Non-Wilson-Fisher kinks of $O(N)$ numerical bootstrap: from the deconfined phase transition to a putative new family of CFTs
Yin-Chen He, Junchen Rong, Ning Su
Abstract
It is well established that the O(N) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>O</mml:mi> <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> Wilson-Fisher (WF) CFT sits at a kink of the numerical bounds from bootstrapping four point function of O(N) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>O</mml:mi> <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> vector. Moving away from the WF kinks, there indeed exists another family of kinks (dubbed non-WF kinks) on the curve of O(N) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>O</mml:mi> <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> numerical bounds. Different from the O(N) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>O</mml:mi> <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> WF kinks that exist for arbitary N <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>N</mml:mi> </mml:math> in 2<d<4 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo><</mml:mo> <mml:mi>d</mml:mi> <mml:mo><</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> dimensions, the non-WF kinks exist in arbitrary dimensions but only for a large enough N>N_c(d) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>></mml:mo> <mml:msub> <mml:mi>N</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>d</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> in a given dimension d <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>d</mml:mi> </mml:math> . In this paper we have achieved a thorough understanding for few special cases of these non-WF kinks, which already hints interesting physics. The first case is the O(4) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>4</mml:mn> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> bootstrap in 2d, where the non-WF kink turns out to be the SU(2)_1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>2</mml:mn> <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) model, and all the SU(2)_{k>2} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>2</mml:mn> <mml:msub> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> WZW models saturate the numerical bound on the left side of the kink. This is a mirror version of the Z_2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>Z</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> bootstrap, where the 2d Ising CFT sits at a kink while all the other minimal models saturating the bound on the right. We further carry out dimensional continuation of the 2d SU(2)_1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi>