Abstract
In this work, we consider a model of a subsystem interacting with a reservoir and study dynamics of entanglement assuming that the overall time-evolution is governed by non-integrable Hamiltonians. We also compare to an ensemble of Integrable Hamiltonians. To do this, we make use of unitary invariant ensembles of random matrices with either Wigner-Dyson or Poissonian distributions of energy. Using the theory of Weingarten functions, we derive universal average time evolution of the reduced density matrix and the purity and compare these results with several physical Hamiltonians: randomized versions of the transverse field Ising and XXZ models, Spin Glass and, Central Spin and SYK model. The theory excels at describing the latter two. Along the way, we find general expressions for exponential n-point correlation functions in the gas of GUE eigenvalues.
Highlights
We study dynamics of information transfer in this system assuming that the overall system is described by non-integrable or integrable Hamiltonians with uniformly random spectral basis, which are modeled by Random Matrix Theory (RMT) from either Wigner-Dyson or Poissonian distributions
We are interested in using Random Matrix Theory (RMT), the techniques of unitary integrals and correlation functions, to gain insights in the statistics of bipartite discrete quantum systems, induced by a Gaussian ensemble of Hamiltonians
A consistent pattern emerges: the I, I -terms decouple from the J, J -terms into a product structure: they do not depend on each other, neither in the contributions due to equation (58), nor in the multiplicative factors that appear from the Schrödinger equation
Summary
The question of emergence of thermal behavior in isolated quantum systems has attracted considerable attention since the birth of quantum mechanics [1]. Assuming some overall symmetries it can be postulated further that the randomized Hamiltonian is described by one of the traditional random matrix ensembles (GUE, GOE, or GSE) This type of setup has been implemented in several recent works [17,18,19,20,21,22,23,24,25,26,27], and most notably, [28]. We are interested in using Random Matrix Theory (RMT), the techniques of unitary integrals and correlation functions, to gain insights in the statistics of bipartite discrete quantum systems, induced by a Gaussian ensemble of Hamiltonians
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