Solving for the integrable boundary states of the ABJM spin chain from $KT$-relations

Technical Summary: Solving for the integrable boundary states of the ABJM spin chain from KT-relations

Problem Statement
The paper addresses the construction of integrable boundary states in the alternating $SU(4)$ spin chain, which describes the planar scalar sector of the three-dimensional $\mathcal{N}=6$ superconformal Chern–Simons-matter (ABJM) theory. While integrable boundary states are well-understood in homogeneous spin chains (defined by the annihilation of odd conserved charges), the alternating structure of the ABJM chain necessitates a formulation based on relations between transfer matrices. Specifically, the authors focus on finding states $|\Psi\rangle$ that satisfy either chiral or achiral integrability conditions, defined by specific relations between the transfer matrices $\tau(u)$ and $\bar{\tau}(u)$ at crossed spectral parameters. The central challenge is to solve the underlying algebraic constraints, known as the KT-relations, for boundary states with $n$-site translational invariance, where the state is built from an elementary $n$-site block.

Methodology
The authors adopt a direct algebraic approach based on the KT-relation, avoiding the assumption that the auxiliary matrix $K(u)$ must satisfy a reflection equation a priori. Instead, they treat the boundary block (denoted as $|\Omega\rangle$, $|\phi\rangle$, or $|\Theta\rangle$ for $n=1, 2, 3$ respectively) and the matrix $K(u)$ as unknowns in a coupled system.

The methodology proceeds as follows:

1. Formulation: The KT-relation is reduced to a fundamental mixed state-operator equation involving the elementary $n$-site block and the matrix $K(u)$.
2. Parity Analysis: The solution strategy is bifurcated based on the parity of $n$:
   • Odd $n$ (1-site, 3-site): The constraints are converted into pure state equations on the quantum space. The authors analyze these equations for both operator-valued solutions (involving an internal space) and $c$-number solutions.
   • Even $n$ (2-site, 4-site): The constraints are reduced to pure operator equations on the space of matrices.
3. Algebraic Techniques: The authors employ group-theoretical arguments, rank analyses of matrix equations at special spectral parameter values (e.g., $u=0, u=-2, u \to \infty$), and the classification of Lie algebra representations (specifically $so(4)$, $sp(4)$, and $sl(4)$) to solve the resulting equations.
4. Verification: For even $n$, the authors check whether the derived $K(u)$ coincides with known solutions of Soliton-Preserving (SP) or Soliton-Non-Preserving (SNP) reflection equations, though they do not assume this initially.

Key Contributions and Results

• 1-Site States:

  • Chiral: The authors construct non-trivial operator-valued solutions where the 1-site block is generated by Clifford algebra generators. The corresponding $K(u)$ is also constructed from these generators and is shown to be invertible.
  • Achiral: It is proven that no non-trivial $c$-number integrable boundary states exist for the achiral 1-site case.

• 2-Site States:

  • Chiral: The analysis demonstrates that only the trivial solution exists for chiral 2-site states.
  • Achiral: Non-trivial $c$-number integrable pairs $(M, K(u))$ are found. The matrix $M$ determines the 2-site block, and $K(u)$ is fixed by $M$. The resulting $K(u)$ is shown to necessarily satisfy the SP-type reflection equation.

• 3-Site States:

  • Chiral: When restricting to $c$-number solutions (without internal spaces) and assuming $K(u)$ satisfies the SNP-type reflection equation, the authors prove that the only solution is the trivial zero state.
  • Achiral: Similarly, when $K(u)$ is chosen from SP-type reflection equation solutions, no non-trivial $c$-number integrable boundary states exist for the 3-site case.

• 4-Site States:

  • Chiral: The solutions are classified based on the constant term $K(0)$ in the Laurent expansion of $K(u)$.
    • In the $so(4)$ case ($(K(0))^t = K(0)$), $K(u)$ is an invertible symmetric constant matrix, and the block is constructed via a specific transformation involving $K(0)$.
    • In the $sp(4)$ case ($(K(0))^t = -K(0)$), $K(u)$ is an invertible antisymmetric constant matrix, yielding a corresponding non-trivial block.
    • In the $sl(4)$ case (generic $K(0)$), only the zero solution exists.
  • Achiral: The 4-site solutions are shown to factorize essentially into the tensor product of two 2-site achiral solutions, reducing the problem to the previously solved 2-site case.

Significance and Claims
The paper claims that its direct approach from the KT-relation offers a broader class of solutions than those obtained solely from reflection equations or their fused counterparts. A primary advantage highlighted is the ability to construct odd-site translationally invariant integrable states (specifically the 1-site chiral case with operator-valued solutions) that cannot be generated by the standard reflection equation method.

The authors note that their results for the 4-site chiral states coincide with those obtained from the fusion construction of reflection matrices in previous literature, serving as a consistency check. They explicitly state that for $n > 1$, they restricted their analysis to $c$-number solutions, leaving the exploration of operator-valued pairs for future work. The paper concludes by suggesting that the constructed states are relevant for computing overlaps with Bethe eigenstates, which are necessary for determining one-point and higher-point correlation functions in defect setups of ABJM theory, and for understanding the effective field-theory interpretation of these boundary conditions in the dual holographic setup.