Absence of nontrivial local conserved quantities in a class of $U(1)$-symmetric spin-1 chains

Abstract: We prove the absence of nontrivial local conserved quantities in a class of $U(1)$-symmetric spin-$1$ chains with nearest-neighbor interactions in which some of the quadrupolar couplings vanish, a class that is not covered by previous studies. Applying the technique of Shiraishi to these systems, we show that, for every model in this class on a periodic chain of $N$ sites, there is no $k$-local conserved quantity for any $3\le k\le N/2$. In particular, for a frustration-free spin-$1$ chain that exhibits spontaneous $U(1)$ symmetry breaking at zero temperature in one spatial dimension, we prove that every local conserved quantity with support up to half of the system size is a linear combination of the identity, the total magnetization $S<sup>z$, and the Hamiltonian itself. This rigorously establishes that, unlike the Heisenberg ferromagnet, the model admits no local order parameter commuting with the Hamiltonian, so that its continuous symmetry breaking is enabled by the frustration-free structure rather than by a conserved order parameter. We also prove the absence of $k$-local conserved quantities for $3\le k\le N/2$ in the periodic Motzkin chain, a frustration-free spin-$1$ chain closely related to the original Motzkin chain, for which spontaneous $U(1)$ symmetry breaking at zero temperature has also been reported.

• The paper proves that, across six patterns of vanishing quadrupolar couplings, no conserved quantity supported on 3 to N/2 sites exists beyond combinations of the identity, 1-local charges, and the Hamiltonian.
• The authors combine Shiraishi’s operator-basis method, a reduction from k-local to 3-local charges, and ten incompatible polynomial constraints to establish nonintegrability when all dipolar couplings are nonzero.
• For the frustration-free chain and periodic Motzkin chain, the results show that U(1) symmetry breaking does not rely on a local conserved order parameter, while the Motzkin model also has a proven (2N+1)-fold degenerate ground space.

Context and motivation

The paper addresses a central question in the study of quantum many-body integrability: which spin chains admit nontrivial local conserved quantities? For nearest-neighbor Hamiltonians, the Grabowski–Mathieu conjecture holds that the existence of a 3-local conserved quantity is both necessary and sufficient for integrability (2608.17548). The Shiraishi technique—expanding $[Q,H]$ in an operator basis and showing that the resulting linear system admits only trivial solutions—has become the standard rigorous tool for proving nonintegrability, and it has been applied to bilinear-biquadratic (BLBQ) chains, mixed-field Ising models, PXP models, and even Hubbard and Holstein systems.

The prior analysis of $U(1)$-symmetric spin-1 chains by Hokkyo, Yamaguchi, and Chiba covered the general nearest-neighbor Hamiltonian written in the dipolar/quadrupolar operator basis $\{E_m\}$, $\{F_m\}$, but only under the assumption that all quadrupolar couplings $f_m$ are nonzero. The present work extends the nonintegrability proof to six patterns of vanishing $f_m$, a class that includes a physically important frustration-free chain exhibiting spontaneous breaking of a continuous symmetry at zero temperature in one dimension—an exception to Coleman's theorem whose mechanism was previously not fully understood.

Main result

The main theorem states that for the general $U(1)$-symmetric spin-1 chain

$$H = \sum_i \Bigl(\sum_{m} e_m E_{m,i}E_{-m,i+1} + \sum_{m} f_m F_{m,i}F_{-m,i+1} + hF_{0,i}\Bigr)$$

on a periodic chain of $N$ sites, with all $e_m \neq 0$ and couplings belonging to any of six specified patterns of vanishing $U(1)$0, there is no $U(1)$1-local conserved quantity for any $U(1)$2. Consequently, every conserved quantity supported on at most half the chain is a linear combination of the identity, 1-local conserved quantities, and $U(1)$3 itself. The bound $U(1)$4 is optimal: powers of trivial conserved quantities such as $U(1)$5 furnish conserved quantities with support length exceeding $U(1)$6.

The motivating special case is the frustration-free chain

$U(1)$7

which corresponds to pattern (i) with $U(1)$8. For this model, the paper proves that every conserved quantity with support up to $U(1)$9 sites is a linear combination of the identity, total magnetization $\{E_m\}$0, and $\{E_m\}$1. This rigorously establishes that the model possesses no local order parameter commuting with the Hamiltonian—a fact previously conjectured but unproven—and therefore that its continuous symmetry breaking cannot be attributed to the ferromagnetic mechanism, where the order parameter itself is conserved.

Proof strategy

The proof proceeds through three stages. First, the authors characterize strictly 2-local quantities $\{E_m\}$2 whose commutator $\{E_m\}$3 contains no length-3 basis strings: they show these are exhausted by scalar multiples of the interaction part $\{E_m\}$4. This is proved case by case for each vanishing pattern, using the "column" graphical notation for commutators introduced by Shiraishi. The argument exploits the absence of cancellation partners when certain interaction terms are missing from $\{E_m\}$5—precisely the difficulty created by vanishing $\{E_m\}$6—and pattern (iv) requires a separate, more delicate treatment involving consistency conditions among multiple generating placements.

Second, this characterization verifies assumption (B) of Hokkyo's reduction theorem, which reduces the analysis of all $\{E_m\}$7-local conserved quantities ($\{E_m\}$8) to that of 3-local quantities $\{E_m\}$9 satisfying $\{F_m\}$0. The reduction further fixes the strictly 3-local part of any such candidate to a specific "doubling" form determined entirely by the coupling constants.

Third, the authors derive ten necessary polynomial conditions $\{F_m\}$1–$\{F_m\}$2 on the couplings for existence of such a 3-local quantity, and show by explicit algebraic manipulation that these conditions are mutually inconsistent whenever all $\{F_m\}$3, in each of the six patterns. The contradictions are obtained by combining subsets of the constraints—for instance, in case (a) ($\{F_m\}$4), the conditions force either $\{F_m\}$5 or $\{F_m\}$6, contradicting the standing assumptions. Finally, the 1-local sector is analyzed directly; for the frustration-free chain, the only surviving 1-local conserved quantity is $\{F_m\}$7.

Physical significance for symmetry breaking

The result resolves an open question about the mechanism of continuous symmetry breaking in one-dimensional frustration-free systems. The Heisenberg ferromagnet evades Coleman's theorem because its order parameter (the magnetization) commutes with the Hamiltonian, protecting long-range order against quantum fluctuations. For the frustration-free chain considered here, the paper proves that no local commuting order parameter exists—the only local conserved quantities are trivial ones. The symmetry breaking must therefore be enabled by the anomalously soft excitations characteristic of gapless frustration-free systems, whose dispersion is quadratic or softer rather than linear as assumed in standard no-go arguments.

The paper also notes that a commuting order parameter does exist beyond the local regime: the ground-space projector $\{F_m\}$8 yields $\{F_m\}$9, which commutes with $f_m$0 but necessarily has support exceeding $f_m$1. The contrast with the ferromagnet is thus sharp: symmetry breaking here is accompanied only by highly nonlocal commuting operators.

Extension to the periodic Motzkin chain

As an independent application, the paper proves absence of $f_m$2-local conserved quantities for $f_m$3 in the area-weighted periodic Motzkin chain, a frustration-free spin-1 chain with projectors onto states $f_m$4, $f_m$5, and $f_m$6. This requires a parallel but distinct analysis because the Motzkin interaction contains cross terms between $f_m$7 and $f_m$8 operators absent from the general Hamiltonian of the main text. The proof verifies the injectivity and 2-local assumptions directly, then shows via Fourier decomposition in momentum sectors that the constraint matrix on candidate coefficients has full rank for all nonzero momenta, while the zero-momentum kernel is one-dimensional and spanned by the Hamiltonian itself. Two independent linear conditions on the remaining free parameters yield a determinant proportional to $f_m$9, nonzero for all $f_m$0, forcing the coefficient of the 3-local part to vanish.

Additionally, the paper constructs explicitly the ground states of the periodic Motzkin chain and proves that it is frustration-free with exactly $f_m$1-fold degenerate ground space, one state per $f_m$2 sector, settling a conjecture of Pronko. The construction uses reversible local moves generating equivalence classes of configurations, showing each $f_m$3 sector forms a single class.

Limitations and open questions

The paper leaves several questions open. The remaining pattern with $f_m$4—the spin-1 XXZ chain with single-ion anisotropy but without biquadratic exchange—is not covered and requires separate analysis. The results apply to periodic boundary conditions; extension to the original (open-boundary) Motzkin chain, whose unique ground state differs from the degenerate periodic case, remains future work, though the authors argue it is natural to expect the same conclusion there. More broadly, a general characterization of continuous symmetry breaking in frustration-free systems lacking a commuting local order parameter is not provided. The classification of conserved quantities beyond $f_m$5, modulo polynomials in $f_m$6 and $f_m$7, is also not addressed.

Conclusion

This work completes the nonintegrability proof for $f_m$8-symmetric spin-1 chains with nearest-neighbor interactions across all patterns of vanishing quadrupolar couplings except one, establishing that the frustration-free chain exhibiting zero-temperature $f_m$9 symmetry breaking in one dimension admits no local conserved quantity beyond the identity, $U(1)$0, and $U(1)$1. The result clarifies that this exception to Coleman's theorem operates through a mechanism fundamentally distinct from the ferromagnetic one, relying on frustration-free spectral structure rather than a conserved order parameter, and provides a second example of this mechanism in the periodic Motzkin chain.