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

TL;DR

This work constructs non-Cardy conformal boundary states in the $SU(n)_1$ WZW CFT by exploiting the conformal embedding $Spin(n)_2 \subset SU(n)_1$, yielding $SO(n)$-symmetric boundary conditions. It identifies lattice realizations of these boundary states as ground states of $SO(n)$ AKLT-type states in the integrable $SU(n)$ ULS spin chain, and leverages integrability to compute the Affleck-Ludwig boundary entropy via exact overlap formulas with the ULS ground state. The authors derive explicit $g$-factors: $g=n^{1/4}$ for odd $n$ and $g=n^{1/4}/\sqrt{2}$ for even $n$, confirming the BCFT predictions and linking symmetry embedding, integrability, and boundary critical phenomena. The results illuminate a concrete bridge between exotic CFT boundary states and microscopic lattice models, with potential extensions to other embeddings and higher-dimensional systems.

Abstract

We construct a class of conformal boundary states in the $\mathrm{SU}(n)_1$ Wess-Zumino-Witten (WZW) conformal field theory (CFT) using the symmetry embedding $\mathrm{Spin}(n)_2 \subset \mathrm{SU}(n)_1$. These boundary states are beyond the standard Cardy construction and possess $\mathrm{SO}(n)$ symmetry. The $\mathrm{SU}(n)$ Uimin-Lai-Sutherland (ULS) spin chains, which realize the $\mathrm{SU}(n)_1$ WZW model on the lattice, allow us to identify these boundary states as the ground states of the $\mathrm{SO}(n)$ Affleck-Kennedy-Lieb-Tasaki spin chains. Using the integrability of the $\mathrm{SU}(n)$ 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.

Figures (4)

• Figure 1: The imaginary-time evolution picture of a composite system consisting of a CFT coupled to massive field theories at both ends (left), and the effective BCFT description in the IR limit (right), where the ground state of the massive field theory flows to the conformal boundary state $|B\rangle$.
• Figure 2: Phase diagrams of the $\mathrm{SO}(n)$ bilinear-biquadratic spin chain for $n=3,4,5,6$. The present work focuses on the ULS point $\theta_{\mathrm{ULS}} = \arctan \frac{1}{n-2}$ and the MPS point $\theta_{\mathrm{MPS}} = \arctan \frac{1}{n}$ in the symmetry-protected topological (SPT) phase. The Reshetikhin point is located at $\theta_{\mathrm{R}} = \arctan \frac{n-4}{(n-2)^2}$, where the SPT phase terminates. (a) For $n=3$, the model reduces to the spin-1 bilinear-biquadratic chain, with the MPS and Reshetikhin points corresponding to the AKLT and Takhtajan-Babujian points Takhtajan1982Babujian1982, respectively. (b) For $n=4$, the model is equivalent to an SO(4)-symmetric spin-1/2 ladder (equivalently, a spin-orbital chain), and the MPS point was first identified by Kolezhuk and Mikeska Kolezhuk1998. (c) For $n=5$, the MPS point was first proposed in an SO(5) ladder model Scalapino1998. (d) For $n=6$, the model is equivalent to an SU(4) spin chain with spins transforming under the six-dimensional self-conjugate representation. The exact ground states at the MPS point are charge-conjugation-symmetry-breaking valence-bond solids Affleck1991c.
• Figure 3: (a) The integral contour $\mathscr{C}=\mathscr{C}_+ + \mathscr{C}_-$ in Eq. \ref{['eq:contour-trick']}, with $\xi \in (0,\frac{1}{2})$. (b) The deformed integral contour $\mathscr{C}'$ in Eq. \ref{['eq:lnovlp-2nd-term-contour']}, also with $\xi \in (0,\frac{1}{2})$.
• Figure 4: The subtracted logarithmic overlap, $\ln |\langle\psi_0|\mathrm{MPS}\rangle | + \alpha N$, is plotted against $1/\ln N$ for $n=3,4,5,6$. The black dashed lines represent linear fits to the data in the range $N\in [800,1200]$. For reference, the exact large-$N$ values, $\ln g = \frac{1}{4}\ln n$, are indicated by solid dots on the vertical axis.