Note on the q = 2 R-para-fermionic SYK model

We investigate the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>q</a:mi> <a:mo>=</a:mo> <a:mn>2</a:mn> </a:math> SYK model with <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>R</c:mi> </c:math> -para-particles ( <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>R</e:mi> <e:mtext>−</e:mtext> <e:msub> <e:mrow> <e:mi>PSYK</e:mi> </e:mrow> <e:mn>2</e:mn> </e:msub> </e:math> ), analyzing its thermodynamics and spectral form factor (SFF) using random matrix theory. The Hamiltonian is quadratic, with coupling coefficients randomly drawn from the Gaussian unitary ensemble. The model displays self-averaging behavior and exhibits an exponential ramp in its SFF dynamics: <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi mathvariant="script">K</g:mi> <g:mo stretchy="false">(</g:mo> <g:mi>t</g:mi> <g:mo stretchy="false">)</g:mo> <g:mo>∼</g:mo> <g:msup> <g:mi>e</g:mi> <g:mrow> <g:msub> <g:mi>C</g:mi> <g:mn>0</g:mn> </g:msub> <g:mi>t</g:mi> </g:mrow> </g:msup> </g:math> . The growth rate <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"> <l:msub> <l:mi>C</l:mi> <l:mn>0</l:mn> </l:msub> </l:math> tends toward either a constant or infinity in the <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"> <n:mi>N</n:mi> <n:mo stretchy="false">→</n:mo> <n:mi>∞</n:mi> </n:math> limit, depending on specific statistics of the model. These results provide novel perspectives on quantum systems with unconventional statistics.