12 Jul 2026
Reading the Exceptional-Point Project V: Complete Notation and Crossover Guide
This post is a self-contained guide to the notation and main calculation in the paper Structured-to-generic crossover at arbitrary-order exceptional points in supersymmetric $\mathcal{PT}$-symmetric arrays.
It explains how the expressions in the manuscript are connected, with special attention to
\[\boxed{ |\delta_c|\asymp \frac{2^{N-2}}{(N-1)!} \frac{|q|^{N/2}}{|\Omega|^{N-1}} }.\]The reader is assumed to know the basic meaning of $\mathcal{PT}$ symmetry and supersymmetry. Everything else needed for this formula is developed below.
1. The Physical Model in One Line
The ideal array Hamiltonian is
\[H_N(\Omega,\gamma) =\Omega J_x+i\gamma J_z.\]It describes $N$ coupled modes. The spin label is
\[j=\frac{N-1}{2},\]so the $J_z$ quantum number takes the values
\[m=-j,-j+1,\ldots,j.\]The nearest-neighbour coupling between sites $n$ and $n+1$ is
\[C_n= \frac{\Omega}{2}\sqrt{n(N-n)}, \qquad n=1,\ldots,N-1.\]The diagonal gain-loss gradient is supplied by $i\gamma J_z$.
2. Master Notation Table
Array and spin labels
| Symbol | Definition | Meaning |
|---|---|---|
| $N$ | $N=2j+1$ | number of modes or sites |
| $j$ | $(N-1)/2$ | spin representation |
| $n$ | $1,\ldots,N$ | physical site index |
| $m$ | $-j,\ldots,j$ | structured-mode label |
| $J_x,J_y,J_z$ | spin-$j$ matrices | algebraic representation of the array |
| $C_n$ | $\frac{\Omega}{2}\sqrt{n(N-n)}$ | coupling of bond $n$ |
Physical control parameters
| Symbol | Definition | Meaning |
|---|---|---|
| $\Omega$ | common coupling scale | controls all designed bonds |
| $\gamma$ | gain-loss scale | controls the imaginary diagonal gradient |
| $J$ | critical value with $\Omega=\gamma=J$ | exceptional-point scale |
| $\epsilon$ | $\Omega=J+\epsilon$ | structured coupling displacement |
| $\delta$ | coefficient of $V_\delta$ | terminal structure-breaking coupling |
| $q$ | $\Omega^2-\gamma^2$ | signed structured distance from the EP |
Polynomial and crossover quantities
| Symbol | Definition | Meaning | ||
|---|---|---|---|---|
| $P_N(E;q,\delta)$ | $\det(EI-H_N-V_\delta)$ | full characteristic polynomial | ||
| $P_N^{(0)}(E;q)$ | $P_N(E;q,0)$ | structured polynomial without terminal link | ||
| $D_{N-2}(E)$ | interior principal determinant | coefficient of $-\delta^2$ | ||
| $A_N(\Omega)$ | $\frac{(N-1)!}{2^{N-2}}\Omega^{N-1}$ | full-chain path coefficient | ||
| $y$ | $E/\sqrt q$ | scaled eigenvalue | ||
| $F_N(y)$ | $\prod_{m=-j}^{j}(y-m)$ | dimensionless structured polynomial | ||
| $\zeta_N$ | $A_N\delta/q^{N/2}$ | crossover variable | ||
| $\delta_c$ | terminal-link crossover scale | value at which $ | \zeta_N | $ is order one |
Eigenvector and feasibility quantities
| Symbol | Definition | Meaning | ||||
|---|---|---|---|---|---|---|
| $r$ | $ | \psi^{\mathsf T}\psi | /(\psi^\dagger\psi)$ | phase rigidity | ||
| $K$ | $r^{-2}$ | Petermann factor | ||||
| $s$ | characteristic eigenvalue displacement | measured spectral splitting | ||||
| $\sigma$ | $s/ | \Omega | $ | dimensionless splitting | ||
| $\eta$ | $\sqrt{ | q | }/ | \Omega | $ | dimensionless distance from the EP |
| $d$ | $ | \delta | / | \Omega | $ | dimensionless terminal coupling |
| $d_c$ | $ | \delta_c | / | \Omega | $ | dimensionless nominal crossover |
3. First Link: From $\Omega,\gamma$ to $q$
The exact structured spectrum is
\[E_m=m\sqrt{\Omega^2-\gamma^2}.\]Defining
\[q=\Omega^2-\gamma^2\]compresses the spectrum to
\[E_m=m\sqrt q.\]The quantity $q$ has dimensions of energy squared, angular frequency squared, or inverse-length squared, depending on the platform.
| Sign of $q$ | Interpretation |
|---|---|
| $q>0$ | real structured spectrum |
| $q=0$ | all structured eigenvalues coalesce |
| $q<0$ | imaginary-conjugate structured levels |
The crossover analysis approaches the exceptional point from
\[q\to0^+.\]4. Second Link: From $q$ to the Structured Polynomial
Without the terminal link,
\[P_N^{(0)}(E;q) = \prod_{m=-j}^{j}(E-m\sqrt q).\]Introduce the scaled eigenvalue
\[E=\sqrt q\,y.\]Every factor becomes
\[E-m\sqrt q = \sqrt q\,(y-m).\]There are $N$ factors, so
\[P_N^{(0)}(E;q) =q^{N/2} \prod_{m=-j}^{j}(y-m).\]Define
\[F_N(y)= \prod_{m=-j}^{j}(y-m).\]Therefore
\[\boxed{ P_N^{(0)}(E;q)=q^{N/2}F_N(y) }.\]This equation separates:
- the dimensional scale $q^{N/2}$,
- the dimensionless root geometry contained in $F_N(y)$.
5. Third Link: The Terminal Perturbation
The structure-breaking perturbation is
\[V_\delta = \delta \left( |1\rangle\langle N| +|N\rangle\langle1| \right).\]It couples the first and last sites. The complete characteristic polynomial is
\[P_N(E;q,\delta) = P_N^{(0)}(E;q) -A_N(\Omega)\delta -\delta^2D_{N-2}(E).\]This identity is exact for $N\ge3$. There are no terms proportional to $\delta^3,\delta^4,\ldots$ because the matrix contains only two entries proportional to $\delta$.
6. Deriving the Path Coefficient $A_N$
The linear term uses one terminal matrix element and the entire nearest-neighbour path through the array.
The product of all designed bonds is
\[\prod_{n=1}^{N-1}C_n = \prod_{n=1}^{N-1} \left[ \frac{\Omega}{2}\sqrt{n(N-n)} \right].\]Separate the common factor:
\[\prod_{n=1}^{N-1}C_n = \left(\frac{\Omega}{2}\right)^{N-1} \left[ \prod_{n=1}^{N-1}n(N-n) \right]^{1/2}.\]Now
\[\prod_{n=1}^{N-1}n=(N-1)!,\]and after replacing $r=N-n$,
\[\prod_{n=1}^{N-1}(N-n) = \prod_{r=1}^{N-1}r =(N-1)!.\]Therefore
\[\left[ \prod_{n=1}^{N-1}n(N-n) \right]^{1/2} =(N-1)!.\]Hence
\[\prod_{n=1}^{N-1}C_n = \frac{(N-1)!}{2^{N-1}} \Omega^{N-1}.\]There are two terminal directions, $1\to N$ and $N\to1$. Multiplying by two gives
\[\boxed{ A_N(\Omega) = \frac{(N-1)!}{2^{N-2}} \Omega^{N-1} }.\]The factorial in the final crossover formula comes from this product of all engineered bonds.
7. Fourth Link: The Dimensionless Crossover Variable
Substitute
\[P_N^{(0)}=q^{N/2}F_N(y)\]into the exact polynomial:
\[q^{N/2}F_N(y) -A_N\delta -\delta^2D_{N-2}(\sqrt q\,y) =0.\]Divide by $q^{N/2}$:
\[F_N(y) = \frac{A_N\delta}{q^{N/2}} + \frac{\delta^2}{q^{N/2}} D_{N-2}(\sqrt q\,y).\]This motivates
\[\boxed{ \zeta_N= \frac{A_N(\Omega)\delta}{q^{N/2}} }.\]The numerator and denominator have the same dimensions:
\[[A_N\delta] = [E]^{N-1}[E] =[E]^N,\]and
\[[q^{N/2}] = ([E]^2)^{N/2} =[E]^N.\]Thus $\zeta_N$ is dimensionless.
8. Why the Quadratic Term Disappears from the Leading Scaling Equation
Hold $\zeta_N$ fixed while $q\to0^+$. From its definition,
\[\delta = \frac{q^{N/2}}{A_N}\zeta_N.\]Therefore
\[\frac{\delta^2}{q^{N/2}} \sim q^{N/2}.\]For fixed $N$ and bounded $y$, the interior determinant remains bounded. Consequently,
\[\frac{\delta^2}{q^{N/2}} D_{N-2}(\sqrt q\,y) \to0.\]The universal leading equation is
\[\boxed{ F_N(y)=\zeta_N }.\]The quadratic term is still included in every finite-$q$ calculation. It disappears only from the leading joint scaling limit.
9. Deriving the Crossover Scale in the Image
The structured regime corresponds to
\[|\zeta_N|\ll1,\]and the terminal-link-dominated regime corresponds to
\[|\zeta_N|\gg1.\]| The crossover occurs when $ | \zeta_N | $ is of order one: |
| Solve for $ | \delta_c | $: |
Using
\[|A_N| = \frac{(N-1)!}{2^{N-2}} |\Omega|^{N-1},\]we obtain
\[\boxed{ |\delta_c| \asymp \frac{2^{N-2}}{(N-1)!} \frac{|q|^{N/2}}{|\Omega|^{N-1}} }.\]This is the expression shown in the manuscript.
10. What the Symbol $\asymp$ Means
The symbol
\[\asymp\]does not mean an exact equality at a uniquely defined physical boundary. Crossovers do not normally have a sharp universal edge.
It means that the two sides have the same leading scale. If one defines the nominal crossover by
\[|\zeta_N|=1,\]then one may use the equality
\[|\delta_c|= \frac{2^{N-2}}{(N-1)!} \frac{|q|^{N/2}}{|\Omega|^{N-1}}.\]Choosing $|\zeta_N|=1/2$ or $2$ changes only an order-one numerical prefactor, not the dependence on $q$, $\Omega$, or $N$.
11. Dimensionless Tolerance Formula
Define
\[\eta= \frac{\sqrt{|q|}}{|\Omega|}, \qquad d= \frac{|\delta|}{|\Omega|}.\]Because
\[|q|^{N/2} = |\Omega|^N\eta^N,\]the nominal crossover becomes
\[\frac{|\delta_c|}{|\Omega|} = \frac{2^{N-2}}{(N-1)!}\eta^N.\]Thus
\[\boxed{ d_c= c_N\eta^N, \qquad c_N= \frac{2^{N-2}}{(N-1)!} }.\]The first few coefficients are
| $N$ | $c_N$ |
|---|---|
| 3 | $1$ |
| 4 | $2/3$ |
| 5 | $1/3$ |
| 6 | $2/15$ |
The power $\eta^N$ makes the structured regime increasingly fragile for larger $N$.
12. Reading the $N=4$ Example
For $N=4$,
\[j=\frac32, \qquad m=-\frac32,-\frac12,\frac12,\frac32.\]Therefore
\[F_4(y) = \left(y^2-\frac94\right) \left(y^2-\frac14\right).\]Expanding,
\[F_4(y) = y^4-\frac52y^2+\frac9{16}.\]The crossover equation is
\[F_4(y)=\zeta_4.\]Treat this as a quadratic equation in $x=y^2$:
\[x^2-\frac52x+\frac9{16}-\zeta_4=0.\]The solutions are
\[y^2= \frac54\pm\sqrt{1+\zeta_4}.\]The inner pair reaches $y=0$ when
\[\frac54-\sqrt{1+\zeta_4}=0.\]Therefore
\[\sqrt{1+\zeta_4}=\frac54,\]and
\[\boxed{ \zeta_4=\frac9{16} }.\]The paper separately checks that this double root has geometric multiplicity one and vanishing phase rigidity. It is therefore a secondary $\mathrm{EP}_2$.
13. Structured and Generic Limits from One Equation
The equation
\[F_N(y)=\zeta_N\]contains both limiting responses.
Small $|\zeta_N|$
The roots remain close to the structured values $y=m$:
\[y_m = m+\frac{\zeta_N}{F_N'(m)}+\cdots.\]Since $E=\sqrt q\,y$,
\[E_m\sim m\sqrt q.\]Large $|\zeta_N|$
| For large $ | y | $, |
Therefore
\[y^N\sim\zeta_N.\]Multiplying by $E=\sqrt q\,y$ gives
\[E^N \sim q^{N/2}\zeta_N = A_N\delta.\]Hence
\[E\sim(A_N\delta)^{1/N}.\]One scaled equation therefore interpolates between the square-root structured spectrum and the generic $N$th-root spectrum.
14. How the Eigenvector Quantities Connect
The paper also uses
\[r= \frac{|\psi^{\mathsf T}\psi|} {\psi^\dagger\psi}, \qquad K=r^{-2}.\]Near the structured exceptional point,
\[r_{jm} \sim \binom{N-1}{j-m}^{-1} \left|\frac{q}{\Omega^2}\right|^{(N-1)/2}.\]For a generic terminal-link unfolding,
\[r_{\mathrm g} \sim \left|\frac{\delta}{\Omega}\right|^{(N-1)/N}.\]At the crossover,
\[|\delta_c| \sim |q|^{N/2},\]so
\[|\delta_c|^{(N-1)/N} \sim |q|^{(N-1)/2}.\]The two rigidity powers therefore match at the same crossover scale.
15. Paper-Reading Roadmap
| Manuscript part | Question answered |
|---|---|
| spin array and $\mathcal{PT}$ symmetry | what is the ideal Hamiltonian? |
| structured perturbations | why does a collective error give a square root? |
| structure-breaking terminal error | why does a terminal link activate the full Jordan chain? |
| exact mixed polynomial | where does $A_N$ come from? |
| crossover theorem | how are $q$ and $\delta$ combined into $\zeta_N$? |
| $N=4$ example | where does the secondary $\mathrm{EP}_2$ occur? |
| phase rigidity and robustness | what happens to the eigenvectors? |
| $N=5$ taxonomy | do other allowed errors give the same exponent? |
| feasibility section | how small must an unwanted terminal link be? |
| appendices | where are the detailed derivations and Maxima checks? |
16. Common Notation Mistakes
| Mistake | Correct reading | ||
|---|---|---|---|
| treating $q$ as an energy | $q$ has dimensions of energy squared | ||
| confusing $\epsilon$ and $\delta$ | $\epsilon$ is structured; $\delta$ breaks the nearest-neighbour profile | ||
| treating $y$ as a measured eigenvalue | $y=E/\sqrt q$ is dimensionless | ||
| treating $\zeta_N$ as a new physical coupling | it is a dimensionless ratio of two competing polynomial terms | ||
| reading $\asymp$ as exact equality | it denotes a crossover scale | ||
| forgetting the factor $(N-1)!$ | it comes from the product of every designed bond | ||
| using the terminal-link theorem for $N=2$ | the structure-breaking terminal link is distinct only for $N\ge3$ | ||
| calling every repeated root an EP | geometric multiplicity must also be checked | ||
| writing $K\sim s^{-2(N-1)}$ with dimensional $s$ | use $\sigma=s/ | \Omega | $ |
17. Final Connection Map
The whole paper can be remembered as the chain
\[(\Omega,\gamma) \longrightarrow q=\Omega^2-\gamma^2 \longrightarrow E=\sqrt q\,y \longrightarrow P_N^{(0)}=q^{N/2}F_N(y),\] \[C_1C_2\cdots C_{N-1} \longrightarrow A_N= \frac{(N-1)!}{2^{N-2}}\Omega^{N-1},\] \[\frac{A_N\delta}{q^{N/2}} \longrightarrow \zeta_N \longrightarrow F_N(y)=\zeta_N,\]and finally
\[|\zeta_N|\sim1 \longrightarrow |\delta_c| \asymp \frac{2^{N-2}}{(N-1)!} \frac{|q|^{N/2}}{|\Omega|^{N-1}}.\]Return to Part I: Non-Hermitian Arrays from First Principles or review Part IV: Petermann Factor from First Principles.
Discussion