SciPost Phys. 13, 014 (2022) ·
published 8 August 2022

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We apply the conformal bootstrap technique to study the $U(1)$ Dirac spin
liquid (i.e. $N_f=4$ QED$_3$) and the newly proposed $N=7$ Stiefel liquid (i.e.
a conjectured 3d nonLagrangian CFT without supersymmetry). For the $N_f=4$
QED$_3$, we focus on the monopole operator and ($SU(4)$ adjoint) fermion
bilinear operator. We bootstrap their single correlators as well as the mixed
correlators between them. We first discuss the bootstrap kinks from single
correlators. Some exponents of these bootstrap kinks are close to the expected
values of QED$_3$, but we provide clear evidence that they should not be
identified as the QED$_3$. By requiring the critical phase to be stable on the
triangular and the kagome lattice, we obtain rigorous numerical bounds for the
$U(1)$ Dirac spin liquid and the Stiefel liquid. For the triangular and kagome
Dirac spin liquid, the rigorous lower bounds of the monopole operator's scaling
dimension are $1.046$ and $1.105$, respectively. These bounds are consistent
with the latest Monte Carlo results.
SciPost Phys. 11, 111 (2021) ·
published 22 December 2021

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We propose a roadmap for bootstrapping conformal field theories (CFTs)
described by gauge theories in dimensions $d>2$. In particular, we provide a
simple and workable answer to the question of how to detect the gauge group in
the bootstrap calculation. Our recipe is based on the notion of
\emph{decoupling operator}, which has a simple (gauge) group theoretical
origin, and is reminiscent of the null operator of $2d$ WessZuminoWitten CFTs
in higher dimensions. Using the decoupling operator we can efficiently detect
the rank (i.e. color number) of gauge groups, e.g., by imposing gap conditions
in the CFT spectrum. We also discuss the physics of the equation of motion,
which has interesting consequences in the CFT spectrum as well. As an
application of our recipes, we study a prototypical critical gauge theory,
namely the scalar QED which has a $U(1)$ gauge field interacting with critical
bosons. We show that the scalar QED can be solved by conformal bootstrap,
namely we have obtained its kinks and islands in both $d=3$ and $d=2+\epsilon$
dimensions.
SciPost Phys. 10, 115 (2021) ·
published 26 May 2021

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It is well established that the $O(N)$ WilsonFisher (WF) CFT sits at a kink
of the numerical bounds from bootstrapping four point function of $O(N)$
vector. Moving away from the WF kinks, there indeed exists another family of
kinks (dubbed nonWF kinks) on the curve of $O(N)$ numerical bounds. Different
from the $O(N)$ WF kinks that exist for arbitary $N$ in $2<d<4$ dimensions, the
nonWF kinks exist in arbitrary dimensions but only for a large enough
$N>N_c(d)$ in a given dimension $d$. In this paper we have achieved a thorough
understanding for few special cases of these nonWF kinks. The first case is
the $O(4)$ bootstrap in 2d, where the nonWF kink turns out to be the $SU(2)_1$
WessZuminoWitten (WZW) model, and all the $SU(2)_{k>2}$ WZW models saturate
the numerical bound on the left side of the kink. We further carry out
dimensional continuation of the 2d $SU(2)_1$ kink towards the 3d $SO(5)$
deconfined phase transition. We find the kink disappears at around $d=2.7$
dimensions indicating the $SO(5)$ deconfined phase transition is weakly first
order. The second interesting observation is, the $O(2)$ bootstrap bound does
not show any kink in 2d ($N_c=2$), but is surprisingly saturated by the 2d free
boson CFT (also called Luttinger liquid) all the way on the numerical curve.
The last case is the $N=\infty$ limit, where the nonWF kink sits at
$(\Delta_\phi, \Delta_T)=(d1, 2d)$ in $d$ dimensions. We manage to write down
its analytical four point function in arbitrary dimensions, which equals to the
subtraction of correlation functions of a free fermion theory and generalized
free theory. An important feature of this solution is the existence of a full
tower of conserved higher spin current. We speculate that a new family of CFTs
will emerge at nonWF kinks for finite $N$, in a similar fashion as $O(N)$ WF
CFTs originating from free boson at $N=\infty$.
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