Convergent Perturbation Theory for the lattice $ϕ^4$-model

2017

Nuclear Physics B

Self Cite

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

Section: Variational Seriesmentioning

confidence: 83%

Section: Introductionmentioning

confidence: 88%

Convergent series for lattice models with polynomial interactions

Ivanov

2017

Nuclear Physics B

Self Cite

“…Recently, the convergent series similar to [18,19] was derived for the real action models defined on finite lattices [25,26]. 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.…”

Section: Introductionmentioning

confidence: 99%

Convergent series for polynomial lattice models with complex actions

2019

Mod. Phys. Lett. A

Lattice models with complex actions are important for the understanding of matter at finite densities, but not accessible by the standard Monte Carlo techniques due to the sign problem. Here we derive a new approach for avoiding the complex action/sign problem, by extending the method of convergent series with a non-Gaussian initial approximation. The main features of the new series are demonstrated on the example of the two dimensional oscillating integral.

“…In recent works [18,3,4] the convergent series for the real and certain complex action models defined on finite lattices was constructed in a rigorous way. It was shown there that the application of the dimensional regularization can be interpreted as an additional re-summation procedure accelerating the series convergence.…”

Section: Introductionmentioning

confidence: 99%

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

Ivanov

2019

Physics Letters B

Recently, a convergent series employing a non-Gaussian initial approximation was constructed and shown to be an effective computational tool for the finite size lattice models with a polynomial interaction. Here we show that the Borel summability is a sufficient condition for the correctness of the convergent series applied to infinite lattice models. We test the numerical workability of the convergent series method by examine one-and two-dimensional φ 4 -infinite lattice models. The comparison of the convergent series computations and the infinite lattice extrapolations of the Monte Carlo simulations reveals an agreement between two approaches.