All-or-nothing dichotomy of local conserved quantities in Hermitian spin chains beyond the parity-symmetric spin-1/2 nearest-neighbor class

A complete classification of integrability is known for spin-1/2 chains with translationally invariant, parity-symmetric (symmetric coupling matrix) nearest-neighbor interactions: every system in this class either is integrable, possessing local conserved quantities of every support size, or has no nontrivial local conserved quantity at all, and within the class the presence of a 3-support conserved quantity determines the presence of all larger ones, confirming the Grabowski–Mathieu conjecture there. Later work sharpened the picture: the dichotomy was proven for isotropic chains of arbitrary spin via a single computable invariant, conservation of the boost-generated 3-site charge (the Reshetikhin condition) was shown to imply integrability for all Hermitian translationally invariant nearest-neighbor chains, and explicit counterexamples in non-Hermitian bosonic chains showed that the all-or-nothing picture is not universal. The open core is the Hermitian problem: whether the dichotomy — equivalently the absence of any Hermitian partially integrable system — extends to the classes not yet covered: parity-asymmetric couplings $J_{αβ} ≠ J_{βα}$, anisotropic spin-$S ≥ 1$ chains in generic fields beyond the classified symmetry subclasses, and interaction ranges longer than next-nearest neighbor; the statement that an arbitrary (not necessarily boost-generated) 3-support conserved quantity implies integrability is the special case of this dichotomy, not a separate demand.

Background

A quantum spin chain is a one-dimensional lattice in which each site carries a finite-dimensional Hilbert space — for spin 1/2, a qubit described by Pauli operators $X_i$, $Y_i$, $Z_i$ — and the Hamiltonian is a sum of terms acting on nearby sites. An operator is a conserved quantity if it commutes with the Hamiltonian; it is local if it is a sum of terms each acting on only $O(1)$ consecutive sites, and it has support $k$ (it is $k$-support, or $k$-local) if each of those terms acts on at most $k$ consecutive sites. For a nearest-neighbor chain the Hamiltonian itself is 2-support and one-site conserved quantities such as the total magnetization are regarded as trivial; following the source work, a local conserved quantity is called nontrivial when its support is at least 3 (more generally, exceeds the interaction range) and it is not a linear combination of, or a polynomial in, the identity, the Hamiltonian, and one-site charges (any power of the Hamiltonian is conserved in every system and must not count). A chain is called integrable in the sense used here if it has infinitely many local conserved quantities — an infinite hierarchy that prevents thermalization, underlies the generalized Gibbs ensemble description of integrable steady states, and produces anomalous transport — and non-integrable if it has no nontrivial local conserved quantity. A partially integrable (intermediate) system would have a finite nonzero number of nontrivial local conserved quantities; no such Hermitian system has ever been exhibited. Grabowski and Mathieu (1995) proposed an integrability test for spin chains with short-range interactions: search, at the lowest nontrivial order, for a conserved quantity supported on three neighboring sites, and conjectured that the outcome is decisive — a system possesses conserved quantities of all support sizes if and only if it possesses a 3-support one. This question is entwined with the Yang–Baxter equation, the consistency condition of the quantum inverse scattering method that solves integrable models: the lowest-order expansion of the Yang–Baxter equation is the Reshetikhin condition, the statement that the specific 3-site operator $Q^{(3)} = [B_H, H]$, built from the Hamiltonian and the boost operator $B_H = Σ_j j h_{j,j+1}$, is conserved. Rigorous non-integrability proofs are far harder than constructing charges: the first, for the XYZ chain in a magnetic field (Shiraishi 2019), introduced an expansion-and-coefficient-comparison technique that was then applied model by model (mixed-field Ising, the PXP Rydberg-blockade chain, and Heisenberg with next-nearest-neighbor coupling). Yamaguchi, Chiba and Shiraishi completed the program for the class $H = Σ_i Σ_{α,β ∈ {x,y,z}} J_{αβ} σ^α_i σ^β_{i+1} + Σ_i Σ_α h_α σ^α_i$ with symmetric coupling matrix $J_{αβ} = J_{βα}$ (this symmetry of $J$ is exactly parity symmetry, that is, invariance of the two-site term under exchange of the two sites): the integrable members are precisely the known ones (free spins, Ising/XY free-fermion families, and Bethe-ansatz-solvable XXZ/XYZ-type models), every other member has no nontrivial local conserved quantity, no partially integrable system exists in the class, and the presence of a 3-support conserved quantity is equivalent to the presence of $k$-support conserved quantities for all $4 ≤ k ≤ N/2$ (here and below $N$ is the number of sites in the chain). The authors explicitly left open whether the Grabowski–Mathieu conjecture holds for more general classes of systems. Subsequent results reshaped this open problem. Shiraishi and Yamaguchi proved the all-or-nothing dichotomy for isotropic ($SU(2)$-symmetric, Heisenberg-type) nearest-neighbor chains of arbitrary spin $S$, the two regimes being separated by a single computable invariant $D^3$; this establishes the Grabowski–Mathieu conjecture within the isotropic class and yields a complete integrability classification for $S ≤ 13.5$. Hokkyo proved that for any translationally invariant Hermitian nearest-neighbor chain, of arbitrary local dimension and without any parity assumption, the Reshetikhin condition implies integrability — the whole mutually commuting boost-generated hierarchy is already encoded in the lowest charge. Conversely, Yamaguchi and Shiraishi constructed explicit counterexamples in non-Hermitian bosonic chains (symmetric nearest-neighbor hopping with arbitrary, possibly non-Hermitian, on-site terms): systems with a 3-local charge but no other nontrivial local charges, and systems with charges of every support except one. All-or-nothing is therefore not a universal law and the Grabowski–Mathieu test is not universally applicable; the question narrows to Hermitian chains. Classification-type results were meanwhile extended to next-nearest-neighbor (zigzag) spin-1/2 chains, to the spin-1 bilinear-biquadratic chain with anisotropic field within a symmetry class assuming translational, $U(1)$, time-reversal and spin-flip symmetry, to certain $U(1)$-symmetric spin-1 chains, and to selected higher-dimensional and electron–phonon models — all case by case. The Hermitian classes still without a dichotomy or classification are: chains whose two-site couplings break parity symmetry ($J_{αβ} ≠ J_{βα}$, as arises with Dzyaloshinskii–Moriya exchange, the antisymmetric component of the two-site coupling), anisotropic spin-$S ≥ 1$ chains outside the classified symmetry subclasses (generic fields and exchange anisotropies), and interaction ranges longer than next-nearest neighbor; in addition, on the integrability side, the Grabowski–Mathieu statement for an arbitrary 3-support conserved quantity — one not generated by the boost operator — has no accepted proof: a 2025 preprint claiming it in general (Zhang) withdrew the claim in its final published version.

Problem Statement

Settle whether the all-or-nothing structure of local conserved quantities is a general law of Hermitian, translationally invariant quantum spin chains with finite-range interactions, or whether a Hermitian partially integrable system exists. Two forms of answer close the problem. (1) Refutation: construct an explicit Hermitian, translationally invariant spin chain with finite-range interactions, lying in a class not already covered (parity-asymmetric two-site couplings $J_{αβ} ≠ J_{βα}$ with or without fields; anisotropic spin-$S ≥ 1$ chains outside the classified symmetry subclasses; or interaction range longer than next-nearest neighbor), that possesses nontrivial (support at least 3 and exceeding the interaction range, and not a linear combination of, or a polynomial in, the identity, the Hamiltonian, and one-site charges) local conserved quantities of some support sizes but not all — for instance a 3-support conserved quantity coexisting with the absence of any 4-support conserved quantity. (2) Affirmation: prove for each of these uncovered classes that every system either possesses nontrivial local conserved quantities of every support exceeding the interaction range up to $N/2$ (for the nearest-neighbor classes this is the range $3 ≤ k ≤ N/2$; infinitely many, i.e. is integrable) or possesses no nontrivial local conserved quantity at all; the statement that an arbitrary, not necessarily boost-generated, 3-support conserved quantity implies integrability in these classes is the special case of this dichotomy — a system carrying such a charge cannot fall in the latter, no-nontrivial-charge branch — and is therefore settled by it rather than being a separate requirement. The single scientific objective uniting the branches is the reach of the Grabowski–Mathieu all-or-nothing principle in Hermitian spin chains: one Hermitian counterexample in any uncovered class resolves it negatively, while the affirmative resolution requires dichotomy proofs covering all three named classes. Partial classifications of single subclasses are valuable progress but do not by themselves close the problem.

References

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