Convergent series for lattice models with polynomial interactions

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.Comment: 25 pages, 14 figure

Convergent series for polynomial lattice models with complex actions

“…Then, (16) and all other steps of the CS construction can be done. The non-perturbative independence on τ of each Z (n) can be proved exactly, as in [26] for the lattice φ 4 -model with the real action. Concluding this section, we would like to note that, the expansion (15) is similar to the Taylor expansion in chemical potential (TE) [32].…”

“…There the restriction to a finite amount of degrees of freedom provided the conditions, enough to carry out the construction mathematically rigorously and to prove the possibility to express coefficients of the convergent series for any model on the finite lattice with the real action and an even degree polynomial interaction as linear combinations of SPT-terms. It was also shown in [26] that the convergent series has a certain variational invariance, which allows one to improve its convergence significantly.…”

Infinite lattice models by an expansion with a non-Gaussian initial approximation

“…Despite that in the considered example the CS terms lose their positivity due to the change of τ = 0 to τ = −1 or to τ = −1/2, one can note that the series nevertheless converges to the correct answer. In the work [3] it is explained by employing an additional regularization of the form…”

“…It happens because the term analogous to γS 3 2 [x] is non-local and the models with local and non-local actions have different scalings at V → ∞. Nevertheless, note that the failure of the regularization (16), (17) prohibits the proof from [3] for infinite volumes, but does not necessary imply the inapplicability of the CS method. Here we study numerically the workability of the CS method investigating the φ 4 -model defined on the one-and two-dimensional infinite lattices.…”

“…Instead, we do a formal substitution V − − > w (as it is prescribed by the CS method) and prove that it leads to the correct answer, independently on V , if Z is Borel-summable. Indeed, considering the asymptotic of large orders [3], one can show that for γ > 0 the series…”