Abstract
In this paper, we use the exactly solvable Sachdev-Ye-Kitaev model to address the issue of entropy dynamics when an interacting quantum system is coupled to a non-Markovian environment. We find that at the initial stage, the entropy always increases linearly matching the Markovian result. When the system thermalizes with the environment at a sufficiently long time, if the environment temperature is low and the coupling between system and environment is weak, then the total thermal entropy is low and the entanglement between system and environment is also weak, which yields a small system entropy in the long-time steady state. This manifestation of non-Markovian effects of the environment forces the entropy to decrease in the later stage, which yields the Page curve for the entropy dynamics. We argue that this physical scenario revealed by the exact solution of the Sachdev-Ye-Kitaev model is universally applicable for general chaotic quantum many-body systems and can be verified experimentally in near future.
Highlights
In this paper, we explore the Sachdev-Ye-Kitaev model [11, 12] with random fourMajorana fermions interactions (SYK4), and this SYK4 model is coupled to a system with random quadratic Majorana fermions couplings (SYK2)
We address the issue of when a Page curve can emerge in entropy dynamics of a system coupled to the environment
We use SYK model as an exactly solvable model to study this problem, the lesson we learn from our model reveals a general physical picture that should be applicable to generic chaotic quantum many-body systems
Summary
We will first discuss situations where the entropy dynamics of our model can recover the Markovian results. When βE = 0, the second term vanishes and the Green’s function of the environment becomes frequency independent, which is equivalent to the Markovian approximation. Under this situation, the standard derivation of the master equation eq (3.1) through a second-order perturbation theory yields the dissipation strength κ = V 2/JE. The entropy is expected to saturate to the maximum value (N/2) log 2 in the long-time limit, which is the same as the Markovian case This can be shown more rigorously using the path-integral formalism by relating the Rényi entropy to the inner product of Kourkoulou-Maldacena pure states [27, 29]. In figure 2(b), one can see that the entropy curve gradually approaches the Markovian result as κ increases
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