Multi-target density matrix renormalization group for 3D CFTs on the fuzzy sphere

TL;DR

The paper addresses the challenge of extracting precise conformal data for 3D CFTs, notably the 3D Ising CFT, where exact diagonalization is limited by exponential Hilbert-space growth. It proposes a hybrid approach that combines fuzzy-sphere regularization with a multi-target DMRG algorithm using a bundled MPS and block Lanczos, exploiting $U(1)$ and $\mathbb{Z}_2$ symmetries and adding $\lambda\vec{L}^2$ to isolate angular-momentum sectors. The key results include computing $24$ low-lying energies at $s=15.5$ ($N=32$) with $D=10000$, extracting six primary operator scaling dimensions, and achieving substantially improved agreement with conformal-bootstrap benchmarks over prior ED results; the critical point is identified via conformal-tower relations among energy levels, e.g., $e_0+1=e_1$ and $e_1+1=e_2$. This framework establishes a scalable route to precision 3D CFT data on the fuzzy sphere and points toward extensions to multi-component or non-Abelian-symmetric CFTs, with a characteristic $\sqrt{N}$ growth of computational cost.

Abstract

The fuzzy sphere regularization provides a powerful framework for studying three-dimensional (3D) conformal field theories (CFTs) by mapping them onto numerically tractable lattice models on the spherical lowest Landau level. However, the system sizes accessible to this method have been limited by the exact diagonalization (ED). In this work, we transcend this limitation by combining the fuzzy sphere regularization with a sophisticated multi-target density matrix renormalization group (DMRG) algorithm. Focusing on the 3D Ising-type model on the spherical lowest Landau level, we calculate the 24 low-lying energies at a larger system size than previously feasible with ED. At criticality, we extract the scaling dimensions of six primary operators, and the results show significantly improved agreement with bootstrap benchmarks compared to previous ED results at smaller sizes. Our approach allows us to efficiently target multiple excited states in larger systems beyond the reach of exact diagonalization. This study establishes the fuzzy sphere regularization combined with advanced DMRG techniques as a powerful and general framework for precision physics in 3D CFTs.

Figures (1)

• Figure 1: Conformal multiplet of several primary operators. Dots show the results from the fuzzy sphere regularization with the truncation $s=15.5$ (corresponding to $N=32$ orbitals) via multi-target DMRG calculations, and lines represent corresponding bootstrap benchmarks.