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arXiv:0708.0238 · Physical Review B

Calculations on the two-point function of the O(N) model

Federico Benitez, Ramón Méndez-Galain, Nicolás Wschebor

Published Jan 31, 2008• 20 citations
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Abstract

The self-energy of the critical three-dimensional $O(N)$ model is calculated. The analysis is performed in the context of the nonperturbative renormalization group by exploiting an approximation which takes into account contributions of an infinite number of vertices. A very simple calculation yields the two-point function in the whole range of momenta, from the UV Gaussian regime to the scaling one. Results are in good agreement with best estimates in the literature for any value of $N$ in all momenta regimes. This encourages the use of this simple approximation procedure to calculate correlation functions at finite momenta in other physical situations.

DOI
10.1103/physrevb.77.024431

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•Sep 3, 2008•0 citations
The self-energy of the critical three-dimensional O(N)O(N)O(N) model is calculated. The analysis is performed in the context of the nonperturbative renormalization group by exploiting an approximation which takes into account contributions of an infinite number of vertices. A very simple calculation yields the two-point function in the whole range of momenta, from the UV Gaussian regime to the scaling one. Results are in good agreement with best estimates in the literature for any value of NNN in all momenta regimes. This encourages the use of this simple approximation procedure to calculate correlation functions at finite momenta in other physical situations.
In a companion paper [Blaizot, Phys. Rev. E 74, 051116 (2006)], we have presented an approximation scheme to solve the nonperturbative renormalization group equations that allows the calculation of the n-point functions for arbitrary values of the external momenta. The method was applied in its leading order to the calculation of the self-energy of the O(N) model in the critical regime. The purpose of the present paper is to extend this study to the next-to-leading order of the approximation scheme. This involves the calculation of the four-point function at leading order, where interesting features arise, related to the occurrence of exceptional configurations of momenta in the flow equations. These require a special treatment, inviting us to improve the straightforward iteration scheme that we originally proposed. The final result for the self-energy at next-to-leading order exhibits a remarkable improvement as compared to the leading order calculation. This is demonstrated by the calculation of the shift DeltaTc, caused by weak interactions, in the temperature of Bose-Einstein condensation. This quantity depends on the self-energy at all momentum scales and can be used as a benchmark of the approximation. The improved next-to-leading order calculation of the self-energy presented in this paper leads to excellent agreement with lattice data and is within 4% of the exact large N result.
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%.
We present the implementation of the Blaizot-Méndez-Wschebor approximation scheme of the nonperturbative renormalization group we present in detail, which allows for the computation of the full-momentum dependence of correlation functions. We discuss its significance and its relation with other schemes, in particular, the derivative expansion. Quantitative results are presented for the test ground of scalar O(N) theories. Besides critical exponents, which are zero-momentum quantities, we compute the two-point function at criticality in the whole momentum range in three dimensions and, in the high-temperature phase, the universal structure factor. In all cases, we find very good agreement with the best existing results.
Critical exponents for the three-dimensional O(n)-symmetric model with n>3 are estimated on the basis of six-loop renormalization-group (RG) expansions. A simple Pad\'e-Borel technique is used for the resummation of the RG series and the Pad\'e approximants [L/1] are shown to give rather good numerical results for all calculated quantities. For large n, the fixed point location gc{\mathit{g}}_{\mathit{c}}gc​ and the critical exponents are also determined directly from six-loop expansions without addressing the resummation procedure. An analysis of the numbers obtained shows that resummation becomes unnecessary when n exceeds 28 provided an accuracy of about 0.01 is adopted as satisfactory for gc{\mathit{g}}_{\mathit{c}}gc​ and the critical exponents. Further, results of the calculations performed are used to estimate the numerical accuracy of the 1/n expansion. The same value n=28 is shown to play the role of the lower boundary of the domain where this approximation provides high-precision estimates for the critical exponents.
Higher{order vertices at zero external momenta for the scalar eld theory, describing the critical behaviour of the Ising model, are studied within the eld{theoretical renormalization group (RG) approach in three dimensions.Dimensionless six{point g6 and eight{point g8 eective coupling constants are calculated in the three{loop approximation.Their numerical values, universal at criticality, are estimated by means of the Pad e and Pad e{Borel summation of the RG expansions found and by putting the renormalized quartic coupling constant equal to its universal xed{point value known from six{loop RG calculations.The values of g 6 obtained are compared with their analogs resulting from the {expansion, Monte Carlo simulations, the Wegner{Houghton equations and the linked cluster expansion series.The eld{theoretical estimates for g 6 are shown to be in a good agreement with each other, diering considerably from the values given by other methods.
The topic of this thesis is the functional renormalization group. We discuss some approximations schemes. Thereafter we apply these approximations to study different fields of condensed matter physics. Generally we have to evaluate an infinite set of vertex functions describing the scattering of particles. These vertex functions get renormalized away from their bare values governed by an infinite hierarchy of flow equations. We cannot expect to actually solve these equations but have to apply a couple of approximations. The aim is to somehow separate relevant contributions from irrelevant ones. One possible scheme opens up if we rescale fields and vertices. Here "relevance" is used in a quantitative way to describe the scaling behaviour of vertices close to a fixed point of the RG. One disadvantage of describing the system in terms of infinitely many vertices is that the majority of these vertices we have to evaluate are not of interest to us. In most cases we are just looking for the self-energy or the two-particle effective interaction. However there might be contributions to the flow of these vertices that are generated by irrelevant vertices. We generally assume that we can express irrelevant vertices in terms of the relevant and marginal ones. Then in turn it should be possible to write the contributions of these irrelevant vertices to the flow of relevant and marginal ones in terms of relevant and marginal vertices as well. We show how this can be achieved by what we term the adiabatic approximation. We now consider weakly interacting bosons at the critical point of Bose-Einstein condensation. As the transition takes place at a finite temperature this temperature defines an effective ultraviolet cut-off. For the investigation of physical properties that depend on momenta smaller than this cut-off it is therefore sufficient to describe the system by a classical field theory. Our central topic here is the self-energy of the bosons and we are able to evaluate it with the full momentum dependence. For small momenta it approaches a scaling form and as the momentum is gradually increased we observe a crossover to the perturbative regime. As a test for the reliability of our expression for the selfenergy we investigate the interaction induced shift of the critical. Our results compare quite satisfactory to the best available estimates for this shift. For the anomalous dimension our approach predicts the correct order of magnitude however with a considerable error. As an improvement we include more vertices into our calculations. Here we observe that our fixed point estimates indeed approach the best known results but this convergence is quite weak. We turn toward systems of interacting fermions. The formulation of the functional renormalization group implicitly requires knowledge of the true Fermi surface of the full interacting system. In general however we can just calculate it a-posteriori from the self-energy. The requirement to flow into a fixed point can be translated into a fine-tuning of the frequency/momentum independent part r_0 of the rescaled 2-point function. We show how this bare value is related to the momentum dependent effective interaction along the complete trajectory of the RG. On the other hand r_0 expresses the difference between the bare and the true Fermi surface. Putting both equations together results into an exact selfconsistency equation for the Fermi surface. We apply our self-consistency equation above to tackle the problem of finding the true Fermi surface of interacting fermions in low dimensions. The most simple non-trivial model with an inhomogeneous Fermi surface is a system of two coupled metallic chains. The process of interband backward scattering leads to a smoothing of the Fermi surface. Of special interest is if the Fermi momenta of the two bands collapse into just one value. We propose the term confinement transition for this behaviour. We bosonize the interband backward scattering by means of a Hubbard-Stratonovich transformation and treat our system as a single channel problem. This bosonization together with the adiabatic approximation allows us to investigate the system even at strong coupling. Within a simple one-loop treatment our method predicts a confinement transition at strong coupling. However taken vertex renormalizations into account we observe that this confinement is destroyed by fluctuations beyond one-loop. Actually we observe how the confined phase can be stabilized by the inclusion of interband umklapp scattering. Thereafter we consider the physically more relevant case of a two-dimensional system of infinitely many coupled metallic chains. Here the Fermi surface consists of two disconnected weakly curved sheets. We are able to repeat the calculations we have performed for our toy model. Within a self-consistent 2-loop calculation indeed signs for a confinement transition at finite coupling strength emerge.