Applying Feynman diagrammatics to non-fermionic strongly correlated models with local constraints might seem generically impossible for two separate reasons: (i) the necessity to have a Gaussian (non-interacting) limit on top of which the perturbative diagrammatic expansion is generated by Wick’s theorem, and (ii) Dyson’s collapse argument implying that the expansion in powers of coupling constant is divergent. We show that for arbitrary classical lattice models both problems can be solved/circumvented by reformulating the high-temperature expansion (more generally, any discrete representation of the model) in terms of Grassmann integrals. Discrete variables residing on either links, plaquettes, or sites of the lattice are associated with the Grassmann variables in such a way that the partition function (as well as all correlation functions) of the original system and its Grassmann-field counterpart are identical. The expansion of the latter around its Gaussian point generates Feynman diagrams. Our work paves the way for studying lattice gauge theories by treating bosonic and fermionic degrees of freedom on equal footing.
Applying Feynman diagrammatics to non-fermionic strongly correlated models with local constraints might seem generically impossible for two separate reasons: (i) the necessity to have a Gaussian (non-interacting) limit on top of which the perturbative diagrammatic expansion is generated by Wick’s theorem, and (ii) Dyson’s collapse argument implying that the expansion in powers of coupling constant is divergent. We show that for arbitrary classical lattice models both problems can be solved/circumvented by reformulating the high-temperature expansion (more generally, any discrete representation of the model) in terms of Grassmann integrals. Discrete variables residing on either links, plaquettes, or sites of the lattice are associated with the Grassmann variables in such a way that the partition function (as well as all correlation functions) of the original system and its Grassmann-field counterpart are identical. The expansion of the latter around its Gaussian point generates Feynman diagrams. Our work paves the way for studying lattice gauge theories by treating bosonic and fermionic degrees of freedom on equal footing.
The application of Feynman’s diagrammatic technique to classical link models with local constraints seems impossible due to (i) the absence of a free Gaussian theory on top of which the perturbative expansion can be constructed, and (ii) Dyson’s collapse argument, rendering the perturbative expansion divergent. However, we show for the classical 3D Ising model how both problems can be circumvented using a Grassmann representation. This makes it possible to obtain an expansion of the spin correlation function and the magnetic susceptibility in terms of the inverse temperature in the thermodynamic limit, through which the values for the critical temperature and critical index g are evaluated within 1.6% and 5.4% of their accepted values, respectively. Our work is a straightforward adaptation of the theory previously developed in an earlier paper.
Many-body perturbation theory (MBPT) is widely used in quantum physics, chemistry, and materials science. At the heart of MBPT is the Feynman diagrammatic expansion, which is, simply speaking, an elegant way of organizing the combinatorially growing number of terms of a certain Taylor expansion. In particular, the construction of the `bold Feynman diagrammatic expansion' involves the partial resummation to infinite order of possibly divergent series of diagrams. This procedure demands investigation from both the combinatorial (perturbative) and the analytical (non-perturbative) viewpoints. In Part II of this two-part series, we approach the analytical investigation of the bold diagrammatic expansion in the simplified setting of Gibbs measures (known as the Euclidean lattice field theory in the physics literature). Using non-perturbative methods, we rigorously construct the Luttinger-Ward formalism for the first time, and we prove that the bold diagrammatic series can be obtained directly via an asymptotic expansion of the Luttinger-Ward functional, circumventing the partial resummation technique. Moreover we prove that the Dyson equation can be derived as the Euler-Lagrange equation associated with a variational problem involving the Luttinger-Ward functional. We also establish a number of key facts about the Luttinger-Ward functional, such as its transformation rule, its form in the setting of the impurity problem, and its continuous extension to the boundary of the domain of physical Green's functions.
It is well known that the partition function of two-dimensional Ising model can be expressed as a Grassmann integral over the action bilinear in Grassmann variables. The key aspect of the proof of this equivalence is to show that all polygons, appearing in Grassmann integration, enter with fixed sign. For three-dimensional model, the partition function can also be expressed by Grassmann integral. However, the action resulting from low-temperature expansion contains quartic terms, which does not allow explicit computation of the integral. We wanted to check - apparently not explored - the possibility that using the high-temperature expansion would result in action with only bilinear terms. (in two dimensions, low-T and high-T expansions are equivalent, but in three dimensions, they differ.) It turned out, however that polygons obtained by Grassman integration are not of fixed sign for any ordering of Grassmann variables on sites. This way, it is not possible to express the partition function of three-dimensional Ising model as a Grassman integral over bilinear action.
DiagrammaticsThe diagrammatic method is one of the most powerful tools of theoretical physics.It allows us to efficiently organize perturbative computations in statistical physics, quantum many-body theory and quantum field theory.The main feature of this method is a representation of individual terms of a perturbative expansion as diagrams (graphs).Diagrams consist of vertices representing terms in the perturbation, lines representing pairings between vertices and, possibly, external legs.There exist several kinds of diagrams.We will try to present them in a systematic way.In Sect.20.1 we present a diagrammatic formalism whose goal is to organize integration of polynomials with respect to a Gaussian measure.This formalism is used extensively in classical statistical physics.It also plays an important role in quantum physics, especially in the Euclidean approach, since many quantum quantities can be expressed in terms of Gaussian integrals over classical variables.We use the term "Gaussian integration" in a rather broad sense.Beside commuting "bosonic" variables, we also consider anti-commuting "fermionic" variables, where we use the Berezin integral with respect to a Gaussian weight.Even in the case of commuting variables, the "Gaussian integral" is not necessarily meant in the sense of measure theory.It denotes an algebraic operation performed on polynomials (or formal power series), which in the case of a positive definite covariance coincides with the usual integral with a Gaussian weight.However, we allow the covariance to be complex, or even negative definite, and do not insist that the operation have a measure theoretic meaning.We distinguish two kinds of spaces on which we perform the integrals: real and complex.As in many other places in our work, we treat these two cases in parallel.Of course, the difference between the real (i.e.neutral), and the complex (i.e.charged) formalism is mainly that of a different notation.In particular, charged lines need to be equipped with an arrow, whereas neutral lines need not.The terminology that we use is inspired by quantum field theory.Therefore, the variables that enter the integral are associated with "particles"; they are divided into "bosons" and "fermions", each subdivided into "neutral" and "charged" particles.In the main part of the chapter we describe the diagram formalism used in quantum many-body physics and quantum field theory.As a preparation, we include a brief Sect.20.2 devoted to the basic terminology of perturbation theory
Abstract Modeling complex systems, like neural networks, simple liquids or flocks of birds, often works in reverse to textbook approaches: given data for which averages and correlations are known, we try to find the parameters of a given model consistent with it. In general, no exact calculation directly from the model is available and we are left with expensive numerical approaches. A particular situation is that of a perturbed Gaussian model with polynomial corrections for continuous degrees of freedom. Indeed perturbation expansions for this case have been implemented in the last 60 years. However, there are models for which the exactly solvable part is non-Gaussian, such as independent Ising spins in a field, or an ideal gas of particles. We implement a diagrammatic perturbative scheme in weak correlations around a non-Gaussian yet solvable probability weight. This applies in particular to spin models (Ising, Potts, Heisenberg) with weak couplings, or to a simple liquid with a weak interaction potential. Our method casts systems with discrete degrees of freedom and those with continuous ones within the same theoretical framework. When the core theory is Gaussian it reduces to the well-known Feynman diagrammatics.
Abstract We explore systems with a large number of fermionic degrees of freedom subject to non-local interactions. We study both vector and matrix-like models with quartic interactions. The exact thermal partition function is expressed in terms of an ordinary bosonic integral, which has an eigenvalue repulsion term in the matrix case. We calculate real time correlations at finite temperature and analyze the thermal phase structure. When possible, calculations are performed in both the original Hilbert space as well as the bosonic picture, and the exact map between the two is explained. At large N , there is a phase transition to a highly entropic high temperature phase from a low temperature low entropy phase. Thermal two-point functions decay in time in the high temperature phase.