<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi mathvariant="double-struck">Z</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:math> stability and continuity in the twisted Eguchi-Kawai model with two-flavor

We investigate the twisted Eguchi-Kawai (TEK) reduced model of four-dimensional <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>S</a:mi> <a:mi>U</a:mi> <a:mo stretchy="false">(</a:mo> <a:mi>N</a:mi> <a:mo stretchy="false">)</a:mo> </a:math> gauge theory in the presence of two-flavor adjoint fermions (adjoint TEK model). Using Monte Carlo simulations with <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>N</e:mi> <e:mo>=</e:mo> <e:mn>121</e:mn> </e:math> , twist parameter <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>k</g:mi> <g:mo>=</g:mo> <g:mn>1</g:mn> </g:math> , hopping parameter <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mrow> <i:mi>κ</i:mi> <i:mo>=</i:mo> <i:mn>0.01</i:mn> <i:mi>–</i:mi> <i:mn>0.03</i:mn> </i:mrow> </i:math> ( <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mo>≪</k:mo> <k:msub> <k:mi>κ</k:mi> <k:mi>c</k:mi> </k:msub> </k:math> ) and inverse ’t Hooft coupling <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mi>b</m:mi> <m:mo>=</m:mo> <m:mn>0.30</m:mn> <m:mi>–</m:mi> <m:mn>0.45</m:mn> </m:math> , we show that heavy adjoint fermions stabilize the <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:mo stretchy="false">(</o:mo> <o:msub> <o:mi mathvariant="double-struck">Z</o:mi> <o:mi>N</o:mi> </o:msub> <o:msup> <o:mo stretchy="false">)</o:mo> <o:mn>4</o:mn> </o:msup> </o:math> center-symmetric vacuum even for the minimal twist satisfying <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"> <t:mi>k</t:mi> <t:mo>/</t:mo> <t:msqrt> <t:mi>N</t:mi> </t:msqrt> <t:mo>&lt;</t:mo> <t:mn>1</t:mn> <t:mo>/</t:mo> <t:mn>9</t:mn> </t:math> , where the <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"> <v:mo stretchy="false">(</v:mo> <v:msub> <v:mi mathvariant="double-struck">Z</v:mi> <v:mi>N</v:mi> </v:msub> <v:msup> <v:mo stretchy="false">)</v:mo> <v:mn>4</v:mn> </v:msup> </v:math> symmetry is spontaneously broken in the absence of adjoint fermions. This result also suggests that the adjoint TEK model with the minimal twist is equivalent to <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"> <ab:mi>S</ab:mi> <ab:mi>U</ab:mi> <ab:mo stretchy="false">(</ab:mo> <ab:mi>N</ab:mi> <ab:mo stretchy="false">)</ab:mo> </ab:math> gauge theory over a broader <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"> <eb:mo stretchy="false">(</eb:mo> <eb:mi>κ</eb:mi> <eb:mo>,</eb:mo> <eb:mi>b</eb:mi> <eb:mo stretchy="false">)</eb:mo> </eb:math> parameter region than the adjoint EK model without twist. We further extend our analysis to a partially reduced model to realize a geometry akin to <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"> <ib:msup> <ib:mi mathvariant="double-struck">R</ib:mi> <ib:mn>3</ib:mn> </ib:msup> <ib:mo>×</ib:mo> <ib:msup> <ib:mi>S</ib:mi> <ib:mn>1</ib:mn> </ib:msup> </ib:math> and study the theory under <lb:math xmlns:lb="http://www.w3.org/1998/Math/MathML" display="inline"> <lb:msup> <lb:mi>S</lb:mi> <lb:mn>1</lb:mn> </lb:msup> </lb:math> compactification with periodic adjoint fermions. Numerical simulations with <nb:math xmlns:nb="http://www.w3.org/1998/Math/MathML" display="inline"> <nb:mi>N</nb:mi> <nb:mo>=</nb:mo> <nb:mn>16</nb:mn> <nb:mi>–</nb:mi> <nb:mn>49</nb:mn> </nb:math> , <pb:math xmlns:pb="http://www.w3.org/1998/Math/MathML" display="inline"> <pb:mi>b</pb:mi> <pb:mo>=</pb:mo> <pb:mn>0.30</pb:mn> <pb:mi>–</pb:mi> <pb:mn>0.46</pb:mn> </pb:math> , and <rb:math xmlns:rb="http://www.w3.org/1998/Math/MathML" display="inline"> <rb:mi>κ</rb:mi> <rb:mo>=</rb:mo> <rb:mn>0.03</rb:mn> <rb:mi>–</rb:mi> <rb:mn>0.16</rb:mn> </rb:math> are consistent with the adiabatic continuity conjecture; with periodic adjoint fermions, the theory remains in a center-symmetric (confined) phase as the <tb:math xmlns:tb="http://www.w3.org/1998/Math/MathML" display="inline"> <tb:msup> <tb:mi>S</tb:mi> <tb:mn>1</tb:mn> </tb:msup> </tb:math> circle size is reduced, in contrast to the deconfining transition observed in the pure TEK model or in the TEK model with antiperiodic adjoint fermions. We present the Polyakov loop measurements and consistency checks supporting these findings.