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The construction of a rooted tree with height m = 6. A fully grown rooted tree corresponds to a single term in the tree expansion which is constructed by recursively applying the definition of the root finding, see equation (20).
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We present the construction and stochastic summation of rooted-tree diagrams, based on the expansion of a root finding algorithm applied to the Dyson-Schwinger equations (DSEs). The mathematical formulation shows superior convergence properties compared to the bold diagrammatic Monte Carlo approach and the developed algorithm allows one to tackle g...
Citations
... Besides, unlike perturbation methods, the HAM provides the great freedom to choose the type of the linear sub-equations [30][31][32] and also the great freedom to choose an initial guess solution. Up to now, several thousands of articles related to the HAM have been published in a wide range of fields including applied mathematics, physics, engineerings, nonlinear mechanics, quantum mechanics, bio-mechanics, astronomy, finances and so on [33][34][35][36][37][38][39][40][41][42][43][44][45][46]. ...
... If this quantum simulation result is not accurate enough, one can further use it as a new guess solution ψ 0 (r, t) to gain a better approximation by quantum simulation, and so on. The key point here is that the HAM can guarantee the convergence of the solution series or the iteration approach by choosing a proper value of the so-called "convergence-control parameter" c 0 , as illustrated in several thousands of HAM publications [5,6,[33][34][35][36][37][38][39][40][41][42][43][44][45][46]. ...
Currently, Jin et al. proposed a quantum simulation technique for any a linear PDE, called Schr\"{o}dingerisation [1-3], which has been successfully applied to solve many non-Hamiltonian linear PDEs. In this paper, the Schr\"{o}dingerisation technique of quantum simulation is expanded to any a nonlinear PDE by means of combining the Schr\"{o}dingerisation technique with the homotopy analysis method (HAM) [4-6] that can transfer any a nonlinear PDE into a series of linear PDEs with convergence guarantee of series solution. In this way, a nonlinear PDE can be solved by quantum simulation using a quantum computer -- yet to be developed in the future. For simplicity, we call it ``the HAM-Schr\"{o}dingerisation quantum algorithm''.
... The HAM has been broadly used and its above-mentioned advantages have been verified and confirmed in thousands of articles by scientists and engineers all over the world [26,27,28,29,30,31,32,33,34,35,36,37,38,39,40]. ...
... So, even for given auxiliary linear operator L and initial guess u 0 , the convergence-control parameter c 0 provides us an additional way to guarantee the convergence of the solution series, which can overcome the limitations of perturbation methods mentioned above, as illustrated below in this paper and other publications [26,27,28,29,30,31,32,33,34,35,36,37,38,39,40]. Substituting the power series (17) into the zeroth-order deformation equation (14) and equating the like-power of q, we have the high-order deformation equation ...
... (A) solution series (17) is expanded in the homotopy parameter q ∈ [0, 1], which has no physical meanings at all. So, the HAM has nothing to do with any small/large physical parameters: it works no matter whether small/large physical parameters exist or not; (B) the HAM provides us great freedom to choose its auxiliary linear operator; (C) the HAM provides us great freedom to choose its initial approximation; (D) the so-called convergence-control parameter c 0 has no physical meanings but can guarantee the convergence of solution series even for high nonlinearity, as illustrated below and verified in many related publications [26,27,28,29,30,31,32,33,34,35,36,37,38,39,40]. ...
The so-called "small denominator problem" was the fundamental problem of dynamics, as pointed out by Poincar\'{e}. Small denominators are found most commonly in the perturbative theory, and the Duffing equation is the simplest example of a non-integrable system exhibiting all problems due to the small denominators. In this paper, using the Duffing equation as an example, we illustrate that the famous "small denominator problems" never appear for arbitrary physical parameters if a non-perturbative technique, namely the homotopy analysis method (HAM), is used. A new HAM-based approach is proposed, which provides us great freedom to directly define the inverse operator of an auxiliary linear operator so that all small denominators can be completely avoided. Convergent series of multiple limit-cycles of the Duffing equation are successfully obtained, although such kind of directly defined inverse operators might be beyond the traditional mathematical theories. Thus, from the viewpoint of the HAM, the famous "small denominator problems" are only artifacts of perturbation methods. Therefore, completely abandoning perturbation methods, one would be never troubled by small denominators. This HAM-based approach has general meanings and can be used to attack many open problems related to "small denominators".
... It can also be advantageous to perform this exact summation by bruteforce enumeration provided the momentum and time variables are chosen appropriately, as successfully demonstrated very recently for the electron gas [70]. A radically different approach would be to work with Schwinger-Dyson equations, for which new algorithms were introduced and applied to bosonic models [71][72][73][74]. ...
We provide a description of a diagrammatic Monte Carlo algorithm for the resonant Fermi gas in the normal phase. Details are given on the diagrammatic framework, Monte Carlo moves, and incorporation of ultraviolet asymptotics. Apart from the self-consistent bold scheme, we also describe a non-self-consistent scheme, for which the ultraviolet treatment is more involved.
... The problem statement and solution is included in the next section. Convergence of the series solution is examined [11][12][13][14][15][16][17][18][19][20]. Effects of physical parameters on velocity as well as temperature profiles is described. ...
This article addresses the effects of heat transfer on magnetohydrodynamic Falkner-Skan wedge flow of a Jeffery fluid. The continuity, momentum and energy balance equations yield the relevant PDE which are transforms to ODE by exploitation of similarity variables. Strength of optimal homotopy series solutions is practiced to solved analytically the transformed ODE model of hydromagnetic Falkner-Skan fluid rheology with heat transfer scenarios. The graphical and numerical illustrations of the result are presented for different interesting flow parameters. Numerical values of Nusselt number are tabulated. It is observed that for the Falkner-Skan rheology, the applied magnetic field acts as a controlling agnet which controls the fluids velocity up to the desired value whereas Debrorah number enhances the fluid velocity.
... After nearly ten years and despite recent and tremendous progress [14][15][16], one may well fear that the combination of an asymptotic/divergent series, even with a mild sign problem, is as prohibitive as the standard approaches. Recently [17], we therefore suggested to use the more flexible Dyson-Schwinger equation (DSE) instead of self-consistent Feynman diagrams [18] to provide a fully self-consistent scheme on the one and two particle level. Furthermore, we extended the homotopy analysis method (HAM) [19,20] to φ 4 field theory in two dimensions (2D) (providing us with more tools to enhance the convergence properties in a systematic way), and showed how the expansion in terms of rooted trees is amenable to a systematic Monte Carlo sampling. ...
... Clearly, this is a radically different way at looking at interacting field theories. However, in our previous work [17] we truncated the DSE at the level of the 6-point vertex. The infinite tower of equations for n-point correlation functions was not solved and differences with the full, exact answer could be seen when the correlation length increases. ...
... The unbiased numerical solution of the DSE is largely unexplored as it was considered to be too complex to be solved even in the simplest cases [23][24][25]. Furthermore, taking into account the functional derivatives deteriorates the convergence properties of the field theory substantially compared to the truncated case considered in Ref. [17]. Here we show how the remaining theory can be brought under control within the HAM as an asymptotic expansion of the HAM deformations in terms of an auxiliary convergence control parameter c 0 (times the 2-point correlation function G) around G = 0, or even as a convergent expansion in the HAM deformations when a shift of G is possible, e.g. by solving the ladder equations. ...
We provide a full and unbiased solution to the Dyson-Schwinger equation illustrated for $\phi^4$ theory in 2D. It is based on an exact treatment of the functional derivative $\partial \Gamma / \partial G$ of the 4-point vertex function $\Gamma$ with respect to the 2-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 $c_0G$, 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 integro-differential equations.
... It was later shown to be applicable to fermionic many-body problems [28,29,30,31,32,12], and to frustrated spin systems [33,34,35]. While the original technique explicitly sampled Feynman diagram topologies, alternative computational approaches are also being developed [36,37,38,39]. Recently, it was shown that Diagrammatic Monte Carlo can be applied to the large-N limit of matrix field theories [40,41]. ...
This thesis contributes to the development of unbiased diagrammatic approaches to the quan-tum many-body problem, which consist in computing expansions in Feynman diagrams toarbitrary order with no small parameter. The standard form of fermionic sign problem - expo-nential increase of statistical error with volume - does not affect these methods as they workdirectly in the thermodynamic limit. Therefore they are a powerful tool for the simulation ofquantum matter.Part I of the thesis is devoted to the unitary Fermi gas, a model of strongly-correlatedfermions accurately realized in cold-atom experiments. We show that physical quantities canbe retrieved from the divergent diagrammatic series by a specifically-designed conformal-Boreltransformation. Our results, which are in good agreement with experiments, demonstrate thata diagrammatic series can be summed reliably for a fermionic theory with no small parameter.In Part II we present a new efficient algorithm to compute diagrammatic expansions to highorder. All connected Feynman diagrams are summed at given order in a computational timemuch smaller than the number of diagrams. Using this technique one can simulate fermions onan infinite lattice in polynomial time. As a proof-of-concept, we apply it to the weak-couplingHubbard model, obtaining results with record accuracy.Finally, in Part III we address the problem of the misleading convergence of dressed dia-grammatic schemes, which is related to a branching of the Luttinger-Ward functional. Afterstudying a toy model, we show that misleading convergence can be ruled out for a large classof diagrammatic schemes, and even for the fully-dressed scheme under certain conditions.
The major obstacle preventing Feynman diagrammatic expansions from accurately solving many-fermion systems in strongly correlated regimes is the series slow convergence or divergence problem. Several techniques have been proposed to address this issue: series resummation by conformal mapping, changing the nature of the starting point of the expansion by shifted action tools, and applying the homotopy analysis method to the Dyson-Schwinger equation. They emerge as dissimilar mathematical procedures aimed at different aspects of the problem. The proposed homotopic action offers a universal and systematic framework for unifying the existing—and generating new—methods and ideas to formulate a physical system in terms of a convergent diagrammatic series. It eliminates the need for resummation, allows one to introduce effective interactions, enables a controlled ultraviolet regularization of continuous-space theories, and reduces the intrinsic polynomial complexity of the diagrammatic Monte Carlo method. We illustrate this approach by an application to the Hubbard model.
Yang–Mills theories are an important building block of the standard model and in particular of quantum chromodynamics. Its correlation functions describe the behavior of its elementary particles, the gauge bosons. In quantum chromodynamics, the correlation functions of the gluons are basic ingredients for calculations of hadrons from bound state equations or properties of its phase diagram with functional methods.
Correlation functions of gluons are defined only in a gauge fixed setting. The focus of many studies is the Landau gauge which has some features that alleviate calculations. I discuss recent results of correlation functions in this gauge obtained from their equations of motions. Besides the four-dimensional case also two and three dimensions are treated, since the effects of truncations, viz., the procedure to render the infinitely large system of equations finite, can be studied more directly in these cases. In four dimensions, the anomalous running of dressing functions plays a special role and it is explained how resummation is realized in the case of Dyson–Schwinger equations.
Beyond the Landau gauge other gauges can provide additional insights or can alleviate the development of new methods. Some aspects or ideas are more easily accessible in alternative gauges and the results presented here for linear covariant gauges, the Coulomb gauge and the maximally Abelian gauge help to refine our understanding of Yang–Mills theories.
We develop numerical tools for diagrammatic Monte Carlo simulations of non-Abelian lattice field theories in the t’Hooft large-N limit based on the weak-coupling expansion. First, we note that the path integral measure of such theories contributes a bare mass term in the effective action which is proportional to the bare coupling constant. This mass term renders the perturbative expansion infrared-finite and allows us to study it directly in the large-N and infinite-volume limits using the diagrammatic Monte Carlo approach. On the exactly solvable example of a large-N O(N) sigma model in D=2 dimensions we show that this infrared-finite weak-coupling expansion contains, in addition to powers of bare coupling, also powers of its logarithm, reminiscent of resummed perturbation theory in thermal field theory and resurgent trans-series without exponential terms. We numerically demonstrate the convergence of these double series to the manifestly nonperturbative dynamical mass gap. We then develop a diagrammatic Monte Carlo algorithm for sampling planar diagrams in the large-N matrix field theory, and apply it to study this infrared-finite weak-coupling expansion for large-N U(N)×U(N) nonlinear sigma model (principal chiral model) in D=2. We sample up to 12 leading orders of the weak-coupling expansion, which is the practical limit set by the increasingly strong sign problem at high orders. Comparing diagrammatic Monte Carlo with conventional Monte Carlo simulations extrapolated to infinite N, we find a good agreement for the energy density as well as for the critical temperature of the “deconfinement” transition. Finally, we comment on the applicability of our approach to planar QCD at zero and finite density.
