Refined cyclic renormalization group in Russian Doll model

TL;DR

This paper investigates the refined cyclic renormalization group behavior in the Russian Doll Model, revealing energy-dependent periodicity, effects of disorder, and providing analytic and numerical insights into its spectral properties.

Contribution

It generalizes the cyclic RG analysis of the Russian Doll Model beyond the wideband limit, including disorder effects and energy-dependent spectral periodicity.

Findings

Periodic spectrum becomes energy-dependent beyond the wideband limit

Disorder affects spectral periodicity only at high energies

Analytic results are supported by exact diagonalization

Abstract

Focusing on Bethe-ansatz integrable models, robust to both time-reversal symmetry breaking and disorder, we consider the Russian Doll Model (RDM) for finite system sizes and energy levels. Suggested as a time-reversal-symmetry breaking deformation of Richardson's model, the well-known and simplest model of superconductivity, RDM revealed an unusual cyclic renormalization group (RG) over the system size $N$, where the energy levels repeat themselves, shifted by one after a finite period in $\ln N$, supplemented by a hierarchy of superconducting condensates, with the superconducting gaps following the so-called Efimov (exponential) scaling. The equidistant single-particle spectrum of RDM made the above Efimov scaling and cyclic RG to be asymptotically exact in the wideband limit of the diagonal potential. Here, we generalize this observation in various respects. We find that, beyond the wideband limit, when the entire spectrum is considered, the periodicity of the spectrum is not constant, but appears to be energy-dependent. Moreover, we resolve the apparent paradox of shift in the spectrum by a single level after the RG period, despite the disappearance of a finite fraction of energy levels. We also analyze the effects of disorder in the diagonal potential on the above periodicity and show that it survives only for high energies beyond the energy interval of the disorder amplitude. Our analytic analysis is supported with exact diagonalization.