Convergent series for lattice models with polynomial interactions

TL;DR

This paper develops methods to construct convergent series for lattice models with polynomial interactions, overcoming the asymptotic nature of standard perturbative expansions, and demonstrates their practical application on the lattice -model.

Contribution

The authors introduce a novel approach combining regularization and new initial approximations to generate convergent series for lattice models with polynomial interactions.

Findings

Convergent series exist for models on finite lattices with polynomial interactions.

The methods are applicable to practical computations, demonstrated on the -model.

Results agree well with Borel re-summation and Monte Carlo simulations.

Abstract

The standard perturbative weak-coupling expansions in lattice models are asymptotic. The reason for this is hidden in the incorrect interchange of the summation and integration. However, substituting the Gaussian initial approximation of the perturbative expansions by a certain interacting model or regularizing original lattice integrals, one can construct desired convergent series. In this paper we develop methods, which are based on the joint and separate utilization of the regularization and new initial approximation. We prove, that the convergent series exist and can be expressed as the re-summed standard perturbation theory for any model on the finite lattice with the polynomial interaction of even degree. We discuss properties of such series and make them applicable to practical computations. The workability of the methods is demonstrated on the example of the lattice $\phi^4$-model. We calculate the operator $\langle\phi_n^2\rangle$ using the convergent series, the comparison of the results with the Borel re-summation and Monte Carlo simulations shows a good agreement between all these methods.