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arXiv:2502.14965 · Physical Review Research

Boundary operator product expansion coefficients of the three-dimensional Ising universality class

Dorian Przetakiewicz, Stefan Weßel, Francesco Parisen Toldin

Published Jul 3, 2025• 2 citations
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Abstract

Recent advances in conformal field theory and critical phenomena have focused on the characterization of boundary or defects in a conformally invariant system. In this Letter, we study the critical behavior of the three-dimensional Ising universality class in the presence of a surface, realizing the ordinary, the special, and the normal universality classes. By combining high-precision Monte Carlo simulations of an improved model, where leading scaling corrections are suppressed, with a finite-size scaling analysis informed by conformal field theory, we determine unbiased, accurate estimates of universal boundary operator product expansion coefficients of experimental relevance. Furthermore, we improve the value of the scaling dimension of the surface field at the special transition by the estimate Δ ̂ σ = 0.3531 ( 3 ) .

DOI
10.1103/qkq6-b6hx

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Recent advances in conformal field theory and critical phenomena have focused on the characterization of boundary or defects in a conformally invariant system. In this Letter, we study the critical behavior of the three-dimensional Ising universality class in the presence of a surface, realizing the ordinary, the special, and the normal universality classes. By combining high-precision Monte Carlo simulations of an improved model, where leading scaling corrections are suppressed, with a finite-size scaling analysis informed by conformal field theory, we determine unbiased, accurate estimates of universal boundary operator product expansion coefficients of experimental relevance. Furthermore, we improve the value of the scaling dimension of the surface field at the special transition by the estimate Δ ̂ σ = 0.3531 ( 3 ) .
We study the critical behavior at the ordinary surface universality class of the three-dimensional O(N)\mathrm{O}(N)O(N) model, bounded by a two-dimensional surface. Using high-precision Monte Carlo simulations of an improved lattice model, where the leading bulk scaling correction is suppressed, and finite-size scaling analysis of the fourth cumulant of the surface magnetization, we obtain precise estimates of the scaling dimension of the surface field operator for N=2,3,4N=2,3,4N=2,3,4. We also determine the fixed-point values of two renormalization-group invariant observables, which characterize the finite-size scaling behavior at the ordinary transition.
Thanks to the impressive progress of conformal bootstrap methods we have now very precise estimates of both scaling dimensions and operator product expansion coefficients for several 3D universality classes. We show how to use this information to obtain similarly precise estimates for off-critical correlators using conformal perturbation. We discuss in particular the $⟨\ensuremath{\sigma}(r)\ensuremath{\sigma}(0)⟩$, $⟨\ensuremath{\epsilon}(r)\ensuremath{\epsilon}(0)⟩$ and $⟨\ensuremath{\sigma}(r)\ensuremath{\epsilon}(0)⟩$ two-point functions in the high and low temperature regimes of the 3D Ising model and evaluate the leading and next to leading terms in the $s={\mathrm{tr}}^{{\mathrm{\ensuremath{\Delta}}}_{t}}$ expansion, where ttt is the reduced temperature. Our results for $⟨\ensuremath{\sigma}(r)\ensuremath{\sigma}(0)⟩$ agree both with Monte Carlo simulations and with a set of experimental estimates of the critical scattering function.
Recently the operator product expansion coefficients of the 3D Ising model universality class have been calculated by studying via Monte Carlo simulation the two-point functions perturbed from the critical point with a relevant field. We show that this method can be applied also when the perturbation is performed with a relevant field coupled to a nonuniform potential acting as a trap. This setting is described by the trap size scaling ansatz, which can be combined with the general framework of the conformal perturbation in order to write down the correlators $⟨\ensuremath{\sigma}(\mathbf{r})\ensuremath{\sigma}(0)⟩$, $⟨\ensuremath{\sigma}(\mathbf{r})\ensuremath{\epsilon}(0)⟩$ and $⟨\ensuremath{\epsilon}(\mathbf{r})\ensuremath{\epsilon}(0)⟩$, from which the operator product expansion coefficients can be estimated. We find ${C}_{\ensuremath{\sigma}\ensuremath{\epsilon}}^{\ensuremath{\sigma}}=1.051(3)$, in agreement with the results already known in the literature, and ${C}_{\ensuremath{\epsilon}\ensuremath{\epsilon}}^{\ensuremath{\epsilon}}=1.32(15)$, confirming and improving the previous estimate obtained in the uniform perturbation case.
We study the special point in the phase diagram of a semi-infinite system where the bulk transition is in the three-dimensional Ising universality class. To this end we performed a finite-size-scaling study of the improved Blume-Capel model on the simple cubic lattice with two different types of surface interaction. In order to check for the effect of leading bulk corrections we also simulated the spin-1/2 Ising model on the simple cubic lattice. We accurately estimated the surface enhancement coupling at the special point of these models. We find yt1=0.718(2){y}_{{t}_{1}}=0.718(2)yt1​​=0.718(2) and yh1=1.6465(6){y}_{{h}_{1}}=1.6465(6)yh1​​=1.6465(6) for the surface renormalization group exponents of the special transition. These results are compared with previous ones obtained by using field-theoretic methods and Monte Carlo simulations of the spin-1/2 Ising model. Furthermore, we study the behavior of the surface transition near the special point, and finally we discuss films with special boundary conditions at one surface and fixed ones at the other.
We study the boundary critical behavior of the three-dimensional Heisenberg universality class, in the presence of a bidimensional surface. By means of high-precision Monte Carlo simulations of an improved lattice model, where leading bulk scaling corrections are suppressed, we prove the existence of a special phase transition, with unusual exponents, and of an extraordinary phase with logarithmically decaying correlations. These findings contrast with naïve arguments on the bulk-surface phase diagram, and allow us to explain some recent puzzling results on the boundary critical behavior of quantum spin models.
Boundary conformal field theory (BCFT) provides a universal framework for critical phenomena in the presence of boundaries. We determine BCFT data for the normal and ordinary boundary universality classes of the 1+11+11+1-dimensional boundaries of the 2+12+12+1-dimensional O(2)O(2)O(2) and O(3)O(3)O(3) Wilson-Fisher fixed points, realized microscopically by a bilayer Heisenberg model on the fuzzy sphere. Using the fuzzy-sphere state-operator correspondence, we obtain boundary operator spectra, identify low-lying boundary primary operators, extract operator-product-expansion (OPE) data, and estimate the boundary central charges for both boundary conditions. For the normal boundary condition, the universal amplitudes aσa_σaσ​ and btb_tbt​ extracted from one- and two-point functions agree quantitatively with Monte Carlo benchmarks where available. For both N=2N=2N=2 and N=3N=3N=3, we find a positive extraordinary-log exponent ααα, providing independent microscopic evidence for extraordinary-log boundary criticality. Our results extend fuzzy-sphere BCFT spectroscopy beyond the Ising universality class to continuous O(N)O(N)O(N) symmetry.