Bootstrapping the simplest deconfined quantum critical point

We study the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>N</a:mi> <a:mo>=</a:mo> <a:mn>3</a:mn> </a:math> case of the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>C</c:mi> <c:msup> <c:mi>P</c:mi> <c:mrow> <c:mi>N</c:mi> <c:mo>−</c:mo> <c:mn>1</c:mn> </c:mrow> </c:msup> </c:math> model, which is a field theory of <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>N</e:mi> </e:math> complex scalars in <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mn>3</g:mn> <g:mi>d</g:mi> </g:math> coupled to an Abelian gauge field with <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mi>S</i:mi> <i:mi>U</i:mi> <i:mo stretchy="false">(</i:mo> <i:mi>N</i:mi> <i:mo stretchy="false">)</i:mo> <i:mo>×</i:mo> <i:mi>U</i:mi> <i:mo stretchy="false">(</i:mo> <i:mn>1</i:mn> <i:mo stretchy="false">)</i:mo> </i:math> global symmetry. Recent evidence suggests the <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:mi>N</o:mi> <o:mo>=</o:mo> <o:mn>2</o:mn> </o:math> theory is not critical, which makes the <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:mi>N</q:mi> <q:mo>=</q:mo> <q:mn>3</q:mn> </q:math> theory the simplest possibility of deconfined quantum criticality. We apply the conformal bootstrap to correlators of charge <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:mi>q</s:mi> <s:mo>=</s:mo> <s:mn>0</s:mn> </s:math> , 1, 2 scalar operators under the <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:mi>U</u:mi> <u:mo stretchy="false">(</u:mo> <u:mn>1</u:mn> <u:mo stretchy="false">)</u:mo> </u:math> symmetry, which gives us access also to <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"> <y:mi>q</y:mi> <y:mo>=</y:mo> <y:mn>3</y:mn> </y:math> , 4 operators. After imposing that only the lowest <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"> <ab:mi>q</ab:mi> <ab:mo>=</ab:mo> <ab:mn>0</ab:mn> </ab:math> , 1, 2 scalar operators are relevant, we find that the bootstrap bounds are saturated by the large <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"> <cb:mi>N</cb:mi> </cb:math> prediction for <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"> <eb:mi>q</eb:mi> <eb:mo>=</eb:mo> <eb:mn>1</eb:mn> </eb:math> , 2, 3, 4 scalar monopole operator scaling dimensions, which were shown earlier to be accurate even for small <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"> <gb:mi>N</gb:mi> </gb:math> , as well as a lattice prediction for the <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"> <ib:mi>q</ib:mi> <ib:mo>=</ib:mo> <ib:mn>0</ib:mn> </ib:math> non-monopole scalar operator. We also predict the scaling dimensions of the lowest spinning monopole operators, which we match to the large charge prediction for spinning operators. This suggests that the critical <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"> <kb:mi>C</kb:mi> <kb:msup> <kb:mi>P</kb:mi> <kb:mn>2</kb:mn> </kb:msup> </kb:math> model is described by this bootstrap bound.