Abstract
We prove the integrability of the two-loop open spin chain Hamiltonian from ABJM determinant like operators given in [1]. By explicitly constructing R-matrices and K-matrices, we successfully obtain the two-loop Hamiltonian from the double row transfer matrices. This proves the integrability of our two-loop Hamiltonian. Based on the vacuum eigenvalues of the transfer matrices, we make a conjecture on the eigenvalues of the transfer matrices for general excited states. Bethe ansatz equations are simply obtained from the analytic conditions at the superficial poles of the eigenvalues.
Highlights
By algebraic construction, we mean that deriving the Hamiltonian from the commutating transfer matrices which serves as the generating function of an infinite number of conserved quantities
We prove the integrability of the two-loop open spin chain Hamiltonian from ABJM determinant like operators given in [1]
Bethe ansatz equations are obtained from the analytic conditions at the superficial poles of the eigenvalues
Summary
We consider the alternating spin chain model with open boundaries which originates from the anomalous dimension matrix of determinant like operators in ABJM theory. The length of the spin chain is 2L with 2L−2 4-dimensional bulk spaces plus two boundary spaces of 3 dimensions. The bulk consists of fundamental representation space of SU(4) labeled as 4 with the basis. A1 = |1 , A2 = |2 , B1† = |3 , B2† = |4 , and anti-fundamental representation space of SU(4) labeled as 4 ̄ with the basis (2.1). Where “ i ” and “ ̄i ” denote the fundamental and anti-fundamental representation spaces of SU(4) respectively and λ ≡ N/k is the ’t Hooft coupling constant of ABJM theory. The operators P and K are defined by the standard elementary matrices eab (with components [eab]ij = δaiδbj) as
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