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arXiv:2601.09245

Matrix product operator representations for the local conserved quantities of the spin-$1/2$ XYZ chain

Kohei Fukai, Kyoichi Yamada

Published Jan 14, 2026• 0 citations
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Abstract

We present explicit matrix product operator (MPO) representations for the local conserved quantities of the spin-$1/2$ XYZ chain. Through these MPO representations, we simplify the coefficients appearing in the local conserved quantities originally derived by one of the authors, and reveal their combinatorial meaning: the coefficients prove to be a polynomial generalization of the Catalan numbers, given by weighted sums over monotone lattice paths. We also prove that a single recurrence relation for the coefficients guarantees the conservation law. We further show that these MPO representations can also be recovered from a product of two eight-vertex transfer matrices built from Baxter's R-matrix, providing the first direct connection between the R-matrix formalism and closed-form expressions for all local conserved quantities. We also obtain a simple Lax operator for the XYZ chain that, unlike Baxter's R-matrix, involves no elliptic functions.

DOI
10.48550/arxiv.2601.09245

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Matrix product operator representations for the local conserved quantities of the spin-1/21/21/2 XYZ chain
We present explicit matrix product operator (MPO) representations for the local conserved quantities of the spin-1/21/21/2 XYZ chain. Through these MPO representations, we simplify the coefficients appearing in the local conserved quantities originally derived by one of the authors, and reveal their combinatorial meaning: the coefficients prove to be a polynomial generalization of the Catalan numbers, given by weighted sums over monotone lattice paths. We also prove that a single recurrence relation for the coefficients guarantees the conservation law. We further show that these MPO representations can also be recovered from a product of two eight-vertex transfer matrices built from Baxter's R-matrix, providing the first direct connection between the R-matrix formalism and closed-form expressions for all local conserved quantities. We also obtain a simple Lax operator for the XYZ chain that, unlike Baxter's R-matrix, involves no elliptic functions.
Matrix-product operators (MPOs) appear throughout the study of integrable lattice models, notably as the transfer matrices. They can also be used as transformations to construct dualities between such models, both invertible (including unitary) and non-invertible (including discrete gauging). We analyse how the local Yang--Baxter integrable structures are modified under such dualities. We see that the Rˇ\check{R}Rˇ-matrix, that appears in the baxterization approach to integrability, transforms in a simple manner. We further argue for a broad class of MPO dualities that the usual Yang--Baxter RRR-matrix should be extended. This extended RRR-matrix satisfies modified algebraic relations, including a modified Yang--Baxter algebra previously identified in the unitary case, that altogether give a local integrable structure underlying the commuting transfer matrices of the dual model. We illustrate these results with two case studies, analysing an invertible unitary MPO and a non-invertible MPO both applied to the canonical XXZ spin chain. The former is the cluster entangler, arising in the study of symmetry-protected topological phases, while the latter is the Kramers--Wannier duality. We show several results for MPOs with exact MPO inverses that are of independent interest.
Abstract Sutherland showed that the XYZ quantum spin-chain Hamiltonian commutes with the eight-vertex model transfer matrix, so that Baxter’s subsequent tour de force proves the integrability of both. The proof requires parametrising the Boltzmann weights using elliptic theta functions and showing they satisfy the Yang-Baxter equation. We here give a simpler derivation of the integrability of the XYZ chain by explicitly constructing an extensive sequence of conserved charges from a matrix-product operator. We show that they commute with the XYZ Hamiltonian with periodic boundary conditions or an arbitrary boundary magnetic field. A straightforward generalisation yields impurity interactions that preserve the integrability. Placing such an impurity at the edge gives an integrable generalisation of the Kondo problem with a gapped bulk. We make contact with the traditional approach by relating our matrix-product operator to products of the eight-vertex model transfer matrix.
Integrable models are characterized by the existence of an extensive number of local conserved quantities.Their existence is closely related to various exact methods, such as the Bethe ansatz and quantum inverse scattering.They also play an important role in nonequilibrium dynamics in the integrable models.Their steady states and transport phenomena exhibit exotic properties, which are also related to this point.Local conserved quantities in a quantum integrable model in one dimension are generated from the transfer matrix of the corresponding classical two-dimensional model.However, it is still non-trivial to obtain their explicit expression.The meaning of "local" here is that each conserved quantity consists of the sum of observables acting on a finite range of sites called its support.We note that the local conserved quantities in this context are not O (1)-body observables in general.Their explicit expressions enable us to analyze theories of nonequilibrium dynamics using the local conserved quantities in detail, for example, the generalized Gibbs ensemble, the Mazur bound, and the generalized hydrodynamics.In this thesis, we have studied the local conserved quantities in the XYZ spin-1/2 chain, which is an integrable model corresponding to the classical eight-vertex model.We have obtained their explicit expression for arbitrary coupling constants of the XYZ chain.The model at the isotropic point called the Heisenberg or XXX chain is an exceptional case in which the expression had been known.We have checked that our results reproduce the known expression for the XXX chain as a special case.To obtain the explicit expression, we have employed the notation of doubling-product operators, which was introduced by Shiraishi to study the non-integrability of the XYZ spin-1/2 chain in a magnetic field.Using this notation, we have found recursion relations of the coefficients among different terms in the conserved quantities in the XYZ chain.Then, we have solved the recursion relations and obtained the explicit expression.We have shown that the local conserved quantity Q k for a fixed support k is essentially unique i ii ABSTRACT if 2 ≤ k ≤ L/2 with L being the number of sites.We have obtained the explicit expression for all {Q k } for 1 ≤ k ≤ L/2 and one Q k for each k with L/2 < k ≤ L.We have also studied the currents of these conserved quantities based on the respective continuity equations.We have obtained their explicit expression and numerically investigated their time evolution.In particular, we have studied the effects of an integrabilitybreaking perturbation on the time evolution of the currents and found that this term drastically changes their long-time behaviors as follows.As for the initial state, we have chosen several energy eigenstates of the XXX chain with a nonzero energy current, which is also known to be conserved.In particular, we have studied the time evolution with the Hamiltonian of the XXZ chain with and without a next-nearest-neighbor interaction and seen the effects of this integrability-breaking perturbation.We have found that, for these initial states, the long-time average of the total currents is nonzero for the integrable cases while zero for the non-integrable cases.We have also studied this behavior analytically and shown that the degeneracy of the energy eigenstates is essential.Our results for the XYZ spin-1/2 chain have revealed the structure of the conserved quantities and currents in detail.They provide materials for the development of algebraic studies in integrable models and will be useful for detailed analyses of nonequilibrium dynamics in these models.
We extend T. Prosen's construction of quasilocal conserved quantities for the XXZ model [Phys. Rev. Lett. 106, 217206 (2011)] to the case of periodic boundary conditions. These quasilocal operators stem from a two-parameter transfer matrix which employs a highest-weight representation of the quantum group algebra inherent in the Yang-Baxter algebra. In contrast with the open chain, where the conservation law is weakly violated by boundary terms, the quasilocal operators in the periodic chain exactly commute with the Hamiltonian and other local conserved quantities.