Abstract
Fixed points in three dimensions described by conformal field theories with \ensuremath{M N}_{m,n} = O(m)^n\rtimes S_nMNm,n=O(m)n⋊Sn global symmetry have extensive applications in critical phenomena. Associated experimental data for m=n=2m=n=2 suggest the existence of two non-trivial fixed points, while the \varepsilonε expansion predicts only one, resulting in a puzzling state of affairs. A recent numerical conformal bootstrap study has found two kinks for small values of the parameters mm and nn, with critical exponents in good agreement with experimental determinations in the m=n=2m=n=2 case. In this paper we investigate the fate of the corresponding fixed points as we vary the parameters mm and nn. We find that one family of kinks approaches a perturbative limit as mm increases, and using large spin perturbation theory we construct a large mm expansion that fits well with the numerical data. This new expansion, akin to the large NN expansion of critical O(N)O(N) models, is compatible with the fixed point found in the \varepsilonε expansion. For the other family of kinks, we find that it persists only for n=2n=2, where for large mm it approaches a non-perturbative limit with \Delta_\phi\approx 0.75Δϕ≈0.75. We investigate the spectrum in the case \ensuremath{M N}_{100,2}MN100,2 and find consistency with expectations from the lightcone bootstrap.
Highlights
Second-order phase transitions display scale-invariant physics and are widely believed to be described by conformal field theories (CFTs), which arise at fixed points of the renormalization group (RG) flow
By tuning mass parameters, the field theory flows under the RG to a fixed point preserving the same global symmetry
We show here that the m−1 corrections can be computed in a perturbative expansion similar to the usual large N expansion of the O(N ) model, where the operator X acts as the Hubbard–Stratonovich auxiliary field
Summary
Second-order phase transitions display scale-invariant physics and are widely believed to be described by conformal field theories (CFTs), which arise at fixed points of the renormalization group (RG) flow. By tuning mass parameters (equivalent to tuning the temperature in experiments), the field theory flows under the RG to a fixed point preserving the same (or larger) global symmetry Methods within this paradigm, such as the expansion [1], produce in many cases values of the critical exponents that match well with experiments; see [2] for an extensive review. 2 Evidence for the existence of further non-perturbative fixed points in the chiral region has been offered [6,7,8], but this has been disputed by other authors [4, 9,10,11,12,13,14] This contradictory set of observations motivated the recent study of MN symmetric theories using the non-perturbative (numerical) conformal bootstrap [15]. For our numerical computations we have used PyCFTBoot [18], qboot [35] and SDPB [36]
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