Perturbative Renormalization and Universality Diagram for Long-Range Quantum Criticality

Abstract

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 $1/r^{d+\ddot{I}ƒ}$ and develop a perturbative renormalization-group expansion around the LR--SR boundary by setting $d=3-\hat{I}\mu$ and $\ddot{I}ƒ=2-\hat{I}´$. In this parametrization, the full interacting LR window $2d/3<\ddot{I}ƒ<2$ becomes $0<\hat{I}´<2\hat{I}\mu /3$, and is therefore perturbatively controlled. A two-loop calculation yields explicit expressions, in terms of $\hat{I}\mu$, $\hat{I}´$, and $n$, for the correlation-length exponent $\hat{I}½$ and for the frequency and momentum anomalous dimensions $\hat{I}{\cdot }_{\ddot{I}}‰$ and $\hat{I}{\cdot }_k$. The resulting exponents reduce to long-range Gaussian scaling at $\ddot{I}ƒ=2d/3$ and to SR quantum Wilson-Fisher scaling in the $\ddot{I}ƒ\to 2$ limit, thereby identifying $\ddot{I}{ƒ}_{\ast }=2$ as the LR--SR boundary within the controlled $3-\hat{I}\mu$ expansion. Combining the RG results with scaling boundaries and classical LR analogies, we propose a $(d,\ddot{I}ƒ)$ 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.

13 pages, 3 figures