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arXiv:2407.00459

The Ising Model on $\mathbb S^2$

Richard C. Brower, Evan K. Owen

Published Jun 29, 2024• 0 citations
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Abstract

We define a 2-dimensional Ising model on a triangulated sphere, $\mathbb S^2$, designed to approach the exact conformal field theory (CFT) in the continuum limit. Surprisingly, the derivation leads to a set of geometric constraints that the lattice field theory must satisfy. Monte Carlo simulations are in agreement with the exact Ising CFT on $\mathbb S^2$. We discuss the inherent benefits of using non-uniform simplicial lattices and how these methods may be generalized for use with other quantum theories on curved manifolds.

DOI
10.48550/arxiv.2407.00459

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The Ising Model on S2\mathbb S^2S2
We define a 2-dimensional Ising model on a triangulated sphere, S2\mathbb S^2S2, designed to approach the exact conformal field theory (CFT) in the continuum limit. Surprisingly, the derivation leads to a set of geometric constraints that the lattice field theory must satisfy. Monte Carlo simulations are in agreement with the exact Ising CFT on S2\mathbb S^2S2. We discuss the inherent benefits of using non-uniform simplicial lattices and how these methods may be generalized for use with other quantum theories on curved manifolds.
I derive a formulation of the 2-dimensional critical Ising model on non-uniform simplicial lattices. Surprisingly, the derivation leads to a set of geometric constraints that a lattice must satisfy in order for the model to have a well-defined continuum limit. I perform Monte Carlo simulations of the critical Ising model on discretizations of several non-trivial manifolds including a twisted torus and a 2-sphere and I show that the simulations are in agreement with the 2d Ising CFT in the continuum limit. I discuss the inherent benefits of using non-uniform simplicial lattices to study quantum field theory and how the methods developed here can potentially be generalized for use with other theories.
Conformal field theory predicts finite-size scaling amplitudes of correlation lengths universally related to critical exponents on sphere-like, semi-finite systems S d − 1 × of arbitrary dimensionality d . Numerical studies have up to now been unable to validate this result due to the intricacies of lattice discretisation of such curved spaces. We present a cluster-update Monte Carlo study of the Ising model on a three-dimensional geometry using slightly irregular lattices that confirms the validity of a linear amplitude-exponent relation to high precision.
We investigate a critical Ising-like model in the curved geometry ${S}^{2}\ifmmode\times\else\texttimes\fi{}{R}^{1}$ obtained by a conformal mapping of the infinite 3D space R3{R}^{3}R3. The incompatibility of regular lattices with this geometry is avoided by use of the anisotropic limit of the lattice Ising model, which renders one of the space coordinates continuous. We determine magnetic and energylike correlation lengths of this model by means of a cluster Monte Carlo algorithm. From these data, and the assumption of conformal invariance, we obtain the magnetic and temperature scaling dimensions as Xh=0.5178(12){X}_{h}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.5178(12)Xh​=0.5178(12) and Xt=1.423(19){X}_{t}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.423(19)Xt​=1.423(19), respectively. These numbers are in a good agreement with the existing results for the 3D Ising universality class.
Numerical studies of phase transitions in statistical and quantum lattice models provide crucial insights into the corresponding Conformal Field Theories (CFTs). In higher dimensions, comparing finite-volume numerical results to infinite-volume CFT data is facilitated by choosing the sphere S^{d-1} S d − 1 as the spatial manifold. Recently, the fuzzy sphere regulator [Zhu et al, Phys. Rev. X 13, 021009 (2023)] has enabled such studies with exact rotational invariance, yielding impressive agreement with known 3D Ising CFT predictions, as well as new results. However, systematic improvements and a deeper understanding of finite-size corrections remain essential. In this work, we revisit the fuzzy sphere regulator, focusing on the original Ising model, with two main goals. First, we assess the robustness of this approach using Conformal Perturbation Theory (CPT), to which we provide a detailed guidebook. We demonstrate how CPT provides a unified framework for determining the critical point, the speed of light, and residual deviations from CFT predictions. Applying this framework, we study finite-size corrections and clarify the role of tuning the model in minimizing these effects. Second, we develop a novel method for extracting Operator Product Expansion (OPE) coefficients from fuzzy sphere data. This method leverages the sensitivity of energy levels to detuning from criticality, providing new insights into level mixing and avoided crossings in finite systems. Our work also includes validation of CPT in a 1+1D Ising model away from the integrable limit.
We formulate conformal mappings between an infinite plane and a spheroid, and one between a semi-infinite plane and a half spheroid. Special cases of the spheroid include the surface of an infinitely long cylinder, of a sphere, and of a flat disc. These mappings are applied to the critical Ising model. For the case of the sphere and the flat disc, we derive analytical expressions for the second and the fourth moments of the magnetization density, and thus for the Binder cumulant. Next, we investigate Ising models on spheroids and half spheroids by means of a continuous cluster Monte Carlo method for simulations in curved geometries. Fixed and free boundary conditions are imposed for half spheroids. The Monte Carlo data are analyzed by finite-size scaling. Critical values of the Binder cumulants and other ratios on the sphere and on the flat disc agree precisely with the exact calculations mentioned above. At criticality, we also sample two- and one-point correlation functions on spheroids on half spheroids, respectively. The magnetic and temperature scaling dimensions, as determined from the Monte Carlo data and the theory of conformal invariance, are in good agreement with exact results.
We consider the transverse field Ising model in (2+1)D, putting 12 spins at the vertices of the regular icosahedron. The model is tiny by the exact diagonalization standards, and breaks rotation invariance. Yet we show that it allows a meaningful comparison to the 3D Ising CFT on \mathbb{R}× S^2 ℝ × S 2 , by including effective perturbations of the CFT Hamiltonian with a handful of local operators. This extreme example shows the power of conformal perturbation theory in understanding finite N effects in models on regularized S^2 S 2 . Its ideal arena of application should be the recently proposed models of fuzzy sphere regularization.