Non-Hermitian Renormalization Group from a Few-Body Perspective

Non-Hermiticity plays a fundamental role in open quantum systems and describes a wide variety of effects of interactions with environments, including quantum measurement. However, understanding its consequences in strongly interacting systems is still elusive due to the interplay between non-perturbative strong correlations and non-Hermiticity. While the Wilsonian renormalization group (RG) method has been applied to tackle this problem, its foundation, based on the existence of the partition function, is ill-defined. In this paper, we establish a microscopic foundation of the non-Hermitian RG method from a few-body perspective. We show that the invariance of the scattering amplitude under RG transformations enables us to rigorously derive the non-Hermitian RG equation, giving a physically transparent interpretation of RG flows. We discuss a detailed structure of such RG flows in a non-relativistic two-body system with inelastic two-body loss, and show its relation to a non-Hermitian quantum scale anomaly. Our analysis suggests that non-Hermitian complex potentials often used in high-energy physics can be interpreted as being caused by quantum measurement, where the detection of elastically scattered particles updates the observer’s knowledge, resulting in a nonunitary state change of the system. We apply our formalism to nuclear physics, find the emergence of a critical semicircle, and show that several nuclei are located near the critical semicircle in the coherent neutron-nucleus scattering. We also propose that the localized dineutron in two-neutron halo nuclei can be interpreted as the quantum measurement effect on the imaginary potential associated with absorption into the core nucleus. Our result bridges different contexts of non-Hermitian systems in high-energy and atomic, molecular, and optical physics, opening an interdisciplinary playground of non-Hermitian few-body physics.

💡 Research Summary

This paper establishes a microscopic foundation for renormalization‑group (RG) analysis of non‑Hermitian quantum systems by focusing on few‑body scattering rather than on the ill‑defined partition function that underlies Wilsonian RG. The authors begin by formulating the dynamics of a two‑particle system with inelastic loss using the Lindblad master equation. By projecting onto the subspace where no loss occurs, they obtain an effective non‑Hermitian Hamiltonian (H_{\rm eff}=H-i g_i\sum_r L_r^\dagger L_r) that governs the non‑unitary time evolution of the surviving particles. The detection of particles that have not been lost constitutes a quantum‑measurement back‑action; it updates the observer’s knowledge and normalizes the density matrix, a process that can be interpreted as Bayesian inference.

The central technical step is to derive the RG flow from the invariance of the scattering T‑matrix under changes of the ultraviolet cutoff. Solving the Lippmann–Schwinger equation for the non‑Hermitian potential (V) yields the T‑matrix (T(E)=V+V(E+i0^+-H_0)^{-1}T(E)), which remains unchanged when the cutoff (\Lambda) is lowered. Differentiating with respect to the logarithmic scale (\ell=\ln(\Lambda_0/\Lambda)) gives a complex beta function (\beta(g)=\mathrm{d}g/\mathrm{d}\ell). The authors compute (\beta(g)) in one, two and three spatial dimensions, revealing distinct flow patterns in the complex coupling plane. In two dimensions, a scale‑invariant contact interaction acquires a non‑Hermitian quantum scale anomaly: the complex coupling runs logarithmically, displaying a form of asymptotic freedom that is absent in the Hermitian case.

The formalism is then applied to nuclear physics. The familiar optical model, which employs a complex potential to describe neutron absorption, is reinterpreted as a measurement process. By evolving the non‑Hermitian coupling, the RG flow approaches a “critical semicircle” in the complex plane—a non‑Hermitian generalization of the unitary limit. Using low‑momentum effective interactions (V_{\rm low,k}), the authors show that several nuclei (e.g., ^{58}Ni, ^{208}Pb) lie near this semicircle when analyzed through coherent neutron‑nucleus scattering data. This suggests a universal non‑Hermitian behavior across disparate nuclear systems.

Finally, the paper addresses two‑neutron halo nuclei such as ^6He and ^11Li. The observed localization of a dineutron pair is explained as a quantum‑measurement effect: when the probability of both neutrons being absorbed by the core is non‑zero, the post‑selection of events without loss projects the surviving pair into a bound‑like state. This provides a novel perspective on halo structure that complements traditional few‑body calculations.

Experimental realizations are discussed for both AMO and nuclear settings. In ultracold atoms, photoassociation‑induced two‑body loss combined with matter‑wave magnification enables controlled implementation of the non‑Hermitian two‑body problem. In nuclear experiments, conditional data selection—keeping only elastic‑scattering events—effectively implements the same projection operator used in the theory.

In summary, the work demonstrates that non‑Hermitian RG can be rigorously derived from scattering‑amplitude invariance, offers a transparent physical interpretation of RG flows as measurement‑induced state updates, and unifies phenomena ranging from quantum scale anomalies to nuclear scattering universality. It opens a new interdisciplinary arena where non‑Hermitian few‑body physics can be explored across atomic, molecular, optical, and high‑energy nuclear contexts.