Exceptional-Point Eigenvalue Monodromy
Follow an eigenvalue through two parameter loops and expose a defective point's singular projectors.
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Exceptional-Point Eigenvalue Monodromy
An exceptional point occurs when both eigenvalues and their corresponding eigenvectors coalesce, making the matrix defective (non-diagonalizable). This module tracks an eigenvalue and its eigenvector around a closed parameter loop in the complex plane, revealing the monodromy: a single 360-degree loop swaps the two eigenvalues, and two full loops (720 degrees) are required to restore the original state.
Worked example: Tracking an eigenvalue around a loop
Consider the matrix A(g) = [[0, 1], [g, 0]], where the parameter g = r * exp(i * phi). The eigenvalues are λ = ±√g. Let's start on the positive branch with r = 0.25 and track λ as phi increases from 0 to 360 degrees.
- At phi = 0, g = 0.25. The eigenvalues are ±0.5. Starting on the positive branch, λ = 0.5. The eigenvectors are (1, 0.5) and (1, -0.5). Their absolute overlap is |1 - 0.25| / (1 + 0.25) = 0.6. The squared Frobenius norm of the spectral projector P+ = 0.5 * (I + A / 0.5) is 0.5 + (0.25 + 4) / 4 = 1.5625.
- As phi increases, g traces a circle of radius 0.25 in the complex plane. The tracked eigenvalue λ = √0.25 * exp(i * phi / 2) traces a semicircle.
- At phi = 360, g returns to 0.25. However, λ = 0.5 * exp(i * 180°) = -0.5. The tracked eigenvalue has swapped to the negative branch!
| phi (degrees) | g | Tracked λ |
|---|---|---|
| 0 | 0.25 | 0.5 |
| 180 | -0.25 | 0.5i |
| 360 | 0.25 | -0.5 |
| 720 | 0.25 | 0.5 |
Experiment: Observe the monodromy and exceptional point
Open the Exceptional-Point Eigenvalue Monodromy module to interactively track the eigenvalues.
- Set Parameter radius to 0.25 and Branch to "Positive start". Observe the two distinct eigenvalues at ±0.5.
- Change Unwrapped degrees from 0 to 360. Watch the tracked eigenvalue move along the upper half of the Riemann surface to arrive at -0.5. The eigenvalues have swapped!
- Change Unwrapped degrees to 720. The tracked eigenvalue completes the second half of the loop and returns to 0.5.
- Set Parameter radius to 0. Watch the two eigenvalue sheets merge at the origin. The matrix becomes a nilpotent Jordan block, and the eigenvectors coalesce.
Model assumptions and limitations
This module visualizes the specific parameterization A(g) = [[0, 1], [g, 0]]. The exceptional point is located precisely at g = 0 (radius r = 0). At this point, the matrix is defective, possessing only a single eigenvalue (0) and a single eigenvector (1, 0). The spectral projectors become singular and their norms diverge as r approaches 0, hence they are marked as "Unavailable" exactly at r = 0.
The visualization uses a continuous unwrapped phase angle phi (0 to 720 degrees). While the principal complex square root has a branch cut, this module intentionally tracks the continuous evolution without reapplying the principal branch after each frame, in order to demonstrate the monodromy correctly. The geometric representation of the Riemann surface is schematic.
Primary references
The concept of non-Hermitian exceptional points and their eigenvalue monodromy is discussed in detail in the literature. For a comprehensive overview, see Physics of Exceptional Points (W. D. Heiss, 2012). The specific 2x2 matrix model and the behavior of the spectral projectors as they approach the exceptional point are standard examples used to illustrate these phenomena.
Experience it interactively
Adjust parameters, observe in real time, and build deep intuition with Riano’s interactive Exceptional-Point Eigenvalue Monodromy module.
Try Exceptional-Point Eigenvalue Monodromy on Riano →