[polyopt] TDVP4all: certified spectral gaps for blockaded Rydberg chains

[hzgpmc]

Team

Team name	TDVP4all
Members	Zhigang Hu

Challenge

Row
Challenge	Can semidefinite-programming spectral-gap certificates be adapted to the Pauli/projector algebra of blockade-constrained Rydberg chains to produce independently checkable gap bounds beyond existing frustration-free hierarchies?
Catalog issue	Addresses #233 — “Certified spectral gaps for blockaded Rydberg chains,” released by Jie Wang and Jin-Guo Liu.
Track	tracks/polyopt/ — the challenge centers on SDP hierarchies and noncommutative polynomial optimization.

Minimal result

This PR implements a sparse Ky Fan SDP workflow for the periodic blockaded
PXP chain

[
H_L(\delta)=\sum_iP_{i-1}X_iP_{i+1}-\delta\sum_i n_i,
\quad P=(I-Z)/2,\quad n=(I+Z)/2,
]

with unit Rabi coefficient, (0=\downarrow), (1=\uparrow), and blockade
(n_i n_{i+1}=0), including the wrap bond. The blockade is expressed through
localizing constraints. Translation and reflection reduce the SDP, while the
target is the multiplicity-counted global (E_1-E_0) in the full constrained
Hilbert space.

For (L=4), global-d2 gives an independently checked positive lower bound
at all 61 points (\delta=0,0.05,\ldots,3.0). Every bound obeys
(\Delta_{\mathrm{cert}}\le\Delta_{\mathrm{ED}}) for the identical setup.
The largest absolute ED-minus-certificate deficit is
(5.4030\times10^{-7}). At the endpoint,

[
\Delta_{\mathrm{cert}}(3)=0.0933579077612491
<\Delta_{\mathrm{ED}}(3)=0.0933580036555446.
]

The scan includes the commonly quoted Ising transition near
(\delta\simeq1.308) in this normalization.

What is included

• exact Pauli/projector algebra and sound blockade localizers;
• sparse, presolved Ky Fan problem construction and symmetry reduction;
• Julia/JuMP/Clarabel numerical solve bridge;
• dyadic PSD reconstruction and exact residual correction;
• standard-library-only independent certificate checkers;
• sparse QuSpin ED oracle;
• an expert-readable report with selected values, exact rational anchor,
  hashes, solver/accounting provenance, and reproduction commands.

Raw solver outputs, certificates, environments, and results/ remain
untracked.

Scope boundary

This is a deliberately minimal finite-(L) submission. It validates the
end-to-end certification workflow, but it does not locate the
thermodynamic transition, certify a thermodynamic-limit gap, or satisfy the
original issue's (N\le20) success gate. In particular, the decreasing
finite-(L) global gap also contains the low-energy splitting associated with
the ordered phase and is not by itself a bulk excitation-gap estimate.

Checks

• git diff --check: passed.
• Local workspace make help: passed.
• Local workspace make verify: passed.
• The complete test suite was not rerun before this urgent PR update, per the
  submitter's explicit instruction; the focused workflow had previously run
  successfully, and the submitted tests remain available under the solution
  directory.