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Full and Unbiased Solution of the Dyson-Schwinger Equation in the Functional Integro-Differential Representation PHYSICAL REVIEW B 98, 195104 (2018) Full and unbiased solution of the Dyson-Schwinger equation in the functional integro-differential representation Tobias Pfeffer and Lode Pollet Department of Physics, Arnold Sommerfeld Center for Theoretical Physics, University of Munich, Theresienstrasse 37, 80333 Munich, Germany (Received 17 March 2018; revised manuscript received 11 October 2018; published 2 November 2018) We provide a full and unbiased solution to the Dyson-Schwinger equation illustrated for φ4 theory in 2D. It is based on an exact treatment of the functional derivative ∂/∂G of the four-point vertex function with respect to the two-point correlation function G within the framework of the homotopy analysis method (HAM) and the Monte Carlo sampling of rooted tree diagrams. The resulting series solution in deformations can be considered as an asymptotic series around G = 0 in a HAM control parameter c0G, or even a convergent one up to the phase transition point if shifts in G can be performed (such as by summing up all ladder diagrams). These considerations are equally applicable to fermionic quantum field theories and offer a fresh approach to solving functional integro-differential equations beyond any truncation scheme. DOI: 10.1103/PhysRevB.98.195104 I. INTRODUCTION was not solved and differences with the full, exact answer could be seen when the correlation length increases. Despite decades of research there continues to be a need In this work, we solve the full DSE by writing them as a for developing novel methods for strongly correlated systems. closed set of integro-differential equations. Within the HAM The standard Monte Carlo approaches [1–5] are convergent theory there exists a semianalytic way to treat the functional but suffer from a prohibitive sign problem, scaling exponen- derivatives without resorting to an infinite expansion of the tially in the system volume [6]. Diagrammatic Monte Carlo successive n-point correlation functions, cf. Refs. [21,22] simulations [7–9] were developed to prevent this, scaling where the DSE has been used to generate new weak coupling exponentially only in the expansion order [10]. However, this expansions. The unbiased numerical solution of the DSE is happens at the expense of substantially worse series conver- largely unexplored as it was considered to be too complex to gence properties [11–13]. After nearly ten years and despite be solved even in the simplest cases [23–25]. Furthermore, recent and tremendous progress [14–16], one may well fear taking into account the functional derivatives deteriorates the that the combination of an asymptotic/divergent series and a convergence properties of the field theory substantially com- mild sign problem is as prohibitive as the standard approaches. pared to the truncated case considered in Ref. [17]. Here we Recently [17], we therefore suggested to use the more flexible show how the remaining theory can be brought under control Dyson-Schwinger equation (DSE) instead of self-consistent within the HAM as an asymptotic expansion of the HAM Feynman diagrams [18] to provide a fully self-consistent deformations in terms of an auxiliary convergence control scheme on the one and two particle level. Furthermore, we parameter c (times the two-point correlation function G) extended the homotopy analysis method (HAM) [19,20]toφ4 0 around G = 0, or even as a convergent expansion in the HAM field theory in two dimensions (2D) (providing us with more deformations when a shift of G is possible, e.g., by solving the tools to enhance the convergence properties in a systematic ladder equations. Although the ideas are illustrated for a 1d way), and showed how the expansion in terms of rooted integral and φ4 theory in 2D the convergence considerations trees is amenable to a systematic Monte Carlo sampling. and the methodology are generically applicable as long as the This expansion is furthermore convenient when dealing with two-point and four-point correlation functions are bounded, multidimensional objects such as the four-point vertex func- and may just as well be applied to the better known Hedin tion. Clearly, this is a radically different way at looking at [26] and parquet equations [27] or to the functional renormal- interacting field theories. However, in our previous work [17], ization group equation [28,29]. we truncated the DSE at the level of the six-point vertex. The The paper is organized as follows. The main ideas and infinite tower of equations for n-point correlation functions results are introduced in Secs. II–IV. In Secs. II and III,we first analyze the toy model (i.e., the 1d integral, cf. Ref. [30]) to illustrate the approach and then proceed in Sec. IV with the full solution of the DSE for the φ4 model in 2D. We provide further technical details, derivations, and information Published by the American Physical Society under the terms of the about the developed algorithm in Appendixes A–C.Wedo Creative Commons Attribution 4.0 International license. Further not include these details in the main text in order to provide distribution of this work must maintain attribution to the author(s) the reader with a clear introduction to our new approach. The and the published article’s title, journal citation, and DOI. appendixes should be consulted in order to judge the validity 2469-9950/2018/98(19)/195104(24) 195104-1 Published by the American Physical Society TOBIAS PFEFFER AND LODE POLLET PHYSICAL REVIEW B 98, 195104 (2018) and broad applicability of the results presented in the main where we denote the derivative with respect to G as , leads text. to the differential equation 3λ λ λ λ [G] = λ − G2 + G42 − G3 + G5. (7) II. FUNCTIONAL CLOSURE 2 2 3 6 The functional closure is most easily demonstrated in the A solution to (4) requires knowledge of the universal func- case of a 0D field theory. Consider the 1D integral tional [G], which can be obtained by solving the differential equation (7). [G] is subsequently used in (4) and solved 1 λ Z[J ] = dφ e−S[φ]+Jφ with S[φ] = kφ2 + φ4, (1) for G for a given inverse of the noninteracting two-point 2 4! correlation function k. The combined system (4) and (7)is where Z[J ] is the generating functional of the n-point corre- typically solved by a fixed point iteration. This gives the lation functions G(n), physical two-point correlation function G. The physical four- point vertex function follows by evaluating the universal 1 1 dnZ[J ] G(n) = dφ e−S[φ]φn = . (2) functional [G] for the physical G.InRef.[17], we have n Z[0] Z[0] dJ J =0 already shown how to solve a realistic FT when only taking the first three terms on the right-hand side of (7) into account, Although in this example the generating functional Z[J ]isa i.e., working with a truncated version of [G], which has a real-valued function and the n-point correlation functions are finite convergence radius. Next, we examine the deterioration real numbers, we will, nevertheless, use the terminology of in convergence properties when taking the derivatives in (7) (quantum) field theory (FT) as there will be no ambiguities. into account. Moreover, we use the shorthand notation G = G(2) for the two-point correlation function. The DSE [31] can be derived by introducing an infinitesimal shift δ in the integration vari- III. VERTEX EQUATION able, φ → φ + δ, and expanding the resulting expression in The exact solution of (7) obtained by the implicit Euler powers of δ. This yields method is shown in Fig. 1(a) for λ = 10. It agrees with inverting the functional G[k] → k[G] and plugging it into dS d φ = Z[J ] = JZ[J ]. (3) [k[G]]. The additional vertical (horizontal) grid line corre- dφ dJ sponds to the physical G() obtained for the model parame- = = The first derivative of the action S with respect to the field φ is ters k 1,λ 10. In case of a FT, (7) takes the form of a = d functional integro-differential equation and the implicit Euler promoted to a differential operator by the substitution φ dJ . The DSE (3) is a definition of the generating functional in method cannot be applied. It is therefore necessary to study terms of a differential equation equivalent to the definition semianalytic solutions to the differential equation (7). = + of Z[J ] through (1). For a realistic FT (3) turns into a The power series expansion [G] λ M n functional integro-differential equation, whereas (1) turns into limM→∞ n=1 cnG has zero convergence radius [32]: a functional integral. bringing (7) into the form of [G] = F [G, [G]], we find Instead of focusing on the solution of (1) we focus on (3). that F is not holomorphic at G = 0, = λ. An expansion Differentiating (3) once with respect to J and setting J = 0 around G = 0 corresponds to approaching the noninteracting afterwards yields, after introducing the connected four-point limit by fixing λ and taking k →∞, i.e., (4) = (4) − 2 correlation function Gc G 3G and the four-point 1 2 λ 4 →∞ − 1 2 − φ − φ k vertex function =−G 4G(4), Gk = dφ φ e 2 4!k2 = 1. (8) c Z λ λ G−1 − k = G − G3. (4) Further details about the analytic structure of the functional 2 6 [G] are given in Appendix A. We observe numerically, This is the first equation in the expansion of the differential see Fig. 1(b), that the power series solution to (7) has the equation (3) into an infinite hierarchy of coupled equations for properties of an asymptotic expansion, i.e., for every small the correlation functions.
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