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arXiv:2410.19683

Schwinger Dyson Summation of Perturbation Expansions

Tom Banks

Published Oct 25, 2024• 0 citations
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Abstract

We introduce a hierarchical system of approximations for summing both conventional perturbation theory and large N vector expansions of models in quantum field theory and condensed matter physics. Each stage of the hierarchy consists of a closed set of nonlinear equations for one particle irreducible correlation functions with no more than K points and captures the perturbation expansion of each of those functions up to some finite order. As K goes to infinity the full Schwinger Dyson hierarchy is approached. We present an argument that for ordinary finite dimensional integrals, the procedure converges to the exact answer in this limit, despite the fact that no finite order of the hierarchy sees effects from non-leading saddle points of the integral. Some potential applications to the calculation of critical exponents, to low dimensional condensed matter systems, and to the phase structure of the homogeneous electron gas are outlined, but no detailed calculations are presented.

DOI
10.48550/arxiv.2410.19683

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We introduce a hierarchical system of approximations for summing both conventional perturbation theory and large N vector expansions of models in quantum field theory and condensed matter physics. Each stage of the hierarchy consists of a closed set of nonlinear equations for one particle irreducible correlation functions with no more than K points and captures the perturbation expansion of each of those functions up to some finite order. As K goes to infinity the full Schwinger Dyson hierarchy is approached. We present an argument that for ordinary finite dimensional integrals, the procedure converges to the exact answer in this limit, despite the fact that no finite order of the hierarchy sees effects from non-leading saddle points of the integral. Some potential applications to the calculation of critical exponents, to low dimensional condensed matter systems, and to the phase structure of the homogeneous electron gas are outlined, but no detailed calculations are presented.
The goal of this thesis is to provide a practical method to calculate, in scalar field theory, accurate numerical values of the renormalized quantities which could be used to test any kind of approximate calculation. We use finite truncations of the Fourier transform of the recursion formula for Dyson's hierarchical model in the symmetric and broken phases to perform high precision calculations of the Green's functions at zero momentum. We use the well-known correspondence between statistical mechanics and field theory in which the large cut-off limit is obtained by letting β reach a critical value βc. We show that the round-off errors on the magnetic susceptibility grow like (βc − β) −1 near criticality. We show that the systematic errors (finite truncation and volume) can be controlled with an exponential precision and reduced to a level lower than numerical errors. We probe the numerical errors made in Renormalization Group (RG) calculations by varying slightly the resealing factor of the fields and resealing back in order to get the same (if there were no round-off errors) magnetic susceptibility. We find a significant departure from Gaussian behavior. In addition, the average value differs from the exact answer by a quantity which is of the same order as the standard deviation. We calculate the critical exponents γ and the subleading exponent Δ of Dyson's model by direct fits of the zero-momentum two-point function calculated with an Ising and a Landau-Ginzburg measure and by linearization about the non-trivial fixed point for O(N) model up to N = 5. We extend the method to the low temperature phase and show that accurate values of the connected one-, two- and three- point functions can be obtained at large volume and in a limited range of constant external field coupled linearly to the field variable. We show that the leading exponents obey the scaling and hyperscaling relations. We discuss the concept of universality in terms of critical exponents, correction amplitudes and the universal coupling constant for different Landau-Ginzburg measures. We compare our accurate numerical results with those obtained from perturbative methods.
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We investigate an approximation to the Schwinger-Dyson (SD) equations of the collective Coulomb field of the large NNN Homogeneous Electron Fluid. The large NNN approximation transforms the infinite SD hierarchy into a set of closed, equations for 1 and 2-pt correlators. In this paper, the dynamics of a toy model -- a small, square Euclidean lattice with periodic boundary conditions -- are considered. The Markov Chain Monte Carlo numerical method evaluated the 1 and 2-pt correlation functions on a 2×22 \times 22×2 and 3×33 \times 33×3 lattice. The derived equations are checked with the correlator values, and an agreement at N∼100N \sim 100N∼100 to order 10−310^{-3}10−3 was found. The agreement can be further strengthened by increasing runs in the Markov Chain Monte Carlo method.
We review recent results concerning the renormalization group (RG) transformation of Dyson's hierarchical model (HM). This model can be seen as an approximation of a scalar field theory on a lattice. We introduce the HM and show that its large group of symmetry simplifies drastically the blockspinning procedure. Several equivalent forms of the recursion formula are presented with unified notations. Rigourous and numerical results concerning the recursion formula are summarized. It is pointed out that the recursion formula of the HM is inequivalent to both Wilson's approximate recursion formula and Polchinski's equation in the local potential approximation (despite the very small difference with the exponents of the latter). We draw a comparison between the RG of the HM and functional RG equations in the local potential approximation. The construction of the linear and nonlinear scaling variables is discussed in an operational way. We describe the calculation of non-universal critical amplitudes in terms of the scaling variables of two fixed points. This question appears as a problem of interpolation between these fixed points. Universal amplitude ratios are calculated. We discuss the large- N limit and the complex singularities of the critical potential calculable in this limit. The interpolation between the HM and more conventional lattice models is presented as a symmetry breaking problem. We briefly introduce models with an approximate supersymmetry. One important goal of this review is to present a configuration space counterpart, suitable for lattice formulations, of functional RG equations formulated in momentum space (often called exact RG equations and abbreviated ERGE).
A new approach to the one-particle Green's functions G\mathit{G}G for finite electronic systems is presented. This approach is based on the diagrammatic perturbation expansions of the Green's function and of the dynamic self-energy part M\mathit{M}M related to G\mathit{G}G via the Dyson equation. The exact summation of the latter expansion is reformulated in terms of a simple algebraic form referred to as algebraic diagrammatic construction (ADC). The ADC defines in a systematical way a set of approximation schemes (nthn\mathrm{th}nth-order ADC schemes) that represent infinite partial summations for M\mathit{M}M and (via the Dyson equation) for G\mathit{G}G being complete through nthn\mathrm{th}nth order of perturbation theory. The corresponding mathematical procedures are essentially Hermitian eigenvalue problems in restricted configuration spaces of unperturbed ionic configurations. Explicit equations for the second-, third-, and fourth-order ADC schemes are derived and analyzed. While the second- and third-order schemes can be viewed as systematic rederivations of previous approximation schemes, the fourth-order ADC scheme represents a complete fourth-order approximation for the self-energy and the one-particle Green's function which was hitherto not available.
For pt.II see ibid., vol.11, p.375 (1978). Exact renormalisation group equations (RGE) are obtained for the d-dimensional n-vector and spin-S Heisenberg versions of Dyson's hierarchical model. In the quantum mechanical case the fixed point and neighbourhood properties of the RGE are shown to be independent of S indicating universality of critical behaviour in spin magnitude. Expansions for critical exponents, analogous to the usual epsilon =4-d expansions, are obtained and the RGE solved numerically to obtain relevant scaling indices. The n to infinity or spherical limit is also discussed and 1/n corrections to critical exponents are calculated to first order in 1/n.