Completeness of the local conserved quantities of the XYZ spin-1/2 chain: the large-support regime and degenerate coupling limits

The periodic XYZ spin-1/2 chain with $J_XJ_YJ_Z\neq0$ possesses an extensive sequence of mutually commuting local conserved quantities $Q_k$ in explicit closed form (Nozawa and Fukai, 2020), and these are unique up to the addition of lower-support conserved quantities only for supports $2\le k\le L/2$. The problem asks for the two missing pieces of the completeness classification: (i) the full classification of $k$-support conserved quantities in the finite-size regime $L/2<k\le L$, where wrap-around of operators on the ring lets commutators of operators of different support cancel each other, so the conditions used in the uniqueness proof become sufficient but not necessary; and (ii) a complete description of all independent local conserved families and their interrelations when one or two coupling constants vanish, where the space of local conserved quantities is known to enlarge. Both components remain open in the literature.

Background

Quantum integrable systems in one dimension are distinguished by an extensive family of mutually commuting conserved quantities that are local, i.e. act only on finitely many consecutive sites. Such local charges underlie exact solvability, they determine the stationary states reached after a quench through the generalized Gibbs ensemble (GGE), a density matrix built from all local conserved quantities, and they supply the continuity equations on which generalized hydrodynamics, the exact hydrodynamic theory of integrable systems constructed from the local charges, is based. For chains solvable by the Bethe ansatz (an exact diagonalization method for interacting one-dimensional quantum models that parameterizes eigenstates by scattering wave functions) the charges are usually generated by expanding the logarithm of the transfer matrix (a commuting family of matrices depending on a spectral parameter) or by iterating a boost operator, an operator whose commutator maps the $n$-th charge to the $(n+1)$-st; but writing down closed-form, fully explicit operator expressions for all charges is a separate and much harder combinatorial problem, solved before 2020 only for the isotropic Heisenberg (XXX) chain.

The XYZ spin-1/2 chain is the most general nearest-neighbor integrable spin chain with periodic boundary conditions, $H=\sum_{i=1}^{L}\left(J_X X_i X_{i+1}+J_Y Y_i Y_{i+1}+J_Z Z_i Z_{i+1}\right),$ where $X_i,Y_i,Z_i$ are Pauli matrices acting on site $i$ and site $L+1$ is identified with site $1$. Nozawa and Fukai (2020) obtained the first closed-form expressions for a complete sequence of local conserved quantities $Q_k$ of this chain at $J_XJ_YJ_Z\neq0$, using a 'doubling product' notation (a way of writing a Pauli string as a product of two-site factors) previously introduced by Shiraishi to prove that adding a magnetic field destroys all local conserved quantities. In their terminology, a $k$-support conserved quantity is a translation-invariant sum of Pauli strings $A^1_iA^2_{i+1}\cdots A^l_{i+l-1}$ of length $l\le k$ whose first and last factors are non-identity Pauli matrices and whose interior factors may be the identity, with at least one string of length exactly $k$, commuting with $H$; $Q_2$ is the Hamiltonian itself. Two $k$-support conserved quantities are identified if they differ by a conserved quantity of strictly smaller maximal support.

Their central structural result is a uniqueness theorem: for $2\le k\le L/2$ and $J_XJ_YJ_Z\neq0$, after fixing normalization the $k$-support conserved quantity is unique up to the addition of lower-support conserved quantities, so the constructed $\{Q_k\}$ exhaust the local charges in that regime; the $Q_k$ mutually commute and remain conserved in the XXZ chain with a longitudinal magnetic field. The restriction $k\le L/2$ is essential to the proof: only then must the $(k+1)$-support pieces of the commutator $[Q_k,H]$ vanish independently, which pins the coefficients one support at a time. For $L/2<k\le L$ the operators in $Q_k$ wrap around the ring, commutators of operators of different support can cancel each other, and the imposed conditions become sufficient but not necessary for $[Q_k,H]=0$; the constructed $Q_k$ are still conserved but may no longer be unique, and no classification of the $k$-support conserved quantities, nor of the linear families arising from such cross-support cancellations, is known in that regime.

Separately, when one or two of the couplings vanish, the space of local conserved quantities is known to enlarge: at $J_Z=0$ the operator $\sum_i(X_iY_{i+1}-Y_iX_{i+1})$ is a 2-support conserved quantity independent of the Hamiltonian, and infinite families of charges for the Ising and XY limits have been known since the 1980s (Grady; Araki; see also Grabowski and Mathieu's analysis of the XXX conservation laws). The Nozawa-Fukai formulas survive these limits after division by the vanishing couplings, but no complete description of the independent conserved families and their interrelations exists there.

The analogous completeness question has since been settled for neighboring models: Nienhuis and Huijgen derived closed-form charges for the closed XXZ chain and conjectured, without proof, that no independent local charges exist beyond them, and Fukai proved completeness for the one-dimensional Hubbard model, uniqueness of the $k$-local conserved quantities up to lower-order additions, while noting that this proof also breaks for $k\ge\lfloor L/2\rfloor$ and that non-interacting (free-fermion) limits possess several independent charge families. The completeness classification of the local conserved quantities of the XYZ chain in the large-support regime and in the degenerate-coupling limits is therefore the remaining gap in this program.

Problem Statement

Within the conventions above (periodic XYZ spin-1/2 chain, $k$-support conserved quantities identified up to the addition of lower-support conserved quantities), determine the complete classification of local conserved quantities in the two regimes left open by Nozawa and Fukai, whose common objective is to decide, for arbitrary chain length $L$ and all values of the couplings, whether the explicitly constructed local conserved quantities of the XYZ chain are all of them.

1. Large-support (finite-size) regime. For $J_XJ_YJ_Z\neq0$ and every chain length $L$, determine the space of $k$-support conserved quantities for $L/2<k\le L$: either prove that the constructed $Q_k$, together with lower-support conserved quantities, exhaust it (so that uniqueness extends beyond $k\le L/2$), or determine all additional linearly independent $k$-support conserved quantities arising from cancellations between commutators of operators of different support, giving explicit expressions and a basis of the solution space for each $k$.

2. Degenerate-coupling limits. For the limits in which exactly one of $J_X,J_Y,J_Z$ is zero and those in which exactly two are zero, give a complete description of all independent local conserved families for arbitrary $L$: their explicit forms, the dimension of the space of $l$-support conserved quantities for each $l$, the position of the Nozawa-Fukai charges (after division by the vanishing couplings) inside this space, and the linear interrelations among the families, including the previously known Ising/XY-limit charges such as $\sum_i(X_iY_{i+1}-Y_iX_{i+1})$ at $J_Z=0$.

A complete answer covers both components, since each is a piece of the same completeness statement for the same chain: no coupling values and no support sizes $2\le k\le L$ may be left unclassified.

References

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