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arXiv:cond-mat/0211338 · Physical Review B

Russian doll renormalization group and superconductivity

André LeClair, José Marı́a Román, Germán Sierra

Published Jan 1, 2004• 57 citations
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Abstract

We show that a simple extension of the standard BCS Hamiltonian leads to an infinite number of BCS eigenstates with different energy gaps and self-similar properties, described by a cyclic renormalization group flow of the BCS coupling constant which returns to its original value after a finite renormalization group time.
DOI
10.1103/physrevb.69.020505

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The recently proposed Russian doll BCS model provides a simple example of a many-body system whose renormalization group analysis reveals the existence of limit cycles in the running coupling constants of the model. The model was first studied using RG, mean field and numerical methods showing the Russian doll scaling of the spectrum, E n ∼ E 0 e −λ n , where λ is the RG period. In this paper we use the recently discovered exact solution of this model to study the low energy spectrum. We find that, in addition to the standard quasiparticles, the electrons can bind into Cooper pairs that are different from those forming the condensate and with higher energy. These excited Cooper pairs can be described by a quantum number Q which appears in the Bethe ansatz equation and has an RG interpretation.
The recently proposed Russian doll BCS model provides a simple example of a many-body system whose renormalization group analysis reveals the existence of limit cycles in the running coupling constants of the model. The model was first studied using RG, mean field and numerical methods showing the Russian doll scaling of the spectrum, En ∼ E0 e−λn, where λ is the RG period. In this paper we use the recently discovered exact solution of this model to study the low energy spectrum. We find that, in addition to the standard quasiparticles, the electrons can bind into Cooper pairs that are different from those forming the condensate and with higher energy. These excited Cooper pairs can be described by a quantum number Q which appears in the Bethe ansatz equation and has an RG interpretation.
This is the second part of the notes to the course on quantum theory of large systems of non-relativistic matter taught by J. Fröhlich at the 1994 Les Houches summer school. It is devoted to a sketchy exposition of some of the beautiful and important, recent results of J.Feldman and E.Trubowitz, and J. Feldman, H. Knörrer, D. Lehmann, J. Magnen, V. Rivasseau and E. Trubowitz. Their results are about a mathematical analysis of non-relativistic many-body theory, in particular of the Landau-Fermi liquid and BCS superconductivity, using Wilson's renormalization group methods and the techniques of the 1/N1/N1/N-expansion. While their work is ultimately aimed at a complete mathematical control (beyond perturbative expansions) of systems of weakly coupled electron gases at positive density and small or zero temperature, we can only illustrate some of their ideas within the context of perturbative solutions of Wilson-type renormalization group flow equations (we calculate leading-order terms in a 1/N1/N1/N-expansion, where NNN is an energy scale) and of one-loop effective potential calculations of the BCS superconducting ground state. Contents: 1. Background material 2. Weakly coupled electron gases 3. The renormalization group flow 4. Spontaneous breaking of gauge invariance, and superconductivity
We consider the Bethe ansatz integrable Russian Doll (RD) model of superconductivity with time-reversal symmetry breaking, which exhibits a cyclic renormalization group. By obtaining an exact solution for the renormalization group flows, we investigate the phase structure in the one-pair sector, which includes localized, fractal, and delocalized phases. We show that the quantum number Q, arising from the Bethe ansatz equations, counts the number of cycles and parametrizes the towers of states. Using the action of the renormalization group on the eigenstates, we demonstrate that Q serves as an order parameter, providing a new mechanism for the formation of the fractal phase in the deterministic systems and an example of the interplay between fractality and cyclic RG.
First, we reformulate the BCS-Bogoliubov theory of superconductivity from the viewpoint of linear algebra. We define the BCS Hamiltonian on C22M\mathbb{C}^{2^{2M}}C22M, where MMM is a positive integer. We discuss selfadjointness and symmetry of the BCS Hamiltonian as well as spontaneous symmetry breaking. Beginning with the gap equation, we give the well-known expression for the BCS state and find the existence of an energy gap. We also show that the BCS state has a lower energy than the normal state. Second, we introduce a new superconducting state explicitly and show from the viewpoint of linear algebra that this new state has a lower energy than the BCS state. Third, beginning with our new gap equation, we show from the viewpoint of linear algebra that we arrive at the results similar to those in the BCS-Bogoliubov theory.
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 N , where the energy levels repeat themselves, shifted by one after a finite period in \ln N 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.