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A spectral-gap classification for flat Schmidt-rank-two chains

Anonymous

Published Sep 28, 2026• 0 citations
Publisher

Abstract

For homogeneous open nearest-neighbour chains with a normalised rank-one local projector whose forbidden vector has exactly two equal nonzero Schmidt probabilities, a written proof candidate classifies gappedness in every finite local dimension. Gaplessness is equivalent to a balanced-marker form and has exact finite-length gap 1−cos(π/N). All other interactions in the class have explicit uniform lower bounds. A complete qutrit short-spectrum fibre has fixed Schmidt probabilities, two- and three-site spectra and ground-space dimension at every length, yet only its endpoint is gapless. The proof distinguishes compatible propagation from an obstruction to four-site saturation. Unrefereed candidate. Producer-side exact and numerical replay corroborates finite statements, not universal analytic proofs. Supplied referee implementation and review are not authenticated unaffiliated reproduction or peer review. No exhaustive priority, experimental or impact claim. Version 1.0.1 repairs the symbolic classifier domain and adds precise antecedent locators and a named full Gram-decomposition lemma with explicit basis appendix. Original prose/data CC0-1.0; original code MIT; third-party exceptions in LICENSES.md.

DOI
10.5281/zenodo.23024568

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For homogeneous open nearest-neighbour chains with a normalised rank-one local projector whose forbidden vector has exactly two equal nonzero Schmidt probabilities, a written proof candidate classifies gappedness in every finite local dimension. Gaplessness is equivalent to a balanced-marker form and has exact finite-length gap 1−cos(π/N). All other interactions in the class have explicit uniform lower bounds. A complete qutrit short-spectrum fibre has fixed Schmidt probabilities, two- and three-site spectra and ground-space dimension at every length, yet only its endpoint is gapless. The proof distinguishes compatible propagation from an obstruction to four-site saturation. Unrefereed candidate. Producer-side exact and numerical replay corroborates finite statements, not universal analytic proofs. Supplied referee implementation and review are not authenticated unaffiliated reproduction or peer review. No exhaustive priority, experimental or impact claim. Version 1.0.1 repairs the symbolic classifier domain and adds precise antecedent locators and a named full Gram-decomposition lemma with explicit basis appendix. Original prose/data CC0-1.0; original code MIT; third-party exceptions in LICENSES.md.
Complete written proofs and two exact replayable checker implementations classify all full-Schmidt-rank equality states without assuming stabilizer structure, classify every two-qudit pure stabilizer score at odd prime dimension, and determine the sharp pure-state ratio p/2 for every prime p congruent to 1 modulo 4. The general rank-deficient equality classification and the sharp ratio for primes congruent to 3 modulo 4, p at least 7, remain open in this record. This is an independently authored preprint with AI-assisted research and self-audit, not external professional peer review or a priority claim. The fixed immutable GitHub Release is https://github.com/mxym/math/releases/tag/ecqc-extremal-classification-v1. AI-assisted research and writing; no external funding. Existing licenses and third-party notices are retained.
We classify the nonnegative finite-size spectral-gap exponents of translation-invariant nearest-neighbour Hamiltonians on the square lattice, with finite local dimension, integer interaction matrices and open boundaries. The gap is measured above the entire ground space on an L×LL\times LL×L box. The existing exponents in Δ(L)=L−z+o(1)\Delta(L)=L^{-z+o(1)}Δ(L)=L−z+o(1) are exactly the nonnegative limit-computable (Δ20\Delta^0_2Δ20​) reals. The finite nonnegative lower/upper exponent pairs are exactly the ordered pairs consisting of a left-Σ20\Sigma^0_2Σ20​ real and a left-Π20\Pi^0_2Π20​ real. In particular, one fixed integer-valued model has exponent equal to the halting probability of any chosen prefix-universal machine. A classical boundary preference fixes a size counter and a computation tableau; a reversible mixed-radix counter supplies the small gap. We bound every sector of the full Hilbert space. The closing modes live on a boundary segment of length O(log⁡L)O(\log L)O(logL), and the local dimension may depend on the prescribed pair. These are finite-box exponents, with no assertion about bulk dispersion or stability of thermodynamic critical exponents.
We consider a family of translation-invariant quantum spin chains with nearest-neighbor interactions and derive necessary and sufficient conditions for these systems to be gapped in the thermodynamic limit. More precisely, let ψψψ be an arbitrary two-qubit state. We consider a chain of nnn qubits with open boundary conditions and Hamiltonian Hn(ψ)H_n(ψ)Hn​(ψ) which is defined as the sum of rank-1 projectors onto ψψψ applied to consecutive pairs of qubits. We show that the spectral gap of Hn(ψ)H_n(ψ)Hn​(ψ) is upper bounded by 1/(n−1)1/(n-1)1/(n−1) if the eigenvalues of a certain two-by-two matrix simply related to ψψψ have equal non-zero absolute value. Otherwise, the spectral gap is lower bounded by a positive constant independent of nnn (depending only on ψψψ). A key ingredient in the proof is a new operator inequality for the ground space projector which expresses a monotonicity under the partial trace. This monotonicity property appears to be very general and might be interesting in its own right. As an extension of our main result, we obtain a complete classification of gapped and gapless phases of frustration-free translation-invariant spin-1/2 chains with nearest-neighbor interactions.
For a fixed Hermitian swap-symmetric nearest-neighbour interaction satisfying leading rigidity, a finite quotient removes all on-site fields and two-site corrections from the three-site almost-conservation equations. A nonzero obstruction excludes exact contiguous charge ranges 3 through floor(N/2), for every periodic N ≥ 6 and arbitrary site-dependent Hermitian fields. Multiplicity-free symmetry evaluates the quotient for the spin-1 bilinear–biquadratic family, giving the exact factor u(u−v)(u+v). The candidate also identifies the residual local commutant and certifies two families missed by a cheaper partial-trace test. Unrefereed candidate with producer-side replay. Universal arguments are not formally verified, and the supplied review is not authenticated external peer review. Quotient consistency is not general integrability; no thermalisation, chaos or quasilocal-charge exclusion is claimed. Version 1.0.1 clarifies predecessor comparisons, support and Hermiticity conventions, adds optional portable rational witnesses and optimized-Python refusal controls. Main theorem statements are unchanged. Original prose/data CC0-1.0; original code MIT. Preserved third-party material retains its own rights; see LICENSES.md.
Preprint of the article "A fixed-level certification dichotomy for an infinite family of frustration-free spin chains" (22 pages, no figures). To every finite loopless directed graph T the paper attaches a rational, range-three, frustration-free spin-chain interaction: arcs of T carry hopping bands for domain walls, missing arcs act as hard local penalties. The main theorem is the exact three-term equivalence: T is acyclic <=> liminf_m gap_m(T) > 0 <=> delta_LTI(5) > 0, where gap_m is the spectral gap of the periodic chain on m sites and delta_LTI(5) is the value of the local translation-invariant (LTI) semidefinite certification hierarchy at level 5, its first sound level. The certification level is fixed while the size of the graph and the local Hilbert-space dimension are unbounded across the family. To the author's knowledge, this is the first fixed-level completeness result for a spectral-gap certification hierarchy over an infinite family of frustration-free chains: a single finite combinatorial invariant, acyclicity, decides both the thermodynamic spectral gap and the success of the certificate at one universal level. The acyclic direction is proved by an explicit rational certificate, a superpotential built from a topological height and three local corrections, giving delta_LTI(5) >= min{1/48, 1/sqrt(12H)} for a DAG of height H. The cyclic direction is made quantitative by a free-fermion analysis of the invariant winding sectors: directed cycles carry a diffusive lowest mode of energy of order 1/m^2, and the finite-volume gaps of the directed 2- and 3-cycles are computed exactly, 2(1-cos(pi/m)) and 2(1-cos(2pi/m)). The relationship with the PVBS chains of Bachmann and Nachtergaele and with earlier gapped-gapless classifications is discussed explicitly. All computer-assisted statements use exact symbolic arithmetic. The complete reproducibility package (symbolic pipeline, generated blocks with exact characteristic polynomials, Sturm data, LDL^T certificates, SHA-256 digests) is archived separately at https://doi.org/10.5281/zenodo.21790411. The manuscript has been submitted to Reports on Mathematical Physics.