Quantum Epidemiology: Operator Growth, Thermal Effects, and SYK

TL;DR

This work advances the understanding of quantum chaos by moving beyond coarse operator-size probes to derive the full finite-temperature operator-growth structure. Using a purification-based framework, it maps operators to states in a doubled Hilbert space, constructs an orthonormal Majorana basis, and defines a size generating function to access all moments of operator size. Applying the method to the SYK model in the large-q limit, the authors obtain a closed-form growth distribution that reveals a simple temperature dependence and a dynamical renormalization of the coupling, tying scrambling slowdown to a thermally renormalized size unit. The finite-temperature epidemic-model interpretation provides an intuitive, quantitative picture of how thermal factors vaccinate a fraction of degrees of freedom and slow down operator growth, with potential implications for holography and chaotic dynamics in fermionic systems.

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.

Figures (10)

• Figure 1: Illustration of the purification procedure that maps operators to states in a doubled Hilbert space. (a) A maximally entangled state $|0\rangle$ (Eq. (\ref{['eq:state0']}) which can be viewed as many EPR pairs between the two systems. (b) The mapping between operator $\mathcal{O}$ and the corresponding state $|\mathcal{O}\rangle$ obtained by applying $\mathcal{O}$ to the left system (see Eq. (\ref{['eq:state for operator O']})).
• Figure 2: (a) The mapping of a Majorana string $\Gamma_{I}$ in Eq. (\ref{['eq:majoranastring']}) to a state in the doubled system. Each fermion operator $\psi_{Li}$ creates a fermion (red dot) while the fermions that are absent in $\Gamma_{I}$ stays in the vacuum state with fermion number $0$ (black dot). (b) Illustration of the relation between average size of operator $\mathcal{O}$ and OTOC.
• Figure 3: Schematic illustration of the size distribution $P_{n}\left[\psi_{1}\left(t\right)\rho^{1/2}\right]$ for the operator $\psi_{1}\left(t\right)\rho^{1/2}$ (black curve) which naturally decomposes into a convolution of a growth distribution $K_{n}^{\beta}\left[\psi_{1}\left(t\right)\right]$ (blue dashed curve) with the size distribution $P_{n}\left[\rho^{1/2}\right]$ of the operator $\rho^{1/2}$ (red dashed curve). This is due to the factorization relation of their respective generating functions (\ref{['eq:Near-Gibbs Generator']}).
• Figure 4: (a) The twisted boundary condition on the imaginary time circle. When $\tau$ crosses $\beta/4$ from below, $\psi(\tau)$ becomes a superposition of $\psi\left(\beta/4+\epsilon\right)$ and $\psi\left(3\beta/4-\epsilon\right)$ (see Eq. (\ref{['eq:Bdry Conds']}). (b) The various symmetry and boundary conditions on the twisted two point function in the $\left(\tau_{1},\tau_{2}\right)$ plane. First, $\mathcal{G}_{\mu}$ is odd under reflections across the red dotted line and even under reflections across the blue dashed lines. Thus, it is sufficient to solve the saddle-point equations in the fundamental domain $0<\tau_{1}-\tau_{2}<\beta/2$ and $\beta/2<\tau_{1}+\tau_{2}<\beta$. The black lines are the locations of the twisting boundary conditions (\ref{['eq:Two-Point Bdry Conds']}), which reduce in the large $q$ limit (\ref{['eq:Large q Bdry Con']}) and divide our fundamental domain into two regions. Region I is where neither of the two fermions have crossed a twist, while in Region II the fermions are on opposite sides of the twist.
• Figure 5: At finite temperature, when an operator such as $\psi_{1}\left(t\right)$ is multiplied to $\rho^{1/2}$, there is a chance that some fermion flavors collide and the size increase is smaller than the size of $\psi_{1}\left(t\right)$ itself.