Piroli, Pozsgay and Vernier (2017) defined an initial state of an integrable lattice model to be integrable if it is annihilated by all local conserved charges odd under space reflection, and proved that this class contains every state built from boundary K-matrices (solutions of the boundary Yang-Baxter reflection equations) via the rotated-channel construction, including matrix product states of arbitrary bond dimension generated by the action of fused transfer matrices. The problem is to complete this classification: prove, or disprove by an explicit finite-bond-dimension counterexample, that the boundary construction exhausts all integrable matrix product states (equivalently, that the annihilation conditions imply the boundary Yang-Baxter-type equations, currently known only under an unproven irreducibility condition); and classify all finite-dimensional operator-valued solutions of these equations for trigonometric (XXZ) and higher-spin chains. Later literature resolved the nested SU(3) boundary interpretation and the factorized-overlap formulas for rational chains, while the exhaustiveness, trigonometric and higher-spin classification questions remain open.

Background

In a global quantum quench, a many-body system is prepared in a simple, far-from-equilibrium initial state and then evolves unitarily under a Hamiltonian whose dynamics is exactly tractable. The natural laboratory is one-dimensional integrable lattice models such as the XXZ spin chain, whose algebraic Bethe ansatz structure provides a commuting family of transfer matrices (traces, over all lattice sites, of products of the two-particle scattering matrix, the RR-matrix) generating local conserved charges QnQ_n; the eigenstates (Bethe states) are parametrized by sets of rapidities (quasi-momenta) {λj}\{\lambda_j\}. The same framework covers the XXX, XXZ and XYZ chains, called rational, trigonometric and elliptic according to whether the RR-matrix depends on the rapidity rationally, trigonometrically or through elliptic (doubly periodic) functions. Because of these extensive conserved charges the system does not thermalize, and the exact late-time steady state can be computed by the quench action method, whose essential input is the overlap between the initial state and the Bethe eigenstates. Before 2017, exact quench-action computations were restricted to a handful of initial states for which such overlaps could be guessed and verified.

Piroli, Pozsgay and Vernier introduced a definition unifying these cases: an initial state is integrable if it is annihilated by all local conserved charges that are odd under space reflection (parity reversal of the chain). They proved that this class contains every state related to an integrable boundary condition in a rotated channel, the lattice analogue of the Ghoshal-Zamolodchikov boundary states of integrable quantum field theory, in which the Euclidean time direction is exchanged with space. On the lattice, integrable boundaries are described by KK-matrices solving the boundary Yang-Baxter (reflection) equation; when the RR-matrix satisfies a crossing relation, a KK-matrix produces a two-site block whose repetition along the chain is an integrable initial state. Acting on such boundary states with fused transfer matrices, whose auxiliary space is a fused (higher-spin) representation, generates integrable matrix product states (MPS), i.e. states of the form tr∏jAj\mathrm{tr}\prod_j A_j with finite-dimensional auxiliary matrices AjA_j (the auxiliary dimension is the bond dimension), of arbitrary bond dimension. For the spin-1/2 XXZ chain every two-site state arises this way and is integrable; for spin s≥1s\ge 1 only a subset does, because the fused KK-matrices carry only three free parameters, too few to parametrize the (2s+1)2(2s+1)^2-dimensional space of two-site blocks. In nested higher-rank models such as the SU(3)SU(3)-invariant Lai-Sutherland chain, solved by a nested Bethe ansatz with two quasiparticle species, the RR-matrix lacks the ordinary crossing relation; the boundary construction then passes to twisted (soliton-non-preserving) reflection equations, whose operator-valued solutions are representations of twisted Yangians, which are deformations of the Yangian symmetry algebra attached to symmetric pairs of Lie algebras such as (SU(N),SO(N))(SU(N),SO(N)). A direct sequel to the 2017 work showed that for matrix product states with a single repeating block the annihilation conditions are equivalent to a linear intertwining relation for the two-site block, called the square root relation because iterating it twice yields the quadratic reflection equation; the square root relation implies the (twisted) reflection equations whenever the dressed matrix product state, obtained by acting with the fused transfer matrices, is irreducible, meaning that its auxiliary-space matrices admit no common proper invariant subspace. This irreducibility condition has so far only been verified numerically, and the equivalence without it is not established.

In every case where exact overlaps are known they are factorized: they vanish unless the Bethe rapidities form opposite pairs {λ,−λ}\{\lambda,-\lambda\} (the pair structure, with exceptions at special non-generic points such as the XX chain at vanishing anisotropy), and are then given by products of one-particle factors and a ratio of Gaudin determinants, the determinants built from the derivatives of the Bethe equations that also give the norms of Bethe states. The 2017 paper left open whether its boundary construction produces all integrable states or only a proper subclass, and how much of this overlap structure follows from the definition itself.

Problem Statement

The problem is to complete the classification of integrable initial states of Bethe-ansatz integrable lattice models, where 'integrable' means annihilated by all local conserved charges odd under space reflection. The common objective is a complete characterization of this class of states; the requested results are its two linked layers, exactly as split by the source paper into a necessity part and a classification part.

  1. Exhaustiveness and necessity. Prove, or disprove by an explicit finite-bond-dimension counterexample, that every translationally invariant matrix product state with finite bond dimension annihilated by all odd charges is generated from boundary two-site blocks by the action of (fused) transfer matrices; equivalently, that the annihilation conditions imply the boundary Yang-Baxter-type equations, which in the rational case means the linear square-root intertwiner relation and the twisted reflection equation. The currently known implication assumes an irreducibility condition on the dressed matrix product states that has only been verified numerically; an analytic proof of the implication without the irreducibility assumption, or a delimitation of the class where exhaustiveness fails that exhibits an explicit finite-bond-dimension counterexample, is required.

  2. Classification of solutions. Enumerate all finite-dimensional operator-valued solutions of the boundary Yang-Baxter-type equations that produce integrable matrix product states. The rational case is organized by the twisted-Yangian symmetric pairs such as (glN,oN)(gl_N,o_N), (glN,spN)(gl_N,sp_N) and (glN,glM⊕glN−M)(gl_N,gl_M\oplus gl_{N-M}) (with oNo_N and spNsp_N the orthogonal and symplectic Lie algebras, and the pairs referring to the symmetric pairs introduced above); a corresponding classification for trigonometric (XXZ) R-matrices, and a complete classification of the integrable two-site blocks for spin s≥1s\ge 1 chains (where the three fused-K parameters cover only a proper subset of the block space), does not exist.

A complete answer consists of the necessity theorem (or explicit counterexample) together with the classification of all finite-dimensional operator-valued solutions of the boundary Yang-Baxter-type equations; the two items are successive layers of one and the same question, namely what the annihilation definition of an integrable quench actually comprises. The factorized-overlap and pair-structure material described in the Background is a source-discussed consequence of this definition and of later work on it, not a further requested deliverable.

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