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arXiv:1004.4834 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Renormalization group improved optimized perturbation theory: Revisiting the mass gap of the O(2N) Gross-Neveu model

Jean-Loı̈c Kneur, A. Neveu

Published Jun 17, 2010• 32 citations
arXivHTMLPDF

Abstract

We introduce an extension of a variationally optimized perturbation method, by combining it with renormalization group properties in a straightforward (perturbative) form. This leads to a very transparent and efficient procedure, with a clear improvement of the nonperturbative results with respect to previous similar variational approaches. This is illustrated here by deriving optimized results for the mass gap of the $O(2N)$ Gross-Neveu model, compared with the exactly known results for arbitrary $N$. At large $N$, the exact result is reproduced already at the very first order of the modified perturbation using this procedure. For arbitrary values of $N$, using the original perturbative information only known at two-loop order, we obtain a controllable percent accuracy or less, for any $N$ value, as compared with the exactly known result for the mass gap from the thermodynamical Bethe Ansatz. The procedure is very general and can be extended straightforwardly to any renormalizable Lagrangian model, being systematically improvable provided that a knowledge of enough perturbative orders of the relevant quantities is available.

DOI
10.1103/physrevd.81.125012

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We introduce an extension of a variationally optimized perturbation method, by combining it with renormalization group properties in a straightforward (perturbative) form. This leads to a very transparent and efficient procedure, with a clear improvement of the nonperturbative results with respect to previous similar variational approaches. This is illustrated here by deriving optimized results for the mass gap of the O(2N)O(2N)O(2N) Gross-Neveu model, compared with the exactly known results for arbitrary NNN. At large NNN, the exact result is reproduced already at the very first order of the modified perturbation using this procedure. For arbitrary values of NNN, using the original perturbative information only known at two-loop order, we obtain a controllable percent accuracy or less, for any NNN value, as compared with the exactly known result for the mass gap from the thermodynamical Bethe Ansatz. The procedure is very general and can be extended straightforwardly to any renormalizable Lagrangian model, being systematically improvable provided that a knowledge of enough perturbative orders of the relevant quantities is available.
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We present a general formalism that allows for the computation of large-order renormalized expansions, effectively doubling the numerically attainable perturbation order of renormalized Feynman diagrams. We show that this formulation compares advantageously to the currently standard techniques due to its high efficiency, simplicity, and broad range of applicability. Our formalism permits to easily complement perturbation theory with non-perturbative information, which we illustrate by implementing expansions renormalized by the addition of a gap or the inclusion of Dynamical Mean-Field Theory. As a result, we present numerically exact results for the square-lattice hole-doped Fermi-Hubbard model in the low-temperature non-Fermi-liquid regime, relevant to study the pseudogap of cuprate superconductors, and show the momentum-dependent suppression of fermionic excitations in the antinodal region.
We present an approximation scheme to solve the nonperturbative renormalization group equations and obtain the full momentum dependence of the n-point functions. It is based on an iterative procedure where, in a first step, an initial ansatz for the n-point functions is constructed by solving approximate flow equations derived from well motivated approximations. These approximations exploit the derivative expansion and the decoupling of high momentum modes. The method is applied to the O(N) model. In leading order, the self-energy is already accurate both in the perturbative and the scaling regimes. A stringent test is provided by the calculation of the shift DeltaTc in the transition temperature of the weakly repulsive Bose gas, a quantity which is particularly sensitive to all momentum scales. The leading order result is in agreement with lattice calculations, albeit with a theoretical uncertainty of about 25%.
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