Abstract
In many-body chaotic systems, the size of an operator generically grows in Heisenberg evolution, which can be measured by certain out-of-time-ordered four-point functions. However, these only provide a coarse probe of the full underlying operator growth structure. In this article we develop a methodology to derive the full growth structure of fermionic systems, that also naturally introduces the effect of finite temperature. We then apply our methodology to the SYK model, which features all-to-all q-body interactions. We derive the full operator growth structure in the large q limit at all temperatures. We see that its temperature dependence has a remarkably simple form consistent with the slowing down of scrambling as temperature is decreased. Furthermore, our finite-temperature scrambling results can be modeled by a modified epidemic model, where the thermal state serves as a vaccinated population, thereby slowing the overall rate of infection.
Highlights
We show that the μ-expansion of Gμ (t) determines the growth distribution induced by multiplying ρ1/2 by ψ (t), and that Gμ (t) is the two-point function for the original theory with a μ-dependent twisted boundary condition
Where the Hamiltonians HL, HR are required to satisfy the condition (HL − HR)|0 = 0. This state is a natural choice for studying thermodynamic properties, because for each operator O, we can consider the corresponding operator Oρ1/2, and its average size will be directly measured by the finite temperature four-point function: δ Oρ1/2 n Oρ1/2 =1−
Determining the system’s full growth distribution amounts to calculating the twisted (3.17) two-point function Gμ followed by inverse transforming in μ (3.12)
Summary
As an operator O (t) evolves in time, it becomes supported along operators of increasing size. This can be inferred from the Heisenberg equation of motion O (t) = i [H, O (t)]. When the Hilbert space is finite-dimensional it is natural to use the Frobenius inner product: OA|OB ≡ Tr(OA† OB). We may expand operators in an orthonormal operator basis, which amounts to inserting a complete set of operators {ΓI }. Note that at this point we have set up a Hilbert space of operators. If the original Hilbert space H has dimension L, the operator Hilbert space is H ⊗ H, with dimension L2
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