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arXiv:2308.03606 · Physical review. B./Physical review. B

Systematic compactification of the two-channel Kondo model. III. Extended field-theoretic renormalization group analysis

Aleksandar Ljepoja, C. J. Bolech, Nayana Shah

Published Jul 8, 2024• 3 citations
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Abstract

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)].

DOI
10.1103/physrevb.110.045110

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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 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.
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.
The $\ensuremath{\beta}$ function of the multichannel Kondo model is calculated exactly in the limit of large spin N and channel number $M=\ensuremath{\gamma}N$, with constant $\ensuremath{\gamma}$. There are no corrections in any finite order of 1/N1/N1/N. One zero is found at a finite coupling strength, showing directly the non-Fermi-liquid behavior of the model. This renormalization group flow allows introduction of a variational principle for the entropy, to obtain the low-temperature thermodynamics. Thus, in particular, the low-temperature thermodynamics of the noncrossing approximation to the Kondo model becomes accessible.
A path integral formulation of the SU(N) Kondo model is explored as a possible theoretical tool for studying generalizations of the model, such as the Kondo lattice, which are difficult to study by other means. Saddle point approximations to the path integral lead to an expansion in powers of 1/N. After developing the basic path integral formalism, the method is applied to three related problems. The first of these is an exactly soluble toy model which, like the Kondo model, shows smooth crossover between weakly mixed configurations at high energy scales and strongly mixed singlet configurations at low energy scales. Simple approximations are found for describing the high and low temperature limits of the model, and the failure of these approximations in the crossover region is interpreted in terms of differences between the high and low energy scales. In chapter 4 the low temperature approximation is used to study the Kondo crossover as a function of magnetic field at zero temperature. Smooth and plausible crossover is found provided the magnetic field is introduced in a way which preserves degeneracy proportional toN, indicating that the failure to describe temperature crossover is not a general limitation of the method. Finally, the same zero temperature equations are generalized in chapter 5 to a model of a magnetic impurity interacting with a gapless electronic band. Using a simple renormalization group argument this model is shown to have a transition as a function of coupling constant at zero temperature. This result is confirmed and extended using the zero temperature path integral equations, and certain special features of the magnetic field crossover are noted.
The pseudogap Kondo problem, describing a magnetic impurity embedded in an electronic environment with a power-law density of states, displays continuous quantum phase transitions between free and screened moment phases. In this paper we employ renormalization group techniques to analytically calculate universal crossover functions, associated to these transitions, for various observables. Quantitative agreement with the results of Numerical Renormalization Group (NRG) simulations is obtained for temperature-dependent static and zero-temperature dynamic quantities, at and away from criticality. In the notoriously difficult realm of finite-temperature low-frequency dynamics, usually inaccessible to both NRG and perturbative methods, we show that progress can be made by a suitable renormalization procedure in the framework of the Callan-Symanzik equations. Our general strategy can be extended to other zero-temperature phase transitions, both in quantum impurity models and bulk systems.
We generalize Emery and Kivelson's (EK) bosonization-refermionization treatment of the 2-channel Kondo model to finite system size and on the EK line analytically construct its exact eigenstates and finite-size spectrum. The latter crosses over to conformal field theory's (CFT) universal non-Fermi-liquid spectrum (and yields the most-relevant operators' dimensions), and further to a Fermi-liquid spectrum in a finite magnetic field. Our approach elucidates the relation between bosonization, scaling techniques, the numerical renormalization group (NRG), and CFT. All CFT Green's functions are recovered with remarkable ease from the model's scattering states.