Non-Hermitian Renormalization Group from a Few-Body Perspective

TL;DR

This work develops a microscopic non-Hermitian renormalization group (RG) framework for strongly interacting few-body systems by enforcing invariance of the two-body scattering amplitude under RG transformations, avoiding reliance on partition functions. The running coupling becomes complex, $g_\Lambda=g_{r,\Lambda}-ig_{i,\Lambda}$, and the exact two-body RG flow is derived, revealing a looped structure in $d=2$ due to a non-Hermitian quantum scale anomaly. The analysis connects measurement backaction to RG evolution, showing how inelastic loss can generate real parts of the coupling and lead to resonance formation, with concrete applications to coherent neutron-nucleus scattering and dineutron correlations in Borromean halo nuclei. Overall, the paper bridges non-Hermitian AMO physics and nuclear physics, offering a unified, few-body perspective on non-Hermitian RG flows and quantum measurement effects with potential experimental verification.

Abstract

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.

Figures (8)

• Figure 1: Schematic illustrations of (a) continuous measurement discussed in AMO physics and (b) conditional data selection in high-energy experiments, through projection $P$ on survival states with inelastic loss. In (a), a monitored system undergoes nonunitary time evolution due to continuous measurement. Even in the absence of the jump process, the state is affected by measurement backaction due to the projector $P$. In (b), after the scattering event, the measurement data are conditionally selected for a specific channel (i.e., elastic scattering channel) which has undergone no jump processes characterized by the projector $P$.
• Figure 2: Non-Hermitian RG flow of the dimensionless complex-valued coupling $U_t=U_{r,t}-iU_{i,t}$ in (a) $d=1$, (b) $d=2$, and (c) $d=3$. The red points denote the fixed points. At $U_{i,t}=0$ (solid thick lines), the RG flow shows $U_{r,t\rightarrow 0}\rightarrow-\infty$ at $U_{r,t}< 0$ in $d=1,2$, corresponding to the emergence of a two-body bound state. Note that the flow to $U_{r,t\rightarrow\infty}=-\infty$ is found for $U_{r,t}<-1$ in $d=3$, where the unstable fixed point $U_{r,t}=-1$ is the unitary limit PhysRevA.75.033608.
• Figure 3: Two-body energy $E=E_{\rm R}-i\Gamma$ under the pure imaginary two-body coupling $U_0=-iU_{i,0}$ in the complex plane for spatial dimensions of $d=1,2,3$.
• Figure 4: Schematic comparison between (a) the conventional resonance that decays into the continuum and (b) the present case with inelastic loss. In case (b), there are possibilities of both $E_{\rm R}>0$ and $E_{\rm R}<0$.
• Figure 5: Two-dimensional two-body energy pole $E=E_{\rm R}-i\Gamma$ obtained by changing $\kappa=U_{i,0}/|U_{r,0}|$ at (a1) strong coupling ($|U_{r,0}|=2$) and at (a2) weak coupling ($|U_{r,0}|=0.02$). In the panels (b1) and (b2), the imaginary coupling dependence of $E_{\rm R}$ and $\Gamma$ on $\kappa$ are plotted at (b1) $|U_{r,0}|=2$ and (b2) $|U_{r,0}|=0.02$.