A / ONE-PARAMETER CROSSINGN=4 CHAIN
A line crosses the branch point.The gap closes at γEP; non-adiabatic transfer and gain/loss decide the final mode.
01 A DYNAMICAL FIELD NOTE
The path through an EP can change the state.
An exceptional point is more than a spectral crossing. Modes coalesce, the gap closes, and time evolution becomes sensitive to how the system moves through parameter space.
ONE POINT. TWO PATHS. DIFFERENT DYNAMICS.
BRANCH STRUCTURE / LOCAL VIEW√z
Two eigenvalue sheetsMeet with one eigenvector
N=4 CHAIN · FIRST EPγEP 0.2736689
COALESCING ENERGYEEP −0.9758053
JORDAN STRUCTUREJ2 one defective pair
CROSSING ≠ ENCIRCLING
PROJECT PROMETHEUS COMPANION DOI10.5281/zenodo.23349491 ↗
WATCH THE EXPLAINER
Exceptional-point dynamics in 40 seconds.
A narrated field note on mode coalescence, crossing versus encircling, and the N=4 slow-ramp result. The loop is an illustrative two-level model; the computed chain result is a separate one-parameter crossing.
FORMAT1080 × 1920 · 30 fps
CASE STUDYN=4 · Sz=0 XXZ
DURATION40.12 seconds · narrated
02 THE SINGULAR POINT
At an EP, both modes
become one.
An ordinary degeneracy brings eigenvalues together while their eigenvectors can remain distinct. At an exceptional point, eigenvalues and eigenvectors coalesce. The matrix becomes defective, and its local eigenvalue branches typically have a square-root form.
LOCAL TWO-LEVEL NORMAL FORM
H(z) = [ 0 1
z 0 ]⇒E±(z) = ±√z
z 0 ]⇒E±(z) = ±√z
The branch point sits at z = 0. One loop around it exchanges the analytic eigenvalue sheets.
The normal form is an illustrative two-level model. The N=4 XXZ results below come from a separate, one-parameter chain calculation.
03 PATH-DEPENDENT EVOLUTION
Cross it—or go around.
The spectrum tells you where the modes are. The trajectory and the elapsed time determine which state survives.
B / CLOSED-LOOP ENCIRCLEMENTTOY NORMAL FORM
PARAMETER PLANE · z
EIGENVALUE PLANE · E
One loop exchanges the sheets.In the adiabatic continuation, the starting eigenstate returns on the other branch. Dynamics can select a different outcome.
LIVE TWO-LEVEL LOOP
Watch the state move.
Integrate i∂tψ = H(z(t))ψ around z=0. This is a pedagogical local model—not the N=4 chain.
θ = 0.00πCCW · one loop
TRAVERSAL
READY TO CIRCLE THE EP
+ endpoint mode—
− endpoint mode—
Move the progress slider or play the loop. Endpoint weights are coordinates in a nonorthogonal right-eigenvector basis, not probabilities.
Illustrative parameters: H(z) = [[0, 1], [z, 0]], z(t) = 0.2 exp(±2πit/T). Positive angle is counterclockwise. A single circuit returns to the same matrix but can return on a different mode branch.
04 THE COMPUTED CHAIN
What the N=4 sweep shows.
A separate, numerical test of a one-parameter crossing—no loop around the EP.
LINEAR RAMP · Sz = 0 SECTORRK4 · ℏ = J = 1
UPWARD: GAIN-MODE SELECTIONDOWNWARD: ~50/50 MODE COORDINATES
REFINED FIRST EPγEP = 0.2736689322coalescing energy EEP = −0.9758052715
The upward sweep picks the gain branch.
For a ramp from γEP−0.04 to γEP+0.04, both initial states increasingly finish in the positive-imaginary-energy mode. At speed 0.002, its right-eigenbasis coordinate weight is 99.83%.
The reverse sweep is not a clean flip.
Downward ramps end near 49.5/50.5 in the lower/upper pair coordinates. The final state still depends strongly on the starting branch and ramp speed; equal coordinates do not mean identical states.
Download sweep data 12 initial-condition runs
01Analytic continuation
Below the EP, the lower real branch continues to the positive-imaginary branch; the upper continues to the negative-imaginary one.
02Time evolution
Non-adiabatic transfer plus differential gain/loss can override that branch mapping. In the slow upward ramp, both starts select gain.
03Do not conflate paths
The chain result crosses a one-dimensional threshold. Clockwise/counterclockwise behavior belongs to a separate encircling protocol.
05 THE SHORT VERSION
Three things to keep.
An EP joins eigenvalues and eigenvectors.
The N=4 case has a second-order Jordan block in the Sz=0 sector.
A path is part of the dynamics.
Crossing and encircling are different protocols. Direction and speed can change the surviving mode.
Gain/loss changes the state norm.
Report the normalization convention alongside any non-Hermitian “population” or mode weight.
NOTATION
EP = exceptional point · γ = staggered imaginary-field strength · z = local complex control parameter · CW/CCW = clockwise/counterclockwise · gain/loss = positive/negative Im(E) for i∂tψ = Hψ.
06 OPEN RECORD
Reproducibility, from source.
The numerical chain result is separate from the illustrative two-level loop. Download the source and data used for the N=4 crossing analysis.
RUN THE N=4 CASE STUDY
Place both Python files in the same directory. With Python 3, NumPy, and Matplotlib installed, run:
python -m pip install numpy matplotlib
python phase_ii_ep_dynamics.pyThe script writes the CSV and plot beside itself. The logged coordinates are right-eigenbasis diagnostics, not Born probabilities.
- 01 / EP SOLVERphase_ii_b_c_bisect.py ↓
- 02 / TIME EVOLUTIONphase_ii_ep_dynamics.py ↓
- 03 / NUMERICAL OUTPUTSphase_ii_ep_dynamics.csv ↓
- 04 / PLOTphase_ii_ep_dynamics.png ↓
- 05 / FULL REPORTphase_ii_ep_dynamics_report.md ↓
PHASE II-F / STATUSThe EP location, Jordan structure, and dynamics results presented here were computed from operator definitions and are reproducible from the artifacts linked below. Phase II-F activation and EP-shift results are separate and remain unreproducible from the accessible public record pending recovery of the original source code.