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FuzzifiED.jl

Since its proposal, the fuzzy sphere regularisation has made significant contribution to the study of 3d CFTs. This Julia package FuzzifiED is aimed at simplifying the numerical calculations on the fuzzy sphere. It facilitates the exact diagonalisation (ED) calculations as well as the density matrix renormalisation group (DMRG) with the help of ITensor. It can also be used for generic fermionic and bosonic models. It can also apply to generic fermionic and bosonic models. This package offers the following features :

  • Versatility : FuzzifiED can reproduce nearly all ED and DMRG results from fuzzy sphere research. It is also designed to be easily adaptable to new models.
  • Usability : With an intuitive Julia interface, writing and understanding the code is straightforward. To help users get started, we provide a collection of examples.
  • Efficiency : FuzzifiED produces results on reasonable system sizes within minutes.
  • Open source : The FuzzifiED codebase is freely available under the MIT License, welcoming reviews and contributions.

A PDF version of the documentation is provided at this link. If you have any questions, please contact Zheng Zhou (周正) at physics@zhengzhou.page.

Installation

To install the package, run the following command in the Julia REPL (read-eval-print loop) (To enter Julia REPL, simply type julia in the command line) 

using Pkg ; Pkg.add("FuzzifiED")

To use the package, include at the start of the Julia script

using FuzzifiED

To obtain the documentation for an interface, type ? followed by the keyword in the Julia REPL, e.g., ?Confs.

Citation

If this package is helpful in your research, we would appreciate it if you mention in the acknowledgement. We have also provided a BibTeX file that includes all the works on the fuzzy sphere works at this link.

Useful information

  • Download Julia at this link.
  • Jupyter Notebook is highly recommended as it allows you to run Julia (and Python) just like running a Mathematica notebook.
  • The package regisitry at Julia General Registry may have some delay. If you encounter trouble at installation, to bring the registry up to date, use Pkg.Registry.update(), or install from the GitHub repos 
using Pkg
Pkg.add(url="https://github.com/FuzzifiED/FuzzifiED_jll.jl")
Pkg.add(url="https://github.com/FuzzifiED/FuzzifiED.jl")

Outline

References

  • [Zhu 2022] Uncovering conformal symmetry in the 3d Ising transition : state-operator correspondence from a quantum fuzzy sphere regularisation, Wei Zhu, Chao Han, Emilie Huffman, Johannes S. Hofmann, and Yin-Chen He, arXiv:2210.13482, Phys. Rev. X 13, 021009 (2023).
  • [Hu 2023Mar] Operator product expansion coefficients of the 3d Ising criticality via quantum fuzzy sphere, Liangdong Hu, Yin-Chen He, and Wei Zhu, arXiv:2303.08844, Phys. Rev. Lett 131, 031601 (2023).
  • [Han 2023Jun] Conformal four-point correlators of the 3d Ising transition via the quantum fuzzy sphere, Chao Han, Liangdong Hu, Wei Zhu, and Yin-Chen He, arXiv:2306.04681, Phys. Rev. B 108, 235123 (2023).
  • [Zhou 2023] The $\mathrm{SO}(5)$ deconfined phase transition under the fuzzy sphere microscope: approximate conformal symmetry, pseudo-criticality, and operator spectrum, Zheng Zhou, Liangdong Hu, Wei Zhu, and Yin-Chen He, arXiv:2306.16435, Phys. Rev. X 14, 021044 (2024).
  • [Lao 2023] 3d Ising CFT and exact diagonalisation on icosahedron : the power of conformal perturbation theory, Bing-Xin Lao, and Slava Rychkov arXiv:2307.02540, SciPost Phys. 15, 243 (2023).
  • [Hu 2023Aug] Solving conformal defects in 3d conformal field theory using fuzzy sphere regularisation, Liangdong Hu, Yin-Chen He, and Wei Zhu, arXiv:2308.01903, Nat. Commun. 15, 3659 (2024).
  • [Hofmann 2024] Quantum Monte Carlo simulation of the 3d Ising transition on the fuzzy sphere, Johannes S. Hofmann, Florian Goth, Wei Zhu, Yin-Chen He, and Emilie Huffman, arXiv:2310.19880, SciPost Phys. Core 7, 028 (2024).
  • [Han 2023Dec] Conformal operator content of the Wilson-Fisher transition on fuzzy sphere bilayers, Chao Han, Liangdong Hu, and Wei Zhu, arXiv:2312.04047, Phys. Rev. B 110, 115113 (2024).
  • [Zhou 2024Jan] The $g$-function and defect changing operators from wavefunction overlap on a fuzzy sphere, Zheng Zhou, Davide Gaiotto, Yin-Chen He, Yijian Zou, arXiv:2401.00039, SciPost Phys. 17, 021 (2024).
  • [Hu 2024] Entropic $F$-function of 3d Ising conformal field theory via the fuzzy sphere regularisation, Liangdong Hu, Wei Zhu, and Yin-Chen He, arXiv:2401.17362.
  • [Cuomo 2024] Impurities with a cusp : general theory and 3d Ising, Gabriel Cuomo, Yin-Chen He, Zohar Komargodski, arXiv:2406.10186.
  • [Zhou 2024Jul] Studying the 3d Ising surface CFTs on the fuzzy sphere, Zheng Zhou, and Yijian Zou, arXiv:2407.15914, SciPost Phys. 18, 031 (2025).
  • [Dedushenko 2024] Ising BCFTs from the fuzzy hemisphere, Mykola Dedushenko, arXiv:2407.15948.
  • [Fardelli 2024] Constructing the infrared conformal generators on the fuzzy sphere, Giulia Fardelli, A. Liam Fitzpatrick, and Emanuel Katz, arXiv:2409.02998.
  • [Fan 2024] Note on explicit construction of conformal generators on the fuzzy sphere, Ruihua Fan, arXiv:2409.08257.
  • [Zhou 2024Oct] A new series of 3d CFTs with $\mathrm{Sp}(N)$ global symmetry on fuzzy sphere, Zheng Zhou, and Yin-Chen He, arXiv:2410.00087.
  • [Voinea 2024] Regularising 3d conformal field theories via anyons on the fuzzy sphere, arXiv:2411.15299.
  • [Han 2025] Quantum phase transitions on the noncommutative circle, Chao Han, and Wei Zhu, Phys. Rev. B 111, 085113 (2025)
  • [Yang 2025] Microscopic study of 3d Potts phase transition via fuzzy sphere regularisation, Shuai Yang, Yan-Guang Yue, Yin Tang, Chao Han, Wei Zhu, and Yan Chen, arXiv:2501.14320

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