Localization in the Dirac spectrum and gauge-field topology

We study localization of the low Dirac modes in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mn>3</a:mn><a:mo>+</a:mo><a:mn>1</a:mn></a:mrow></a:math> dimensional pure SU(3) gauge theory at zero and nonzero imaginary <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>θ</c:mi></c:math> angle, with the aim of better characterizing the relation between low-mode localization and topological features of gauge theories. We show that the mobility edge observed in the deconfined phase at <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>θ</e:mi><e:mo>=</e:mo><e:mn>0</e:mn></e:math> is present also at nonzero <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>θ</g:mi></g:math>, appearing exactly at the deconfinement transition. We find that the mobility edge is affected by topology only indirectly, through its effects on the ordering of the Polyakov loop. Moreover, the change in the mobility edge is strongly correlated with the change in the Polyakov-loop expectation value, both as one moves along the critical line, and as one departs from it toward higher temperatures. This further strengthens the connection between low-mode localization and deconfinement, showing in particular the key role played by the ordering of the Polyakov loop.