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arXiv:cond-mat/0306159 · Physical Review B

Potential flow of renormalization group in quasi-one-dimensional systems

Wei Chen, Ming-Shyang Chang, Hsiu‐Hau Lin, Darwin Chang, Chung‐Yu Mou

Published Nov 12, 2004• 11 citations
arXivHTMLPDF

Abstract

We address the issue why the phase diagrams for quasi-one-dimensional systems are rather simple, while the renormalization-group equations behind the scene are nonlinear and messy looking. The puzzle is answered in two steps---we first demonstrate that the complicated coupled flow equations are simply described by a potential $V({h}_{i})$, in an appropriate basis for the interaction couplings ${h}_{i}$. The renormalization-group potential is explicitly constructed by introducing the Majorana fermion representation. The existence of the potential prevents chaotic behaviors and other exotic possibilities such as limit cycles. Once the potential is obtained, the ultimate fate of the flows is described by a special set of fixed-ray solutions and the phase diagram is determined by Abelian bosonization. Extension to the strong-coupling regime and comparison with the Zamolodchikov $c$ theorem are discussed at the end.

DOI
10.1103/physrevb.70.205413

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We address the issue why the phase diagrams for quasi-one-dimensional systems are rather simple, while the renormalization-group equations behind the scene are nonlinear and messy looking. The puzzle is answered in two steps---we first demonstrate that the complicated coupled flow equations are simply described by a potential V(hi)V({h}_{i})V(hi​), in an appropriate basis for the interaction couplings hi{h}_{i}hi​. The renormalization-group potential is explicitly constructed by introducing the Majorana fermion representation. The existence of the potential prevents chaotic behaviors and other exotic possibilities such as limit cycles. Once the potential is obtained, the ultimate fate of the flows is described by a special set of fixed-ray solutions and the phase diagram is determined by Abelian bosonization. Extension to the strong-coupling regime and comparison with the Zamolodchikov ccc theorem are discussed at the end.
We studied the correlated quasi-one-dimensional systems by one-loop renormalization group techniques in weak coupling. In contrast to conventional g-ology approach, we formulate the theory in terms of bilinear currents and obtain all possible interaction vertices. Furthermore, the one-loop renormalization group equations are derived by operator product expansions of these currents at short length scale. It is rather remarkable that these coupled non-linear equations, after appropriate rescaling, can be casted into potential flows. The existence of what we nicknamed “RG potential ” provides a natural explanation of the emergent symmetry enhancement in ladder systems. Further implications arisen from the RG potential are also discussed at the end. Quasi-one-dimensional (Q1D) systems have attracted extensive attentions from both experimental and theoretical aspects. 1),2) Due to the low dimensionality, strong quantum fluctuations often give rise to surprising behaviors, which are rather different from our intuitions built in higher dimensions. It is then exciting to explore
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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.
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Renormalization group limit cycles and more chaotic behavior may be commonplace for quantum Hamiltonians requiring renormalization, in contrast to experience based on classical models with critical behavior, where fixed points are far more common. We discuss the simplest quantum model Hamiltonian identified so far that exhibits a renormalization group with both limit cycle and chaotic behavior. The model is a discrete Hermitian matrix with two coupling constants, both governed by a nonperturbative renormalization group equation that involves changes in only one of these couplings and is soluble analytically.
In the previous chapter, the ferromagnetic Ising model provided a simple example of a phase diagram with an associated fixed point structure of the renormalization group flows. There were stable fixed points corresponding to low and high temperature phases, and a critical fixed point controlling the behaviour of critical Ising Systems. However, more realistic systems often have more complicated phase diagrams, and therefore a richer fixed point structure. In this chapter we study some of these examples, and show how, even with a rather qualitative description of renormalization group flows, it is possible to understand phase diagrams from the renormalization group viewpoint. More importantly, when more than one non-trivial fixed point is present, the question arises as to which is the dominant one in a particular region of the phase space. The renormalization group answers this question through the theory of cross-over behaviour . The existence of such phenomena, whereby different fixed points may influence the properties of the same system on different length scales, is totally absent in mean field treatments. Ising model with vacancies As a first example, consider a generalisation of the Ising model in which the spin variables s ( r ) may take the value 0 as well as ±1. This may be viewed as the classical version of a quantum spin-1 magnet, or as a lattice gas of magnetic particles, with | s ( r )| playing the role of the occupation number.