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arXiv:2308.03569 · Physical review. B./Physical review. B

Systematic compactification of the two-channel Kondo model. I. Consistent bosonization-debosonization approach and exact comparisons

Aleksandar Ljepoja, C. J. Bolech, Nayana Shah

Published Jul 8, 2024• 3 citations
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Abstract

Capitalizing on recent work, which clarifies the consistent use of bosonization-debosonization methods to study Kondo-type quantum impurity models even in nonequilibrium settings, we revisit the compactification procedure of the two-channel Kondo model (by which it is rewritten more ``compactly'' using the single-channel version of the model and exploiting separation and duality between spin and charge) and uncover some hidden approximations that could limit its range of validity. This complements and extends, for two or any even number of channels, beyond previous work on the Toulouse limit of these models, and reinforces the need for the use of an extended framework in these calculations. We carry out a number of exact comparisons between the different models and show that keeping track of the so-called consistency factors leads to full agreement between the compactified and original versions of the model.

DOI
10.1103/physrevb.110.045108

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Capitalizing on recent work, which clarifies the consistent use of bosonization-debosonization methods to study Kondo-type quantum impurity models even in nonequilibrium settings, we revisit the compactification procedure of the two-channel Kondo model (by which it is rewritten more ``compactly'' using the single-channel version of the model and exploiting separation and duality between spin and charge) and uncover some hidden approximations that could limit its range of validity. This complements and extends, for two or any even number of channels, beyond previous work on the Toulouse limit of these models, and reinforces the need for the use of an extended framework in these calculations. We carry out a number of exact comparisons between the different models and show that keeping track of the so-called consistency factors leads to full agreement between the compactified and original versions of the model.
Following up on the systematic compactification of the two-channel Kondo model [and its multichannel extensions; see Ljepoja et al., Phys. Rev. B 110, 045108 (2024)] and the demonstration of its validity over the past proposal of compactification, we resort to a study of scaling using Anderson's simple poor man's procedure to carry out a comparative study of these two and the original model. By doing so we unveil a universal agreement among the three models in how they flow upon scaling, and suggest the general limits of such a concordance. In this way we further elucidate the conditions under which the standard simplifications implicit in many bosonization-based mappings (particularly of quantum impurity models) can be used reliably, and when the consistent bosonization-debosonization approach is needed.
We carry out a field-theoretical renormalization group procedure based on the Callan-Symanzik equation to calculate the detailed flow for the (multi) two-channel Kondo model and its compactified versions. In doing so, we go beyond the universal terms in the beta function we obtained using poor man's scaling [see Phys. Rev. B 110, 045109 (2024)] and culminate our analysis of how the compactified versions of the model fare against the original one. Among other results, we explore the large-channel-number limit and extend our considerations to the finite temperature crossover region. Moreover, we gain insights into the contradistinction between the consistent vs conventional bosonization-debosonization formalisms, thereby advancing our understanding on multiple fronts. In particular, we make use of renormalization-flow arguments to further justify the consistent refermionization of the parallel Kondo interaction we presented earlier [see Phys. Rev. B 110, 045108 (2024)].
Following the development of a scheme to bosonize and debosonize consistently [N. Shah and C. J. Bolech, Phys. Rev. B 93, 085440 (2016)], we present in detail the Toulouse-point analytic solution of the two-lead Kondo junction model. The existence and location of the solvable point is not modified, but the calculational methodology and the final expressions for observable quantities change markedly as compared to the existent results. This solvable point is one of the remarkably few exact results for nonequilibrium transport in correlated systems. It yields relatively simple analytical expressions for the current in the full range of temperature, magnetic field, and voltage. It also shows precisely, within the limitations of the Toulouse fine-tuning, how the transport evolves depending on the relative strengths of interlead and intralead Kondo exchange couplings ranging from weak to strong. Thus its improved understanding is an important stepping stone for future research.
The bosonization of the S=1/2 anisotropic two-channel Kondo model is shown to yield two equivalent representations of the original problem. In a straight-forward extension of the Emery-Kivelson approach, the interacting resonant level model previously derived by the Anderson-Yuval technique is obtained. In addition, however, a compactified ``(\ensuremath{\sigma},\ensuremath{\tau})'' description is also found. The strong coupling fixed point of the (\ensuremath{\sigma},\ensuremath{\tau}) model was originally postulated to be related to the intermediate coupling fixed point of the two-channel Kondo model. The equivalence of the \ensuremath{\sigma},\ensuremath{\tau} model to the two-channel Kondo model is formally established. A summary of what one may learn from a simple study of these different representations is also given.
This thesis consists of two parts. In part one we discuss the Kondo model. First we point out the differences in physical properties between the one- and multi-channel Kondo model using a renormalization group analysis. We also point out the challenges in obtaining exact solutions. Then we give an overview of two approaches, bosonization and spin chain, for finding exact solution of the one-, two- and four-channel Kondo model. Using bosonization we review a mapping from the one- and two-channel model to the exactly solvable resonant level model for a specific coupling strength. Following previous work, we derive the Kondo effect for a spin chain construction with XX and Ising coupling, showing the low energy behavior to be equivalent to the two- or four-channel Kondo model. We also discuss why one, two and four channels are special and we point out the difficulties in extending these approaches to other multi-channel Kondo models.\nThen in part two we investigate survival of the topological edge effects of the Kitaev chain (Majorana edge modes) after a quantum quench. We determine a stability region for the edge modes after chemical potential and superconductivity quenches, by analyzing the wave functions using two measure: inverse participation ratio and overlap. We see that this region does not cover the full topological phase in parameter space. The Majoranas already disappear before the quenched system has reached the phase transition.
We adapt the strong-coupling method first used in the one-channel Kondo model to develop a simple description of the spin- 12\frac{1}{2}21​ two-channel Kondo model with channel anisotropy. Our method exploits spin-charge decoupling to develop a compactified Hamiltonian that describes the spin excitations. The structure of the fixed-point Hamiltonian and quasiparticle impurity S matrix are incompatible with a Fermi-liquid description.