The Effective Field Theory Origin of the Fuzzy Sphere Method
Abstract
Fuzzy sphere Hamiltonians for (2+1)D conformal field theories are usually written directly on the lowest Landau level. We introduce a framework organizing them as descendants of local ``mother theories'' on S2×ℝ. We start with the example of Ising mother theory built from two charged Dirac fermions coupled to a real mediator ϕ. Integrating out ϕ and the higher Landau levels produces the fuzzy Hamiltonian, while integrating out the fermions gives a conventional local ϕ effective field theory flowing to the Ising point. Mild modifications of the same theory reach the Gaussian point (tricritical route) and the Lee-Yang continuation. Exploring the space of mother theories, we next promote ϕ to a complex field Φ and construct a new example, whose descendant is an easy-plane ferromagnet on the fuzzy sphere. Its spontaneous symmetry breaking phase gives a pseudo-Goldstone relativistic scalar, similar to the model introduced by Y.-C. He, while its critical point is non-relativistic, providing the first example of z=2 scaling on the fuzzy sphere. Remarkably, it admits exact one-magnon solutions, and the z=2 critical point can be found analytically. We then outline a rich landscape of further applications of our approach.
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BibTeX
@misc{dedushenko2026effective,
title = {{The Effective Field Theory Origin of the Fuzzy Sphere Method}},
author = {Mykola Dedushenko},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38702},
eprint = {2610.38702},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38702},
}
RIS
TY - UNPB
TI - The Effective Field Theory Origin of the Fuzzy Sphere Method
AU - Mykola Dedushenko
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38702
UR - https://arxiv.org/abs/2610.38702
AB - Fuzzy sphere Hamiltonians for (2+1)D conformal field theories are usually written directly on the lowest Landau level. We introduce a framework organizing them as descendants of local ``mother theories'' on S^2×ℝ. We start with the example of Ising mother theory built from two charged Dirac fermions coupled to a real mediator ϕ. Integrating out ϕ and the higher Landau levels produces the fuzzy Hamiltonian, while integrating out the fermions gives a conventional local ϕ effective field theory flowing to the Ising point. Mild modifications of the same theory reach the Gaussian point (tricritical route) and the Lee-Yang continuation. Exploring the space of mother theories, we next promote ϕ to a complex field Φ and construct a new example, whose descendant is an easy-plane ferromagnet on the fuzzy sphere. Its spontaneous symmetry breaking phase gives a pseudo-Goldstone relativistic scalar, similar to the model introduced by Y.-C. He, while its critical point is non-relativistic, providing the first example of z=2 scaling on the fuzzy sphere. Remarkably, it admits exact one-magnon solutions, and the z=2 critical point can be found analytically. We then outline a rich landscape of further applications of our approach.
ER -