Monte Carlo approach to the conformal bootstrap

We introduce an approach to find approximate numerical solutions of truncated bootstrap equations for Conformal Field Theories (CFTs) in arbitrary dimensions. The method is based on a stochastic search via Metropolis algorithm guided by action $S$ which the logarithm single scalar field correlator. While conformal methods semi-definite programming put rigorous exclusion bounds CFTs, this looks solutions, correspond local minima $S$, when present, and can be even far from extremality region. By protocol we that if no constraint operator scaling dimensions imposed, has minimum, corresponding Free Theory. If fix external dimension, however, encounter studied with our approach. Imposing conserved stress-tensor, $\mathbf{Z}_2$ symmetry one relevant scalar, identify two regions where are present. When projected $(\Delta_\sigma, \Delta_{\epsilon})$-plane, $\sigma$ $\epsilon$ being lightest exchanged operators, these essentially coincides line found previous studies. other region along generalized free theories $d = 2$ below both 3$ 4$. empirically prove some associated known theories, including $2d$ $3d$ Ising Yang-Lee model.