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arXiv:2508.13114 · SciPost Physics

SO($n$) Affleck-Kennedy-Lieb-Tasaki states as conformal boundary states of integrable SU($n$) spin chains

Yueshui Zhang, Ying-Hai Wu, Meng Cheng, Hong-Hao Tu

Published Mar 2, 2026• 1 citation
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Abstract

We construct a class of conformal boundary states in the \mathrm{SU}(n)_1 S U ( n ) 1 Wess-Zumino-Witten (WZW) conformal field theory (CFT) using the symmetry embedding \mathrm{Spin}(n)_2 \subset \mathrm{SU}(n)_1 S p i n ( n ) 2 ⊂ S U ( n ) 1 . These boundary states are beyond the standard Cardy construction and possess \mathrm{SO}(n) S O ( n ) symmetry. The \mathrm{SU}(n) S U ( n ) Uimin-Lai-Sutherland (ULS) spin chains, which realize the \mathrm{SU}(n)_1 S U ( n ) 1 WZW model on the lattice, allow us to identify these boundary states as the ground states of the \mathrm{SO}(n) S O ( n ) Affleck-Kennedy-Lieb-Tasaki spin chains. Using the integrability of the \mathrm{SU}(n) S U ( n ) ULS model, we analytically compute the corresponding Affleck-Ludwig boundary entropy using exact overlap formulas. Our results unveil intriguing connections between exotic boundary states in CFT and integrable lattice models, thus providing deep insights into the interplay of symmetry, integrability, and boundary critical phenomena.

DOI
10.21468/scipostphys.20.3.067

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SO(nnn) Affleck-Kennedy-Lieb-Tasaki states as conformal boundary states of integrable SU(nnn) spin chains
We construct a class of conformal boundary states in the \mathrm{SU}(n)_1 S U ( n ) 1 Wess-Zumino-Witten (WZW) conformal field theory (CFT) using the symmetry embedding \mathrm{Spin}(n)_2 \subset \mathrm{SU}(n)_1 S p i n ( n ) 2 ⊂ S U ( n ) 1 . These boundary states are beyond the standard Cardy construction and possess \mathrm{SO}(n) S O ( n ) symmetry. The \mathrm{SU}(n) S U ( n ) Uimin-Lai-Sutherland (ULS) spin chains, which realize the \mathrm{SU}(n)_1 S U ( n ) 1 WZW model on the lattice, allow us to identify these boundary states as the ground states of the \mathrm{SO}(n) S O ( n ) Affleck-Kennedy-Lieb-Tasaki spin chains. Using the integrability of the \mathrm{SU}(n) S U ( n ) ULS model, we analytically compute the corresponding Affleck-Ludwig boundary entropy using exact overlap formulas. Our results unveil intriguing connections between exotic boundary states in CFT and integrable lattice models, thus providing deep insights into the interplay of symmetry, integrability, and boundary critical phenomena.
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We investigate the entanglement properties of the valence-bond-solid (VBS) state defined on two-dimensional lattices, which is the exact ground state of the Affleck-Kennedy-Lieb-Tasaki model. It is shown that the entanglement entropy obeys an area law and the nonuniversal prefactor of the leading term is strictly less than ln2\mathrm{ln}2ln2. The analysis of entanglement spectra for various lattices reveals that the reduced density matrix associated with the VBS state is closely related to a thermal density matrix of a holographic spin chain, the spectrum of which is reminiscent of that of the spin-1/2 Heisenberg chain. This correspondence is further supported by comparing the entanglement entropy in the holographic spin chain with conformal field theory predictions.
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We examine the effective field theory of the Bethe ansatz integrable Heisenberg antiferromagnetic spin chains. It shows that the quantum critical theories for the integer spin-S chains should be characterized by the SO(3)level-S Wess-Zumino-Witten model, and classified by the third cohomology group H3(SO(3),Z)=ZH^{3}(SO(3),Z)=ZH3(SO(3),Z)=Z. Depending on the parity of spin S, this integer classification is further divided into two distinct universality classes, which are associated with two completely different conformal field theories: the even-S chains have gapless bosonic excitations and the odd-S chains have both bosonic and fermionic excitations. We further show that these two classes of critical states correspond to the boundary states of two distinct topological phases in two dimension, which can be described by two-dimensional doubled SO(3) topological Chern-Simons theory and topological spin theory, respectively.
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