Numerical estimation of structure constants in the three-dimensional Ising conformal field theory through Markov chain uv sampler

Herdeiro and Doyon [Phys. Rev. E 94, 043322 (2016)] introduced a numerical recipe, dubbed uv sampler, offering precise estimations of the conformal field theory (CFT) data of the planar two-dimensional (2D) critical Ising model. It made use of scale invariance emerging at the critical point in order to sample finite sublattice marginals of the infinite plane Gibbs measure of the model by producing holographic boundary distributions. The main ingredient of the Markov chain Monte Carlo sampler is the invariance under dilation. This paper presents a generalization to higher dimensions with the critical 3D Ising model. This leads to numerical estimations of a subset of the CFT data—scaling weights and structure constants—through fitting of measured correlation functions. The results are shown to agree with the recent most precise estimations from numerical bootstrap methods [Kos, Poland, Simmons-Duffin, and Vichi, J. High Energy Phys. 08 (2016) 036].

Figure 1

Measured lattice correlators. (a) Graph of the $\langle {\sigma }_{\mathbf{0}}^L{\sigma }_{\mathbf{n}}^L\rangle$ correlations as a function of the separation $\left|\mathbf{n}\right|\in [1,50]$. The power-law behavior seems manifest. The gray line represents the most accurate power-law fit. This is solid proof that the sample is very close to the planar expectation. (b) Graph of the $\langle {\varepsilon }_{\mathbf{0}}^L{\varepsilon }_{\mathbf{n}}^L\rangle -{\langle {\varepsilon }^L\rangle }^2$ connected two-point function for separations $\left|\mathbf{n}\right|\in [1,25]$. As much as for ${\sigma }^L{\sigma }^L$ correlations, the power-law behavior is explicit. This observable seems to show more microscopic excess of correlations for the smallest values of the separation. (c) Graph of the connected part $\langle {\sigma }_{-\mathbf{k}}^L{\varepsilon }_{\mathbf{0}}^L{\sigma }_{\mathbf{k}}^L\rangle -\langle {\varepsilon }^L\rangle \langle {\sigma }_{-\mathbf{k}}^L{\sigma }_{\mathbf{k}}^L\rangle$ as a function of $\left|\mathbf{k}\right|\in [3,35]$. With the two ${\sigma }^L$ insertions diametrically opposed when sitting on the ${\varepsilon }^L$ insertion point, CFT predicts a $\propto C_{\varepsilon \sigma \sigma }{\left|\mathbf{k}\right|}^{-2{\mathrm{\Delta }}_{\sigma }-{\mathrm{\Delta }}_{\varepsilon }}$ profile. (d) Graph of the connected part $\langle {\varepsilon }_{-\mathbf{k}}^L{\varepsilon }_{\mathbf{0}}^L{\varepsilon }_{\mathbf{k}}^L\rangle -\langle {\varepsilon }^L\rangle \left(\langle {\varepsilon }_{-\mathbf{k}}^L{\varepsilon }_{\mathbf{0}}^L\rangle +\langle {\varepsilon }_{-\mathbf{k}}^L{\varepsilon }_{\mathbf{k}}^L\rangle +\langle {\varepsilon }_{\mathbf{0}}^L{\varepsilon }_{\mathbf{k}}^L\rangle \right)-{\langle {\varepsilon }^L\rangle }^3$ as a function of $\left|\mathbf{k}\right|\in [5,14]$. Conformal symmetry constrains the decay to be as $C_{\varepsilon \varepsilon \varepsilon }{N}_{\varepsilon }^3{2}^{-{\mathrm{\Delta }}_{\varepsilon }}{\left|\mathbf{k}\right|}^{-3{\mathrm{\Delta }}_{\varepsilon }}$.