View source for Absolute X-distribution and self-duality

===Motivation===

Various models of QCD vacuum predict that it is dominated by excitations
that are predominantly self-dual or anti self-dual. In this work we look at
the tendency for self-duality in the case of pure-glue SU(3) gauge theory
using the overlap-based definition of the field-strength tensor. To gauge
this tendency, we use the absolute X-distribution method which is
designed to quantify the ''dynamical'' tendency for polarization for
arbitrary random variables that can be decomposed in a pair of orthogonal
subspaces.

===Ensembles===

$$
\def\fm {\,{\tt fm}}
\def\MeV {\,{\tt MeV}}
\def\GeV {\,{\tt GeV}}
\def\degC{\,{^\circ{\tt C}}}
\def\degK{\,{\tt K}}
$$

{| class="wikitable"  style="text-align:center"
|-
! Ensemble !! Size !! Volume !! $$N_\text{config}$$ dmatrix !! $$N_\text{config}$$ eigenmodes !! Lattice spacing !! Iwasaki $$\beta$$ !! Location
|-
| $$E_1$$ || $$8^4$$ || $$(1.32\fm)^4$$ || 0? || 100 || $$0.165\fm$$ || $$2.295$$ || dlx:scratch/overlap_eigensystem
|-
| $$E_2$$ || $$12^4$$ || $$(1.32\fm)^4$$ || 400 || 97(3) || $$0.110\fm$$ || $$2.53$$|| dlx:scratch/overlap_eigensystem dlx:scratch/topo12/dmatrices
|-
| $$E_3$$ || $$16^4$$ || $$(1.32\fm)^4$$ || 200 || 99(1) || $$0.0825\fm$$ || $$2.725$$ || dlx:scratch/overlap_eigensystem dlx:scratch/topo16/dmatrices
|-
| $$E_8$$ || $$20^4$$ || $$(1.32\fm)^4$$ || 80 || 0 || $$0.066\fm$$ || $$2.892$$(?) || dlx:scratch/topo20/dmatrices
|-
| $$E_4$$ || $$24^4$$ || $$(1.32\fm)^4$$ || 39(1) || 96(4) || $$0.055\fm$$ || $$3.0375$$ || dlx:scratch/overlap_eigensystem dlx:scratch/topo24/dmatrices
|-
| $$E_7$$ || $$32^4$$ || $$(1.32\fm)^4$$ || 19(1) || 0 ||$$0.041\fm$$ || $$3.278$$(?) || dlx:scratch/topo32
|-
| $$E_5$$ || $$16^4$$ || $$(1.76\fm)^4$$ || 0 || 99(1) || $$0.110\fm$$ || $$2.53$$ || dlx:scratch/overlap_eigensystem
|-
| $$E_6$$ || $$32^4$$ || $$(1.76\fm)^4$$ || 20 || 20 || $$0.055\fm$$ || $$3.0375$$ || qcd.phys.gwu.edu:scratch/topo32new
|-
| $$E_9$$ || $$12^3\times6$$ || $(1.32\fm)^3\times0.66\fm$ || 200 || 100 || $0.110\fm$ || $2.53$ || pyramid:scratch/iwasaki_configs/conf126 gpucluster:scratch/iwasaki_configs/conf126
|}

The number of configurations in each ensemble is given by the number of dmatrix files computed. The number in parentheses in the $N_\text{config}$ column is the number of corrupted dmatrix files in that ensemble. The $\beta$ parameter determination is discussed [[Lattice_spacing#Lattice_spacing_for_Wilson_action|here]]. The values that are marked with a question mark are inferred using the lattice spacing, whereas the other values are based on the following e-mail exchange with Jianbo Zhang:

 <nowiki>
Date: Thu, 04 Mar 2004 09:11:43 +1030
From: Jianbo Zhang <jzhang@physics.adelaide.edu.au>
To: Ivan Horvath <horvath@pa.uky.edu>
Subject: Re: 8^4 configs

Hi Ivan,

   I used the table which Frank. Lee give me,  the table presents the
\beta value and corresponding lattice spacing.    and using string
tension.   You can ask Frank Lee
for details.

  for 8^4  , I use 16 processors ( 2,2,2,2),   \beta = 2.295
  for 12^4, Iuse 27 processors ( 1,3,3,3),    \beta = 2.53
  for 16^4, Iuse  64 processors ( 2,2,4,4) ,   \beta = 2.725
  for 24^4, I use  108 processors ( 2,2,6,6),   \beta = 3.0375
BTW,   12^4 50 configurations have finished and 16^4 now have 20 configs.

Jianbo
</nowiki>

===To do===

Some of the configurations seem corrupt. For now we will remove them from the analysis but they need to be fixed. The following configurations have problems:

* dmatrix '''g3232iwa041_364.Dmatrix''' from ensemble $E_7$. The total charge for this dmatrix file is -8.5689925884e-01, which is significantly different from an integer value. For the other dmatrices the difference is of the order $O(10^{-4})$. Running the ''get_fmunu_components_find0'' code on this dmatrix reports 16477 exact zero entries for $F_{\mu\nu}$ which is most likely due to the fact that these entries are corrupted. We need to identify the position of these entries and re-run the calculation of the dmatrix for these points.
* dmatrix '''g2424iwa055_188.Dmatrix''' from ensemble $E_4$ has charge -2.5969878024e+00 and ''get_fmunu_components_find0'' reports two exact zeros. No other configuration in this ensemble has a charge so different from an integer and exact zeros for $F_{\mu\nu}$.

===Papers===

* A. Alexandru and I. Horváth, ''How Self-Dual is QCD?'' , Physics Letters B (2011) [http://arxiv.org/abs/1110.2762 arXiv:1110.2762].
* A. Alexandru, T. Draper, I. Horvath, and T. Streuer, ''The Analysis of Space-Time Structure in QCD Vacuum II: Dynamics of Polarization and Absolute X-Distribution'', Annals of Physics (2011) [http://arxiv.org/abs/1009.4451 arXiv:1009.4451].
* A. Alexandru, T. Draper, I. Horvath, and T. Streuer, ''Absolute Measure of Local Chirality and the Chiral Polarization Scale of the QCD Vacuum'', PoS LATTICE2010 (2010) 082, [http://arxiv.org/abs/arXiv:1010.5474 arXiv:1010.5474].
* T. Draper, A. Alexandru, Y. Chen, S.-J. Dong, I. Horvath, et. al., ''Improved measure of local chirality'', Nucl.Phys.Proc.Suppl. 140 (2005) 623–625, [http://arxiv.org/abs/hep-lat/0408006 hep-lat/0408006].
* C. Gattringer, ''Testing the self-duality of topological lumps in SU(3) lattice gauge theory'', Phys. Rev. Lett. 88 (2002) 221601, [http://arxiv.org/abs/hep-lat/0202002 hep-lat/0202002].
* I. Horvath et. al., ''Local chirality of low-lying Dirac eigenmodes and the instanton liquid model'', Phys. Rev. D66 (2002) 034501, [http://arxiv.org/abs/hep-lat/hep-lat/0201008 hep-lat/0201008].
* I. Horvath, N. Isgur, J. McCune, and H. B. Thacker, ''Evidence against instanton dominance of topological charge fluctuations in QCD'', Phys. Rev. D65 (2002) 014502, [http://arxiv.org/abs/hep-lat/hep-lat/0102003 hep-lat/0102003].