Localization-delocalization transition at weak coupling in two-color matrix QCD

We numerically investigate the matrix model of two-color one-flavor adjoint QCD (matrix- <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:msubsup> <a:mrow> <a:mi>QCD</a:mi> </a:mrow> <a:mrow> <a:mn>2</a:mn> <a:mo>,</a:mo> <a:mn>1</a:mn> </a:mrow> <a:mrow> <a:mi>adj</a:mi> </a:mrow> </a:msubsup> </a:mrow> </a:math> ) in the weak coupling regime (small <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>g</c:mi> </c:math> ) and in the chiral limit. The Yang-Mills potential has two distinct gauge invariant minima: one at <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msub> <e:mi>A</e:mi> <e:mi>i</e:mi> </e:msub> <e:mo>=</e:mo> <e:mn>0</e:mn> </e:math> and the other at <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:msub> <g:mi>A</g:mi> <g:mi>i</g:mi> </g:msub> <g:mo>=</g:mo> <g:mfrac> <g:msub> <g:mi>σ</g:mi> <g:mi>i</g:mi> </g:msub> <g:mrow> <g:mn>2</g:mn> <g:mi>g</g:mi> </g:mrow> </g:mfrac> </g:math> . We show that when the chiral chemical potential <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mi>c</i:mi> <i:mo>≤</i:mo> <i:mfrac> <i:mn>3</i:mn> <i:mn>2</i:mn> </i:mfrac> </i:math> there is a quantum phase transition (QPT) at <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:msubsup> <k:mi>g</k:mi> <k:mn>0</k:mn> <k:mo>*</k:mo> </k:msubsup> <k:mo>≃</k:mo> <k:mn>0.143</k:mn> </k:math> : for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mi>g</m:mi> <m:mo>&lt;</m:mo> <m:msubsup> <m:mi>g</m:mi> <m:mn>0</m:mn> <m:mo>*</m:mo> </m:msubsup> </m:math> , the ground-state wave function is localized near <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:msub> <o:mi>A</o:mi> <o:mi>i</o:mi> </o:msub> <o:mo>=</o:mo> <o:mn>0</o:mn> </o:math> , while for <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:mi>g</q:mi> <q:mo>&gt;</q:mo> <q:msubsup> <q:mi>g</q:mi> <q:mn>0</q:mn> <q:mo>*</q:mo> </q:msubsup> </q:math> , the ground state is delocalized over the gauge configuration space. The transition between these two phases is singular, with the ground state at <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:msubsup> <s:mi>g</s:mi> <s:mn>0</s:mn> <s:mo>*</s:mo> </s:msubsup> </s:math> being distinctly different from that of <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:msubsup> <u:mi>g</u:mi> <u:mn>0</u:mn> <u:mo>*</u:mo> </u:msubsup> <u:mo>±</u:mo> <u:mo stretchy="false">|</u:mo> <u:mi>ε</u:mi> <u:mo stretchy="false">|</u:mo> </u:math> . At <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"> <y:msubsup> <y:mi>g</y:mi> <y:mn>0</y:mn> <y:mo>*</y:mo> </y:msubsup> </y:math> , we show that the square of the chromoelectric field vanishes, strongly suggesting that the system is in a “dual superconductor” phase. Numerical evidence shows that the localization-delocalization phenomenon holds for the first and second excited states as well, leading us to conjecture that there are an infinite number of isolated singular points <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"> <ab:msubsup> <ab:mi>g</ab:mi> <ab:mn>0</ab:mn> <ab:mo>*</ab:mo> </ab:msubsup> <ab:mo>&gt;</ab:mo> <ab:msubsup> <ab:mi>g</ab:mi> <ab:mn>1</ab:mn> <ab:mo>*</ab:mo> </ab:msubsup> <ab:mo>&gt;</ab:mo> <ab:msubsup> <ab:mi>g</ab:mi> <ab:mn>2</ab:mn> <ab:mo>*</ab:mo> </ab:msubsup> <ab:mo>&gt;</ab:mo> <ab:mo>⋯</ab:mo> </ab:math> accumulating to <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"> <cb:mi>g</cb:mi> <cb:mo>=</cb:mo> <cb:mn>0</cb:mn> </cb:math> . For <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"> <eb:mi>c</eb:mi> <eb:mo>=</eb:mo> <eb:mn>1</eb:mn> </eb:math> , the model formally possesses <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"> <gb:mi mathvariant="script">N</gb:mi> <gb:mo>=</gb:mo> <gb:mn>1</gb:mn> </gb:math> supersymmetry. We show that in the localized phase (i.e., for <jb:math xmlns:jb="http://www.w3.org/1998/Math/MathML" display="inline"> <jb:mi>g</jb:mi> <jb:mo>&lt;</jb:mo> <jb:msubsup> <jb:mi>g</jb:mi> <jb:mn>0</jb:mn> <jb:mo>*</jb:mo> </jb:msubsup> </jb:math> ) the supermultiplet structure is disrupted and supersymmetry is spontaneously broken.