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Fixed point stability and decay of correlations

Ettore Vicari·Jean Zinn-Justin
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Abstract

In the framework of the renormalization-group theory of critical phenomena, a quantitative description of many continuous phase transitions can be obtained by considering an effective $Φ^4$ theories, having an N-component fundamental field $Φ_i$ and containing up to fourth-order powers of the field components. Their renormalization-group flow is usually characterized by several fixed points. We give here strong arguments in favour of the following conjecture: the stable fixed point corresponds to the fastest decay of correlations, that is, is the one with the largest values of the critical exponent $η$ describing the power-law decay of the two-point function at criticality. We prove this conjecture in the framework of the $ε$-expansion. Then, we discuss its validity beyond the $ε$-expansion. We present several lower-dimensional cases, mostly three-dimensional, which support the conjecture. We have been unable to find a counterexample.

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Ettore Vicari, Jean Zinn-Justin. 2006-11-14. Fixed point stability and decay of correlations. https://doi.org/10.1088/1367-2630%2F8%2F12%2F321

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