---
title: 'ABJM Spin Chain: Integrability and Fusion'
url: https://www.emergentmind.com/topics/aharony-bergman-jafferis-maldacena-spin-chain
type: topic
---

# ABJM Spin Chain: Integrability and Fusion

The Aharony–Bergman–Jafferis–Maldacena (ABJM) spin chain refers to an integrable quantum spin chain arising in the planar limit of ABJM theory, a three-dimensional superconformal Chern–Simons-matter theory with OSp$(6|4)$ symmetry. The theory's two-loop planar dilatation operator in the scalar SU(4) subsector is mapped to an alternating spin chain of length $2L$, with odd sites in the fundamental $\mathbf{4}$ and even sites in the anti-fundamental $\bar{\mathbf{4}}$ of SU(4). The spin chain possesses intricate integrable properties, with distinct physical and mathematical structures appearing in its fundamental alternating and fused (nearest-neighbor) incarnations, and supports the explicit construction of chiral integrable boundary states relevant for defect CFT observables in the AdS/CFT correspondence [2410.05987] [2602.01697].

## 1. Algebraic Bethe Ansatz and the Alternating ABJM Spin Chain

The planar ABJM “alternating” spin chain is constructed via the algebraic Bethe ansatz, with the quantum space
$$
\mathcal H = \bigotimes_{l=1}^{2L} V_l, \qquad V_{2j-1}\cong\mathbf{4}, \quad V_{2j}\cong\bar{\mathbf{4}},
$$
where odd sites transform under the SU(4) fundamental and even sites under the anti-fundamental. Integrability is encoded through R-matrices that interchange or trace SU(4) indices:
\[
R_{ab}(u) = u\,\mathbf{1}_{ab} + P_{ab}, \qquad R_{a\bar b}(u) = -(u+2)\,\mathbf{1}_{a\bar b} + K_{a\bar b},
\]
where $P$ is the permutation and $K$ the trace operator. Two monodromy matrices,
\[
T_0(u) = R_{0,1}(u)\cdots R_{0,2L}(u),\quad
\widehat{T}_{\bar0}(u) = R_{\bar0,1}(u)\cdots R_{\bar0,2L}(u),
\]
lead to transfer matrices $\tau(u)$ and $\widehat\tau(u)$ whose commutativity encodes conserved quantities arising from Yang–Baxter integrability [2410.05987][2602.01697].

## 2. Fusion Procedure and the Fused Model

The non-regularity of the fundamental R-matrix prevents the construction of a genuine nearest-neighbor Hamiltonian. To circumvent this, the fusion procedure is introduced: every pair of adjacent sites $(2j-1,2j)$ is merged to form a 16-dimensional vector space $V_{2j-1}\otimes V_{2j}\cong\mathbf{4}\otimes\bar{\mathbf{4}}$. The fused R-matrix,
\[
R^{\rm (fus)}_{(aa),(bb)}(u) = R_{a,b}(u)\,R_{a,\bar b}(u)\,R_{\bar a,b}(u)\,R_{\bar a,\bar b}(u),
\]
acts between fused quantum or auxiliary spaces and satisfies a regularity condition,
\[
R^{\rm (fus)}_{(aa),(bb)}(0) = 4\,P_{13}\,P_{24},
\]
yielding a nearest-neighbor model via a fused transfer matrix $t^{\rm(fus)}(u) = \Tr_{(00)} \mathcal{T}_{(00)}(u) = \tau(u)\widehat{\tau}(u)$ [2410.05987].

## 3. Hamiltonian, Boost Operator, and Local Charges

The Hamiltonian of the fused ABJM chain is obtained from the logarithmic derivative of the transfer matrix at the regular point $u=0$:
\[
H = i\,\frac{d}{du}\ln t^{\rm (fus)}(u)\Big|_{u=0}
= \sum_{j=1}^L H_{(2j-1,2j),(2j+1,2j+2)},
\]
with the two-site local density expressed as
\[
H_{(2j-1,2j),(2j+1,2j+2)} = h_{2j-1,2j,2j+1} + h_{2j,2j+1,2j+2},
\]
where
\[
h_{j,j+1,j+2} = P_{j,j+2} - K_{j+1,j+2} P_{j,j+2} - P_{j,j+2} K_{j,j+1}.
\]
The fused model, being regular, allows the definition of a boost operator,
\[
B = -\sum_{k=1}^{L-1} k\,H_{(2k-1,2k),(2k+1,2k+2)},
\]
which generates higher local charges recursively:
\[
Q_{n+1} = [B, Q_n],\qquad Q_2 \equiv -iH, \quad Q_1 = -i\ln t^{\rm(fus)}(0).
\]
Such a boost operator is absent in the original alternating chain, highlighting the structural advantage of the fused construction [2410.05987].

## 4. Open Boundary Conditions and Chiral Integrable States

Integrable open boundary conditions in the ABJM spin chain are introduced via boundary K-matrices $K^\pm(u)$ acting on the fused auxiliary space. The open-chain double-row transfer matrix,
\[
t_{\rm open}(u) = \Tr_{(00)}\left[K^+_{(00)}(u)\, \mathcal T_{(00)}(u)\, K^-_{(00)}(u)\, \mathcal T_{(00)}^{-1}(-u)\right],
\]
yields, via logarithmic differentiation,
\[
H_{\rm open} = H_{\rm bulk} + H_{\rm left} + H_{\rm right},
\]
where the boundary terms factorize into Kronecker-delta structures up to model-dependent coefficients. The boundary K-matrices satisfy (fused) reflection equations, enabling a systematic construction of chiral integrable boundary states [2410.05987][2602.01697].

Chiral boundary states $|\mathcal{B}\rangle$ within the ABJM chain are defined by annihilation under all odd conserved charges. These “chiral” states are constructed by solving the condition
\[
\tau(u)\,|\mathcal B\rangle = \Pi\,\tau(u)\,\Pi\,|\mathcal B\rangle,
\]
with $\Pi$ the site-reversal permutation, or equivalently for $\widehat\tau(u)$ [2602.01697].

## 5. Reflection Equations, Fusion of Boundary States, and Matrix Product States

To generate higher-rank chiral integrable boundary states, the standard fusion procedure is applied to the K-matrices:
\[
K_{(12)}(u) = \tilde K_2(u)\,R_{12}(2u)\,\tilde K_1(u),
\]
and by repeating this process $n$ times, one constructs $n$-fused K-matrices $K^{(n)}(u)$ solving fused reflection equations. Each solution yields a $2n$-site block state,
\[
|\Phi^{(n)}(u)\rangle = [K^{(n)}(u)]^{i_1\ldots i_n}_{j_1\ldots j_n} |i_n,\ldots,i_1, j_n,\ldots,j_1\rangle \in V^{\otimes 2n},
\]
forming translationally invariant matrix product states (MPS) under repetition across the chain. All such states satisfy the chiral integrability condition imposed by the transfer matrices [2602.01697].

## 6. Overlaps, Bethe States, and Gaudin Determinants

For the special case $n=2$, the four-site block MPS $|\Psi_4\rangle = |\Phi^{(2)}(-1)\rangle^{\otimes L/2}$ possesses explicitly computable overlaps with Bethe eigenstates characterized by three rapidity sets:
\[
\langle\Psi_4|\mathbf u,\mathbf w,\mathbf v\rangle =
(-2)^{L/2} \prod_{i=1}^{N_u/2} u_i(u_i-\tfrac{i}{2}) \prod_{j=1}^{N_v/2} v_j(v_j-\tfrac{i}{2}) \prod_{k=1}^{N_w/2} \frac{w_k}{w_k+\tfrac{i}{2}} \times \det G_+,
\]
where $G_+$ denotes a sector of the Gaudin matrix arising under chiral pairing of rapidities, and the construction extends to various block sizes and dressing procedures. These formulas underpin the exact computation of one-point functions in certain defect CFT contexts [2602.01697].

## 7. Classification and Applications of Chiral Integrable Subspaces

Numerical investigation of the space of chiral integrable boundary states reveals its rapid growth with system size: for $L=2$, the space has dimension $196$, and for $L=3$, $616$. Simple product ansätze or four-site MPS together span only subspaces of the full space, implying the existence of yet-undiscovered integrable states beyond those generated by standard fusion procedures. This structure underpins applications in AdS/CFT, including one-point observables in defect CFTs such as Wilson lines or domain walls, and provides a foundation for the study of quantum quenches and boundary correlation functions in nested SU(4) integrable systems [2602.01697].

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For a comprehensive algebraic and analytic treatment, refer to "The Fused Model of Alternating Spin Chain from ABJM Theory" [2410.05987] and "Chiral Integrable Boundary States of ABJM Spin Chain from Reflection Equations" [2602.01697].

Source: https://www.emergentmind.com/topics/aharony-bergman-jafferis-maldacena-spin-chain