Abstract
We discuss recent developments for exact reformulations of lattice field theories in terms of worldlines and worldsheets. In particular we focus on a strategy which is applicable also to non-abelian theories: traces and matrix/vector products are written as explicit sums over color indices and a dual variable is introduced for each individual term. These dual variables correspond to fluxes in both, space-time and color for matter fields (Abelian color fluxes), or to fluxes in color space around space-time plaquettes for gauge fields (Abelian color cycles). Subsequently all original degrees of freedom, i.e., matter fields and gauge links, can be integrated out. Integrating over complex phases of matter fields gives rise to constraints that enforce conservation of matter flux on all sites. Integrating out phases of gauge fields enforces vanishing combined flux of matter-and gauge degrees of freedom. The constraints give rise to a system of worldlines and worldsheets. Integrating over the factors that are not phases (e.g., radial degrees of freedom or contributions from the Haar measure) generates additional weight factors that together with the constraints implement the full symmetry of the conventional formulation, now in the language of worldlines and worldsheets. We discuss the Abelian color flux and Abelian color cycle strategies for three examples: the SU(2) principal chiral model with chemical potential coupled to two of the Noether charges, SU(2) lattice gauge theory coupled to staggered fermions, as well as full lattice QCD with staggered fermions. For the principal chiral model we present some simulation results that illustrate properties of the worldline dynamics at finite chemical potentials.
Highlights
Dualization techniques have recently received quite some attention in the lattice community as a possible way to solve complex action problems caused by finite density or topological terms
In order to develop the Abelian color flux and cycle concepts in this paper we discuss them for three different systems: the SU(2) principal chiral model with chemical potentials coupled to two of the conserved charges, the SU(2) lattice gauge theory coupled to staggered fermions, as well as full QCD with staggered fermions
In this contribution we discuss our recent results for the dual representation of lattice field theories with non-abelian symmetries, using the Abelian color cycle / Abelian color flux (ACC/ACF) approaches
Summary
Dualization techniques have recently received quite some attention in the lattice community as a possible way to solve complex action problems caused by finite density or topological terms (see, e.g., the reviews [1,2,3,4,5]). In all cases a suitable representation was found such that after strong coupling expansion of local Boltzmann factors the original degrees of freedom could be integrated out in closed form. This is a strategy that led to interesting worldline representations for several non-abelian systems and is the basis for the approach reviewed here. In the so-called Abelian color flux and Abelian color cycle approaches one writes all traces and matrix/vector products as sums over color indices (or more generally over ’internal indices’) and introduces a dual variable for each individual contribution At this stage of the dualization the dual variables are the expansion indices used for the Taylor series of the individual Boltzmann factors. In order to develop the Abelian color flux and cycle concepts in this paper we discuss them for three different systems: the SU(2) principal chiral model with chemical potentials coupled to two of the conserved charges, the SU(2) lattice gauge theory coupled to staggered fermions, as well as full QCD with staggered fermions
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