Numerical Tests of Conjectures of Conformal Field Theory for 3D Systems

Wolfhard Janke E, Martin Weigel

W. Zhu, Chao Han, Emilie Huffman +2 more

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.

P. Surànyi

Al.B. Zamolodchikov, Al.B. Zamolodchikov

Cristian Voinea, Ruihua Fan, Nicolas Regnault +1 more

The recently introduced “fuzzy-sphere” method has enabled accurate numerical regularizations of certain three-dimensional (3D) conformal field theories (CFTs). The regularization is provided by the noncommutative geometry of the lowest Landau level filled by electrons, such that the charge is trivially gapped due to the Pauli exclusion principle at filling factor ν = 1 , while the electron spins encode the desired CFT. Successful applications of the fuzzy sphere to paradigmatic CFTs, such as the 3D Ising model, raise an important question: How finely tuned does the underlying electron system need to be? Here, we show that the 3D Ising CFT can also be realized at electron fillings. In such cases, the CFT spectrum is intertwined with the charge-neutral spectrum of the underlying fractional quantum Hall state—a feature that is trivially absent in the previously studied ν = 1 case. Remarkably, we show that the mixing between the CFT spectrum and the fractional quantum Hall spectrum is strongly suppressed within the numerically accessible system sizes. Moreover, we demonstrate that the CFT critical point is unaffected by the exchange statistics of the particles and by the nature of topological order in the charge sector. Our results set the stage for the fuzzy-sphere exploration of conformal critical points between topologically ordered states.

Zheng Zhou, Yin-Chen He

The quest to discover new 3D CFTs has been intriguing for physicists. For this purpose, fuzzy sphere regularization that studies interacting quantum systems defined on the lowest Landau level on a sphere has emerged as a powerful tool. In this Letter, we discover a series of new CFTs with global symmetry Sp(N) in the fuzzy sphere models that are closely related to the SO(5) deconfined phase transition, and are related to a Sp(N)/[Sp(M)×Sp(N-M)] nonlinear sigma model with a Wess-Zumino-Witten term. We numerically verify the emergent conformal symmetry by observing the integer-spaced conformal multiplets, the quality of conformal generators, and studying the finite-size scaling of the conformality. We discuss possible candidates for these newly discovered CFTs, the most plausible ones being Chern-Simons-matter theories which have N flavors of gapless bosons or fermions coupled to a non-Abelian [viz. Sp(1), Sp(2), etc.] Chern-Simons gauge field. Our Letter provides new avenues for studying interacting CFTs in 3D, possibly facilitating the nonperturbative study of critical gauge theories and previously undiscovered CFTs.