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Register a new userBootstrapping the O(N) Vector Models
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We study the conformal bootstrap for 3D CFTs with O(N) global symmetry. We obtain rigorous upper bounds on the scaling dimensions of the first O(N) singlet and symmetric tensor operators appearing in the $phi_i times phi_j$ OPE, where $phi_i$ is a fundamental of O(N). Comparing these bounds to previous determinations of critical exponents in the O(N) vector models, we find strong numerical evidence that the O(N) vector models saturate the bootstrap constraints at all values of N. We also compute general lower bounds on the central charge, giving numerical predictions for the values realized in the O(N) vector models. We compare our predictions to previous computations in the 1/N expansion, finding precise agreement at large values of N.
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We use numerical bootstrap techniques to study correlation functions of a traceless symmetric tensors of $O(N)$ with two indexes $t_{ij}$. We obtain upper bounds on operator dimensions for all the relevant representations and several values of $N$. We discover several families of kinks, which do not correspond to any known model and we discuss possible candidates. We then specialize to the case $N=4$, which has been conjectured to describe a phase transition in the antiferromagnetic real projective model $ARP^{3}$. Lattice simulations provide strong evidence for the existence of a second order phase transition, while an effective field theory approach does not predict any fixed point. We identify a set of assumptions that constrain operator dimensions to a closed region overlapping with the lattice prediction. The region is still present after pushing the numerics in the single correlator case or when considering a mixed system involving $t$ and the lowest dimension scalar singlet.
We compute critical exponents of O(N) models in fractal dimensions between two and four, and for continuos values of the number of field components N, in this way completing the RG classification of universality classes for these models. In d=2 the N-dependence of the correlation length critical exponent gives us the last piece of information needed to establish a RG derivation of the Mermin-Wagner theorem. We also report critical exponents for multi-critical universality classes in the cases N>1 and N=0. Finally, in the large-N limit our critical exponents correctly approach those of the spherical model, allowing us to set N~100 as threshold for the quantitative validity of leading order large-N estimates.
A variety of lattice discretisations of continuum actions has been considered, usually requiring the correct classical continuum limit. Here we discuss weird lattice formulations without that property, namely lattice actions that are invariant under most continuous deformations of the field configuration, in one version even without any coupling constants. It turns out that universality is powerful enough to still provide the correct quantum continuum limit, despite the absence of a classical limit, or a perturbative expansion. We demonstrate this for a set of O(N) models (or non-linear $sigma$-models). Amazingly, such weird lattice actions are not only in the right universality class, but some of them even have practical benefits, in particular an excellent scaling behaviour.
We study the conformal bootstrap constraints for 3D conformal field theories with a $mathbb{Z}_2$ or parity symmetry, assuming a single relevant scalar operator $epsilon$ that is invariant under the symmetry. When there is additionally a single relevant odd scalar $sigma$, we map out the allowed space of dimensions and three-point couplings of such Ising-like CFTs. If we allow a second relevant odd scalar $sigma$, we identify a feature in the allowed space compatible with 3D $mathcal{N}=1$ superconformal symmetry and conjecture that it corresponds to the minimal $mathcal{N}=1$ supersymmetric extension of the Ising CFT. This model has appeared in previous numerical bootstrap studies, as well as in proposals for emergent supersymmetry on the boundaries of topological phases of matter. Adding further constraints from 3D $mathcal{N}=1$ superconformal symmetry, we isolate this theory and use the numerical bootstrap to compute the leading scaling dimensions $Delta_{sigma} = Delta_{epsilon} - 1 = .58444(22)$ and three-point couplings $lambda_{sigmasigmaepsilon} = 1.0721(2)$ and $lambda_{epsilonepsilonepsilon} = 1.67(1)$. We additionally place bounds on the central charge and use the extremal functional method to estimate the dimensions of the next several operators in the spectrum. Based on our results we observe the possible exact relation $lambda_{epsilonepsilonepsilon}/lambda_{sigmasigmaepsilon} = tan(1)$.
We study the conformal bootstrap for a 4-point function of fermions $langlepsipsipsipsirangle$ in 3D. We first introduce an embedding formalism for 3D spinors and compute the conformal blocks appearing in fermion 4-point functions. Using these results, we find general bounds on the dimensions of operators appearing in the $psi times psi$ OPE, and also on the central charge $C_T$. We observe features in our bounds that coincide with scaling dimensions in the Gross-Neveu models at large $N$. We also speculate that other features could coincide with a fermionic CFT containing no relevant scalar operators.
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