12 Jul 2026
Reading the Exceptional-Point Project II: Exceptional Points from First Principles
This is Part II of the project guide, but it is written to be read independently. A matrix is non-Hermitian when $H\neq H^\dagger$. Such a matrix can have complex eigenvalues and nonorthogonal eigenvectors. The two-site example below supplies all the additional background needed here.
Part I develops the broader array interpretation and introduces the exact spectrum
\[E_m=m\sqrt{\Omega^2-\gamma^2}.\]We now derive what happens when the square root vanishes.
1. Ordinary Degeneracy Versus Exceptional Point
Two eigenvalues are degenerate when they have the same numerical value. That statement alone does not tell us what happens to the eigenvectors.
For a Hermitian matrix, a twofold degeneracy normally still has two independent eigenvectors. For example,
\[H_{\mathrm d}= \begin{pmatrix} 0&0\\ 0&0 \end{pmatrix}\]has eigenvalue zero twice, but both
\[\begin{pmatrix}1\\0\end{pmatrix}, \qquad \begin{pmatrix}0\\1\end{pmatrix}\]are independent eigenvectors.
An exceptional point is stronger:
- eigenvalues coalesce,
- eigenvectors also coalesce,
- the matrix becomes defective and cannot be diagonalized.
For a two-state problem this is an $\mathrm{EP}_2$. If $N$ states and their eigenvectors coalesce, it is an $\mathrm{EP}_N$.
2. The Two-Site Array at the Critical Point
Take
\[H_2= \frac12 \begin{pmatrix} i\gamma&\Omega\\ \Omega&-i\gamma \end{pmatrix}, \qquad q=\Omega^2-\gamma^2.\]The eigenvalues are
\[E_\pm=\pm\frac{\sqrt q}{2}.\]At $\Omega=\gamma=J$,
\[q=0, \qquad E_+=E_-=0.\]The critical matrix is
\[H_{\mathrm{EP}} =\frac{J}{2} \begin{pmatrix} i&1\\ 1&-i \end{pmatrix}.\]To find an eigenvector, solve
\[H_{\mathrm{EP}}\psi=0.\]The first row gives
\[i\psi_1+\psi_2=0,\]so
\[\psi_2=-i\psi_1.\]Both rows give the same condition. There is therefore only one independent eigenvector, for example
\[\psi_{\mathrm{EP}}= \begin{pmatrix} 1\\ -i \end{pmatrix}.\]The algebraic multiplicity is two, but the geometric multiplicity is one. This is the defining signature of an $\mathrm{EP}_2$.
3. Jordan Form
A diagonalizable matrix can be written as
\[H=SDS^{-1},\]where $D$ is diagonal. At an exceptional point, the correct canonical form is a Jordan block:
\[H_{\mathrm{EP}} =S \begin{pmatrix} E_{\mathrm{EP}}&1\\ 0&E_{\mathrm{EP}} \end{pmatrix} S^{-1}.\]After shifting $E_{\mathrm{EP}}$ to zero,
\[L_2= \begin{pmatrix} 0&1\\ 0&0 \end{pmatrix}\]satisfies
\[L_2^2=0, \qquad L_2\neq0.\]This is nilpotency of order two.
For an $\mathrm{EP}_N$, the canonical nilpotent block is
\[L_N= \begin{pmatrix} 0&1&0&\cdots&0\\ 0&0&1&\cdots&0\\ \vdots&&\ddots&\ddots&\vdots\\ 0&\cdots&0&0&1\\ 0&\cdots&\cdots&0&0 \end{pmatrix},\]with
\[L_N^N=0, \qquad L_N^{N-1}\neq0.\]The paper proves that its critical $N$-site Hamiltonian is similar to one such block.
4. Generalized Eigenvectors
One ordinary eigenvector is not enough to span the space at an $\mathrm{EP}_N$. The missing vectors are replaced by a Jordan chain:
\[H_{\mathrm{EP}}\chi_0=0,\] \[H_{\mathrm{EP}}\chi_1=\chi_0,\] \[H_{\mathrm{EP}}\chi_2=\chi_1,\]and so on. The vector $\chi_0$ is the coalesced eigenvector. The remaining $\chi_r$ are generalized eigenvectors.
This chain is why a perturbation can produce fractional powers. A perturbation may need to propagate through the entire chain before it closes the spectral equation.
5. Why an $\mathrm{EP}_N$ Produces an $N$th Root
At an ideal $\mathrm{EP}_N$, the characteristic polynomial is
\[P_0(E)=E^N.\]Suppose a perturbation of strength $\delta$ adds a nonzero constant carrier:
\[P(E,\delta)=E^N-A\delta+\cdots.\]To leading order,
\[E^N=A\delta.\]Therefore the $N$ roots are
\[E_k=(A\delta)^{1/N} \exp\left(\frac{2\pi i k}{N}\right), \qquad k=0,\ldots,N-1.\]Hence
\[|E_k|\propto|\delta|^{1/N}.\]This is the generic $\mathrm{EP}_N$ response. The fractional powers form a Puiseux series rather than an ordinary Taylor series.
6. Why the Paper Also Finds a Square Root
The existence of an $\mathrm{EP}_N$ does not mean that every perturbation must produce an $N$th root. The exponent depends on the direction of perturbation in matrix space.
The structured array has
\[E_m=m\sqrt q.\]If
\[\Omega=J+\epsilon, \qquad \gamma=J,\]then
\[q=(J+\epsilon)^2-J^2 =2J\epsilon+\epsilon^2.\]For small $\epsilon$,
\[E_m\sim m\sqrt{2J\epsilon} \propto\epsilon^{1/2}.\]The exponent remains one half for every $N$ because this perturbation stays inside the specially correlated spin-array family.
By contrast, the terminal link
\[V_\delta= \delta\left( |1\rangle\langle N| +|N\rangle\langle1| \right)\]breaks the nearest-neighbour structure and activates the full Jordan chain:
\[E_k\propto\delta^{1/N}.\]| Perturbation | Leading response |
|---|---|
| structured change of $\Omega$ | $\epsilon^{1/2}$ |
| terminal structure-breaking link | $\delta^{1/N}$ |
The paper is about the competition between these two laws.
7. Repeated Root Is Not Automatically an Exceptional Point
A repeated root satisfies
\[P(E_*)=0, \qquad P'(E_*)=0.\]But this only proves algebraic degeneracy. To prove an exceptional point, one must also show that the eigenspace has insufficient dimension.
For a double root:
| Nullity of $E_*I-H$ | Interpretation |
|---|---|
| 2 | two independent eigenvectors; not an $\mathrm{EP}_2$ |
| 1 | one eigenvector; defective $\mathrm{EP}_2$ |
The paper uses a nonzero cofactor to show that the positive $N=4$ secondary double root has matrix rank three and nullity one. Its phase rigidity also vanishes. Together these facts prove that it is a genuine secondary $\mathrm{EP}_2$.
8. What to Remember
| Term | Meaning |
|---|---|
| algebraic multiplicity | number of times an eigenvalue occurs in the characteristic polynomial |
| geometric multiplicity | number of independent eigenvectors |
| defective matrix | geometric multiplicity is too small |
| Jordan chain | eigenvector plus generalized eigenvectors |
| $\mathrm{EP}_N$ | one eigenvalue and one eigenvector formed from $N$ coalescing states |
| Puiseux response | fractional-power eigenvalue expansion |
The compact logic is
\[\text{one Jordan chain of length }N \quad\Longrightarrow\quad E^N\sim\delta \quad\Longrightarrow\quad E\sim\delta^{1/N}.\]Continue with Part III: Phase Rigidity from First Principles.
Discussion