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      The Renormalization Group and Two Dimensional Multicritical Effective Scalar Field Theory


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          Direct verification of the existence of an infinite set of multicritical non-perturbative FPs (Fixed Points) for a single scalar field in two dimensions, is in practice well outside the capabilities of the present standard approximate non-perturbative methods. We apply a derivative expansion of the exact RG (Renormalization Group) equations in a form which allows the corresponding FP equations to appear as non-linear eigenvalue equations for the anomalous scaling dimension \(\eta\). At zeroth order, only continuum limits based on critical sine-Gordon models, are accessible. At second order in derivatives, we perform a general search over all \(\eta\ge.02\), finding the expected first ten FPs, and {\sl only} these. For each of these we verify the correct relevant qualitative behaviour, and compute critical exponents, and the dimensions of up to the first ten lowest dimension operators. Depending on the quantity, our lowest order approximate description agrees with CFT (Conformal Field Theory) with an accuracy between 0.2\% and 33\%; this requires however that certain irrelevant operators that are total derivatives in the CFT are associated with ones that are not total derivatives in the scalar field theory.

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          The Exact Renormalisation Group and Approximate Solutions

          We investigate the structure of Polchinski's formulation of the flow equations for the continuum Wilson effective action. Reinterpretations in terms of I.R. cutoff greens functions are given. A promising non-perturbative approximation scheme is derived by carefully taking the sharp cutoff limit and expanding in `irrelevancy' of operators. We illustrate with two simple models of four dimensional \(\lambda \varphi^4\) theory: the cactus approximation, and a model incorporating the first irrelevant correction to the renormalized coupling. The qualitative and quantitative behaviour give confidence in a fuller use of this method for obtaining accurate results.

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            19 October 1994
            Custom metadata
            SHEP 94/95-04, CERN-TH.7403/94
            Phys.Lett. B345 (1995) 139-148
            Note added on "shadow operators". Version to be published in Phys. Lett. B
            hep-th cond-mat hep-lat


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