Bootstrapping Deconfined Quantum Tricriticality

The paradigmatic example of deconfined quantum criticality is the Neel to valence bond solid phase transition. The continuum description of this transition is 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>2</a:mn></a:math> case of the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mrow><c:mi>C</c:mi><c:msup><c:mrow><c:mi>P</c:mi></c:mrow><c:mrow><c:mi>N</c:mi><c:mo>−</c:mo><c:mn>1</c:mn></c:mrow></c:msup></c:mrow></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 3D coupled to an Abelian gauge field with <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi mathvariant="normal">S</g:mi><g:mi mathvariant="normal">U</g:mi><g:mo stretchy="false">(</g:mo><g:mi>N</g:mi><g:mo stretchy="false">)</g:mo><g:mo>×</g:mo><g:mi mathvariant="normal">U</g:mi><g:mo stretchy="false">(</g:mo><g:mn>1</g:mn><g:mo stretchy="false">)</g:mo></g:math> global symmetry. Lattice studies and duality arguments suggest the global symmetry of the <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:mi>C</p:mi><p:msup><p:mi>P</p:mi><p:mn>1</p:mn></p:msup></p:math> model is enhanced to SO(5). We perform a conformal bootstrap study of SO(5) invariant fixed points with one relevant SO(5) singlet operator, which would correspond to two relevant <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:mi mathvariant="normal">S</r:mi><r:mi mathvariant="normal">U</r:mi><r:mo stretchy="false">(</r:mo><r:mn>2</r:mn><r:mo stretchy="false">)</r:mo><r:mo>×</r:mo><r:mi mathvariant="normal">U</r:mi><r:mo stretchy="false">(</r:mo><r:mn>1</r:mn><r:mo stretchy="false">)</r:mo></r:math> singlets, i.e., a tricritical point. We find that the bootstrap bounds are saturated by four different predictions from the large <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:mi>N</ab:mi></ab:math> computation of monopole operator scaling dimensions, which were recently shown to be very accurate even for small <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"><cb:mi>N</cb:mi></cb:math>. This suggests that the Neel to valence bond solid phase transition is described by this bootstrap bound, which predicts that the second relevant singlet has dimension <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"><eb:mo>≈</eb:mo><eb:mn>2.36</eb:mn></eb:math>.