[WIP] Split CTMRG

[Yue-Zhengyuan]

This PR implements the split-CTMRG introduced in

Naumann, J., Weerda, E. L., Eisert, J., Rizzi, M., & Schmoll, P. (2025). Variationally optimizing infinite projected entangled-pair states at large bond dimensions: A split corner transfer matrix renormalization group approach. Physical Review B, 111(23), 235116. (arXiv)

It is initially designed for bilayer networks (e.g. ground state iPEPS and purified thermal iPEPO) to reduce computational cost.

TODO

• Expectation values and pre-conditioning are still handled by first converting the split-CTMRG environment back to an ordinary CTMRG environment.
• For optimization, only FixedPointGradient can be used with SplitCTMRG. ImplicitGradient is not supported.
• Only iPEPS contraction is supported. Purified thermal iPEPO case may be deferred to later work.

Not TODO

Later there is a three-layer generalization (https://arxiv.org/html/2506.19339v1) that handles PEPS-PEPO-PEPS sandwich useful for 3D networks. But for now we focus on the bilayer case.

[codecov]

Codecov Report

❌ Patch coverage is 89.86711% with 61 lines in your changes missing coverage. Please review.

Files with missing lines	Patch %	Lines
src/environments/split_ctmrg_environments.jl	81.64%	29 Missing ⚠️
src/algorithms/ctmrg/split.jl	88.46%	9 Missing ⚠️
...lgorithms/optimization/implicit_differentiation.jl	87.50%	6 Missing ⚠️
src/algorithms/ctmrg/projectors/split.jl	95.68%	5 Missing ⚠️
src/algorithms/ctmrg/gaugefix.jl	90.00%	3 Missing ⚠️
src/algorithms/ctmrg/gaugefix_split.jl	94.28%	2 Missing ⚠️
src/utility/lq.jl	86.66%	2 Missing ⚠️
src/algorithms/ctmrg/ctmrg.jl	85.71%	1 Missing ⚠️
.../algorithms/expectation_value/expectation_value.jl	50.00%	1 Missing ⚠️
src/algorithms/expectation_value/network_value.jl	0.00%	1 Missing ⚠️
... and 2 more

Files with missing lines	Coverage Δ
src/PEPSKit.jl	100.00% <ø> (ø)
...orithms/contractions/ctmrg/enlarge_corner_split.jl	100.00% <100.00%> (ø)
src/algorithms/contractions/ctmrg/gaugefix.jl	100.00% <100.00%> (ø)
...algorithms/contractions/ctmrg/renormalize_split.jl	100.00% <100.00%> (ø)
src/algorithms/contractions/ctmrg/split_env.jl	100.00% <100.00%> (ø)
src/algorithms/ctmrg/projectors/projectors.jl	87.50% <100.00%> (ø)
src/algorithms/optimization/peps_optimization.jl	97.16% <100.00%> (+0.16%)	⬆️
src/networks/infinitesquarenetwork.jl	64.89% <100.00%> (+5.59%)	⬆️
src/utility/qr.jl	82.75% <100.00%> (+0.61%)	⬆️
src/utility/retractions.jl	100.00% <100.00%> (ø)
... and 12 more

... and 6 files with indirect coverage changes

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