Necessary Condition for Emergent Symmetry from the Conformal Bootstrap

Abstract

We use the conformal bootstrap program to derive necessary conditions for emergent symmetry enhancement from discrete symmetry (e.g. Zn) to continuous symmetry (e.g. U (1)) under the renormalization group flow. In three dimensions, in order for Z2 symmetry to be enhanced to U (1) symmetry, the conformal bootstrap program predicts that the scaling dimension of the order parameter field at the infrared conformal fixed point must satisfy ∆1 > 1.08. We also obtain the similar conditions for Z3 symmetry with ∆1 > 0.580 and Z4 symmetry with ∆1 > 0.504 from the simultaneous conformal bootstrap analysis of multiple four-point functions. Our necessary conditions impose severe constraints on many controversial physics such as the chiral phase transition in QCD, the deconfinement criticality in Néel-VBS transitions and anisotropic deformations in critical O(n) models. In some cases, we find that the conformal bootstrap program dashes hopes of emergent symmetry enhancement proposed in the literature.

Compensating strong coupling with large charge

“…It is remarkable that the formulae obtained for the anomalous dimensions of charged operators agree extremely well with their numerical (non-perturbative, in principle) values obtained in [10] (see also [11,12]), even for small values of the global charge (see section five). This seems to imply that the analytic expressions in [1] for anomalous dimensions are such that the terms proportional to positive powers of J 1/2 are universal.…”

Emergence and spontaneous breaking of approximate O(4) symmetry at a weakly first-order deconfined phase transition

“…But if so, the lengthscale required to see it seems to be inaccessible at present: we can access the 'pseudocritical' regime but not the 'spin flop' regime at larger scales. (It is also conceivable that the ultimate fate of the models with SO(5) symmetry is something else, though this is constrained by conformal bootstrap [26,59,60].) By contrast, we show here that the phenomenon of a weakly first order transition with emergent approximate O(4) symmetry can be seen in models for deconfined criticality with four order parameter components, or rather in at least one such model.…”

autoboot: a generator of bootstrap equations with global symmetry

“…At this point, it is straightforward to convert it into a form understandable by a semi-definite program solver spdb. We implemented a function toCboot[...] which constructs a Python program which uses cboot [46]. After saving the Python program to a file, some minor edits will be necessary to set up the gaps in the assumed spectrum etc.…”

“…There are also various computing tools developed to perform the numerical bootstrap more easily and more efficiently. For example, we now have a dedicated semi-definite programming solver sdpb [33], and two Python interfaces to sdpb, namely PyCFTBoot [45] and cboot [46]. There is also a Julia implementation JuliBootS [90], and also a Mathematica package to generate 4d bootstrap equations of arbitrary spin [62].…”