Exploring the infrared landscape of the SYK model

A bstract We analyse a class of SYK models whose Hamiltonian is the sum of two SYK Hamiltonians with different numbers of fermions q, $$ \overset{\sim }{q} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>q</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> in each interaction. We consider both Euclidean and Lorentzian probes of the quantum system in the large N limit. In the strong coupling phase, the entropy provides a diagnostic of the thermal renormalisation group flow. Under certain conditions, two parametrically separated regimes of near-conformal behaviour emerge. The first reproduces the standard linear-in-temperature scaling characteristic of the single SYK model. The system then flows to another near-fixed point whose entropy scaling depends on the ratio $$ n=q/\overset{\sim }{q} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mi>q</mml:mi> <mml:mo>/</mml:mo> <mml:mover> <mml:mi>q</mml:mi> <mml:mo>~</mml:mo> </mml:mover> </mml:math> . For n &lt; 3 / 2, the entropy exhibits anomalous, stronger-than-linear scaling in temperature. At n = 3 / 2, there is an additional logarithmic enhancement. Using conformal perturbation theory, we argue that in the infrared regime of the SYK model, there may exist disordered conformal operators with dimensions 1 &lt; ∆ ≤ 3 / 2. In Lorentzian signature, we study the out-of-time-ordered correlator and show that these deformed theories exhibit near-maximal chaos in both regimes (when they exist). We comment on the relation between the anomalous scalings found here and those observed in certain near-extremal black holes in two and higher dimensions.