A plane defect in the 3d O(N) model

Abstract

It was recently found that the classical 3d O (N) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="true" form="prefix">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="true" form="postfix">)</mml:mo> </mml:mrow> </mml:math> model in the semi-infinite geometry can exhibit an “extraordinary-log” boundary universality class, where the spin-spin correlation function on the boundary falls off as \langle \vec{S}(x) \cdot \vec{S}(0)\rangle \sim \frac{1}{(\log x)^q} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="false" form="prefix">〈</mml:mo> <mml:mover> <mml:mi>S</mml:mi> <mml:mo accent="true">→</mml:mo> </mml:mover> <mml:mrow> <mml:mo stretchy="true" form="prefix">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="true" form="postfix">)</mml:mo> </mml:mrow> <mml:mo>⋅</mml:mo> <mml:mover> <mml:mi>S</mml:mi> <mml:mo accent="true">→</mml:mo> </mml:mover> <mml:mrow> <mml:mo stretchy="true" form="prefix">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy="true" form="postfix">)</mml:mo> </mml:mrow> <mml:mo stretchy="false" form="postfix">⟩</mml:mo> <mml:mo>∼</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:msup> <mml:mrow> <mml:mo stretchy="true" form="prefix">(</mml:mo> <mml:mo>log</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="true" form="postfix">)</mml:mo> </mml:mrow> <mml:mi>q</mml:mi> </mml:msup> </mml:mfrac> </mml:mrow> </mml:math> . This universality class exists for a range 2 &amp;#8804;N &amp;lt;N_c <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>N</mml:mi> <mml:mo>&lt;</mml:mo> <mml:msub> <mml:mi>N</mml:mi> <mml:mi>c</mml:mi> </mml:msub> </mml:mrow> </mml:math> and Monte-Carlo simulations and conformal bootstrap indicate N_c &amp;gt; 3 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mi>N</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:mo>&gt;</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> . In this work, we extend this result to the 3d O (N) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="true" form="prefix">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="true" form="postfix">)</mml:mo> </mml:mrow> </mml:math> model in an infinite geometry with a plane defect. We use renormalization group (RG) to show that in this case the extraordinary-log universality class is present for any finite N &amp;#8805;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> . We additionally show, in agreement with our RG analysis, that the line of defect fixed points which is present at N = &amp;#8734; <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:mi>∞</mml:mi> </mml:mrow> </mml:math> is lifted to the ordinary, special (no defect) and extraordinary-log universality classes by 1/N <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mi>/</mml:mi> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> corrections. We study the “central charge” a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>a</mml:mi> </mml:math> for the O(N) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mrow> <mml:mo stretchy="true" form="prefix">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="true" form="postfix">)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> model in the boundary and interface geometries and provide a non-trivial detailed check of an a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>a</mml:mi> </mml:math> -theorem by Jensen and O’Bannon. Finally, we revisit the problem of the O (N) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline">

Topics & Concepts