ABJM Spin Chain Model

• ABJM spin chain model is the integrable realization of planar ABJM theory, mapping gauge-invariant single-trace operators onto an alternating SU(4) chain.
• The model is derived at two loops using a range-three Hamiltonian and nested Bethe ansatz techniques to compute anomalous dimensions and capture integrable dynamics.
• Extensions include a full OSp(6|4) supersymmetric formulation, open-chain constructions with giant gravitons, flavors, Wilson loops, and integrable boundary states via matrix product states.

The ABJM spin chain model is the integrable spin-chain realization of planar ABJM theory, most directly in the scalar sector where gauge-invariant single-trace operators are mapped to an alternating $SU(4)$ chain with odd sites in the fundamental $\mathbf 4$ and even sites in the anti-fundamental $\bar{\mathbf 4}$. In a broader sense, the term also encompasses its lift to the full $OSp(6|4)$ super-spin-chain, open-chain versions associated with giant gravitons, flavors, domain walls, and Wilson loops, as well as boundary-state formulations in which defect one-point functions and certain three-point functions are computed as overlaps between Bethe eigenstates and matrix product states (0901.1142, Gombor et al., 2022, Kristjansen et al., 2021).

1. Two-loop origin and alternating geometry

ABJM theory is the three-dimensional $\mathcal N=6$ superconformal Chern–Simons matter theory with gauge group

$$U(N)_k\times U(N)_{-k}, \qquad \lambda=\frac{N}{k}.$$

In planar perturbation theory, scalar single-trace operators of the form

$$\operatorname{tr}\!\left(Y^{A_1}Y^\dagger_{A_2}\cdots Y^{A_{2L-1}}Y^\dagger_{A_{2L}}\right)$$

map to an alternating chain of length $2L$ with Hilbert space

$(\mathbf 4\otimes \bar{\mathbf 4})^{\otimes L}.$

The alternation is structural rather than optional: odd sites carry $\mathbf 4$, even sites carry $\mathbf 4$0, a feature that distinguishes ABJM from the homogeneous $\mathbf 4$1 SYM chains (Gombor et al., 2022, Kristjansen et al., 2021).

In this sector the planar two-loop dilatation operator is the alternating-chain Hamiltonian

$\mathbf 4$2

equivalently

$\mathbf 4$3

Here $\mathbf 4$4 is the permutation operator on like-representation sites and $\mathbf 4$5 is the trace operator between neighboring $\mathbf 4$6 and $\mathbf 4$7 sites. The local interaction is therefore intrinsically three-site, or range-three in the original alternating description (Gombor et al., 2022, Yang et al., 2021, 0901.1142).

A standard pseudovacuum is

$\mathbf 4$8

corresponding to $\mathbf 4$9. In ABJ theory the same operator structure survives at two loops with $\bar{\mathbf 4}$0 replaced by $\bar{\mathbf 4}$1; parity-violating contributions cancel so the difference between ABJM and ABJ is invisible at this order apart from that coupling replacement (0901.1142).

2. $\bar{\mathbf 4}$2-matrices, transfer matrices, and nested Bethe ansatz

The integrable structure is encoded by alternating monodromies and transfer matrices. The literature uses equivalent but convention-dependent $\bar{\mathbf 4}$3-matrices. One common convention writes

$\bar{\mathbf 4}$4

while another writes

$\bar{\mathbf 4}$5

These define two commuting transfer matrices, one with fundamental and one with anti-fundamental auxiliary space, and their logarithmic derivatives generate the conserved charges, including the two-loop Hamiltonian (Gombor et al., 2022, Yang et al., 2021, Bai et al., 2019).

In the scalar $\bar{\mathbf 4}$6 chain the spectrum is described by three sets of Bethe roots, conveniently denoted $\bar{\mathbf 4}$7, $\bar{\mathbf 4}$8, and $\bar{\mathbf 4}$9. In the conventions of the nested Bethe ansatz,

$OSp(6|4)$0

$OSp(6|4)$1

$OSp(6|4)$2

with

$OSp(6|4)$3

The $OSp(6|4)$4- and $OSp(6|4)$5-roots are momentum-carrying, whereas the middle-node roots $OSp(6|4)$6 are auxiliary. The momentum-rapidity relation is

$OSp(6|4)$7

and the anomalous dimension takes the form

$OSp(6|4)$8

in the scalar-sector conventions where only $OSp(6|4)$9 and $\mathcal N=6$0 carry energy (Yang et al., 2021).

Baxter polynomials,

$\mathcal N=6$1

enter both the transfer-matrix eigenvalues and the exact overlap formulas for integrable boundary states. For paired configurations the Gaudin norm matrix factorizes into parity-even and parity-odd blocks, a feature that later becomes central in ABJM overlap formulas (Gombor et al., 2022, Kristjansen et al., 2021).

3. The full $\mathcal N=6$2 spin chain

The scalar alternating $\mathcal N=6$3 chain is only the simplest closed subsector. The full two-loop ABJM spin chain is governed by the superalgebra $\mathcal N=6$4. Its construction proceeds by first building an integrable Hamiltonian for an $\mathcal N=6$5 subsector and then lifting it to $\mathcal N=6$6 through the matching of irreducible representation projectors in the alternating tensor products of chiral and antichiral singletons (0901.1142).

In the full Bethe ansatz one introduces five sets of roots,

$\mathcal N=6$7

with $\mathcal N=6$8 and $\mathcal N=6$9 momentum-carrying and the remaining three auxiliary. The cyclicity condition is

$$U(N)_k\times U(N)_{-k}, \qquad \lambda=\frac{N}{k}.$$0

and the anomalous dimension is

$$U(N)_k\times U(N)_{-k}, \qquad \lambda=\frac{N}{k}.$$1

This extension reproduces the fermionic two-loop Hamiltonian and matches explicit field-theory computations, including parity-paired multiplets and their degeneracies (0901.1142).

Representation-theoretically, the elementary fields furnish singleton modules of $$U(N)_k\times U(N)_{-k}, \qquad \lambda=\frac{N}{k}.$$2, so the ABJM chain is an alternating chain of chiral and antichiral singletons rather than only a bosonic $$U(N)_k\times U(N)_{-k}, \qquad \lambda=\frac{N}{k}.$$3 magnet. This broader perspective is essential for later constructions involving fermionic dualities, superdeterminants of Gaudin matrices, and full-theory overlap formulas (0901.1142, Kristjansen et al., 2021).

4. Open chains, giant gravitons, flavors, and Wilson loops

Open ABJM spin chains arise when the alternating bulk is terminated by physical boundaries. Determinant-like operators dual to open strings attached to a maximal giant graviton D4-brane wrapping a $$U(N)_k\times U(N)_{-k}, \qquad \lambda=\frac{N}{k}.$$4 produce an open alternating chain with reduced boundary Hilbert spaces: the left boundary excludes $$U(N)_k\times U(N)_{-k}, \qquad \lambda=\frac{N}{k}.$$5, the right boundary excludes $$U(N)_k\times U(N)_{-k}, \qquad \lambda=\frac{N}{k}.$$6. The resulting two-loop Hamiltonian contains the standard ABJM bulk terms plus explicit projector-valued boundary interactions, and one-particle excitations have dispersion

$$U(N)_k\times U(N)_{-k}, \qquad \lambda=\frac{N}{k}.$$7

at two loops (Chen et al., 2018, Bai et al., 2019).

The algebraic proof of integrability for this giant-graviton chain uses projected operator-valued $$U(N)_k\times U(N)_{-k}, \qquad \lambda=\frac{N}{k}.$$8-matrices and double-row transfer matrices. Because the boundary spaces are three-dimensional while bulk spaces are four-dimensional, ordinary c-number boundary matrices are insufficient. The transfer matrices commute,

$$U(N)_k\times U(N)_{-k}, \qquad \lambda=\frac{N}{k}.$$9

and their logarithmic derivatives reproduce the physical open-chain Hamiltonian up to normalization and additive constants (Bai et al., 2019).

The asymptotic all-loop generalization is formulated in terms of centrally extended $$\operatorname{tr}\!\left(Y^{A_1}Y^\dagger_{A_2}\cdots Y^{A_{2L-1}}Y^\dagger_{A_{2L}}\right)$$0 bulk symmetry and residual boundary $$\operatorname{tr}\!\left(Y^{A_1}Y^\dagger_{A_2}\cdots Y^{A_{2L-1}}Y^\dagger_{A_{2L}}\right)$$1. The right-boundary reflection matrix takes the diagonal form

$$\operatorname{tr}\!\left(Y^{A_1}Y^\dagger_{A_2}\cdots Y^{A_{2L-1}}Y^\dagger_{A_{2L}}\right)$$2

with scalar factor fixed by boundary unitarity and boundary crossing. The associated open Bethe-Yang equations describe A- and B-particles propagating between the two reflecting ends (Chen, 2019).

A different open-chain realization appears in planar flavored ABJM theory, where mesonic operators with fundamental matter at the endpoints yield an open alternating chain with nontrivial boundary scattering. Coordinate Bethe ansatz gives anti-diagonal reflection matrices in the original flavored model, while a later classification showed that, within a broad boundary-coupling ansatz, only two integrable classes exist at each end: trivial diagonal reflection and the flavored anti-diagonal solution (Bai et al., 2017, Wu, 2017). A subsequent fused-model analysis argued that the flavored open chain does not fit the universal fused boundary patterns and may therefore fail to be integrable in the stricter algebraic-Bethe-ansatz sense, despite earlier coordinate-Bethe-ansatz evidence (Bai et al., 2024).

Operator insertions on a $$\operatorname{tr}\!\left(Y^{A_1}Y^\dagger_{A_2}\cdots Y^{A_{2L-1}}Y^\dagger_{A_{2L}}\right)$$3-BPS Wilson line lead to yet another open ABJM chain. In that system the exact bulk $$\operatorname{tr}\!\left(Y^{A_1}Y^\dagger_{A_2}\cdots Y^{A_{2L-1}}Y^\dagger_{A_{2L}}\right)$$4-matrix is the standard ABJM $$\operatorname{tr}\!\left(Y^{A_1}Y^\dagger_{A_2}\cdots Y^{A_{2L-1}}Y^\dagger_{A_{2L}}\right)$$5-invariant one, while the reflection factors distinguish type-A and type-B magnons. The model admits a $$\operatorname{tr}\!\left(Y^{A_1}Y^\dagger_{A_2}\cdots Y^{A_{2L-1}}Y^\dagger_{A_{2L}}\right)$$6-system and BTBA description, and the leading finite-size correction reproduces the one-loop cusp anomalous dimension

$$\operatorname{tr}\!\left(Y^{A_1}Y^\dagger_{A_2}\cdots Y^{A_{2L-1}}Y^\dagger_{A_{2L}}\right)$$7

(Correa et al., 2023).

5. Matrix product states, defect observables, and overlap formulas

One of the most distinctive developments in the ABJM spin chain is the appearance of integrable boundary states described by matrix product states (MPS). For the $$\operatorname{tr}\!\left(Y^{A_1}Y^\dagger_{A_2}\cdots Y^{A_{2L-1}}Y^\dagger_{A_{2L}}\right)$$8-BPS co-dimension-one domain wall, a classical scalar profile induces an MPS of bond dimension $$\operatorname{tr}\!\left(Y^{A_1}Y^\dagger_{A_2}\cdots Y^{A_{2L-1}}Y^\dagger_{A_{2L}}\right)$$9, with $2L$0 equal to the jump in gauge-group rank across the wall. The defect breaks

$2L$1

and for $2L$2 the MPS degenerates into a valence-bond state. The hallmark of integrability is an achiral parity condition on Bethe roots,

$2L$3

which constrains the states with nonvanishing overlap (Kristjansen et al., 2021).

For these domain-wall states, tree-level one-point functions reduce to normalized overlaps. In the $2L$4 scalar-sector case the formula is

$2L$5

where $2L$6 is the parity-factorized Gaudin superdeterminant. The same framework extends, via fermionic dualities, to the full $2L$7 chain (Kristjansen et al., 2021).

The arbitrary-bond-dimension generalization gives a closed formula for the normalized overlap with a paired Bethe state: $2L$8 Here $2L$9 is understood with possible zero roots removed, and the overlap in the regulator limit is finite only when $(\mathbf 4\otimes \bar{\mathbf 4})^{\otimes L}.$0. Since defect one-point functions are directly proportional to these overlaps, the formula determines the tree-level one-point data of scalar operators in the domain-wall background (Gombor et al., 2022).

A closely related MPS technology was developed for three-point functions involving determinant and sub-determinant operators. In that setting the structure constants become overlaps between Bethe eigenstates and boundary MPSs, with Gaudin-like determinant formulas and the same characteristic parity constraints on admissible Bethe roots (Yang et al., 2021). The status of generic sub-determinant boundary states, however, became a point of revision: evidence for a whole family of integrable sub-determinant MPSs was later challenged by an explicit analysis concluding that the giant-graviton MPS is integrable only for two special values, $(\mathbf 4\otimes \bar{\mathbf 4})^{\otimes L}.$1 and $(\mathbf 4\otimes \bar{\mathbf 4})^{\otimes L}.$2, including the maximal giant graviton (Yang et al., 2021, Yang, 2022).

6. Fusion, boost operators, and chiral boundary states

A major algebraic reformulation of the closed ABJM chain is the fused model. By combining adjacent sites into

$(\mathbf 4\otimes \bar{\mathbf 4})^{\otimes L}.$3

the original range-three alternating Hamiltonian is repackaged as a nearest-neighbor chain on length-$(\mathbf 4\otimes \bar{\mathbf 4})^{\otimes L}.$4 fused sites. The fused $(\mathbf 4\otimes \bar{\mathbf 4})^{\otimes L}.$5-matrix is regular,

$(\mathbf 4\otimes \bar{\mathbf 4})^{\otimes L}.$6

the fused transfer matrix is

$(\mathbf 4\otimes \bar{\mathbf 4})^{\otimes L}.$7

and the standard boost operator exists,

$(\mathbf 4\otimes \bar{\mathbf 4})^{\otimes L}.$8

so higher local charges can be generated recursively by

$(\mathbf 4\otimes \bar{\mathbf 4})^{\otimes L}.$9

This makes explicit why the ABJM model is integrable despite appearing as a medium-range chain in the unfused variables (Bai et al., 2024).

A general fusion procedure for open $\mathbf 4$0 chains later showed that the ABJM alternating model itself can be obtained by antisymmetric triple fusion of the fundamental $\mathbf 4$1, producing the anti-fundamental $\mathbf 4$2. In the $\mathbf 4$3 application, the mixed $\mathbf 4$4-matrices arise as restrictions of fused Yang $\mathbf 4$5-matrices, and the construction yields three classes of anti-fundamental boundary reflection matrices on the fused auxiliary space (Bai, 25 Jul 2025).

For several years the explicitly known integrable ABJM boundary states were achiral, in the sense that nonzero overlaps required mixed-type pairing such as $\mathbf 4$6. This changed with the identification of the first chiral integrable boundary states in the alternating $\mathbf 4$7 chain. They satisfy the untwisted relation

$\mathbf 4$8

equivalently

$\mathbf 4$9

and therefore impose same-type parity pairing,

$\mathbf 4$00

The explicit solutions include two-site and four-site repeating basis states such as $\mathbf 4$01 and $\mathbf 4$02, with distinctness conditions on the site labels (Liu et al., 4 Jul 2025).

A subsequent reflection-equation and fusion framework generalized these constructions to $\mathbf 4$03-site chiral integrable MPSs and proposed exact overlap formulas for the four-site case in terms of scalar prefactors times $\mathbf 4$04, or in normalized form

$\mathbf 4$05

The same analysis also showed that the known chiral constructions are not exhaustive: the numerically determined chiral integrable subspace has dimension $\mathbf 4$06 for $\mathbf 4$07 and $\mathbf 4$08 for $\mathbf 4$09, exceeding the span of the presently constructed families (Liu et al., 2 Feb 2026).

Taken together, these developments show that the ABJM spin chain model is no longer only the original two-loop alternating scalar chain. It is an expanding integrable hierarchy comprising closed and open chains, super-spin-chain lifts, fused reformulations, and a growing zoo of achiral and chiral boundary states whose exact overlap formulas connect spectral data to observables in defects, giant gravitons, and Wilson-loop backgrounds (Gombor et al., 2022, Bai et al., 2024, Liu et al., 4 Jul 2025).