Abstract
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.
- Received 22 November 2022
- Revised 7 February 2023
- Accepted 28 March 2023
- Corrected 30 November 2023
DOI:https://doi.org/10.1103/PhysRevX.13.021009
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Corrections
30 November 2023
Correction: A factor of in Eq. (14) was missing two times and has been inserted. The subscript to in Eqs. (A7), (A9), and (A11) was incorrect and has been fixed.
Popular Summary
Phase transitions and related critical phenomena originate from collective behavior of interacting particles, the study of which is generally challenging in statistical and condensed matter physics. Interestingly, if a novel conformal symmetry—a symmetry that can change shapes while maintaining angles—emerges at the phase transition point, numerous insights into critical behaviors can be inferred by using conformal field theory. This approach has been very successful in 2D systems, however, the emergence and appearance of conformal symmetry in higher dimensions is rarely studied, mainly due to unavoidable mathematical or conceptual obstructions. Here, we propose and design an innovative theoretical scheme to study the 3D phase transition on a spherical geometry, wherein the physics of conformal symmetry becomes transparent.
Specifically, we study a quantum Ising transition—a phase transition in a mathematical model of ferromagnetism—defined on a fuzzy (noncommutative) sphere. In this setting, the space coordinates are noncommuting, resulting in a fuzzy geometry akin to the uncertainty principle in quantum mechanics. Our findings reveal the telltale signs of emergent conformal symmetry at the transition, and, notably, we achieve a rich understanding of the 3D Ising phase transition at an astonishingly low computational cost. For example, our model with just four spins produces new results that were unavailable in previous numerical simulations involving millions of spins.
This methodology opens a new avenue for the study of critical phenomena in nature and indicates a new connection between physics (conformal field theory) and mathematics (noncommutative geometry).
