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

We construct a class of conformal boundary states in the \mathrm{SU}(n)_1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mstyle mathvariant="normal"> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi> </mml:mstyle> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>n</mml:mi> <mml:msub> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:math> Wess-Zumino-Witten (WZW) conformal field theory (CFT) using the symmetry embedding \mathrm{Spin}(n)_2 \subset \mathrm{SU}(n)_1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mstyle mathvariant="normal"> <mml:mi>S</mml:mi> <mml:mi>p</mml:mi> <mml:mi>i</mml:mi> <mml:mi>n</mml:mi> </mml:mstyle> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>n</mml:mi> <mml:msub> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>⊂</mml:mo> <mml:mstyle mathvariant="normal"> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi> </mml:mstyle> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>n</mml:mi> <mml:msub> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:math> . These boundary states are beyond the standard Cardy construction and possess \mathrm{SO}(n) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mstyle mathvariant="normal"> <mml:mi>S</mml:mi> <mml:mi>O</mml:mi> </mml:mstyle> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> symmetry. The \mathrm{SU}(n) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mstyle mathvariant="normal"> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi> </mml:mstyle> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> Uimin-Lai-Sutherland (ULS) spin chains, which realize the \mathrm{SU}(n)_1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mstyle mathvariant="normal"> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi> </mml:mstyle> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>n</mml:mi> <mml:msub> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:math> WZW model on the lattice, allow us to identify these boundary states as the ground states of the \mathrm{SO}(n) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mstyle mathvariant="normal"> <mml:mi>S</mml:mi> <mml:mi>O</mml:mi> </mml:mstyle> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> Affleck-Kennedy-Lieb-Tasaki spin chains. Using the integrability of the \mathrm{SU}(n) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mstyle mathvariant="normal"> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi> </mml:mstyle> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> 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.