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arXiv:2504.07019 · Physical Review Letters

Non-Hermitian Numerical Renormalization Group: Solution of the Non-Hermitian Kondo Model

Phillip C. Burke, Andrew K. Mitchell

Published Nov 12, 2025• 6 citations
arXivHTMLPDF

Abstract

Non-Hermitian (NH) Hamiltonians describe open quantum systems, nonequilibrium dynamics, and dissipative processes. Although a rich range of single-particle NH physics has been uncovered, many-body phenomena in strongly correlated NH systems have been far less well studied. The Kondo effect, an important paradigm for strong correlation physics, has recently been considered in the NH setting. Here, we develop an NH generalization of the numerical renormalization group and use it to solve the NH Kondo model. Our nonperturbative solution applies beyond weak coupling, and we uncover a nontrivial phase diagram. The method is showcased by application to the NH pseudogap Kondo model, which we show supports a completely novel phase with a genuine NH stable fixed point and complex eigenspectrum. Our NH numerical renormalization group code, which can be used in regimes and for models inaccessible to, e.g., perturbative scaling and Bethe ansatz, is provided open source.

DOI
10.1103/19td-1k9s

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Non-Hermitian (NH) Hamiltonians describe open quantum systems, nonequilibrium dynamics, and dissipative processes. Although a rich range of single-particle NH physics has been uncovered, many-body phenomena in strongly correlated NH systems have been far less well studied. The Kondo effect, an important paradigm for strong correlation physics, has recently been considered in the NH setting. Here, we develop an NH generalization of the numerical renormalization group and use it to solve the NH Kondo model. Our nonperturbative solution applies beyond weak coupling, and we uncover a nontrivial phase diagram. The method is showcased by application to the NH pseudogap Kondo model, which we show supports a completely novel phase with a genuine NH stable fixed point and complex eigenspectrum. Our NH numerical renormalization group code, which can be used in regimes and for models inaccessible to, e.g., perturbative scaling and Bethe ansatz, is provided open source.
Non-Hermitian physics in open quantum many-body systems provides novel opportunities for the discovery of exotic quantum phenomena unexpected in Hermitian systems. A previous study of the non-Hermitian Kondo problem in ultracold atoms reports reversion of renormalization group flows which violates the ggg theorem and produces an unusual quantum phase transition. In this work, we study the effect of electron-electron interactions by considering the non-Hermitian Kondo problem in a Luttinger liquid. By performing a perturbative renormalization group analysis to two-loop order, we find that the interplay between non-Hermitian Kondo couplings and electron-electron interactions can produce a pair of complex fixed points. Complex fixed points have often been discussed in an attempt to understand the extremely long correlation length of Hermitian systems with weakly first-order transitions. Here, we show that complex fixed points arise naturally and can be physically realized in open quantum systems. We discuss consequences of the complex fixed points and future directions.
Non-Hermitian Hamiltonians, as effective models, capture phenomena such as energy dissipation and nonunitary evolution in open quantum systems. New phases and phenomena appear that are not present in their Hermitian counterparts. Such a Hamiltonian, the non-Hermitian Kondo model, has been used to describe inelastic scattering between mobile and confined atoms in an optical lattice [M. Nakagawa, N. Kawakami, and M. Ueda, Phys. Rev. Lett. 121, 203001 (2018)]. Using a combination of Bethe ansatz and perturbative calculation, the authors argued that this model has two distinct phases: the Kondo and the non-Kondo phases, where the impurity is screened and unscreened, respectively. We show, however, that a novel phase termed $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{YSR}$ emerges between the Kondo and unscreened phases. Characterized by two RG invariants: a generalized Kondo temperature (TK{T}_{K}TK​) and a loss strength parameter ($\ensuremath{\alpha}$), the system exhibits three distinct phases. In the increasing order of losses, they are: the Kondo phase ($0<\ensuremath{\alpha}<\ensuremath{\pi}/2$), the $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{YSR}$ phase ($\ensuremath{\pi}/2<\ensuremath{\alpha}<3\ensuremath{\pi}/2$), and the local moment phase ($\ensuremath{\alpha}>3\ensuremath{\pi}/2$). Notably, phase transition driven by dissipation occurs across $\ensuremath{\alpha}=\ensuremath{\pi}/2$, where both energetics and different timescales associated with loss play roles.
In the early 1970s, Wilson developed the concept of a fully nonperturbative renormalization group transformation. When applied to the Kondo problem, this numerical renormalization group (NRG) method gave for the first time the full crossover from the high-temperature phase of a free spin to the low-temperature phase of a completely screened spin. The NRG method was later generalized to a variety of quantum impurity problems. The purpose of this review is to give a brief introduction to the NRG method, including some guidelines for calculating physical quantities, and to survey the development of the NRG method and its various applications over the last 30 years. These applications include variants of the original Kondo problem such as the non-Fermi-liquid behavior in the two-channel Kondo model, dissipative quantum systems such as the spin-boson model, and lattice systems in the framework of the dynamical mean-field theory.
Renormalization group methods become natural for quantum many-body systems where there is an emergence of new energy scales like as Kondo scale in case of Kondo model. It was the Numerical Renormalization Group which gave the full quantitative understanding of the strong coupling fixed point of Kondo model and also of Anderson Impurity model. However, it was Andersons poor man scaling method which gave the analytical insights into the scaling behaviour of quantum impurity models. That makes poor man scaling method very important, and hence it has been used extensively in carrying out renormalization studies of quantum impurity models. This method has been generalized and also applied to other quantum many-body Hamiltonians. In this paper, we give a detailed presentation of poor man scaling method and apply it to study the scaling behaviour of Kondo model and Anderson impurity model as well. Though there are many reviews of perturbative renormalization of Kondo model, in this paper we have also presented a detailed discussion about scaling analysis of Anderson impurity model. We also present another method called flow equation method which is a generalization of poor man scaling method and has recently been applied to study the quantum many-body dynamics both in equilibrium and out-of equilibrium. We also present a numerical procedure to extract Kondo scale from flow equations of Kondo model. The importance of this procedure is that it can be applied to calculate the Kondo scale for situations where poor mans scaling method can not be applied.
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.
The unusual low temperature behaviour of metals and metal alloys, such as heavy fermion systems, Mott insulators and unconventional superconductors, has been a central topic of scientific investigation for over half a century. Due to strong correlations between the many particles, the solution to most theoretical models of these systems is difficult because standard perturbation methods break down. In certain cases, non-perturbative renormalization group approaches can be used to access the low energy behaviour and take into account the effects of strong correlations. However, these methods do not work for several important and interesting classes of model. These include lattice models with finite dimension, models with impurities of high degeneracy and steady state transport through quantum dots. The aim of this thesis is therefore to develop a new and general renormalization group approach that will lend itself to wide application in models where existing techniques fail. The method explored here utilises renormalized perturbation theory (RPT) in conjunction with scaling equations and collective excitations. It is tested on the single impurity Anderson model, the paradigmatic model used to understand magnetic impurities within a non-magnetic host metal. The Anderson model is simple yet exhibits interesting features due to strong correlation, such as a single renormalized low energy scale called the Kondo temperature. This model is also well understood as it has been solved using the numerical renormalization group (NRG), which provides ample numerical data against which this new technique is compared. Finally, a preliminary study is given for this method when it is applied to a wider class of models.