Perturbative Renormalization and Universality Diagram for Long-Range Quantum Criticality

Experimental progress in quantum simulators highlights the role of long-range (LR) interactions in reshaping quantum criticality and stabilizing exotic phases beyond the short-range (SR) paradigm. We study ferromagnetic long-range quantum O(n) models with interactions decaying as 1/r^d+σ and develop a perturbative renormalization-group expansion around the LR--SR boundary by setting d=3-ε and σ=2-δ. In this parametrization, the full interacting LR window 2d/3<σ<2 becomes 0<δ<2ε/3, and is therefore perturbatively controlled. A two-loop calculation yields explicit expressions, in terms of ε, δ, and n, for the correlation-length exponent ν and for the frequency and momentum anomalous dimensions η_ω and η_k. The resulting exponents reduce to long-range Gaussian scaling at σ=2d/3 and to SR quantum Wilson-Fisher scaling in the σ→ 2 limit, thereby identifying σ_*=2 as the LR--SR boundary within the controlled 3-ε expansion. Combining the RG results with scaling boundaries and classical LR analogies, we propose a (d,σ) universality diagram for ferromagnetic long-range quantum O(n) criticality and use it as an organizing framework for the phase diagram of long-range quantum spin chains.