[Paper Review] Boundary non-local charges from the open spin chain

Anastasia Doikou|arXiv (Cornell University)|Feb 24, 2004

Spectral Theory in Mathematical Physics5 citations

TL;DR

This paper derives boundary non-local charges from the open XXZ spin chain with non-diagonal and special diagonal boundaries, showing they commute with the transfer matrix and, crucially, with all generators of the blob algebra in the XXZ representation. The key contribution is establishing algebraic commutation relations that reveal hidden symmetries in integrable open quantum spin chains, extending field-theoretic non-local charges to lattice models via algebraic Bethe ansatz techniques and reflection algebra structures.

ABSTRACT

The $N$ site open XXZ quantum spin chain with a right non-diagonal boundary and special diagonal left boundary is considered. The boundary non-local charges originally obtained from a field theoretical viewpoint, for the sine Gordon model on the half line, are recovered from the spin chain point of view. Furthermore, the symmetry of the open spin chain is exhibited. More specifically, we show that certain non--local charges commute with the transfer matrix of the open spin chain, depending on the choice of boundary conditions. In addition, we show explicitly that for a special choice of the left boundary, one of the non--local charges, in a particular representation, commutes with each one of the generators of the blob algebra, and hence with the corresponding local Hamiltonian.

Motivation & Objective

• To recover boundary non-local charges—originally derived in field theory for the sine-Gordon model on the half-line—within the framework of the open XXZ spin chain.
• To demonstrate that certain non-local charges commute with the transfer matrix of the open spin chain, depending on the choice of left boundary conditions.
• To show that, for a specific diagonal left boundary, one non-local charge commutes with all generators of the blob algebra in the XXZ representation.
• To establish that the corresponding local Hamiltonian commutes with the boundary non-local charge, thereby revealing a new symmetry structure.
• To generalize the method to higher-rank algebras and provide a foundation for studying non-local charges in models with non-diagonal boundaries not yet analyzed in field theory.

Proposed method

• Construct the open XXZ spin chain using the generalized quantum inverse scattering method (QISM), employing R-matrices and K-matrices satisfying the Yang-Baxter and reflection equations.
• Use the asymptotic behavior of the monodromy matrix T(λ) as λ → ∞ to extract boundary non-local charges via the coproduct of generators of the reflection algebra.
• Employ linear intertwining relations between the transfer matrix and non-local charges to derive algebraic exchange relations that ensure commutativity under specific boundary conditions.
• Utilize the evaluation representation of Uq(ŝl2) to map algebraic structures to the XXZ spin chain, enabling explicit computation of commutation relations.
• Prove that the non-local charge Q(N)₁ commutes with all blob algebra generators h(Ul) by verifying the commutation relations using the representation (2.12) and the structure of the R-matrix.
• Express the open spin chain Hamiltonian in terms of blob algebra generators and verify that it commutes with the non-local charge Q(N)₁ via the derived algebraic relations.

Experimental results

Research questions

• RQ1 Can boundary non-local charges from the sine-Gordon model on the half-line be recovered from a lattice spin chain model?
• RQ2 Under what boundary conditions do non-local charges commute with the transfer matrix of the open XXZ spin chain?
• RQ3 Does a specific non-local charge commute with all generators of the blob algebra in the XXZ representation?
• RQ4 Can the local Hamiltonian of the open spin chain be shown to commute with the boundary non-local charge through algebraic means?
• RQ5 Can this method be generalized to models associated with higher-rank quantum groups like Uq(ŝln)?

Key findings

• Boundary non-local charges derived from the asymptotic expansion of the monodromy matrix T(λ) as λ → ∞ are shown to commute with the transfer matrix of the open XXZ spin chain for a specific choice of left boundary.
• For a special diagonal left boundary, the non-local charge Q(N)₁ commutes with each generator h(Ul) of the blob algebra in the XXZ representation, as proven via direct algebraic computation using the representation (2.12).
• The local Hamiltonian H, expressed in terms of blob algebra generators, commutes with the non-local charge Q(N)₁, as a consequence of the commutation relations (4.27) and (4.30).
• The commutation relations (4.27) are established algebraically and are independent of the choice of representation, making them robust across different realizations.
• The method provides a systematic algebraic framework to derive and verify symmetries in open integrable spin chains with non-diagonal boundaries.
• The results suggest a generalization to higher-rank algebras such as Uq(ŝln), where similar non-local charges may be constructed from solutions of the reflection equation via the Hecke algebraic approach.