Exact Diagonalization, Matrix Product States and Conformal Perturbation Theory Study of a 3D Ising Fuzzy Sphere Model

TL;DR

This work investigates a 3D Ising fuzzy sphere regulator with Conformal Perturbation Theory (CPT) to address finite-size effects and extract CFT data from finite-volume spectra. By combining exact diagonalization and matrix product state methods, the authors quantify how near-critical lattices on S^2 encode the 3D Ising CFT data and develop a minimal CPT framework to determine the critical point, speed of light, and leading irrelevant perturbations. They introduce a novel OPE-coefficient extraction method from detuning the microscopic model, revealing new coefficients and offering insight into level mixing and avoided crossings. The study validates CPT against a 1+1D Ising example and demonstrates the method’s potential for precise CFT data extraction, while also identifying finite-size, curvature, and mixing effects that warrant further refinement. Overall, the fuzzy sphere/CPT approach provides a concrete, scalable path to connect finite-volume spectra to the infinite-volume Ising CFT and related universality classes, with practical implications for conformal bootstrap benchmarks and numerical studies of critical phenomena.

Abstract

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 $S^{d-1}$ as the spatial manifold. Recently, the fuzzy sphere regulator in Ref. [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.

Figures (12)

• Figure 1: Weyl transformation from $\mathbb{R}^d$ to the cylinder $S^{d-1}\times \mathbb{R}$.
• Figure 2: Preparing ket states on $\mathbb{R}^d$ (left) and on the cylinder (right).
• Figure 3: Preparing bra states on $\mathbb{R}^d$ (left) and on the cylinder (right).
• Figure 4: The matrix element (left), expressed as an integrated correlator on the cylinder where the bra and ket states are prepared by inserting operators ${\cal O}_{\rm cyl}$ at infinite future and infinite past (center), and as an integrated CFT three-point function on $\mathbb{R}^d$ (right).
• Figure 5: Comparison of the ED energy gaps with the 2D Ising CFT spectrum and with the 2D Ising spectrum corrected with the help of a CPT model. Circles and full colored lines: ED energy gaps for the Hamiltonian \ref{['eq:Ham_1DTFI']} for $N=26$ rescaled by $R/v$ and plotted as a function of $h_x$ for $h_z=1$. Horizontal gray lines: the exact 2D Ising CFT spectrum. Squares and dashed colored lines: the 2D Ising spectrum corrected by the Reinicke CPT model with couplings $g_\varepsilon$, $g_{T\widetilde{T}}$, $g_{T^2+\widetilde{T}^2}$ obtained by fit to $T,\sigma,\partial\sigma,\varepsilon,\partial\varepsilon$ levels. The vertical dashed line denotes the location of the critical point, determined from the equation $g_\varepsilon=0$. The CPT model describes the ED energy gaps very well, despite the visible deviations of the ED energy gaps from the 2D Ising CFT values. The CPT and ED agreement is so good that the dashed lines are essentially invisible, hidden behind solid lines, for all the levels except $\partial^2\varepsilon$. Accordingly the circles are always inside the squares.

Theorems & Definitions (1)

• Remark 5.1