Canonical simulations with worldlines: An exploratory study in ϕ24 lattice field theory

Abstract

In this letter we explore the perspectives for canonical simulations in the worldline formulation of a lattice field theory. Using the charged φ 4 field in two dimensions as an example we present the details of the canonical formulation based on worldlines and outline algorithmic strategies for canonical worldline simulations. We discuss the steps for converting the data from the canonical approach to the grand canonical picture which we use for cross-checking our results. The canonical approach presented here can easily be generalized to other lattice field theories with a worldline representation. Introductory remarksImplementing a clean ab-initio calculation of lattice QCD at finite density is one of the great open challenges in the field. The reason for the difficulties is the fact that at non-zero chemical potential µ the action S becomes complex and the Boltzmann factor exp(−S) cannot be used as a weight factor in a Monte Carlo simulation. Among the different approaches for overcoming that so-called complex action problem are canonical simulations where one works at fixed net baryon number, i.e., at a fixed density. The key challenge for implementing the canonical strategy in QCD is the projection to the desired quark or baryon number: in the conventional path integral representation the quark number is not an integer valued observable and the fermion determinant has to be decomposed into temporal winding classes for the gauge loops it consists of. Various strategies for that task can be found in the literature. They either use different expansions of the fermion determinant or Fourier transformation of the determinant with respect to imaginary chemical potential (see [1] for some examples). However, so far the results for canonical lattice QCD are restricted to small volumes and low densities [1].In recent years an alternative formulation based on worldlines was found and explored for several lattice field theories (see [2] for reviews given at the yearly lattice conference series on that topic). In the worldline (or dual) formulation the partition function Z is exactly rewritten in terms of new variables, such that Z becomes a sum over configurations of worldlines. The corresponding weights for the worldline configurations are real and positive also at non-zero chemical potential and the complex action problem is overcome completely in such a worldline representation.

Regressive and generative neural networks for scalar field theory

“…(A3), the weight of any field configuration is real and positive, thus the system can be simulated using standard Monte-Carlo algorithms. Specifically, we employ a worm algorithm [26,40,41], which is capable of automatically satisfying the divergence constraint ∇ ⋅ k = 0 on all lattice sites. Differentiating the partition function (A3) we obtain the representations (3)-(4) of the operators in terms of the integers k and .…”

Investigation of Complex ϕ4 Theory at Finite Density in Two Dimensions Using TRG

“…This model exhibits the so-called Silver Blaze phenomenon in which bulk observables do not depend on the chemical potential below the critical point. Since it is directly related to the imaginary part of the action, various methods that could overcome the sign problem, such as the complex Langevin approach [19], the thimble method [20][21][22], and the worldline representation [23,24], have been used to study the model. In case of TRG, it is not straightforward to apply the algorithm to the scalar field theory because the tensor indices are given by the field variable which takes any real or complex number and numerical computation is not directly applied to such an infinite dimensional tensor.…”

“…If the Taylor expansion of the hopping term is used instead of the character expansion, another dual formulation is obtained[24].…”

Kramers–Wannier duality and worldline representation for the SU(2) principal chiral model

“…In the third step of (8) we have inserted the path integral measure D[U ] and the explicit parameterization (5) for the U x and the Haar integration measure (we have dropped the indices x of θ x , α x and β x for better readability). The integrals in (8) can be solved in closed form: The integrals over α and β give rise to Kronecker deltas for the integer valued combinations of the j ab x,ν in the respective exponents, i.e., these intergrals generate constraints for these combinations at each site x. The integrals over θ give rise to beta-functions that can be simplified as fractions of factorials, since the exponents of cos θ and sin θ are odd (this follows from the constraints).…”

“…Thus the particle numbers are defined as integers for individual configurations in the worldline representation, while in the conventional lattice representation they are non-integer functionals of the fields U x corresponding to some lattice discretization of the expressions (3). This property opens the door towards a clean and straightforward implementation of a canonical simulations in the worldline representation which might be a more powerful approach in some parameter ranges (compare [8]).…”