Fuzzy Sphere Meets Conformal Bootstrap 2025

2 juin 2025, 09:00 → 6 juin 2025, 17:00 Europe/Paris

Centre de conférences Marilyn et James Simons (Le Bois-Marie)

Fuzzy Sphere Meets Conformal Bootstrap 2025

Since 2023, the fuzzy sphere regulator has become a valuable tool in analyzing 2+1D (quantum) phases transitions and comparing with CFT. After the first edition of the "Fuzzy Sphere Meets Conformal Bootstrap" workshop at the SCGP, Stony Brook (link to https://scgp.stonybrook.edu/archives/41863), this second meeting will bring together condensed matter and high energy physicists working on numerical methods, conformal field theory, and related areas.

For its second edition, the Fuzzy Sphere Meets Conformal Bootstrap workshop, organized by Emilie Huffman (Perimeter Institute), Slava Rychkov (IHES), Yin-Chen He (YITP, Stony Brook University), Liam Fitzpatrick (Boston University) will take place on June 2-6, 2025.

This workshop is supported by the Simons Collaboration on the Non-Perturbative Bootstrap

 

lun. 2 juin

09:00 Registration and welcome coffee

1 Deconfined Quantum Criticality from SO(5) to O(4)

Through the fuzzy sphere regularization, we present the renormalization group flow of field operators of deconfined quantum criticality from SO(5) to O(4).

Orateur: Wei ZHU (Westlake University)

10:30 Coffee break

2 Multiple Scaling Dimensions from Operator Covariance in Monte Carlo Simulations

In classical and quantum Monte Carlo simulations of critical systems, the focus is often on the standard critical exponents, which correspond to the relevant scaling dimensions of the underlying continuum field theory. I will discuss a method by which multiple scaling dimensions can be computed efficiently bu diagonalizing a covariance matrix. While such an approach is common for extracting low-lying levels in the spectrum of a Hamiltonian, I show that the eigenvalues also disentangle scaling dimensions in the regime where they exhibit power-law decays. I will illustrate the method with the 2D and 3D classical Ising models, the Blume-Capel model at its tricritical point, as well as 1D quantum systems. The method should be useful for further studying what appears to be an SO(5) "deconfined" multi-critical point in the setting of J-Q models.

Orateur: Anders SANDVIK (Boston University)

Vidéo

12:00 Lunch

3 Fuzzy Sphere, Conformal Generators, and Ising Field Theory

In this talk, I will present recent results on constructing conformal generators on the fuzzy sphere for the 3D Ising conformal field theory. Our approach builds translations and special conformal transformations directly from the Hamiltonian density and as a result requires additional tuning to suppress contributions from irrelevant operators and their descendants. I will show how these improved generators can be used to accurately isolate primary operators. In the second part of the talk, I will discuss deformations by the lowest Z2-odd relevant operator, which defines the so-called Ising Field Theory. This deformation is especially rich and well studied in two dimensions, and I will present new results that emerge in the 3D setting.

Orateur: Giulia FARDELLI (Boston University)

Vidéo

15:00 Coffee break

4 Conformal Perturbation Theory, ED and MPS simulations for 3D Ising and O(2) Fuzzy Sphere Models

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 [1] 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.

Orateur: Andreas LÄUCHLI (École Polytechnique Fédérale de Lausanne (EPFL))

Vidéo

mar. 3 juin

09:00 Registration and welcome coffee

5 SO(8) Unified Theory of 2+1 Dimensional Dirac Fermions and the Dark Side of the Gross-Neveu Model

Dirac fermions appear as the low-energy quasiparticle excitations in many condensed matter systems of current interest. They are often weakly interacting, as in graphene for example, but at intermediate interactions they can also exhibit numerous ordered ground states by suffering continuous phase transitions at which gapless fermions play an essential role. I will discuss a unified theory of leading order parameters for Dirac systems in 2+1 dimensions based on their underlying O(8) symmetry. The interacting field theory with the general O(M) symmetry is nothing but the cellebrated Gross-Neveu model. The O(M) symmetry-preserving transition of the Gross-Neveu model for one sign of the interaction coupling is reasonably well understood; I will argue, however, that for the opposite sign of the same coupling the theory could have a previously unsuspected transition at which O(M) spontaneously breaks to O(M/2) x O(M/2). The subtle problem of the universality class of this transition in which the order parameter is a second-rank real tensor coupled to relativistic fermions will be addressed.

Orateur: Igor HERBUT (Simon Fraser University)

Vidéo

6 How the SO(5) k=1 WZW near-CFT3 explains the underdoped regime of the cuprate high temperature superconductors

There is now much theoretical evidence that quantum spin liquids proximate to the Neel state on the square lattice are associated with the near-CFT3 described by the SO(5) non-linear sigma model with a level k=1 Wess-Zumino-Witten term. I will use a method of ancilla qubits to describe the confining states obtained when such an antiferromagnet is doped. Key features of the near-CFT3 help explain numerous experiments on the cuprates including (i) d-wave superconductivity with 4 nodal quasiparticles with anisotropic velocities, (ii) quantum oscillations in the charge-ordered states, (iii) higher energy RIXS spectra, and (iv) recent observations of the Yamaji effect in the pseudogap.

Orateur: Subir SACHDEV (Harvard University)

Vidéo

10:30 Coffee break

7 O(3) CFT from Truncated Quantum Rotors

We introduce a general method for investigating O(N) criticality with truncated quantum rotor models, specialising in the O(3) model on a fuzzy sphere. Applying conformal perturbation theory, we locate the critical point, compute scaling dimensions via the state-operator correspondence, and extract operator product expansion (OPE) coefficients from finite-size effects, essentially turning noise into signal. Our investigation reveals a range of new operators, including an O(3) pseudoscalar Lorentz pseudovector, which resolves a long-standing puzzle in the critical exponents of dimerised antiferromagnets. Furthermore, the appearance of the stress-energy tensor and the conserved Noether current in the spectrum validate the presence of conformal symmetry

Orateur: Arjun DEY (Paul Scherrer Institut / École Polytechnique Fédérale de Lausanne)

Vidéo

8 Regularizing 3D Conformal Field Theories via Anyons on the Fuzzy Sphere

In the recently introduced "fuzzy sphere" method, numerical regularization of certain 3D conformal field theories (CFTs) is achieved through the non-commutative geometry of the lowest Landau level filled by electrons; here, the charge is trivially gapped due to the Pauli exclusion principle at filling factor v=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? In this talk, I will show that the 3D Ising CFT can also be realized at fractional electron fillings, unaffected by the exchange statistics of the particles or the nature of topological order in the charge sector. In such cases, the CFT spectrum is intertwined with the charge-neutral spectrum of the underlying fractional quantum Hall (FQH) state -- a feature that is trivially absent in the previously studied v=1 case. Remarkably, the mixing between the CFT and the FQH spectra is strongly suppressed within the numerically-accessible system sizes.

Orateur: Cristian VOINEA (University of Leeds)

Vidéo

12:00 Lunch

9 Magnetised Bounds for Conformal Field Theories

External probes of conformal field theories (CFTs) introduce novel observables and provide new insights into their structure. Well-known examples include CFTs in curved space, at finite temperature, or with a non-zero chemical potential. A key tool in this exploration is the low-energy effective action, which can be systematically constructed using symmetry principles. In this talk, we will focus on three-dimensional CFTs with a global U(1) symmetry coupled to a background magnetic field. Assuming that the magnetic field drives the CFT to a gapped phase, we will examine the associated effective action and use it to define two-point functions of the conserved current and stress-energy tensor. Leveraging dispersive arguments, we will derive constraints on the effective action coefficients and explore their implications for physically relevant observables.

Orateur: Andreas STERGIOU (King's College London)

Vidéo

15:00 Coffee break

10 Free Real Scalar CFT on Fuzzy Sphere: Spectrum, Algebra and Wavefunction Ansatz

Orateur: Yin-Chen HE (Stony Brook University)

mer. 4 juin

09:00 Registration and welcome coffee

11 Bootstrapping Line Defects in CFTs: Applications to 3d Ising Model

Orateur: Max METLISKI (Massachusetts Institute of Technology)

Vidéo

10:30 Coffee break

12 An Upper Bound on the Thermal Mass Gap at Criticality

An intrinsic quantity characterizing a scale-invariant quantum field theory (SFT) is its thermal mass gap, controlling its correlation length at finite temperature. By scale-invariance, this is a pure number in units of the temperature. For 2+1d SFTs, one way to probe the thermal mass gap is via the Casimir energy on a two-torus. Using a certain positivity assumption on the torus Casimir energy that I will motivate, in addition to extra constraints coming from modular invariance, I will show that the thermal mass gap is bounded from above within the class of theories satisfying the positivity constraint, independent of any other theory-dependent data. As a by-product, I will obtain bounds on the shape-dependence of the torus Casimir energy and will study a solution saturating the bounds.

Orateur: Ryan LANZETTA (Perimeter Institute)

Vidéo

12:00 Lunch

jeu. 5 juin

09:00 Registration and welcome coffee

13 Quantization by LLL Projection and BCFT from Fuzzy Hemisphere

Orateur: Mykola DEDUSHENKO (Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS))

Vidéo

10:30 Coffee break

14 Fermion Monte Carlo: DQCP and defects

We provide an overview of fermion models of deconfined quantum critical points (DQCPs) that can be simulated with fermion Monte Carlo methods. The growing evidence that DQCPs are weakly first-order transitions implies the existence of a tuning parameter that can render them strongly first-order. We provide an explicit realization of this phenomenon in the context of the Su-Schrieffer-Heeger model. The DQCP is a realization of an emergent anomaly, which implies the existence of fermionic edge states on open manifolds. Using a realization of the DQCP between a quantum spin Hall insulator and an s-wave superconductor, we provide numerical evidence for this confirming this picture. Finally, we touch on defects realized by spin chains in metals and semimetals, as well as on the simulation of continuum field theories.

Orateur: Fakher ASSAAD (University of Würzburg)

Vidéo

12:00 Lunch

15 Lee-Yang on the Fuzzy Sphere – Introduction

This talk is divided into two parts.

In the first part, we review the Lee-Yang theorem and the associated Lee-Yang conformal field theory (CFT). We then introduce a Fuzzy Sphere regulator to study the 3D Lee-Yang CFT. The model is constructed by deforming the Ising model on the Fuzzy Sphere with a purely imaginary longitudinal magnetic field, leading to a quantum phase transition.

In the second part, we present the main results of arXiv:2505.07655, where the critical point of the model is identified with the 3D Lee-Yang CFT. We describe how to tune the model and show that the lowest-lying states of the Hamiltonian align well with the expected CFT spectrum. We discuss Fuzzy Sphere estimates for the scaling dimension ∆ϕ of the lowest primary operator and interpret small deviations from the expected CFT values in terms of leading irrelevant operators. Finally, we show that the Fuzzy Sphere results are consistent with the best available five-loop ϵ-expansion estimates.

Orateur: Joan ELIAS-MIRO (International Centre for Theoretical Physics)

Vidéo

16 Yang-Lee on the fuzzy sphere: spectrum and finite-size scaling (REMOTE)

In this short talk, I will present the spectrum of the Yang-Lee conformal field theory. Here we utilize use the conformality of the energy spectrum and its consistency with conformal perturbation theory for determining critical points that requires no a priori knowledge of CFT scaling dimensions. We also discuss a new finite-size scaling on the fuzzy sphere that allows us to extract conformal data more reliably, and compare it with the conventional analysis using the (1+1)-dimensional Yang-Lee problem as an example. Our results show broad agreement with previous Monte-Carlo and conformal bootstrap results. We also uncover one previously unknown primary operator and several operator product expansion coefficients.

Orateur: Ruihua FAN (University of California, Berkeley)

Vidéo

15:00 Coffee break

17 Yang-Lee Criticality in Various Dimension (REMOTE)

Yang-Lee criticality is the simplest non-Hermitian conformal field theory. The model was first reported as a phase transition of Ising model in imaginary longitudinal magnetic field more than half a centry ago. Since then, many qualitative and quantitative properties of YL criticality have been studied, remarkably, including the fact that the model can be described in Landau-Ginzburg scheme with a scalar $i\phi^3$ theory in $D<6$ and the fact that the 2D version is an exactly solvable minimal model. In higher dimensions, the model lacks the same level of understanding as the Ising criticality due to its non-Hermitian nature. We report a new study of 3D YL criticality as a phase transition of Fuzzy Sphere model, which facilitates a direct survey of many quantities such as the spectrum and OPE coefficient to high precision. These quantitative results show a beautiful agreement with conformal symmetry and previous estimates from $(6-\epsilon)$ expansion and conformal bootstrap. We also study YL model in 3D and 4D regular polytopes and obtained qualitative agreement with other methods.

Orateur: Yuan XIN (Carnegie Mellon University)

Vidéo

18 Lee-Yang on the Fuzzy Sphere - Results and Summary

In the first part, we review the Lee-Yang theorem and the associated Lee-Yang conformal field theory (CFT). We then introduce a Fuzzy Sphere regulator to study the 3D Lee-Yang CFT. The model is constructed by deforming the Ising model on the Fuzzy Sphere with a purely imaginary longitudinal magnetic field, leading to a quantum phase transition.

In the second part, we present the main results of arXiv:2505.07655, where the critical point of the model is identified with the 3D Lee-Yang CFT. We describe how to tune the model and show that the lowest-lying states of the Hamiltonian align well with the expected CFT spectrum. We discuss Fuzzy Sphere estimates for the scaling dimension ∆ϕ of the lowest primary operator and interpret small deviations from the expected CFT values in terms of leading irrelevant operators. Finally, we show that the Fuzzy Sphere results are consistent with the best available five-loop ϵ-expansion estimates.

Orateur: Joan ELIAS-MIRO (International Centre for Theoretical Physics)

Vidéo

ven. 6 juin

09:00 Registration and welcome coffee

19 Numerical Simulations of Boundary Critical Phenomena

Recent years have seen a significant advance in our understanding of critical systems in the presence of boundaries and defects.
This progress has been driven especially by developments in conformal field theory and in condensed matter physics.
In the talk, I review some of these advances, focusing on Monte Carlo studies.
In the first part I discuss the boundary critical behavior of the three-dimensional O(N) model, and the physics of the recently discovered extraordinary-log phase.
Next, I consider a class of dimerized quantum spin models, whose critical behavior at an edge is not fully understood.
In the final part, I present recent Monte Carlo calculations of the boundary operator product expansion of the three-dimensional Ising universality class, for all known surface universality classes.

Orateur: Francesco PARISEN-TOLDIN (Rheinisch-Westfälische Technische Hochschule Aachen)

Vidéo

10:30 Coffee break

20 Constructing Critical Gauge Theories on the Fuzzy Sphere : Non-Linear σ-Model and Chern-Simons Matter Theories

Gauge theories are not only the IR descriptions of many exotic critical phases and transitions, but also one of the few known methods for constructing interacting CFTs in 3d. Obtaining their conformal data has been a rewarding pursuit. In this talk, I present a construction of critical gauge theories on the fuzzy sphere through an intermediate step of the non-linear σ-model (NLSM) with Wess-Zumino-Witten (WZW) term. I demonstrate how fuzzy sphere models can be matched with NLSM-WZW, and how NLSM-WZW can correspond to critical gauge theories. As a concrete example, I focus on the discovery of a series of Sp(N)-symmetric CFTs corresponding to the NLSM on symplectic Grassmannians, whose most natural candidates are Chern-Simons theories coupled to critical scalar/fermion fields. I present the numerical evidence for emergent conformal symmetry and discuss their operator contents. I will also briefly highlight some of our other progress on the fuzzy sphere, including conformal defects, boundaries and our numerical package FuzzifiED.

Orateur: Zheng ZHOU (Perimeter Institute)

Vidéo

12:00 Lunch

21 Critical Gauge Theories on the Fuzzy Sphere: a QMC Perspective

The deconfined quantum critical point (DQCP) exemplifies a phase transition beyond the Landau paradigm, yet its true nature remains debated. We investigate a generalisation of the SO(5) non-linear sigma model with a Wess-Zumino-Witten term—a possible effective description of the DQCP—using the recently proposed fuzzy sphere regularization. This method offers a powerful lens to probe the model’s critical behavior. Employing quantum Monte Carlo simulations of the generalized Sp(N)-symmetric model, we find evidence of conformal symmetry for N ≥ 4 . For N = 2, corresponding to SO(5), our results indicate spontaneous symmetry breaking, consistent with a pseudo-critical regime identified in earlier studies.

Orateur: Johannes HOFMAN (Max-Planck-Institut für Physik komplexer Systeme)

Vidéo

15:00 Coffee break

22 Fuzzy Sphere Meets Entanglement Bootstrap

I will introduce the entanglement bootstrap, a program to understand and extract the universal data of quantum many body states from local entanglement information. Since this universal data is usually encoded in a quantum field theory, the program can teach us about QFT. The program proceeds by identifying conditions satisfied by the reduced density matrix of a ball for a given class of fixed-point states. So far it has been successful for states with liquid topological order, and for CFT in 1+1 dimensions. The current frontier is 2+1d conformal field theory, where the fuzzy sphere is very useful.

Orateur: John McGreevy (University of California, San Diego)

Vidéo