourhood, depending on how we approach the degenerate point. The odd-order EPs can be understood as a mixture of higher- and lower-valued even-order EPs. Such an anomalous behaviour is related to the irregular topology of the EPs as the subspace of the Hamiltonians in question, which is a unique feature of the Jordan blocks. The enhanced eigenvector sensitivity can be described by observing how the quantum distance to the target eigenvector converges to zero. In order to capture the eigenvector-coalescence, we provide an algebraic method to describe the conditions for the existence of these EPs. This complements previous studies based on resultants and discriminants, and unveils heretofore unexplored structures of higher-order exceptional degeneracy.

An n th -order exceptional point (EP n ) [33][34][35][36][37][38][39][40] appears when the Jordan decomposition of the matrix contains an n-dimensional (with n > 1) Jordan block J n (E) along its diagonal, at the eigenvalue E. Near an EP 2 , the dispersion varies as a square root, viz., δE ∼ |δq|, where |δq| characterizes the deviation from the EP in the momentum space spanned by the vector q.The derivative of the dispersion diverges at the EP, implying that the change in eigenvalue becomes more and more sensitive as we approach the EP.Such a sensitivity is further enhanced at a higher-order EP n (n > 2), because now an n th -order root sensitivity (i.e., δE ∼ |δq| 1/n ) can appear in the vicinity of the EP n for generic situations [37,38,40].The eigenvalue overlap at higher-order EPs can be captured by equations involving discriminants [40] or resultants [38].However, another important and unique property of an EP, namely the coalescence of eigenstates, remains elusive under this approach.Moreover, the space spanned by the exceptional degeneracy is not a closed subspace of the parameter space of the corresponding matrix [41].In fact, this space has a finer topological structure beyond the solutions captured by continuous functions (such as the discriminants and resultants) of the matrix.
In this paper, we use an algebraic method to classify the higher-order EPs according to their eigenvectorcoalescence.We focus on the nature of the higher-order EPs that can appear in two-dimensional (2d) systems in the presence of a sublattice symmetry [cf.Fig. 1(a)], and determine how their eigenstates collapse.The main results are summarized in Fig. 1(b) and Table I.Remarkably, according to our classification, all EP n 's can be categorized into two types.A regular EP n exhibits a typical n-fold eigenvector-coalescence, while a mixed-type EP n can exhibit different eigenvector-coalescence depending on how our Hamiltonian is approaching it in the parameter space.
The model is implemented by considering N flavours of fermions, living on a bipartite lattice, whose creation operators are given by c α 1 † and c α 2 † (α ∈ [1, N ]).The degrees of freedom for the two sublattices have been distinguished by the subscripts 1 and 2. The sublattice symmetry ensures that the Hamiltonian H obeys P H P = −H [42,43], with the operator P acting as c α .This is a very natural condition when the Hamiltonian contains only hoppings from sublattice 1 to sublattice 2. Examples of such Hamiltonians include solvable spin liquid models, such as the Kitaev spin liquid [44] (corresponding to N = 1), and the Yao-Lee SU (2) spin liquid [45] (corresponding to N = 3).In Hermitian systems, the sublattice symmetry can be viewed as the product of time-reversal transformation and particle-hole transformation of fermions, which translates to a chiral symmetry [42].In the momentum space, a generic non-Hermitian Hamiltonian with the sublattice symmetry can be brought to the block off-diagonal form: where B and B are N × N matrices.In order to demonstrate our results in closed analytical forms, we will focus on the N = 2 case, where the system can be described by 4 × 4 matrices.We will characterize our EPs based on the nilpotency of Jordan blocks in the generalized eigenspace.To explain the terminologies, let us consider the example of an EP 3 .Near an EP 3 , we have a three-dimensional Jordan block, and the Hamiltonian can be expressed as E 1 1 0 0 0 . . .0 E 1 1 0 0 . . .0 0 E 1 0 0 . . .0 0 0 E 2 0 . . .0 0 0 0 E 3 . . . . . . . . . . . .0 0 . . .
where E 1 is a three-fold degenerate eigenvalue with only one linearly independent eigenvector proportional to e 1 = V (1, 0, . . . ) T .The generalized eigenspace L E1 of E 1 includes two other vectors, viz., e 2 = V (0, 1, 0, . . . ) T , and e 3 = V (0, 0, 1, 0, . . . ) T , such that (H − E 1 ) is nilpotent in L E1 .In other words, (H − E 1 ) e 3 = e 2 , (H − E 1 ) e 2 = e 1 , and (H − E 1 ) e 1 = 0. Intuitively, this EP 3 is interpreted as the singular point where the three eigenvectors of E 1 collapse into one.According to the Jordan decomposition, we denote the point q = q * as a simple EP 3 , if all the Jordan blocks belonging to the other eigenvalues E 2 , E 3 , . . .are trivial (i.e., one-dimensional).If the Hamiltonian has more than one eigenvalue whose Jordan block is nontrivial (i.e., has dimension greater than unity), we denote the point q = q * as a compound EP.The paper is organized as follows.In Sec.II, we discuss the sublattice symmetry and the nature of the EPs, which is the main result of this paper.Sec.III focusses on the properties of various types of EPs and the analytical solutions of eigenvectors in their neighbourhoods.In Sec.IV, we use a quantum distance to characterize the eigenvector folding near an EP, and explain the enhanced eigenvector sensitivity in terms of the unique subspace topology for non-Hermitian matrices.Sec.V deals with some explicit realizations of the systems discussed, and also touches upon the predictions for generic N -values.We conclude with a summary and outlook in Sec.VI.Appendices A-E show the details of the mathematical derivations of various results mentioned in the main text.

II. Sublattice symmetry and the EP parameter space
The sublattice symmetry makes the characteristic polynomial of the Hamiltonian even in the eigenvalue E, as captured by the relation det(E − H) = det[P (E − H) P ] = det(E + H), where we use the fact that the dimension of H is even.The eigenvalues of H thus always come in pairs of {E, −E}.A natural choice of basis under the sublattice symmetry is to group the upper and lower components of the eigenstates as ψ and χ, respectively.With this choice, the eigenvalue problem is reduced to the following equations: The above indicates that if (ψ T , χ T ) T is an eigenvector for the eigenvalue E, (ψ T , −χ T ) T is an eigenvector for −E.Besides eigenvalues, the sublattice symmetry also imposes the constraint that nondegenerate eigenvectors should appear in pairs.
The above pairing relation for eigenvectors at E and −E also applies to generalized eigenvectors, which include other linearly independent vectors in the generalized eigenspaces L E and L −E , apart from the eigenvectors.If we take two generalized eigenvectors (ψ 1 , χ 1 ) and (ψ 2 , χ 2 ), corresponding to an eigenvalue E having a nontrivial Jordan block, then T is also in the generalized eigenspace of E. By applying P to this equation, one can verify that (H + E) The coalescence of eigenvectors for a four-band model near a regular EP3 (blue oval disc) and a mixed-type EP3 (red oval disc).Near the regular EP3, three eigenvectors out of the four are collapsing to a single eigenvector at the EP.Near the mixed-type EP3, how the eigenvectors coalesce strongly depends on the path chosen to approach the EP.There can be two-fold, three-fold, and four-fold eigenvector-coalescence for the three different paths indicated by the dash-dotted, solid, and dashed lines, respectively.When sublattice symmetry is imposed, the three-fold eigenvector-coalescence is forbidden.
According to the above analysis, the degeneracy of the system should be distinguished depending on whether it involves a zero or nonzero eigenvalue E as follows: (1) If all the eigenvalues are nonzero (i.e., E = 0), the lower component is linearly related to the upper component as χ = −i B (q) ψ(q)/E.The problem is then entirely determined by the 2 × 2 matrix B(q) • B (q).If E is an eigenvalue where two eigenvectors coalesce at the momentum q = q * , then −E shows an identical behaviour.Hence, the exceptional degeneracy for E = 0 must be a compound EP, always appearing as a doublet of EP 2 's.
(2) If E = 0 is an eigenvalue with algebraic multiplicity l, the corresponding eigenvector is obtained from the kernels of the two matrices, i.e., those ψ and χ which satisfy B(q) χ = 0 and B (q) ψ = 0.The eigenvectors are given by (ψ T , 0) T and (0, χ T ) T .Assuming that the numbers of solutions to the two equations are dim(ker B) = m and dim(ker B ) = n, respectively, we can construct (m + n) distinct eigenvectors.Hence, the order of the EP can range from 2 to (l + 1 − m − n).
From the two possible cases, we find that the E = 0 situation gives us the richest EP structure, and hence, this will be the focus of the rest of this paper.Denoting the eigenvalues of B(q) • B (q) for N = 2 as λ, the dispersion can be generically written as λ ∼ |δq| or λ ∼ |δq| 1/2 , in the vicinity of the EP, where δq = q − q * .According to Eq. (3), the dispersion then takes the form E ∼ |δq| 1/2 or E ∼ |δq| 1/4 .
After defining the model, our goal is to work out the Hamiltonian along with the eigenvectors at E = 0, as well as the nontrivial generalized eigenspace L 0 .At an n thorder EP, a series of vectors {e 0 , e 1 , e 2 , . . ., e n } satisfies the chain equations H e j = e j−1 , with e 0 denoting the null vector and e 1 the eigenvector (for E = 0).When there is no symmetry, the corresponding parameter space of the Hamiltonian, denoted by EP n , can be figured out easily using the standard methods [41] (cf.Appendix C).However, in the presence of sublattice symmetry, employing the standard formalism usually becomes complicated, because it is difficult to find out all the matrices that commute with both the Jordan decomposition and the symmetry transformation.To avoid this issue, we instead employ the decomposition of each eigenstate as e j = (ψ T j , χ T j ) T , such that the condition for the existence of a higher-order EP simplifies to (i B χ j , −i B ψ j ) = (ψ j−1 , χ j−1 ).As we have already shown that e 1 is related to the kernels of B and B , the chain equations can be solved step by step.The condition for the existence of an EP requires a series of relations between the images {im(B), im(B )} and the kernels {ker(B), ker(B )}.From these algebraic relations, we can explicitly work out EP n .The results are summarized in Table I (with the derivation shown in Appendix A).We can clearly infer that the results in Table I cannot be obtained from solutions of some simple continuous equations derived from the Hamiltonian.Hence, a non-Hermitian system exhibits a much richer structure for degeneracies, compared to a Hermitian degeneracy, as observed in the case of no symmetry [41].

III. The eigenvector structures of different types of
EPs for SU (2) In the following subsections, we discuss the properties of various possible EP n 's in great detail, especially focussing on the analytic solutions for the eigenvectors.The system with N ≥ 2 can host both EP 2 's and higher-order EPs, which we discuss below on a case-by-case basis for Different types of EPs for N = 2 N = 2.We denote the location of an EP by q = q * , and use δq = q − q * to parametrize the momentum coordinates in the vicinity of this point.The angle between δq and the q x -axis is denoted as θ.In other words, near the degenerate point, we parametrize the momentum by δq = |δq| (cos θ x + sin θ ŷ).The real parts of the eigenvalues around various kinds of EPs are shown schematically in Fig. 2. The explicit derivations for the eigenvectors of the higher-order EPs have been worked out in Appendix B.
A. Lowest-order EPs EP 2 's are obtained where there is an SU (2) symmetry relating the two flavours of fermions.Hence, there must be a 2 × 2 sub-Hamiltonian that describes a single fermion flavour, and is similar to a 2d Jordan block at the EP.The full Hamiltonian in Eq. (1) at q = q * is therefore similar to a matrix with two J 2 (0) Jordan blocks in the diagonal: V H(q * ) V −1 = diag{J 2 (0), J 2 (0)}.On the other hand, the SU (2) symmetry among the two fermion flavours requires the off-diagonal blocks, B(q) and B (q), to be proportional to the identity matrix.Hence, at q = q * , B(q * ) = 0 (also see the first column of Table I).This is a doublet of EP 2 's and, to leading powers in δq, the off-diagonal matrices can then be approximated as where c is a constant.Without any loss of generality, we can parametrize v(θ) = v x cos θ +i v y sin θ 1 , with v x and v y being its real and imaginary parts, respectively.The eigenvalues of the resulting Hamiltonian are ± c v(θ) |δq|, each 1 One can perform a linear coordinate transformation (δqx, δqy) → (δq x , δq y ), such that B(q) → B(q ) I 2 ⊗ v δq is holomorphic in the complex coordinate defined as δq ≡ δq x + i δq y .

B. Highest-order EPs
The system supports higher-order EPs once we couple the two different fermion flavours together, and break the SU (2) symmetry.EP 4 's are the highest-order EPs that can appear, because we have a four-band system.
Because of the sublattice symmetry, the eigenvalues come in pairs of {E, −E} -this implies that the EP 4 can only appear at E = 0. Since we require all the eigenvectors to collapse into one at the EP 4 , with E = 0 being a fourfold degenerate eigenvalue, this brings about several restrictions.First of all, λ = 0 must be a two-fold degenerate eigenvalue of the 2 × 2 matrix B(q * ) • B (q * ).Secondly, this matrix product can have only one linearly independent eigenvector.Following the discussion in Sec.II, the zero-energy eigenvectors of the Hamiltonian are given by the kernels of B and B .The single-eigenvector condition thus requires that the total dimension of the kernels, dim[ker B(q * )] + dim[ker B (q * )], be equal to 1. Without any loss of generality, we can assume dim[ker B(q * )] = 1 and dim[ker B (q * )] = 0.If we denote the zero-energy eigenstate of B(q * ) as χ 1 , the four-dimensional generalized eigenspace L 0 of H(q * ) has the first vector e 1 proportional to (0, χ T 1 ) T .The details of sorting out this generalized eigenspace have been explained in Appendix A.
The EP 4 Hamiltonian at q * is similar to a fourdimensional Jordan block, i.e., V H(q * ) V −1 = J 4 (0).We present a concrete example, which follows the forms shown in the second column of Table I, by turning on the minimal number of non-Hermitian hoppings.To leading power in |δq|, where b j and b j are constants, and v j (θ) and v j (θ) are functions of the angle arg(δq x + i δq y ).More precisely, we assume that these parameters contain O(|δq|) corrections, so that we do not lose crucial terms when expanding our eigenvalues and eigenvectors in powers of |δq|.Using Eq. ( 3), the eigenvalues and the eigenstates are given by (more details can be found in Appendix B) respectively.
The eigenvalues vanish as |δq| [cf.Fig. 2(b)], while the four eigenvectors converge to (0, 0, 1, 0) T , right at the EP.Although the dispersions scale as square roots (rather than quartic roots) around the EP 4 , the typical behaviour of an EP 4 involving the eigenvectorcoalescence into a single one is observed.
We would like to emphasize that the EP 4 here does not exhibit a quartic-root dispersion.This is expected as an EP n can exhibit arbitrary m th -order root singularity, where m ≤ n [33,40], or even dispersions that cannot be expressed as root functions [46].In Appendix D, we show an example where a singularity in the form of a root of quartic order is realized in our four-band sublattice-symmetric system.

C. Odd-order EPs
As we have shown in Sec.II, the sublattice symmetry requires the dispersion near an EP at E = 0 to scale as δE ∼ |δq| 1/(2p) , with p ∈ Z + .In addition, the sublattice symmetry also restricts the ways in which E = 0 eigenvectors coalesce.These conditions seem to obstruct an odd-order EP.However, through an explicit construction of an EP 3 for the N = 2 four-band model, we will show that a somewhat anomalous EP 3 can exist.Although the generic case is expected to exhibit a cube-root dispersion around the singularity, a sublattice symmetry forces it to have a square-root-dispersion [37], which is indeed found to be the case here.We also find that the way the eigenvectors coalesce with one another depends on the path chosen to approach the EP 3 (while a regular EP 3 has three eigenvectors collapsing together for any path).The EP 3 here is anomalous and different from the usual scenarios.
Because of the sublattice symmetry, a zero eigenvalue can appear only with an even algebraic multiplicity.Hence, for the N = 2 case, the existence of an EP 3 with E = 0 requires that its algebraic multiplicity must be four.The degenerate point is thus an EP 3 plus an accidental zeroenergy eigenstate.According to our symmetry analysis, the total dimension of the kernels for B(q * ) and B(q * ) is m + n = 2.If m = 2 and n = 0, the matrix B(q * ) is identically zero, and B (q * ) can be brought to a diagonal matrix via a transformation matrix V. Applying the transformation matrix diag(V, V) to H(q * ) then brings it explicitly to a form similar to Eq. ( 4).Hence, either (m = 2, n = 0) or (m = 0, n = 2) gives a doublet of EP 2 's.An EP 3 can emerge only when m = n = 1.Now we look at a specific example.According to Table I, an EP 3 appears when V H(q * ) V −1 = diag{J 3 (0), 0}, and There are two linearly independent eigenvectors at E = 0, which are proportional to e 1 = (0, 0, 1, 0) T and e 2 = (b 2 , −b 1 , 0, 0) T , proving that it is not an EP 4 .From the Jordan decomposition, we find that e 1 belongs to a generalized eigenspace of dimension three, such that e 1 = H(q * ) ẽ2 and ẽ2 = H(q * ) ẽ3 , with ẽ2 = (1/b 1 , 0, 0, 0) T and ẽ3 = (0, 0, 0, 1/(b 2 b 1 )) T .Hence, this is an EP 3 accidentally coinciding with a zero-energy eigenvector.
To investigate how the symmetry constraints play out in this case, we explicitly show how the eigenvectors behave in the vicinity of this EP 3 .As the sublattice symmetry forbids the three eigenvectors folding together, they show an anomalous behaviour, which is in-between the coaslescence features of the eigenvectors of EP 2 and EP 4 .This is the reason why the eigenvector-coalescence depends on the path chosen while approaching q * .Whenever an EP is anisotropic [47], the eigenvectors indeed exhibit an enhanced path-dependent sensitivity.

Path 1
We approach the EP along the q x -direction (i.e., q y = 0 along this path), assuming that all deviations are linear, in which case the expansion looks like As before, we implicitly assume that the variables {b j , b j } and {v j (0), v j (0)} can contain O(|δq x |) corrections.The product determines the eigenvalue E 2 and χ [cf.Eq. ( 3 .Therefore, all the four eigenvectors coalesce to e 1 = (0, 0, 1, 0) T at q = q * .In comparison, there is no eigenvector converging to the eigenvector e 2 at q * .Although this EP is of order three, its singularity behaviour along the q x -path is similar to a typical EP 4 .

Path 2
The eigenvectors exhibit a typical EP 2 behaviour if v 3 (θ) and v 4 (θ) vanish for some angle θ, which can be obtained by imposing an additional symmetry to these parameters.For convenience, we choose the direction of approach to the EP in this case to be along the q y -direction, and set v 3 (π/2) = v 4 (π/2) = 0.The off-diagonal matrices take the forms: and their product is given by One of its eigenvalues of the product matrix vanishes as T , respectively.In this situation, the two eigenvectors (±ψ T 1 , χ T 1 ) converge to e T 1 , while the other two eigenvectors (±ψ T 2 , χ T 2 ) go to two other linearlyindependent vectors, which we denote as e T 3 and e T 4 .Hence, along this path, the eigenvectors behave as a single eigenvector of an EP 2 plus two linearly-independent accidental zero-energy eigenvectors.

IV. Irregular subspace topology of the EPs
In this section, we will formulate a way to quantitatively characterize the overlap of eigenvectors, following which we will illustrate the origin of the anomalous behaviour of the odd-order EPs under sublattice symmetry.The conclusion that comes out of this set-up is that eigenvector-coalescence is not actually a point-like property of the EP itself, but it depends on how the Hamiltonian looks like in its neighborhood.In fact, we will see that for our example of N = 2, the EP 3 under sublattice symmetry can in fact be understood as the point at which the parameter spaces of EP 4 and EP 2 intersect.This feature comes from the subspace topology of EP n , as a subspace of all 4 × 4 matrices M 4 (C).
When analyzing the coalescence of eigenvectors, it can be ambiguous if we directly compare them, because eigenvectors are equivalent upto phases.In order to characterize unambiguously how the states coalesce near regular EPs and mixed-type EPs, it is most convenient to introduce the quantum distance D [48], such that Clearly, D 2 (u, u ) is invariant under U (1) × U (1) transformations, i.e., under the change of the phases of u and u .
Here the states are normalized as u | u = u | u = 1, and || • || is the usual norm •|• of a quantum state.Using u to denote the eigenvectors at the EP at q = q * , and u to denote the states away from q * , D is a function of (q − q * ).D 2 is positive-definite, and vanishes only when u and u differ by a phase (i.e., when u and u denote the same quantum state).Hence, D 2 (u, e j ) can be used to describe unambiguously how the eigenvectors are approaching their target eigenvectors at the EP.
Since the eigenstates e j 's at the EP (i.e., at E = 0) are invariant under the sublattice symmetry, the two nondegenerate eigenstates (±ψ, χ), related by the sublattice symmetry, have the same D 2 value with e j .In Fig. 3, we show how the eigenvectors approach the ones at the EPs, as q approaches q * .In all the cases, the four nondegenerate states fall into two classes: each corresponding to a sublattice symmetry-related pair.Let us denote the two pairs of eigenvectors as {u 1 , u 2 } and {u 3 , u 4 }.For the EP 4 , D 2 is computed from e 1 (which is the sole linearly-independent eigenvector right at the EP) and each of the four nondegenerate eigenvectors, and it goes to zero as we approach the EP 4 .However, things are more complicated for the EP 3 , and in fact the behaviour of D 2 corroborates the results obtained in Sec.III C. Approaching the EP along the Path 1 of Sec.III C (with q y = 0), for all i ∈ [1, 4], D 2 (u i , e 1 ) goes to zero, while D 2 (u i , e 2 ) remains nonvanishing.On the other hand, if one approaches the EP along the Path 2 (with q x = 0), D 2 (u 1 , e 1 ) and D 2 (u 2 , e 1 ) go to zero, while D 2 (u 3 , e j ) and D 2 (u 3 , e j ) remain nonzero for both j = 1 and j = 2.
The anomalous behaviour of the eigenvectors near the EP 3 can be explained by its mixed nature.This is a very special property of a Jordan decomposition when the diagonal of a Jordan block coincides with some other eigenvalue(s).An EP with such a Jordan block is qualitatively different from an EP whose Jordan block has a nonzero gap with other eigenvalues.We denote the latter as regular EPs.The space EP 3 comprises two sets, namely, the set U 1 of regular EP 3 and the set U 2 of mixed-nature EP 3 .
To illustrate the possible structures around an EP 3 , we consider a 4 × 4 matrix M with no particular symmetry.Such a matrix has a 16 (complex) dimensional parameter space M 4 (C).The most common matrices in this space are those which are the non-singular ones featuring nondegenerate eigenvalues.The EPs are repre-sented by matrices with singularity, and they form lowerdimensional subspaces of M 4 (C).The dimension of the parameter space EP n decreases as n becomes larger (see Appendix C).For an EP 3 with Jordan decomposition M EP3 = diag{J 3 (0), 0}, one can easily verify that, within EP 2 , there is a sequence of points whose limit is This implies that, in any neighbourhood of M EP3 , we can always find points belonging to EP 2 .In particular, when the matrices representing nondegenerate eigenvalues are close enough to the matrices representing EP 2 's, two of the four eigenvectors of our non-Hermitian Hamiltonian should also come close to each other (see Fig. 4).
In addition to the above limit, we can find another limit by tuning the parameters of the matrix containing the Jordan block of the EP 3 , such that the EP 3 of M EP3 is now FIG. 4. Schematic depiction of the location of a mixed-type EP3 in the parameter space of a non-Hermitian matrix.The white (uncoloured) region in the parameter space represents matrices with nondegenerate eigenvalues.They are dense and their parameter space has the highest dimensionality.In the absence of any symmetry, the dimension of the EPn space decreases as n increases.The mixed-type EP3 appears as the intersection point of the EP2 (light blue cube), EP3 (gray surface), and EP4 (green line).When the sublattice symmetry is imposed, the regular EP3 surface (gray region) is forbidden, and the mixed-type EP3 can be approached only via the neighbourhood of either EP4 (dotted line) or EP2 (dashed line).This leads to two different ways of eigenvector-coalescence, which are shown by the collapse of directed arrows against "Path qx" and "Path qy" (corresponding to Path 1 and Path 2 of Sec.III C, respectively).
a limiting point of an EP 4 .This can be seen from lim This result is much more counter-intuitive than the coincidence with the EP 2 case, because EP 4 has a lower dimension than EP 3 , and the region M ( ) EP4 in the neighbourhood of M EP3 is usually very small.However, the neighbourhood of EP 4 comprising all matrices representing nondegenerate eigenvalues is not small.These matrices then can have a large overlap with the nondegenerate neighbourhood of M EP3 .As a result, all paths through this intersecting region will show a behaviour characteristic of a four-fold eigenvector-coalescence.The arguments above show that a mixed-type EP can appear as a common limit point of lower-and higher-order EPs, which implies that such an exceptional degeneracy cannot form a closed subspace in M 4 (C) by simply combining certain higher-order EPs.This anomalous behaviour of the odd-order EPs is absent in Hermitian systems.In the parameter space of a Hermitian matrix H herm , if we denote the space with an n-fold degenerate eigenvalue E as HD n (E), the space ∪ n≥m HD n (E) is given by the zeros of the resultants (R) or discriminants (D), i.e., by R(E) = 0 [38] or D[H herm (E)] = 0 [40].These equations involve continuous functions in M 4 (C), and hence their solutions constitute a closed subspace of M 4 (C).This means that the limit of a series Hermitian degeneracy HD n can only end in some HD m (m ≥ n).As a result, only higher-order degeneracy can be the limit of a lower-order degeneracy, but not the other way around.Therefore, for Hermitian matrices, there is no mixed-type degeneracy.
In summary, the enhanced eigenvector sensitivity can be understood intuitively in the following way (see also Fig. 4).The different directions of approaching q * in the Brillouin zone can be mapped to approaching the EP 3 through different tracks in the space of matrices representing nondegenerate eigenvalues.Due to the sublattice symmetry, it is forbidden to approach the EP 3 through the neighbourhood of the matrices representing a regular EP 3 .Consequently, the sublattice symmetry-restricted EP 3 can only be reached through the neighbourhoods of EP 2 and EP 4 .Of course in those neighbourhoods, either two or four eigenvectors coalesce together, leading to the anomalous behaviour of the eigenvectors of the EP 3 .

V. Lattice realizations and expectations for generic N -values
Examples of fermionic Hamiltonians with sublattice symmetry include solvable spin liquid models, such as the Kitaev spin liquid [12,17,44] (corresponding to N = 1), and the Yao-Lee SU (2) spin liquid [45] (corresponding to N = 3).The N = 2 model studied in this paper can be embedded in the Yao-Lee model.There, the low-energy physics is described by N = 3 flavours of Majorana fermion operators, with the Hamiltonian consisting of only nearestneighbour hoppings amongst fermions of the same flavour.
In order to produce the higher-order degeneracies discussed in this paper, we need to introduce terms which couple different flavours [thus breaking the SU (N ) symmetry] -these can be generated by terms σ α,i τ x i τ x j σ β,j (with i = j) in terms of the original spin operators.The details have been outlined in Appendix E.
Setting N = 3, one can get EPs up to sixth order, which is then expected to display a richer eigenvector sensitivity.For this case, an EP 5 can exist where a five-dimensional Jordan block becomes degenerate with another band.Near this EP 5 , the coalescence of eigenvectors can be four-fold or six-fold.Moreover, since two-fold coalescence is also permitted by the sublattice symmetry, there exists paths along which the eigenvectors collapse like they do in the vicinity of an EP 2 .Consequently, such an EP 5 has a higher degree of eigenvector sensitivity, making it possible to have more knobs to tune quantum states.
For a generic value of N , in order to obtain an N -fold compound EP 2 , or a highest-order simple EP 2N , the algebraic conditions are simply obtained by replacing the expressions for N = 2 by the appropriate N -value.More specifically, the N -fold EP 2 is SU (N )-invariant, and is obtained by choosing B as a diagonal matrix vanishing at q * , while B (q * ) remains nonzero.As for EP 2N , we need dim(ker B) + dim(ker B ) = 1 for generic N as well.Additionally, in order to ensure that all the 2N linearlyindependent eigenvectors coalesce to a single one, we need to impose the condition B • B ∼ J 2N (0), which can alternatively be represented as ker(B • B ) m = im(B • B ) 2N −m (with 0 < m < 2N ).For EPs with orders between 2 and 2N , the analysis becomes more complicated.Mixed-type odd-order EPs will exist at E = 0, analogous to the EP 3 of the N = 2 case that we have explicitly studied.Although the dimensions of the kernels can be worked out in a way similar to that shown in Table I, the image and kernel relations need to be figured out on a case-by-case basis, and closed-form expressions for the eigenvectors might involve extremely complicated calculations.Nevertheless, the generic topological relations between higher-order EPs remain valid.

VI. Summary and outlook
In this paper, we have explored the emergence of higherorder EPs in two-dimensional four-band non-Hermitian systems, with a sublattice symmetry.Such systems are relevant to non-Hermitian extensions of solvable spin liquid models.The sublattice symmetry forces the eigenvalues to appear in pairs of {E, −E}, and the dispersion around an EP is restricted to be an even root of the deviation in the momentum space.We have explicitly computed how the eigenvectors collapse at an EP, and found an anomalous behaviour for odd-order EPs.Based on the analytical solvability of a four-band system, we have shown that the collapse of the eigenvectors depends on the specific path of approaching an EP 3 .The behaviour is anomalous in the sense that it is in contradiction with the intuition that n eigenvectors always coalesce together at an EP n .In fact, the number of collapsing eigenvectors for a mixed-type oddorder EP is an even number smaller or greater than n, which is caused by the presence of the sublattice symmetry.Intuitively, this unconventional feature can be understood from the fact that there is a restriction in the parameter space of EP 3 due to the sublattice symmetry, and this unusual EP 3 can be approached only via the neighbourhoods of EP 2 's and EP 4 's.
Using the notion of a quantum distance, we have further explored the behaviour of the eigenvectors near the mixedtype EP 3 .We have found that the eigenvectors do not necessarily converge to those of a regular EP 3 , especially when we are approaching it from a neighbourhood of EP 2 .The quantum distance to the eigenvectors at the mixedtype EP 3 can change abruptly if we slightly perturb the approaching process.It is already known that the nonunitary evolution under a non-Hermitian Hamiltonian leads to a shorter quantum distance [49,50], which can play a role in state preparation.Hence, we expect that the anomalous behaviour near higher-order EPs will significantly enhance this effect, and lead to novel applications exploiting the features we have discovered through our analysis.
The enhanced eigenvector sensitivity for the mixed-type EP 3 s is a reminiscence of generic counter-intuitive features specific to non-Hermitian systems (i.e., these are absent in the corresponding Hermitian counterparts).A very wellknown example is the non-Hermitian skin effect [51][52][53][54][55][56][57][58][59], where a very small change in the boundary conditions brings about remarkable modifications to the spectrum.The mixed nature of the odd-order EPs also generalizes the notion of the recently-studied non-defective EPs [60], where a Hermitian degeneracy mixes with the usual EP 2 .
A promising future research direction is to explore analogous unconventional EPs in three-dimensional systems with appropriate symmetry constraints.The extended dimensionality is expected to provide a richer parameter space for the characterization of generic EPs [61].Another significant direction is to investigate the role of the higherorder EPs, especially the odd-order ones with anomalous behaviour, in designing non-Hermitian topological sensors [32].Due to higher-order singular behaviour near a regularly behaved higher-order EP, the sensors based on such EPs are expected to show greater sensitivity than an EP 2 , and the existence of mixed-type EPs may enable us to tune the sensitivity by tuning the parameter space [62].

A. Exceptional degeneracy under sublattice symmetry
When a symmetry is imposed, the standard method for obtaining the EP parameter space (see Appendix C) can be very complicated to employ in practice.Therefore, we adopt a more direct way to find the EP parameter space under sublattice symmetry, which employs the algebraic connections between B and B as linear transformation operators.In this appendix, we demonstrate this method for the N = 2 case, where we can obtain closed-form expressions.We use C × to represent the set of all complex numbers z = 0. We also introduce the notation J n to denote the set of nondegenerate matrices commuting with the Jordan block J n .In fact, J n is given by all upper-triangular translational-invariant matrices [41] 2 .
To get the SU (2)-invariant doublet of EP 2 's, the Hamiltonian is determined by B or B with a second-order EP.Hence, the corresponding parameter space EP 2 is given by GL(2)/J 2 .
For EP 4 , we notice that dim(ker B) + dim(ker B ) = 1, according to the discussions in the main text.Assuming that dim(ker B) = 1, dim(ker B ) = 0, without any loss of generality, B is invertible.As we have shown in Sec.III C, the matrix B • B must be similar to J 2 (0), which means There can be two scenarios according to whether B is diagonalizable or nondiagonalizable: 1.When B is diagonalizable, let the eigenvectors of B be χ 1 and χ 2 .We choose χ 1 ∈ ker B. In order to have Since χ 2 is an eigenvector with a nonzero eigenvalue, this is equivalent to (B • B ) χ 2 = 0, implying that B χ 2 ∈ ker B. Switching to the basis formed by χ 1 and χ 2 , we get In order to ensure that B invertible, we need b 2 b 3 = 0.
2. When B is not diagonalizable, it is equal to J 2 (0) in a basis formed by two linearly independent vectors χ 1 and χ 2 , still with χ 1 ∈ ker B.Here also, we only need to have (B • B • B) χ 2 = 0, which is now equivalent to (B • B ) χ 1 = 0.This tells us that B χ 1 ∝ χ 1 , i.e., χ 1 is also an eigenvector of B .Switching to the basis formed by χ 1 and χ 2 , we get The invertibility of B requires that b 1 b 4 = 0. Therefore, we find that the parameter space EP 4 comprises two sets: The Z 2 part in either set comes from the symmetry under B ↔ B .
The space EP 3 , as shown in Sec.III C, is restricted to obey dim(ker B) = dim(ker B ) = 1.Basic linear algebra then tells us that their corresponding image dimensions are also equal to one, i.e., dim(im B) = dim(im B ) = 1.Let the corresponding eigenvectors be χ 1 and ψ 1 , such that B χ 1 = 0 and B ψ 1 = 0.It is straightforward to verify that (0, χ T 1 ) T and (ψ T 1 , 0) T are eigenvectors of H.We assume that (0, χ T 1 ) T belongs to a generalized eigenspace of dimension three.Hence, there exists a vector ( We can choose ψ 2 to be in the subspace complementary to that of ψ 1 (i.e., ψ 2 ∈ {C 2 − (ker B )}) and set χ 2 = 0.In order to form a three-dimensional generalized eigenspace, we need a third linearly-independent vector (ψ 3 , χ 3 ), such that H (ψ T 3 , χ T 3 ) T = (ψ T 2 , 0) T .From this relation, we have ψ 3 ∈ ker B and im B = ker B , which enforces the condition ψ 2 ∈ im B -therefore we can choose ψ 3 = 0 and χ 3 ∈ (ker B) ⊥ .One can verify that the four vectors -(0, χ T 1 ) T , (ψ T 1 , 0) T , (ψ T 2 , 0) T , and (0, χ T 3 ) T -that we have just constructed, are linearly-independent.To summarize, once the matrix B is fixed, the image of B also gets fixed, and ker B must be different from im B. Since dim(ker B ) = 1, the matrix B is determined by B up to a nonzero vector (characterizing the ratio between the first and second columns of B ).The matrix B can be built from two linearly dependent row vectors: because its kernel is one-dimensional.Here at least one of p 1 and p 2 is nonzero and so are u 1 , u 2 .The kernel of B is generated by the vector (u 2 , −u 1 ) T , and its image is generated by (p 1 , p 2 ) T .According to the relations between B and B , we have where (p 1 , p 2 ) is not collinear with (p 1 , p 2 ).We observe that all the pairs (u 1 , u 2 ), (p 1 , p 2 ), and (p 1 , p 2 ) exclude the origin (0, 0).Since B is invariant under the transformations u i → z u i , p i → p i /z, and p i → p i /z, its parameter space is represented by

B. Solutions for eigenvectors near an EP
In this appendix, we work out the explicit expressions for the eigenvalues and eigenvectors near the EP 4 and EP 3 studied in Sec.III.Near the EP 4 , the off-diagonal submatrices of the Hamiltonian take the forms: to leading order in the powers of |δq|.Their product matrix is given by with eigenvalues The four eigenvalues E of the Hamiltonian are therefore given by ± √ λ 1 and ± √ λ 2 .The (unnormalized) eigenvectors of and hence are seen to converge to (1, 0) at the EP.Using the relation ψ a = i B χ a /E for E = 0, we deduce that , giving the four eigenvectors of the Hamiltonian as (±ψ T a , χ T a ) T .Clearly, these four vectors collapse to e 1 = (0, 0, 1, 0) T , as described in the main text.
As for the EP 3 , since the exact expression is quite complicated, we only show the leading order terms.For Path 1, where all deviations from the EP are linear, we have Since the eigenvalues of the product matrix are the eigenvalues of the Hamiltonian are of O( |δq x |).The corresponding eigenvectors are given by According to the relation Overall, the four eigenvectors (±ψ T a , χ T a ) T are seen to collapse to e 1 = (0, 0, 1, 0) T , resulting in the EP 3 behaving as a typical EP 4 , as far as the eigenvector-coalescence is concerned.
When we consider Path 2 for approaching the EP 3 , the off-diagonal matrices are given by leading to In order to figure out the eigenspace of an n × n matrix D, it boils down to finding a nondegenerate matrix V ∈ GL n (C), such that V D V −1 is equal to a block diagonal matrix M d = diag{J i1 (E 1 ), J i2 (E 2 ), . . .}.All information about exceptional degeneracy is encoded in M d .Let us denote the matrices commuting with M d as S d , which may also be called the stabilizer of M d under the action of GL n (C).The possible distinct matrices sharing the same exceptional structure are then given by the orbit GL n (C)/S d .Thus, for a given M d , GL n (C)/S d is the parameter space of the EP at the energy (E 1 , E 2 . . .).
Let us now demonstrate how the parameter space of an EP looks like by focussing on the case of n = 4.All 4 × 4 complex matrices form a 16-dimensional complex space M 4 (C) = C 16 .The parameter space of an EP is thus a (topological) subspace of this C 16 and, compared to Hermitian degeneracies, the space of an exceptional degeneracy has a much richer structure.The constructions for the various possible cases are shown below: 1. We first consider the scenario when all eigenvalues are degenerate, which consists of the highest-order EP, with the corresponding parameter space denoted as EP 4 [41].Using the notations introduced in Appendix A, the Jordan block for the exceptional degeneracy is given by J 4 (E), and the EP 4 is described by C × GL 4 (C)/J 4 [where the first C corresponds to the complex eigenvalue E of J 4 (E)]; its complex dimension is 4 2 + 1 − 4 = 13.The stabilizer J 4 is composed of polynomials of J n (0), with the condition that the coefficient of I 4 is nonzero.The space EP 4 is not simply connected, and is homotopically equivalent to SU (4)/Z n , where Z 4 is the cyclic group formed by all fourth-order roots of unity [41] -this implies that EP 4 has a nontrivial topology.A major difference from the degeneracies of Hermitian matrices stems from the fact that the transformation group GL 4 (C), unlike the unitary group, is neither a closed subspace of C 16 [it is an open subspace as the pre-image of det(M 4 ) = 0], nor compact.Additionally, the parameter space of an EP at a given energy is not closed, as we have already shown in the main text.This is in sharp contrast with the parameter space of highestorder Hermitian degeneracy.The latter is given by C, which is contractible, simply-connected, and closed in C 16 .It is described by matrices of the from E × I 4 .The degeneracy parameter space at a given energy is simply a point.
2. An EP 3 is of intermediate order, and the space EP 3 in C 16 is represented by The parameters E 1 and E 2 form the space C 2 .In order to work out EP 3 , we need to quotient out those V commuting with diag{J 3 (E 1 ), E 2 }.
To do so, first we rewrite diag{J 3 (E 1 ), E 2 } as E 1 I 4 + diag{J 3 (0), E 2 − E 1 }.Since I 4 commutes with any matrix, the problem is now reduced to finding the matrices commuting with diag{J 3 (0), E 2 − E 1 }, which we denote as S. The block form of S should satisfy with S 1 representing a 3 × 3 matrix and S 4 denoting a complex number.When E 1 = E 2 , we must have S 2 = 0 and S 3 = 0.For E 1 = E 2 , they can be nonvanishing.The results are summarized as 1 s 4 = 0; if 1 s 4 = 0 .(C2) , which is of complex dimension 15, where the last Z 2 comes from the general linear transformations that merely exchange E 2 and E 3 .

D. EP4 with quartic-root singularity around it
The EP 4 example provided in the main text has a squareroot dispersion near the degeneracy.Here, we provide an example of a different EP 4 which features a branch cut with quartic-root singularity.
As shown in Table .I, the requirement for the existence of an EP 4 is to have B • B proportional to a 2 × 2 Jordan block, with at least one of the individual matrices (i.e., B or B ) being non-invertible.To get a fourth-order root for the dispersion of the Hamiltonian, the eigenvalues of B • B should have a square-root dispersion.The typical form of B • B then needs to be a Jordan matrix with a linear term ∼ |δq| for the lower-left component.According to this logic, we can consider the forms: In each position where the matrix element is put to zero, we have neglected possible O |δq| terms, as they give a higher-order dispersion as explained in the main text.The product matrix is then given by An example of N = 3 flavours of fermions with sublattice symmetry is provided by the SU (2) spin liquid model by Yao and Lee [45].We use two of its flavours to realize the exceptional points discussed in the main text.The Hamiltonian in this decorated honeycomb lattice [cf.Fig. 1 (a)] is given by where the indices i and j label the triangles, and σ i,α denotes the vector spin-1/2 operator at site α ∈ {1, 2, 3} of the i th triangle.Furthermore, S i = σ i,1 + σ i,2 + σ i,3 is the total spin operator of the i th triangle.The coupling constant J is the strength of the intra-triangle spin-exchange, while J λ describes the inter-triangle couplings on the λtype link.There are three different types of links, x-, y-, and z− links, represented by red, green and blue ones in Fig. 1 respectively.Since S 2 i , S j = 0 and S 2 i , τ λ j = 0, the operator S 2 i commutes with the Hamiltonian for all i.Hence, the total spin of each triangle is a good quantum number, which we can use to subdivide the Hilbert space.
Just like the case of Kitaev's model on the honeycomb lattice [44], we first introduce the Majorana fermion repre-sentations for the Pauli matrices σ i,α and τ β i as follows: where η α i and d α i are Majorana fermion operators (i.e., η α i † = η α i and d α i † = d α i ).The Hilbert space is enlarged in the Majorana representation and the physical states are those invariant under a Z 2 gauge transformation.Using the above notation, we can reexpress the Hamiltonian ĤY L as where on the λ-type link ij , and Q is the projection operator on the physical states.Because u ij , Ĥ = 0 and [u ij , u i j ] = 0, the eigenvalues (which take the values ±1) of the u ij 's are good quantum numbers.From its form, it is clear that ĤeY L describes three flavours of Majorana fermions, coupled with the background Z 2 gauge fields denoted by u ij .One can verify that ĤeY L is invariant under the local Z 2 gauge transformation, which takes η α i → Λ i η α i and u ij → Λ i u ij Λ j , with Λ i = ±1.In addition to the Z 2 gauge symmetry, the system has a global SO(3) symmetry, which rotates among the three flavours of Majorana fermions, and is a consequence of the SU (2) symmetry of the original spin model.
Each Majorana flavour c α has a Hamiltonian identical to the single Majorana flavour in Kitaev's honeycomb model [44], and hence the Yao-Lee model effectively gives us three copies of the Kitaev model.The spectrum of the Majorana fermions is gapless, while the Z 2 gauge field has a finite gap from the flux-fee configuration given by u ij = 1.The low-energy theory of the SU (2) model is thus captured by setting u ij = 1, leading to the momentum-space Majorana Hamiltonian where Here, c α denotes the Fourier transform of a real-space η αoperator, and the subscripts 1 and 2 refer to the two sublattice sites A and B of the honeycomb lattice.Furthermore, the unit cell vectors of the triangular lattice, generating the honeycomb lattice, have been labelled by r 1 and r 2 .For notational convenience, we also introduce a third vector defined by r 3 = r 1 − r 2 .
Before constructing lattice Hamiltonians harbouring higher-order EPs, let us first review the second-order EPs obtained in a non-Hermitian extension of the Kitaev model, studied in Ref. [12].The momentum-space Hamiltonian takes the form: where the spin-spin coupling constants are tuned to complex values, parametrized as J 1 = |J 1 | exp(i φ 1 ) and J 2 = |J 2 | exp(i φ 2 ), and J 3 (with φ 1 , φ 2 , and J 3 constrained to be real numbers).The Dirac points of the Majorana fermion dispersion for φ 1 = φ 2 = 0 morph into EPs, as nonzero values of φ 1 and φ 2 are turned on [12], and are located at where q = (q 1 , q2 ) are the coordinates of the momentum vector in the reciprocal lattice space, in the basis of the reciprocal lattice vectors.The second equation fixes the signs in the first.The exceptional nature stems from complex J α 's due to the fact that Ã * (q) = Ã(−q).There are pairs of EPs connected by Fermi arcs, and are thus robust against perturbations.
The SO(3)-extension of Eq. (E6), as shown in Eq. (E5), has six bands, and thus has the possibility to host higherorder EPs.To start with, we can tune the J α 's into complex numbers, as illustrated above.However, this results only in a triplet of EP 2 's, each arising from one flavour of the Majorana fermions.In order to obtain higher-order EPs, we need to break the SO(3)-symmetry by introducing couplings between the three flavours in various ways, and/or using different values of the J λ 's for the three flavours.For instance, for nearest-neighbour couplings between different flavours, the relevant spin operators take the form: σ α i . . .τ β i . . .σ γ j . . .τ λ j . . ., with i and j here denoting the indices of the nearest-neighbour sites.As a concrete example, the operator i exp(i q • r 1 ) c α (−q) c β (q) (with α = β) translates into σ α,i τ 1 i τ 1 j σ β,j .In order to have a non-Hermitian behaviour, we choose J 1 = J exp(i φ), and J 2 = J 3 = J, where J and φ are real parameters.The EPs are assumed to appear at q = q * , as before.We introduce the functions g(q) = exp(i q • r 1 + i φ) + exp(q * • r 2 ) + 1, and h(q) = exp(i q * • r 1 − i φ) + exp(i q • r 2 ) + 1.One can verify that g(q * ) = h(−q * ) = 0. Since both of these represent nearestneighbour hoppings, they can be constructed via the spin operators as described in the earlier paragraph.
To realize an EP 4 , one way is to consider the form: A(q) = diag B(q), Ã0 (q) , B(q) =   Ã(q) z 1 g(−q) + z 2 h(q) 0 Ã (q)   , (E8) which affects only the couplings among the operators c 1 1 (q), c 2 1 (q), c 1 2 (q), and c 2 2 (q).Here, z 1 and z 2 are the coupling constants for the h(q) and g(−q) hoppings.For the flavour α = 2, we have used a different coupling Ã (q), which is obtained by adding Ã(q) to h(q) or g(−q).Note that, in the low-energy Majorana fermion model, we have B (−q) = B T (q) due to the particle-hole symmetry.
The coupling Ã0 = 2 J (0) 1 e i q•r1 + J (0) 2 e i q•r2 + J (0) 3 corresponds to the flavour α = 3, and can be composed of a real set of values for the J (0) λ 's (as in the Hermitian case), as the 2 × 2 block of this flavour does not take part in the exceptional physics corresponding to the 4 × 4 block that we are tying to construct.
In order to realize an EP 3 , we need to make the couplings c 2 1 c 1 2 and c 2 1 c 2 2 anisotropic around the EP.This can be done by combining functions related by some type of crystal symmetry.Let us assume that the function f (q) vanishes linearly in δq near q * .Then, we can find another function f (q x , 2q * y − q y ), which is the mirror reflection of f (q x , q y ) with respect to q * .Near q * , the combined function f (q x , q y ) + f (q x , 2q * y − q y ) has a vanishing first-order derivative along the q y -direction, while its leading order Taylor expansion along the q x -direction is still linear, resulting in the desired anisotropy.Using these functions, we can now construct the Hamiltonian of the Majoranas as A(q) = diag B(q), Ã0 (q) , B(q) =   Ã(q) f 1 (q) + f 1 (q x , 2q * y − q y ) z 1 g(q) + z 2 h(−q) Ã (q) + Ã (q x , 2q * y − q y )   ,

(E9)
where f 1 = z 1 g(−q) + z 2 h(q), and B (−q) = B T (q).The coupling Ã0 can be constructed from real J (0) λ 's, similar to the EP 4 case.However, we immediately realize that the mirror-symmetric part of Ã (q) [i.e., Ã (q x , 2q * y − q y )], added to the original Hermitian spin Hamiltonian, is not perturbatively small.Hence, the above construction may create flux-excitations in the corresponding spin model (so that we are no longer in the zero flux state).Nevertheless, for a purely fermionic model, this construction will work without involving such issues.

FIG. 1 .
FIG. 1.(a) Decorated honeycomb lattice model of fermions with N = 3 flavours [45], also dubbed as the "Yao-Lee" model (see Appendix E).The system has a sublattice symmetry when only nearest-neighbour hoppings are included in the Hamiltonian.The fermions are labelled by their sublattice indices A and B, together with their flavour index α ∈ {1, 2, 3} on each sublattice site.(b)The coalescence of eigenvectors for a four-band model near a regular EP3 (blue oval disc) and a mixed-type EP3 (red oval disc).Near the regular EP3, three eigenvectors out of the four are collapsing to a single eigenvector at the EP.Near the mixed-type EP3, how the eigenvectors coalesce strongly depends on the path chosen to approach the EP.There can be two-fold, three-fold, and four-fold eigenvector-coalescence for the three different paths indicated by the dash-dotted, solid, and dashed lines, respectively.When sublattice symmetry is imposed, the three-fold eigenvector-coalescence is forbidden.

FIG. 2 .
FIG. 2. Real parts of the eigenvalues E for the cases of different types EPs when N = 2: (a) Re[E] for a doublet of EP2's, where each eigenvalue is doubly degenerate.(b) Re[E] for an EP4, where all different energy eigenvalues coalesce at one singular point.(c) Re[E] for an EP3, for which the eigenvectors are sensitive to how the point of singularity is approached in the Brillouin zone.We choose the EP to be anisotropic.The scaling of E around the EP can take different forms along different directions.

)
The leading order expansion for an eigenvalueλ of B • B goes as λ b 2 b 1 b 4 v 3 (θ) |δq|.As the energy goes as √ λ, we obtain a quartic-root behaviour in the vicinity of the EP 4 .E. Lattice realizations for N = 2 through the Yao-Lee model

TABLE I .
Explanation of the conditions for the existence of different types of EPs when N = 2.The forms of the matrices B and B at the degenerate point q = q * are shown.Since there is an obvious symmetry under the exchange B ↔ B , every case displayed in the table has a B ↔ B partner.The parameters in the bottom row need to further satisfy (1) b = 0 in the first column; (2) det(B ) = 0 and B = 0 in the second column; (3) |u1| 2 + |u2| 2 = 0, |p1| 2 + |p2| 2 = 0, |p 1 | 2 + |p 2 | 2 = 0, and p 1 p2 − p 2 p1 = 0 in the third column.
where U 1 and U 2 are two disjoint sets, with complex dimensions 14 and 11, respectively.The space U 1 consists of all matrices withE 1 = E 2 , i.e., U 1 = Conf 2 (C) × GL 4 (C)/(C × × J 3 ) [where Conf 2 (C) is the second configuration space comprising all pairs {E 1 ∈ C, E 2 ∈ C} with E 1 = E 2 ].U 1 characterizes all regular EP 3 's, where they exhibit the typical eigenvector-coalescence features, since there is a gap between the Jordan block and other levels.On the other hand, the space U 2 accounts for the caseE 1 = E 2 in the set {E 1 , E 2 }, and is given by C × GL 4 (C)/[C 2 × C × × J 3 ].3.The remaining exceptional degeneracy relevant to ourdiscussions is EP 2 .The space EP 2 also contains those EPs that are of a mixed nature.But for the sake of simplicity, we neglect them, focussing only on regular EP 2 's.In this case, the Hamiltonian matrix takes the form diag[J 2 (E 1 ), E 2 , E 3 ], with distinct eigenvalues E 1 , E 2 , and E 3 .The corresponding stabilizer turns out to be diag[S 1 , s 2 , s 3 ], with S 1 ∈ J 2 .As a result, the regular part of EP 2 is given by GL 4