# Dynamical crossing of the N=4 exceptional point

**Project Prometheus companion DOI:** [https://doi.org/10.5281/zenodo.23349491](https://doi.org/10.5281/zenodo.23349491)

## Model and protocol

The simulation uses the supplied open-chain spin-1/2 XXZ Hamiltonian at `J=1`, `Delta=0.5`, and `b=J`, restricted to the conserved `Sz_tot=0` sector. It evolves the right state according to

\[
i\,\partial_t|\psi(t)\rangle=H(\gamma(t))|\psi(t)\rangle,\qquad \hbar=1,
\]

with a linear ramp from \(\gamma_{\rm EP}-0.04\) to \(\gamma_{\rm EP}+0.04\), or in reverse. The refined EP is \(\gamma_{\rm EP}=0.27366893221371\), and the coalescing energy is \(E_{\rm EP}=-0.97580527153601\). The simulation starts in each of the two EP-pair right eigenstates: the lower/upper real-energy eigenstate below the EP, or the positive-/negative-imaginary-energy (gain/loss) eigenstate above it. Sweep speeds are 0.04, 0.01, and 0.002, giving total ramp durations 2, 8, and 40 in the chosen units.

RK4 integration is used with a maximum step of `0.001`. The state is rescaled after each integration step to avoid overflow; the accumulated logarithmic rescaling gives \(\log\|\psi\|\) for a unit-norm initial state. Results were stable when the slowest-ramp step was reduced from `0.002` to `0.0005`.

## Results

The table's branch weights are normalized squared coordinates in the **instantaneous right-eigenvector basis**, not Born probabilities; that basis is nonorthogonal. The “ray fidelity” is the ordinary normalized Hilbert-space overlap squared between the two final states produced from the two possible initial EP-pair states. `log norm` is the accumulated non-unitary amplitude change.

### Upward ramp: unbroken to broken phase

| Speed | Initial branch | Final gain-mode weight | Final loss-mode weight | Final-state ray fidelity between starts | log norm |
|---:|---|---:|---:|---:|---:|
| 0.04 | lower | 0.681703 | 0.318297 | 0.785867 | +0.059973 |
| 0.04 | upper | 0.681703 | 0.318297 | 0.785867 | +0.065632 |
| 0.01 | lower | 0.944533 | 0.055467 | 0.964665 | +0.513678 |
| 0.01 | upper | 0.944533 | 0.055467 | 0.964665 | +0.510393 |
| 0.002 | lower | 0.998260 | 0.001740 | 0.999992 | +2.603548 |
| 0.002 | upper | 0.998260 | 0.001740 | 0.999992 | +2.594172 |

The two initial branches become increasingly aligned with the **gain mode** as the sweep is slowed. The leading Puiseux coefficient is \(c_{1/2}=+0.9185063i\): analytic branch continuation maps the lower real branch to the positive-imaginary branch, while the upper real branch maps to the negative-imaginary branch. The dynamics does **not** preserve that mapping for both initial states: even the initial upper branch ends predominantly in the gain mode, rather than the loss mode. That is a dynamical branch-selection/flip relative to analytic continuation, driven by differential amplification and nonadiabatic transfer near the gap closure.

### Downward ramp: broken to unbroken phase

| Speed | Initial branch | Final lower-mode weight | Final upper-mode weight | Final-state ray fidelity between starts | log norm |
|---:|---|---:|---:|---:|---:|
| 0.04 | gain | 0.495082 | 0.504918 | 0.747588 | +0.290356 |
| 0.04 | loss | 0.495082 | 0.504918 | 0.747588 | -0.384618 |
| 0.01 | gain | 0.495262 | 0.504738 | 0.152699 | +0.972360 |
| 0.01 | loss | 0.495264 | 0.504736 | 0.152699 | -1.673152 |
| 0.002 | gain | 0.495267 | 0.504733 | 0.921202 | +1.813639 |
| 0.002 | loss | 0.495266 | 0.504734 | 0.921202 | -1.330464 |

The reverse ramp does not end in a clean lower- or upper-eigenstate selection in this protocol: the endpoint coordinates are nearly 50/50 for all tested speeds. However, equal coordinate weights do **not** imply identical states. The gauge-invariant final-state fidelity between the two initial branches is speed-dependent and is especially low (about 0.153) at speed 0.01. Non-Hermitian amplification also makes the norm outcome strongly dependent on whether the sweep starts in the gain or loss eigenstate.

## Interpretation and scope

- This one-parameter crossing exhibits **direction-dependent, nonadiabatic state selection**, not a topological chiral state flip from encircling an EP. No loop in a two-parameter space was performed.
- The upward slow-ramp result is robust in this tested interval: both starting branches approach the positive-imaginary gain mode. The downward crossing instead yields a nearly balanced pair-coordinate decomposition, with phase-sensitive final states.
- The EP closes the instantaneous gap, so the ordinary Hermitian adiabatic intuition does not apply. A slower ramp gives the system more time in the broken-phase region, where gain/loss separates the branches exponentially.
- The results depend on the chosen endpoints, linear ramp, initial eigenvectors, sector, and parameters. “State-flipping” is therefore not a universal binary outcome without specifying a protocol and a state diagnostic.
- All numbers are finite-dimensional numerical results for the Hamiltonian as coded. They are not experimental observations. Eigenbasis coordinate weights are not probabilities, and the reported log norm is model-dependent under non-Hermitian evolution.

## Reproducibility

- `phase_ii_ep_dynamics.py` runs the sweeps and writes the CSV and plot.
- `phase_ii_ep_dynamics.csv` contains all per-initial-state numerical outcomes.
- `phase_ii_ep_dynamics.png` visualizes endpoint mode selection and net amplification/attenuation.

To reproduce the table, install Python 3 with NumPy and Matplotlib, place `phase_ii_b_c_bisect.py` beside `phase_ii_ep_dynamics.py`, and run `python phase_ii_ep_dynamics.py`. The script writes the CSV and PNG beside itself.