Implementing Wilsonian renormalization group transformations in an iterative way, we develop a nonperturbative field theoretical framework for strongly coupled quantum theories, which takes into account all-loop quantum corrections organized in the $1/N$ expansion. Here, $N$ represents the flavor number of strongly correlated quantum fields. The resulting classical field theory is given by an effective Landau-Ginzburg theory for a local order parameter field, which appears in one-dimensional higher spacetime. We confirm the nonperturbative nature of this field theoretical framework for the Kondo effect. Intriguingly, we show that the recursive Wilsonian renormalization group method can explain nonperturbative thermodynamic properties of an impurity, consistent with Bethe ansatz for the whole temperature region.
Implementing Wilsonian renormalization group transformations in an iterative way, we develop a nonperturbative field theoretical framework for strongly coupled quantum theories, which takes into account all-loop quantum corrections organized in the 1/N expansion. Here, N represents the flavor number of strongly correlated quantum fields. The resulting classical field theory is given by an effective Landau-Ginzburg theory for a local order parameter field, which appears in one-dimensional higher spacetime. We confirm the nonperturbative nature of this field theoretical framework for the Kondo effect. Intriguingly, we show that the recursive Wilsonian renormalization group method can explain nonperturbative thermodynamic properties of an impurity, consistent with Bethe ansatz for the whole temperature region.
We construct an emergent holographic dual description in the Luttinger-Ward functional approach, where the renormalization group (RG) flows of collective bi-local fields appear manifestly in the bulk effective action with an emergent extra dimension. This holographic dual effective field theory reproduces 1/N quantum corrections in a self-consistent manner when we take the UV limit in the bulk effective action. Going into the IR regime in the extra dimension, we observe that a partial class of the field theoretic 1/N, 1/N2, ... quantum corrections are resummed in the all-loop order and reorganized to form a holographic dual effective field theory in a large N fashion living on the one-higher dimensional spacetime. In this study, we apply this theoretical framework into an Sachdev-Ye-Kitaev (SYK) model. Taking the large N limit in the holographic dual effective field theory, we obtain nonlinearly coupled second-order bulk differential equations of motion for the three bi-local order-parameter fields of fermion self-energy, Green's function, and polarization function. Here, both UV and IR boundary conditions are derived self-consistently from the boundary effective action. We solve these highly intertwined nonlinear differential equations based on the so called matching method. Our ansatz for the bi-local order-parameter fields coincide with the conformally invariant solution of the field theoretic large N limit in the UV limit, but their overall coefficients RG−flow along the extra dimensional space, respectively, reflecting effects of higher-order quantum corrections. As a result, we find an insulating behavior, where the self-energy diverges at IR. ...
The Kondo problem, which describes the interaction of a spin s magnetic impurity with a free Fermi gas, is a classic example of strongly coupled physics. Historically, the problem has been solved by Wilson's numerical renormalization group and later by Bethe ansatz. In this paper, we present an alternate analytic solution of the Kondo problem that combines an expansion in 1/s with the renormalization group. We study both the case of an impurity interacting with a single channel K=1 of fermions in the s→∞ limit and the case with K channels in the double-scaling limit K→∞, s→∞, K/s fixed. Our approach allows us to describe analytically intermediate scales of the Kondo problem at large s and compute thermodynamic observables such as the impurity entropy and susceptibility. We find these observables to agree with the Bethe ansatz results. We also compute the impurity spectral function, finite temperature resistivity and the Kondo screening cloud profile, properties that are not easily accessible from Bethe ansatz. Notably, in the regime K>2s we access the "non-Fermi-liquid" overscreened fixed point of the multichannel Kondo problem.
Boundary phases and thermodynamics of the Kondo spin-s chain: from overscreened Kondo to boundary-bound states
We study a spin-21 impurity coupled to the boundary of a strongly correlated spin-s Takhtajan--Babujian chain, an integrable model whose low-energy physics is described by a perturbed SU(2)2s Wess--Zumino--Witten conformal field theory. While boundary conformal field theory determines the low-energy universality class of the weak-coupling regime, exact Bethe Ansatz methods reveal a sequence of boundary quantum phase transitions in which impurity-bound states emerge and reorganize the Hilbert space into multiple excitation towers built on distinct boundary configurations. This tower restructuring provides the organizing principle for a rich boundary phase diagram extending beyond the conventional Kondo regime. Weak antiferromagnetic coupling realizes the overscreened 2s-channel Kondo universality class, whereas stronger couplings generate localized boundary modes and qualitatively new screening mechanisms. To describe the resulting thermodynamics, we develop a generalized thermodynamic Bethe Ansatz framework that captures the multi-tower structure across all regimes. The impurity entropy reproduces the boundary conformal field theory prediction in the overscreened Kondo regime but develops pronounced nonmonotonic temperature dependence once boundary-bound states appear, in quantitative agreement with large-scale finite-temperature matrix-product-operator simulations. Complementary dynamical calculations reveal sharp threshold features in the impurity spectral function that directly track the underlying tower structure. Together, boundary conformal field theory, exact Bethe Ansatz, generalized thermodynamic Bethe Ansatz, and tensor-network simulations provide a unified description of impurity screening, boundary-bound-state formation, and excitation-tower reconstruction in a correlated spin-s chain.
We present a nonequilibrium nonperturbative field theory for the Kondo effect in strongly interacting quantum dots at finite temperatures. Unifying the slave-boson representation with the Keldysh field integral, an effective Keldysh action is derived and explored in the vicinity of the zero slave-bosonic field configuration. The theory properly reflects the essential features of the Kondo physics and at the same time significantly simplifies a field-theoretic treatment of the phenomenon, avoiding complicated saddle-point analysis or 1/N expansions, used so far. Importantly, our theory admits a closed analytical solution which explains the mechanism of the Kondo effect in terms of an interplay between the real and the imaginary parts of the slave-bosonic self-energy. It thus provides a convenient nonperturbative building block, playing the role of a ``free propagator,'' for more advanced theories. We finally demonstrate that already this simplest possible field theory is able to correctly reproduce experimental data on the Kondo peak observed in the differential conductance, correctly predicts the Kondo temperature, and, within its applicability range, has the same universal temperature dependence of the conductance as the one obtained in numerical renormalization group calculations.
We propose a model of a strongly-interacting two-impurity Kondo system based on the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence, also known as holography. In a Landau Fermi Liquid, the single-impurity Kondo effect is the screening of an impurity spin at low temperature T . The two-impurity Kondo model then describes the competition between the Kondo interaction and the Heisenberg interaction between two impurity spins, also called the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. For spin-1/2 impurities, that competition leads to a quantum phase transition from a Kondo-screened phase to a phase in which the two impurity spins screen one another. Our holographic model is based on a (1 + 1)-dimensional CFT description of the two-impurity Kondo model, reliable for two impurities with negligible separation in space. We consider only impurity spins in a totally anti-symmetric representation of an SU(N ) spin symmetry. We employ a large-N limit, in which both Kondo and RKKY couplings are double-trace, and both Kondo and inter-impurity screening appear as condensation of single-trace operators at the impurities’ location. We perform the holographic renormalization of our model, which allows us to identify the Kondo and RKKY couplings as boundary conditions on fields in AdS. We numerically compute the phase diagram of our model in the plane of RKKY coupling versus T , finding evidence for a quantum phase transition from a trivial phase, with neither Kondo nor inter-impurity screening, to a non-trivial phase, with both Kondo and anti-ferromagnetic inter-impurity screening. More generally we show, just using SU(N ) representation theory, that ferromagnetic correlations must be absent at leading order in the large-N limit. Our holographic model may be useful for studying many open problems involving strongly-interacting quantum impurities, including for example the Kondo lattice, relevant for describing the heavy fermion compounds.
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.