Sparsity dependence of Krylov state complexity in the SYK model

We study the Krylov state complexity of the Sachdev-Ye-Kitaev (SYK) model for <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>N</a:mi><a:mo>≤</a:mo><a:mn>28</a:mn></a:math> Majorana fermions with <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>q</c:mi></c:math>-body fermion interaction with <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>q</e:mi><e:mo>=</e:mo><e:mn>4</e:mn></e:math>, 6, 8 for a range of sparse parameter <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>k</g:mi></g:math> that controls the number of remaining terms in the original SYK model after sparsification. The critical value of <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>k</i:mi></i:math> below which the model ceases to be holographic, denoted <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msub><k:mi>k</k:mi><k:mi>c</k:mi></k:msub></k:math>, has been subject of several recent investigations. Using Krylov complexity as a probe, we find that the peak value of complexity does not change as we increase <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>k</m:mi></m:math> beyond <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>k</o:mi><o:mo>≥</o:mo><o:msub><o:mi>k</o:mi><o:mi>min</o:mi></o:msub></o:math> at large temperatures. We argue that this behavior is related to the change in the holographic nature of the Hamiltonian in the sparse SYK-type models such that the model is holographic for all <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>k</q:mi><q:mo>≥</q:mo><q:msub><q:mi>k</q:mi><q:mi>min</q:mi></q:msub><q:mo>≈</q:mo><q:msub><q:mi>k</q:mi><q:mi>c</q:mi></q:msub></q:math>. Our results provide a novel way to determine <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:msub><s:mi>k</s:mi><s:mi>c</s:mi></s:msub></s:math> in SYK-type models.