[Paper Review] Retrieve the Bethe states of quantum integrable models solved via off-diagonal Bethe Ansatz

Xin Zhang, Yuanyuan Li+4 more|arXiv (Cornell University)|Jul 20, 2014

Algebraic structures and combinatorial models4 citations

TL;DR

This paper presents a systematic method to retrieve Bethe-type eigenstates for quantum integrable models without U(1) symmetry, using the inhomogeneous T-Q relation from the off-diagonal Bethe Ansatz (ODBA). By constructing a reference state from the T-Q relation and employing an orthogonal Hilbert space basis, the authors derive well-defined homogeneous limits of Bethe states for the XXZ spin torus and open XXX spin-1/2 chains, resolving a key gap in ODBA framework by enabling full eigenstate reconstruction from spectral data.

ABSTRACT

Based on the inhomogeneous T-Q relation constructed via the off-diagonal Bethe Ansatz, a systematic method for retrieving the Bethe-type eigenstates of integrable models without obvious reference state is developed by employing certain orthogonal basis of the Hilbert space. With the XXZ spin torus model and the open XXX spin-1/2 chain as examples, we show that for a given inhomogeneous T-Q relation and the associated Bethe Ansatz equations, the constructed Bethe-type eigenstate has a well-defined homogeneous limit.

Motivation & Objective

• To address the open problem of reconstructing Bethe-type eigenstates from the spectrum obtained via off-diagonal Bethe Ansatz (ODBA), particularly for models lacking U(1) symmetry.
• To develop a systematic procedure that retrieves eigenstates from the inhomogeneous T-Q relation, which is derived without prior knowledge of the reference state.
• To demonstrate that the constructed Bethe states possess a well-defined homogeneous limit, crucial for connecting to standard integrable models.
• To validate the method on two archetype models: the XXZ spin torus and the open XXX spin-1/2 chain with generic boundary fields.

Proposed method

• Constructs a complete orthogonal basis (both left and right) of the Hilbert space to facilitate state reconstruction.
• Uses the inhomogeneous T-Q relation as the central input, derived via ODBA, to infer the underlying reference state that is otherwise unknown in non-U(1) symmetric models.
• Applies operator product identities to derive the T-Q relation, which encodes the spectrum and enables eigenstate reconstruction.
• Employs the reference state to build Bethe-type eigenstates through iterative application of creation operators in the orthogonal basis.
• Ensures consistency by verifying that the constructed eigenstates satisfy the original T-Q relation and yield the correct eigenvalues.
• Demonstrates the homogeneous limit by taking the inhomogeneous parameters to zero, showing convergence to standard Bethe states.

Experimental results

Research questions

• RQ1 How can Bethe-type eigenstates be systematically retrieved from the inhomogeneous T-Q relation in integrable models without U(1) symmetry?
• RQ2 What is the role of the reference state in the reconstruction of eigenstates when it is not explicitly known in models like the XXZ spin torus or open XXX chain?
• RQ3 Can the constructed Bethe states from the ODBA framework be consistently taken to the homogeneous limit, and does this limit recover known results?
• RQ4 How does the orthogonal basis of the Hilbert space facilitate the reconstruction of eigenstates from spectral data alone?
• RQ5 Is the conjectured form of the open XXX spin-1/2 chain eigenstate (based on T-Q relation) provably correct using this method?

Key findings

• The method successfully retrieves Bethe-type eigenstates for the XXZ spin torus model using only the inhomogeneous T-Q relation and an orthogonal Hilbert space basis.
• The constructed eigenstates for the XXZ spin torus model have a well-defined homogeneous limit, confirming consistency with the standard integrable model.
• For the open XXX spin-1/2 chain with generic boundary fields, the method constructs eigenstates that satisfy the T-Q relation and exhibit the correct homogeneous limit.
• The reference state is reconstructed from the T-Q relation, resolving a key obstacle in ODBA where such states are typically unknown a priori.
• The scalar product between off-shell states and the constructed eigenstate is consistent with the T-Q relation, validating the method’s correctness.
• The approach provides a rigorous proof of the conjecture on the open XXX spin-1/2 chain eigenstate, confirming its validity via the homogeneous limit.