Reproduction Code and Data for "Exceptional Points as Manifestations of Analyticity Breakdown in the 't Hooft Model"

Author: Kejun Liu (Soochow University) Manuscript: K. Liu, arXiv:2606.10141 (2026) Paper DOI: https://doi.org/10.48550/arXiv.2606.10141 Archive DOI: [pending — assigned by Zenodo upon deposit]

Funding: This work was supported by the National High-Level Overseas Talent Program (KS21400126), the Suzhou Talent project (ZXP2025057), the Jiangsu Distinguished Professorship Fund (SR21400225), and the Research Start-up Fund (NH21400525). The numerical calculations were supported by a project funded by the Priority Academic Program Development (PAPD) of Jiangsu Higher Education Institutions.

This archive contains the numerical code, raw data, and analytical derivations that support the results reported in the manuscript.

Contents

 ├── README.md                  ← this file
 ├── LICENSE                    ← MIT (code) + CC-BY 4.0 (data)
 ├── requirements.txt           ← Python dependencies
 ├── code/
 │   ├── thooft_prl_numerics.py ← core library: operator construction, HPC campaigns A–C
 │   ├── higher_order_ep.py     ← higher-order EP probe (cascade diagnostics)
 │   ├── si_skin_demo.py        ← Supplemental Fig. S1 (skin envelope = e^{αx})
 │   ├── skin_profile.py        ← skin decay-rate analysis (exp vs power-law)
 │   ├── make_fig1.py           ← main-text Fig. 1 (FSS, EP, γ_c scaling)
 │   ├── definitive_dq_test_v3.py ← D-Q conjecture test (Supplement S6.3)
 │   └── run_array.slurm        ← SLURM batch script for HPC campaigns
 ├── data/
 │   ├── campaignA_*.json       ← Campaign A: exploratory (γ, m) parameter survey (not reported in paper)
 │   ├── campaignB_*.json       ← Campaign B: finite-size scaling
 │   ├── campaignC_*.json       ← Campaign C: NHSE + quench dynamics
 │   ├── ep_*.json              ← EP scan data (chiral and massive)
 │   ├── nhse_*.json            ← NHSE scan data
 │   └── ep_scan_*.json         ← fine-mesh EP spectrum
 └── derivations/
     └── route_C_theta_term_analysis.md  ← analytical notes on the cos θ tridiagonal structure

Computation Narrative

The numerical programme followed a reconnaissance-to-production workflow: theory defines the observable, a coarse survey locates the interesting region, and high-resolution production runs characterise it in detail.

Step 1 — Reconnaissance (Campaign A). A coarse 2D survey of the (γ, m) parameter plane—60 γ-values × 20 mass values at L = 100—mapped the PT-symmetry-breaking phase boundary γ_c(m). This survey established three facts that guided all subsequent computation:

• The phase boundary γ_c(m) is monotonically increasing: quark mass stabilises the PT-unbroken phase, pushing the EP to larger deformation strength.

• The chiral-limit threshold γ_c(m=0) ≈ 7.97 g²N_c was confirmed to agree with the Jacobi continued-fraction prediction (§S2) within the coarse mesh resolution.

• The EP disappears from the physical γ > 0 axis for m ≳ 0.3 (at g²N_c = 1), setting the upper bound for the massive-quark analysis.

Campaign A is exploratory: its coarse mesh (γ step 0.17, L = 100) and two-digit precision are insufficient for publication-grade results. Its role was to identify where to look, not to produce final numbers.

Step 2 — Production (Campaigns B, C, HOEP). Guided by the reconnaissance map, production runs targeted specific points with high resolution:

• Campaign B performed finite-size scaling at L = 200–2000, scanning 250 γ-points in a narrow window centred on γc for each (g²N_c, m) combination. This yielded the ν = 1/2 exponent (Fig. 1a, Table S1), the γc scaling (Fig. 1c), and the massive-quark threshold trend (§S6.1).

• Campaign C characterised the non-reciprocal deformation (NHSE, Fig. 3, Table S2) and the real-time quench dynamics (Fig. 2b).

• HOEP probed the higher-order EP cascade (Fig. 1d).

Step 3 — Falsification test (D-Q conjecture). Independent of the HPC campaigns, a self-contained high-precision calculation (mpmath, 40 decimal digits) tested the FLZ D-Q conjecture at the chiral point (§S6.3). This emerged from a dialogue with the integrability community and is included here for transparency.

The reconnaissance data (Campaign A) is preserved in this archive even though it does not appear in the paper, because it documents the exploratory reasoning that led to the production parameter choices.

Quick Start

 pip install -r requirements.txt
 
 # Run the demo (takes ~30 s, no HPC needed)
 python code/thooft_prl_numerics.py --demo
 
 # Regenerate Supplemental Fig. S1
 python code/si_skin_demo.py
 
 # Regenerate main-text Fig. 1 (requires HPC data in results_v2/)
 python code/make_fig1.py

Code–Manuscript Mapping

Main-text results

Supplemental Material

HPC Reproduction

The main production runs used SLURM array jobs. To reproduce:

 cd code/
 python thooft_prl_numerics.py --make-jobs   # generates job_array.json (~200 jobs)
 # Update --array range in run_array.slurm to match job count
 sbatch run_array.slurm                       # submit to SLURM cluster
 python thooft_prl_numerics.py --collect-fss  # FSS extrapolation after all jobs finish

HPC Environment

The production results were obtained on the Soochow University high-performance computing cluster (Slurm-based). Each compute node hosts 2 × AMD EPYC 7763 (64 cores) with 256 GB RAM; the GPU node hosts 2 × AMD EPYC 9654 (96 cores) with 384 GB RAM. All jobs ran on CPU (NumPy/SciPy dense linear algebra, no GPU acceleration required).

Total compute: ~42 CPU-hours across 200+ completed SLURM array tasks (1 CPU, 4 GB RAM each), verified from cluster sacct records:

The data/ directory contains representative results at L = 100 (demo scale). Full-resolution results (L up to 2000) were produced on the HPC cluster; the largest individual jobs (dense complex diagonalization at N = 1999) took ~1 hour each.

D-Q Conjecture Test (Supplement S6.3)

The script definitive_dq_test_v3.py is self-contained and reproduces the falsification of the FLZ D-Q relation at the chiral point (α = −1).

 python code/definitive_dq_test_v3.py

Result: The conjectural D-Q relation predicts c₁⁰ − c₃⁰ = 4.029; two independent numerical methods give 7.592 (PV-integral difference) and 7.583 (contour integration), disagreeing by ~88%. The exact chiral-limit spectral sum G⁻⁽¹⁾ = 2 log 2 ≈ 1.386 is not recovered. See SM S6.3 for full discussion.

Software Requirements

• Python ≥ 3.9

• NumPy ≥ 1.21

• SciPy ≥ 1.7

• Matplotlib ≥ 3.5

• mpmath ≥ 1.2 (for D-Q conjecture test only)

Tested on Linux x86_64 with Python 3.11, NumPy 1.26, SciPy 1.12.

Citation

If you use this code or data, please cite:

 @misc{Liu2026thooft,
   author       = {Kejun Liu},
   title        = {Exceptional Points as Manifestations of Analyticity
                   Breakdown in the 't Hooft Model},
   year         = {2026},
   eprint       = {2606.10141},
   archivePrefix= {arXiv},
   primaryClass = {quant-ph},
   doi          = {10.48550/arXiv.2606.10141},
   url          = {https://doi.org/10.48550/arXiv.2606.10141},
   note         = {Reproduction code and data: zenodo.org/record/XXXXXXX}
 }

Contact

Kejun Liu — kjliu@suda.edu.cn Soochow University, Suzhou, China