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arXiv:0911.0166 · Physical Review B

Hierarchy of relevant couplings in perturbative renormalization group transformations

Hong-Yan Shih, Wen-Min Huang, Sze‐Bi Hsu, Hsiu‐Hau Lin

Published Mar 31, 2010• 6 citations

Abstract

The phase diagram for the interacting fermions in weak coupling is described by the perturbative renormalization group equations. Due to the lack of analytic solutions for these coupled nonlinear differential equations, it is rather subtle to tell which couplings are relevant or irrelevant. We propose a powerful classification scheme to build up the hierarchy of the relevant couplings by a scaling Ansatz found numerically. To demonstrate its superiority over the conventional classification for the relevant couplings, we apply this scheme to a controversial phase transition in the two-leg ladder and show that it should be a nontrivial crossover instead. The scaling Ansatz we propose here can classify the relevant couplings in hierarchical order without any ambiguity and can improve significantly how we interpret the numerical outcomes in general renormalization group methods.

DOI: 10.1103/physrevb.81.121107

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The phase diagram for the interacting fermions in weak coupling is described by the perturbative renormalization group equations. Due to the lack of analytic solutions for these coupled nonlinear differential equations, it is rather subtle to tell which couplings are relevant or irrelevant. We propose a powerful classification scheme to build up the hierarchy of the relevant couplings by a scaling Ansatz found numerically. To demonstrate its superiority over the conventional classification for the relevant couplings, we apply this scheme to a controversial phase transition in the two-leg ladder and show that it should be a nontrivial crossover instead. The scaling Ansatz we propose here can classify the relevant couplings in hierarchical order without any ambiguity and can improve significantly how we interpret the numerical outcomes in general renormalization group methods.

Competitions between different interactions in strongly correlated electron systems often lead to exotic phases. Renormalization group is one of the powerful techniques to analyze the competing interactions without presumed bias. It was recently shown that the renormalization group transformations to the one-loop order in many correlated electron systems are described by potential flows. Here we prove several rigorous theorems in the presence of renormalization-group potential and find the complete classification for the potential flows. In addition, we show that the relevant interactions blow up at the maximal scaling exponent of unity, explaining the puzzling power-law Ansatz found in previous studies. The above findings are of great importance in building up the hierarchy for relevant couplings and the complete classification for correlated ground states in the presence of generic interactions.

In previous studies, we proposed a scaling ansatz for electron-electron interactions under renormalization group transformation. With the inclusion of phonon-mediated interactions, we show that the scaling ansatz, characterized by the divergent logarithmic length

{l}_{d}

and a set of renormalization-group exponents, also works rather well. The superconducting phases in a doped two-leg ladder are studied and classified by these renormalization-group exponents as demonstration. Finally, nontrivial constraints among the exponents are derived and explained.

A renormalization group transformation (RGT) that permits us to analyze the stability of fermionic systems to various perturbations in any number of dimensions is developed. An RGT that leaves invariant the free fermion system (on or off a lattice) is defined and interactions are classified as relevant, irrelevant or marginal. It is shown how the RGT automatically considers competing instabilities simultaneously, in contrast to mean field theory, which focuses on just one. It is shown that at weak coupling only the BCS coupling is relevant for a spherical Fermi surface. Both Landau theory and the Kohn-Luttinger result are viewed in the light of the RG.

A renormalization group transformation (RGT) that permits us to analyze the stability of fermionic systems to various perturbations in any number of dimensions is developed. An RGT that leaves invariant the free fermion system (on or off a lattice) is defined and interactions are classified as relevant, irrelevant or marginal. It is shown how the RGT automatically considers competing instabilities simultaneously, in contrast to mean field theory, which focuses on just one. It is shown that at weak coupling only the BCS coupling is relevant for a spherical Fermi surface. Both Landau theory and the Kohn-Luttinger result are viewed in the light of the RG.

Using a functional renormalization-group method, we obtain the phase diagram of the two-leg ladder system within the Holstein-Hubbard model, which includes both electron-electron and electron-phonon interactions. Our renormalization-group technique allows us to analyze the problem for both weak and strong electron-phonon couplings. We show that, in contrast to results from conventional weak-coupling studies, electron-phonon interactions can dominate electron-electron interactions, and retardation effects play an important role on the phase diagram.

A renormalization group transformation (RGT) that permits us to analyze the stability of fermionic systems to various perturbations in any number of dimensions is developed. An RGT that leaves invariant the free fermion system (on or off a lattiice) is defined and interactions are classified as relevent, irrelevent or marginal. It is shown how the RGT automatically considers competing instabilities simultaneously, in contrast to mean field theory, which focuses on just one. It is shown that at weak coupling only the BCS coupling is relevent unless there is nesting. Both Landau theory and the Kohn-Luttinger argument are discussed in this context.