<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi mathvariant="script">N</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math> SYK models with dynamical bosons and fermions

We study a class of Sachdev, Ye, and Kitaev (SYK) models with <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi mathvariant="script">N</a:mi><a:mo>=</a:mo><a:mn>2</a:mn></a:math> supersymmetry, described by <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:mi>N</d:mi></d:math> fermions in chiral Fermi multiplets, as well as <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:mi>α</f:mi><f:mi>N</f:mi></f:math> first-order bosons in chiral multiplets. The interactions are characterized by two integers <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"><h:mo stretchy="false">(</h:mo><h:mi>p</h:mi><h:mo>,</h:mo><h:mi>q</h:mi><h:mo stretchy="false">)</h:mo></h:math>. We focus on the large <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:mi>N</l:mi></l:math> and low energy limit of these models. Despite the presence of dynamical bosons, we find conformal behavior akin to the standard SYK model. We use <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:mi mathvariant="script">I</n:mi></n:math>-extremization of a Witten index to study the supersymmetric solutions. In particular, we find an exact expression for the entropy, which matches the numerical solutions to the Schwinger-Dyson equations. We further solve the model both in the large <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>p</q:mi></q:math> and large <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mrow><s:mi>p</s:mi></s:mrow></s:math>, <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mrow><u:mi>q</u:mi></u:mrow></u:math> limits. Numerically, we verify our analytical results and obtain estimates for the Schwarzian coupling in the near zero-temperature limit. We also study the low-lying spectrum of operators to determine the parameter ranges where the Schwarzian mode dominates the IR dynamics. Lastly, we study out-of-time-ordered correlators to show that the model is maximally chaotic.