Equivalence of the three integrability notions for matrix product states, the reducible/non-invertible boundary intertwiners, and the spectra of the double-row quantum transfer matrices

The boundary-integrability approach to integrable matrix product states (MPS) of Pozsgay, Piroli and Vernier relates three a priori distinct notions for one-site invariant MPS of the rational $\mathfrak{gl}_N$- and $\mathfrak{o}_N$-symmetric spin chains treated there: (A) the integrability condition $t(u)|\Psi\rangle=\tilde t(u)|\Psi\rangle$; (B) the linear "square-root relation" intertwining the two transfer-matrix dressings of the MPS through an operator-valued matrix $K(u)$; and (C) the twisted boundary Yang-Baxter equation for $K(u)$, i.e. the statement that $K(u)$ is a representation of a twisted Yangian. The source proved the implications only under irreducibility or complete-reducibility assumptions on the dressed MPS (verified numerically in a few examples), while the scalar reference state satisfies all three conditions without complete reducibility, leaving open under which minimal hypotheses the three notions coincide. The overlap formulas that motivated part of the program have since been derived by independent algebraic-Bethe-ansatz methods based on the KT-relation (the intertwining of $K(u)$ with the monodromy matrix), but those works assume the reflection-algebra structure and the irreducibility of the $K$-matrix. The surviving open core is: the minimal-hypotheses equivalence theorem; analytic irreducibility criteria for dressed and doubly dressed MPS; the classification and boundary-algebra interpretation of the reducible, non-invertible intertwiners; the explicit fusion hierarchy and decomposition into irreducible twisted-Yangian representations; and the spectra and eigenstates of the double-row quantum transfer matrices built with the operator-valued $K$-matrices.

Background

Quantum integrable spin chains in one dimension, such as the XXX Heisenberg chain and its rational $\mathfrak{gl}_N$-, $\mathfrak{o}_N$- and $\mathfrak{sp}_N$-symmetric generalizations, possess an infinite family of commuting conserved charges generated by a transfer matrix $t(u)=\mathrm{Tr}\,T(u)$, where the monodromy matrix $T(u)$ obeys the RTT relation $R_{12}(u-v)\,T_1(u)T_2(v)=T_2(v)T_1(u)\,R_{12}(u-v)$ with an $R$-matrix solving the Yang-Baxter equation. Matrix product states (MPS) are many-body states $|\Psi\rangle=\sum \mathrm{Tr}\,\omega_{i_L}\cdots\omega_{i_1}|i_1\cdots i_L\rangle$ built from a finite set of matrices $\{\omega_j\}$ acting on an auxiliary "bond" space; one-site invariant MPS repeat the same tensor at every site and include the Néel-type and Clifford-algebra states relevant for quench dynamics, as well as the states encoding one-point functions of defect operators in the AdS/dCFT correspondence (dCFT: defect conformal field theory, a conformal field theory on a space with a codimension-one defect).

For quantum quenches, the operative notion is an "integrable" initial state: a state annihilated by all parity-odd conserved charges, $Q_{2k+1}|\Psi\rangle=0$, which for two-site shift-invariant states (one-site invariant MPS in particular) is exactly equivalent in every finite volume to the eigenvalue relation $t(u)|\Psi\rangle=\tilde t(u)|\Psi\rangle$, where $\tilde t(u)$ is the transfer matrix of the reflected monodromy (Piroli, Pozsgay and Vernier). Such states have non-vanishing overlaps only with Bethe eigenstates whose rapidities come in pairs $\{u,-u\}$ ("pair structure"), which is what makes their post-quench steady states accessible through the Quench Action method (a variational principle that fixes the post-quench steady state from the overlaps with the Bethe eigenstates).

The second ingredient is boundary integrability. An integrable boundary of a spin chain or a two-dimensional integrable quantum field theory is described by a $K$-matrix solving the (twisted) boundary Yang-Baxter (BYB) equation, the reflection equation introduced by Sklyanin and by Ghoshal and Zamolodchikov; a $K$-matrix guarantees commutativity of Sklyanin double-row transfer matrices. The twisted BYB (built with a transposed $R$-matrix, corresponding to "soliton non-preserving" boundaries that scatter each particle into its conjugate representation) is solved by $K$-matrices forming representations of twisted Yangians $Y(\mathfrak{g},\mathfrak{h})$, the coideal subalgebras of the Yangian of $\mathfrak{gl}_N$ attached to a symmetric pair $(\mathfrak{g},\mathfrak{h})$ of Lie algebras such as $(\mathfrak{sl}_N,\mathfrak{so}_N)$; the finite-dimensional irreducible representations of twisted Yangians were classified by Molev through a projection ("dressing") construction.

Pozsgay, Piroli and Vernier connected the two pictures. Dressing the one-site blocks $\omega_j$ by the Lax operators of the direct and the reflected monodromies produces two sets of matrices $A_j$ and $B_j$; the integrability condition states that the two resulting dressed MPS coincide, and if the dressed MPS are completely reducible this yields an invertible intertwiner $K(u)$ with $A_jK(u)=K(u)B_j$, called the "square-root relation" (sq.r.r.) because it amounts to half of the steps of the twisted BYB. The sq.r.r. with invertible $K(u)$ implies the integrability condition, and if the doubly dressed MPS are irreducible then $K(u)$ also solves the twisted BYB. The three characterizations of integrable MPS are therefore logically linked only under irreducibility assumptions, which the source verified numerically in concrete examples; the scalar (bond-dimension-one) reference state of the $SU(N)$-invariant chain satisfies all three conditions without complete reducibility, with a non-invertible $K$. Finally, the twisted BYB permits the construction of commuting double-row transfer matrices, called Quantum Transfer Matrices (QTMs) in the source (transfer matrices of auxiliary two-dimensional models that encode the thermodynamics of the quantum chain), which carry the operator-valued $K(u)$ as a boundary condition acting on the internal space and which underlie partition functions, Loschmidt amplitudes (the return amplitudes of the post-quench time evolution) and thermodynamic quantities for quenches from these MPS.

Problem Statement

The problem is to replace the case-by-case sufficient conditions established by Pozsgay, Piroli and Vernier with a complete, minimal and operative theory of the boundary-integrability characterization of integrable matrix product states. Its linked deliverables, as posed in the source's Conclusions, are:

(1) Equivalence theorem. Determine the precise, minimal hypotheses under which the three notions - (A) the integrability condition $t(u)|\Psi\rangle=\tilde t(u)|\Psi\rangle$ (equivalently $Q_{2k+1}|\Psi\rangle=0$), (B) the square-root relation $A_jK(u)=K(u)B_j$, i.e. the existence of a non-zero operator-valued intertwiner $K(u)$ between the dressed matrices $A_j$ and $B_j$ (the invertibility of $K(u)$, which complete reducibility of the dressed MPS guarantees in the cases treated by the source, is one of the hypotheses whose necessity is to be determined and is not part of the definition of (B)), and (C) the twisted boundary Yang-Baxter equation for $K(u)$ - are logically equivalent for one-site invariant MPS of the rational $\mathfrak{gl}_N$- and $\mathfrak{o}_N$-symmetric spin chains treated in the source (the analogous questions for the symplectic, nested, graded and trigonometric deformations are natural extensions that remain equally open, but they are not additional deliverables). In particular: can the irreducibility and complete-reducibility assumptions be weakened, given that the scalar reference state of the $SU(N)$-invariant chain satisfies all three conditions without complete reducibility? Does every MPS satisfying the integrability condition admit an operator-valued $K(u)$ solving the square-root relation? Does every solution of the twisted BYB that factorizes at $u=0$ (initial condition $\psi_{jk}(0)=\omega_j\omega_k$ up to a scalar factor, on the components of $K(u)$) also solve the square-root relation? Completing this item requires the full if-and-only-if characterization: a set of hypotheses under which each of (A), (B) and (C) implies the others, with every hypothesis necessary in the sense that no proper subset of the stated hypotheses suffices; the three directional questions just listed give acceptable partial resolutions of (1) but do not by themselves complete it.

(2) Analytic irreducibility. Prove analytic criteria, in place of the numerical spot checks of the source, that establish for given one-site blocks $\omega_j$ whether the dressed and doubly dressed MPS are irreducible for generic spectral parameters.

(3) Reducible and non-invertible cases. Classify the cases with invariant subspaces and non-invertible intertwiners, determine their boundary-algebra meaning, and develop the related general theory of the reducibility of fused representations of twisted Yangians, for which no general results exist. Completing this item requires: (i) the classification of the reducible and non-invertible intertwiners for the one-site invariant MPS of the rational $\mathfrak{gl}_N$- and $\mathfrak{o}_N$-symmetric chains treated in the source, i.e. for the block families those chains give rise to; and (ii) a necessary-and-sufficient reducibility criterion for the fused twisted-Yangian representations (what makes such a representation reducible, and what its invariant subspaces and irreducible constituents are), together with the boundary-algebra interpretation of the resulting non-invertible intertwiners. Classifying isolated families, or covering only special cases, does not complete it.

(4) Fusion hierarchy and decomposition. Determine the explicit fusion hierarchy of the solutions and the explicit decomposition of the dressed solutions into irreducible twisted-Yangian representations, including the concrete projector $\Pi$ of eq. (4.15) of the source.

(5) Double-row QTM spectra. Compute the spectrum and the eigenstates of the double-row Quantum Transfer Matrices constructed with the new operator-valued boundary $K$-matrices, which define new integrable models with boundary degrees of freedom and underlie thermodynamic and steady-state computations for quenches from these MPS.

Completion standard: items (1)-(5) are independent, separately completable sub-deliverables, each with the completion boundary stated in the corresponding item, and none subsumes another; a resolution of any one of them is a valid partial resolution, and the task as posed is fully resolved only when all five are addressed. The three sub-questions embedded in (1) are facets of the single equivalence theorem (1), not separate deliverables.

Deliverables (2)-(4) are the representation-theoretic substance on which the equivalence theorem (1) rests, and (5) is its constructive payoff: once $K(u)$ is known to satisfy the twisted BYB under minimal hypotheses, the double-row transfer matrices are guaranteed to commute and to define exactly solvable boundary models. The physical overlap formulas that motivated part of this program have meanwhile been obtained by independent algebraic-Bethe-ansatz methods and do not by themselves answer any of the five items.

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