


Volume 20 No 20 (2022)
Download PDF
TOPOLOGICAL CHARGE OF LQCD WITH QCDLAB
Dafina Xhako , Niko Hyka , Enkela Nocka
Abstract
Computing the solution of the equation of motion from the Yang-Mills Lagrangianrequires too
much computational power. Therefore, computing the time evolution ofobservables at high
temperatures is currently not feasible. However, it is possible tocompute expectation values of
observables in statistical physics according to the pathintegral formulation in Euclidean
spacetime. The theory behind that is called latticeQCD. In this formulation spacetime is
discretized in a finite volume by introducing afinite lattice spacing a such that the path integral
has finite dimensions. This allowsus to compute observables numerically on computers using
statistical Monte-Carlotechniques. Computing observables this way results in some uncertainties.
To controlthese discretization effects, a continuum extrapolation a → 0 is done in a final step. In
Quantum Chromo Dynamics (QCD) there are fascinating connectionsbetween chiral symmetry
in the quark sector and topological properties inthe gluon sector.There is a well-known theorem
that relates the zero modes of the operator to the background topology of the calibration field,
called the ASIT theorem (Atiyah-Singer Index Theorem). The ASIT theorem states that the
topological charge of the calibration field configuration is equal to the difference between the
number of zero modes with positive and negative chirality. This relation helps to relate
properties of fermionic quantities to the topology of the lattice calibration field. In the lattice it is
possible to calculate the topological charge for each configuration of the calibration field.
Another important use of the ASIT theorem is the fact that the lattice eigenvectors of the real
eigenvalued Diracoperator can be interpreted as the lattice counterparts of zero modes in
continuum theory. The study of the distribution of eigenvalues of the lattice Wilson-Dirac
operator in the presence of calibration fields generated in simulations has a number of
motivations. The density of small eigenvalues of the Dirac operator is related to spontaneous
chiral symmetry breaking. For this reason we bring in this paper our calculation of topological
charge of lattice QCD using QCDLAB
Keywords
.
Copyright
Copyright © Neuroquantology
Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Articles published in the Neuroquantology are available under Creative Commons Attribution Non-Commercial No Derivatives Licence (CC BY-NC-ND 4.0). Authors retain copyright in their work and grant IJECSE right of first publication under CC BY-NC-ND 4.0. Users have the right to read, download, copy, distribute, print, search, or link to the full texts of articles in this journal, and to use them for any other lawful purpose.