Abstract
Implementing Wilsonian renormalization group transformations in an iterative way, we develop a non-perturbative field theoretical framework, which takes into account all-loop quantum corrections organized in the $1/N$ expansion, where $N$ represents the flavor number of 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 claim that such all-loop quantum corrections are introduced into an equation of motion for the order parameter field through the evolution in the emergent extra dimension. Based on this non-perturbative theoretical framework, we solve the Kondo effect, where the quantum mechanics problem in the projective formulation is mapped into a Landau-Ginzburg field theory for the hybridization order parameter field with an emergent extra dimension. We confirm the non-perturbative nature of this field theoretical framework. Intriguingly, we show that the Wilsonian renormalization group method can explain non-perturbative thermodynamic properties of an impurity consistent with the Bethe ansatz solutions. Finally, we speculate how our non-perturbative field theoretical framework can be connected with the AdS$_{d+2}$/CFT$_{d+1}$ duality conjecture.
Highlights
Non-Fermi liquid physics near the metallic quantum criticality [1,2], the nature of metal-insulator transitions [3], the emergence of exotic quantum liquids in the vicinity of heavy-fermion quantum criticality [4,5], and the physics of rare events in strongly disordered systems [6] are all beyond the perturbative theoretical framework
We claim that this nonperturbative physics would be encoded into our effective Landau-Ginzburg description for the Kondo effect beyond the 1=N framework, where the evolution of the hybridization order parameter is given by the diffusion equation in the emergent spacetime with an extra dimension
We derived an effective field theory of a hybridization order parameter field for the Kondo effect, living on the spacetime with an extra dimension given by Wilsonian renormalization group transformations
Summary
Non-Fermi liquid physics near the metallic quantum criticality [1,2], the nature of metal-insulator transitions [3], the emergence of exotic quantum liquids in the vicinity of heavy-fermion quantum criticality [4,5], and the physics of rare events in strongly disordered systems [6] are all beyond the perturbative theoretical framework. Corrections through the extra dimension, organized in the 1=N expansion [25], where N represents the flavor number of strongly correlated quantum fields Applying this recursive Wilsonian renormalization group method to the Kondo problem [26], we succeed in describing the crossover regime from a weakly correlated local moment fixed point at high temperature to a strongly coupled local Fermi-liquid fixed point at low temperature in a nonperturbative way, where the characteristic energy scale is given by the Kondo temperature. Based on this effective classical field theory in ð1 þ 1ÞD, we investigate thermodynamic properties of this Kondo problem for the range of whole temperatures We compare both specific heat and spin susceptibility for the impurity dynamics from our emergent holographic description with Bethe ansatz results, discussed in Sec. III B.
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