Studying Conformal Field Theories in Three Dimensions with the Fuzzy Sphere

Abstract

Conformal field theory (CFT) is one of the central topics of modern physics. CFT has provided important insights into various aspects of theoretical physics. In condensed-matter physics, it has produced useful predictions about the critical phenomena. Many classical and quantum phase transitions are conjectured to have emergent conformal symmetry in the infra-red (IR). In high-energy physics and quantum field theories, CFT also plays important role in understanding string theory, duality with gravitational theories on anti-de Sitter (AdS) space, and renormalisation group flow structure. In 2D, the infinite-dimensional conformal algebra has made many theories exactly solvable. However, going to the higher dimensions, the CFTs are less well-studied due to a much smaller conformal group, although some of the conformal data are determined at high precision by conformal bootstrap.

The fuzzy-sphere regularisation has emerged as a new powerful method to study 3D CFTs. This method involves studying quantum systems on a sphere that is `fuzzy' (non-commutative) due to a magnetic monopole at its centre. It offers distinct advantages including the exact preservation of rotation symmetry, the direct observation of emergent conformal symmetry and the efficient extraction of conformal data. In the fuzzy-sphere method, the state-operator correspondence plays an essential role. Specifically, there is a one-to-one correspondence between the eigenstates of the critical Hamiltonian on the sphere and the CFT operators, where the energy gaps are proportional to the scaling dimensions. The power of this approach has been first demonstrated in the context of the 3D Ising transition, where the presence of emergent conformal symmetry has been convincingly established. Since its proposal, various studies have greatly expanded its horizon, including accessing the conformal data of the 3D Ising CFT, developing and applying techniques to improve the numerical precision, realising various 3D CFTs at integer filling, exploring fractional quantum Hall (FQH) transitions, and studying the conformal defects, boundaries and lower-dimensional CFTs.

In this thesis, I first review fuzzy-sphere regularisation and the numerical methods applied to it, and focus on three aspects of my own work: constructing 3D interacting CFTs from critical gauge theories and non-linear sigma models, extracting universal data for conformal defects and boundaries, and accessing criticality at fractional quantum Hall (FQH) transitions.

First, we realise the critical gauge theories on the fuzzy sphere with the help of non-linear sigma models (NLSMs) with Wess-Zumino-Witten (WZW) terms. The NLSM captures help us match the symmetry and anomaly of fuzzy-sphere model with those of critical gauge theories with dynamical gauge fields coupled to critical matter. Our first target is the deconfined quantum critical point (DQCP), a paradigmatic beyond-Landau transition, featuring emergent SO(5) symmetry and field-theory dualities. We realise it on a 4-flavour fuzzy-sphere model and provide numerical evidence for approximate conformal symmetry and pseudo-criticality. We further generalise it into a new series of parity-breaking 3D CFTs with Sp(N) global symmetry, whose candidate theories include Chern-Simons matter theories.

Second, we study conformal defects and boundaries, where defect or boundary deformations trigger RG flows to interacting defect or boundary fixed points (dCFTs/bCFTs) with reduced conformal symmetry. The bulk-defect/boundary interactions generate rich phenomena, advancing understanding of topological phases, confinement of gauge theories, quantum gravity, entanglement, and experiment. We introduce a computational strategy revealing a wealth of defect conformal data, including the first non-perturbative computation of an RG-monotonic quantity called $g$-function, via overlaps of different defect configurations; its power is shown for the pinning-field defect on the fuzzy sphere. On the other hand, we realise the ordinary and extraordinaryIsing boundary CFTs on the fuzzy sphere and reported the operator spectrum and OPE coefficients.

Third, we extend fuzzy sphere to CFTs arising from fractional quantum Hall transitions. Many such transitions admit effective descriptions in terms of Chern-Simons-matter theories and dualities, and are motivated by experiments in Moiré materials. We present a minimal example described by a complex critical scalar coupled to U(1)_2 Chern-Simons gauge field, realised as a transition between a ν_f=2 fermionic integer quantum Hall state and a ν_b=1/2 bosonic fractional quantum Hall state. We show that the transition is continuous and governed by a conformal field theory with SO(3) global symmetry. The operator spectrum contains only one relevant Lorentz scalar, the SO(3) singlet with scaling dimension Δ_S=1.52(18), which tunes the transition. Finally, we construct free-Majorana-fermion CFTs on the fuzzy sphere using a set-up of boson-fermion mixture. On the phase diagram, we observe two continuous transitions described respectively by a free Majorana fermion and a gauged Ising CFT. We numerically confirm the emergent conformal symmetry through the operator spectrum and the two-point correlation function of the local Majorana fermion. We further establish a correspondence between the fuzzy-sphere models and the field-theory Lagrangians, and extend it to an interacting fermionic CFT --- the super-Ising theory with emergent super-conformal symmetry.qu