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Journal Club for Condensed Matter Physics

Sharpening the Conformal Symmetry of the 3D Ising Transition by Fuzzy Sphere Simulations

Published Jan 31, 2023• 0 citations
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Abstract

Uncovering conformal symmetry in the 3D Ising transition: State-operator correspondence from a fuzzy sphere regularization Authors: Wei Zhu, Chao Han, Emilie Huffman, Johannes S. Hofmann, and Yin-Chen He arXiv:2210.13482; DOI: 10.48550/arXiv.2210.13482 Recommended with a commentary by Ashvin Vishwanath, Harvard University |View Commentary (pdf)| This commentary may be cited as: DOI: 10.36471/JCCM_January_2023_02 https://doi.org/10.36471/JCCM_January_2023_02

DOI
10.36471/jccm_january_2023_02

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Uncovering conformal symmetry in the 3D Ising transition: State-operator correspondence from a fuzzy sphere regularization Authors: Wei Zhu, Chao Han, Emilie Huffman, Johannes S. Hofmann, and Yin-Chen He arXiv:2210.13482; DOI: 10.48550/arXiv.2210.13482 Recommended with a commentary by Ashvin Vishwanath, Harvard University |View Commentary (pdf)| This commentary may be cited as: DOI: 10.36471/JCCM_January_2023_02 https://doi.org/10.36471/JCCM_January_2023_02
The 3D Ising transition, the most celebrated and unsolved critical phenomenon in nature, has long been conjectured to have emergent conformal symmetry, similar to the case of the 2D Ising transition.Yet, the emergence of conformal invariance in the 3D Ising transition has rarely been explored directly, mainly due to unavoidable mathematical or conceptual obstructions.Here, we design an innovative way to study the quantum version of the 3D Ising phase transition on spherical geometry, using the "fuzzy (noncommutative) sphere" regularization.We accurately calculate and analyze the energy spectra at the transition, and explicitly demonstrate the state-operator correspondence (i.e., radial quantization), a fingerprint of conformal field theory.In particular, we identify13 parity-even primary operators within a high accuracy and two parity-odd operators that were not known before.Our result directly elucidates the emergent conformal symmetry of the 3D Ising transition, a conjecture made by Polyakov half a century ago.More importantly, our approach opens a new avenue for studying 3D conformal field theories by making use of the state-operator correspondence and spherical geometry.
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.
Conformal field theory (CFT) offers a modern viewpoint for understanding phase transitions. However, directly accessing the conformal algebra and microscopically uncovering the emergent conformal symmetry, especially in higher dimensions, remains a significant challenge. Motivated by recent advances in revealing CFT features via the fuzzy two-sphere, here we generalize this approach to higher dimensions and aim to expose the conformality at the (3+1)-D quantum critical point. We demonstrate this framework by investigating quantum phase transitions belonging to the Ising and Yang-Lee universality classes in a (3+1)-D quantum model (equivalent to a classical four-dimensional system), realized via Landau level projection on the fuzzy three-sphere. By computing the energy spectra at criticality, we explicitly verify the state-operator correspondence, a hallmark of conformal invariance. Together with prior advances, this work establishes a new pathway for the microscopic study of emergent conformality in higher-dimensional phase transitions.