SciPost Submission Page
The Motzkin Spaghetto
by Zhao Zhang, Olai B. Mykland
Submission summary
| Authors (as registered SciPost users): | Zhao Zhang |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2506.02103v3 (pdf) |
| Date submitted: | Dec. 19, 2025, 10:49 a.m. |
| Submitted by: | Zhao Zhang |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
While highly entangled ground states of gapless local Hamiltonians have been known to exist in one dimension, their two-dimensional counterparts were only recently found, with rather sophisticated interactions involving at least four neighboring degrees of freedom. Here, we show that similar bipartite entanglement properties can be realized on a square lattice with anisotropic interactions in four different quadrants. The interaction to generate such entanglement is much simpler than the previous constructions by coupling orthogonal arrays of highly entangled chains. The new construction exhibits an entanglement phase transition with different scalings of entanglement entropy at the critical point and in the lowly entangled phase, and faster decay of the spectral gap in the highly entangled phase. The tensor network representation of the new ground state consists of tensors with lower rank, while preserving a global geometry similar to that of the original networks.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Weaknesses
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Arguing that a 1D model wrapped on a 2D space-filling curve is actually 2D.
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Sections 4 and 5 only repeat existing results as if they were new.
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Arguing that accidental correlations between remote sections of a 1D chain is a sign of a 2D phase of matter (antiferromagnetism).
Report
While the entanglement entropy of the ground states of gapped Hamiltonians satisfy an area law, this is not necessarily the case for gapless systems, which usually pick up logarithmic corrections to the area law. In 1D, however, there are several models (such as the rainbow chain and the Motzkin chain) whose ground states are volume-law entangled. In higher dimensions, similarly simple (i.e. isotropic and up-to-two-body interacting) models are not known.
In this paper, Zhang and Mykland attempt to produce such a model in 2D by wrapping a one-dimensional Motzkin chain on the sites or edges of a square lattice in a double spiral pattern. They show that, if viewed as a 2D model, it indeed satisfies a volume law of entanglement for a number of straight cuts going through the middle of the square.
However, there is no conceivable sense in which this model is two-dimensional. The Hamiltonian only contains terms along the original Motzkin chain, so nothing stops simply unwrapping the spiral back into a 1D chain. The authors try to explain away this fact in a couple of ways:
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Since the spiral turns regularly in different directions, it has connections going in all directions on the square lattice. In an actual two-dimensional model, each site would have connections going in several different directions.
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The wrapped chain is "antiferromagnetically ordered in the radial direction". This is simply a function of the original chain having a gradient from positive to negative spin on the two ends, which happen to be interleaved in the double spiral. Antiferromagnetism is a phase of matter, so these correlations should be stable against finite perturbations, in particular against introducing slight ferromagnetic couplings along the now-empty bonds. I am sceptical the "antiferromagnetism" at hand would survive such a perturbation.
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A tensor network representation, morally similar to MERA, of the ground state is provided, underlining the volume-law scaling. This is again a known 1D representation of the Motzkin chain ground state wrapped up into a spiral, without any additional couplings that would make it genuinely 2D.
The only result in the paper that isn't a trivial repurposing of known results for the 1D Motzkin chain is the calculation of the entanglement entropy for the 2D entanglement cuts that slice the chain into a number of small segments instead of a simple chain bipartition. Computing the entanglement entropy for such an unusual bipartition is technically challenging, but the result is as expected given the known volume-law half-chain entanglement.
In summary, the paper completely fails to propose a genuinely two-dimensional system with a volume-law entangled ground state, which is its stated purpose. Since the volume-law entanglement in the original chain is a function of a superexponentially small gap, a rather fine-tuned property, I am not optimistic it would persist under finite perturbations along the radial direction, which would be needed to make the model two-dimensional. Unless the authors convincingly demonstrate the opposite, I cannot see any point in publishing this paper either in SciPost or elsewhere.
Recommendation
Reject