Perturbative and nonperturbative studies of CFTs with MN global symmetry

Abstract

Fixed points in three dimensions described by conformal field theories with \ensuremath{M N}_{m,n} = O(m)^n\rtimes S_n <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>M</mml:mi> <mml:mi>N</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>O</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>m</mml:mi> <mml:msup> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>⋊</mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:math> global symmetry have extensive applications in critical phenomena. Associated experimental data for m=n=2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> suggest the existence of two non-trivial fixed points, while the \varepsilon <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>ε</mml:mi> </mml:math> expansion predicts only one, resulting in a puzzling state of affairs. A recent numerical conformal bootstrap study has found two kinks for small values of the parameters m <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>m</mml:mi> </mml:math> and n <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>n</mml:mi> </mml:math> , with critical exponents in good agreement with experimental determinations in the m=n=2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> case. In this paper we investigate the fate of the corresponding fixed points as we vary the parameters m <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>m</mml:mi> </mml:math> and n <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>n</mml:mi> </mml:math> . We find that one family of kinks approaches a perturbative limit as m <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>m</mml:mi> </mml:math> increases, and using large spin perturbation theory we construct a large m <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>m</mml:mi> </mml:math> expansion that fits well with the numerical data. This new expansion, akin to the large N <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>N</mml:mi> </mml:math> expansion of critical 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> models, is compatible with the fixed point found in the \varepsilon <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>ε</mml:mi> </mml:math> expansion. For the other family of kinks, we find that it persists only for n=2 <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:mn>2</mml:mn> </mml:mrow> </mml:math> , where for large m <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>m</mml:mi> </mml:math> it approaches a non-perturbative limit with \Delta_\phi\approx 0.75 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">

Topics & Concepts