Easy bootstrap for the 3D Ising model: a hybrid approach of the lightcone bootstrap and error minimization methods

TL;DR

This paper introduces a hybrid bootstrap method combining lightcone and error minimization techniques to efficiently approximate the critical properties of the 3D Ising model without positivity constraints, achieving results close to the most precise bounds.

Contribution

It presents a novel, computationally inexpensive hybrid bootstrap approach that accurately estimates critical exponents of the 3D Ising model using minimal crossing constraints.

Findings

Accurately estimates critical scaling dimensions close to rigorous bounds.

Uses only 10 low-derivative crossing constraints for the analysis.

Achieves results comparable to high-cost methods with simple analytic approximations.

Abstract

As a simple lattice model that exhibits a phase transition, the Ising model plays a fundamental role in statistical and condensed matter physics. The Ising transition is realized by physical systems, such as the liquid-vapor transition. Its continuum limit also furnishes a basic example of interacting quantum field theories and universality classes. Motivated by a recent hybrid bootstrap study of the quantum quartic oscillator, we revisit the conformal bootstrap approach to the 3D Ising model at criticality, without resorting to positivity constraints. We use at most 10 nonperturbative crossing constraints at low derivatives from the Taylor expansion around a crossing symmetric point. The high-lying contributions are approximated by simple analytic formulae deduced from the lightcone singularity structure. Surprisingly, the low-lying properties are determined to good accuracy by this computationally very cheap approach. For instance, the results for the two relevant scaling dimensions $(\Delta_\sigma,\Delta_\epsilon)\approx (0.518153,1.41278)$ are close to the most precise rigorous bounds obtained at a much higher computational cost.