Grassmannization of classical models

Abstract

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 (noninteracting) 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 Grassmannfield 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.

Grassmannization of the 3D Ising Model

“…The details of the algorithm are discussed in [9]. The output of the code, which generates and evaluates all possible Feynman diagrams for the spin correlator up to order 10, is a table shown in Table 1.…”

“…The approach proposed by Pollet et al [9] consists of representing the partition function of the 3D Ising model in terms of pairs of Grassmann variables. We do not repeat the development of the formalism here but only discuss the differences of the 3D case compared to the 2D case [9], and introduce notation to make the discussion self-contained.…”

“…The approach proposed by Pollet et al [9] consists of representing the partition function of the 3D Ising model in terms of pairs of Grassmann variables. We do not repeat the development of the formalism here but only discuss the differences of the 3D case compared to the 2D case [9], and introduce notation to make the discussion self-contained. Given a simple cubic lattice, one assigns a set of discrete variables to the links (or, bonds, b), {α b }, with a link factor f ({α b }) depending on the occupancy number of the link, and one defines a site factor g j = g({α b } j ) that depends only on the links attached to the jth site.…”

“…One of us (LP) proposed in Ref. [9], together with M. Kiselev, N. V. Prokof'ev and B. V. Svistunov, a theory in which (pairs of) Grassmann variables [10] represent all degrees of freedom of the high-temperature series expansion of the classical partition function (or, more generally, any discrete representation of the system can be used). The resulting expansion is a Feynman series for the spin correlation function, from which the susceptibility can be computed.…”

“…Since the series is found to be convergent down to the critical temperature T c , the convergence radius of the power series gives therefore access to the critical temperature, T c , and the critical index, γ. The results presented here have been obtained following the Grassmannization recipe developed by Pollet et al [9] and by generalizing the code written by Pollet from the 2D case to the 3D case.…”