arXiv:2009.10416v1 [quant-ph] 22 Sep 2020 Thermalization of isolated quantum many-body system and entanglement Prasenjit Deb, ∗ Pratik Ghosal, † Pratapaditya Bej, ‡ and Abhishek Banerjee § Department of Physics and Center for Astroparticle Physics and Space Science, Bose Institute, Bidhan Nagar Kolkata - 700091, India. Abstract Thermalization of an isolated quantum system has been a non-trivial problem since the early days of quantum mechanics. In generic isolated systems, non-equilibrium dynamics is expected to result in thermalization, indicating the emergence of statistical mechanics from quantum dynamics. However, what feature of many-body quantum system facilitates quantum thermalization is still not well understood. Here we revisit this problem and show that introduction of entanglement in the system gives rise to thermalization, and it takes place at the level of individual eigenstate. We also show that the expectation value in the energy eigenstate of each subsystem is close to the canonical average. PACS numbers: I. INTRODUCTION A prerequisite for statistical mechanics is the maxi- mization of entropy in a system at thermal equilibrium. In other words, when a system gets thermalized one can find out the values of corresponding physical ob- servables and thermodynamic functions from its statis- tical description or representative ensemble. However, an isolated quantum many-body system initialized in a pure state remains pure during unitary evolution, and in this sense it has zero entropy. Then, what is the mechanism through which such a quantum system, whose initial state is pure, gets thermalized and quantum sta- tistical mechanics emerges from it? Thermalization of an isolated quantum system and emergence of statisti- cal ensembles from its unitary time evolution has been a fascinating problem since the early days of quantum mechanics[1–5]. In the classical scenario, the assumption of ergodicity leads to statistical mechanics. However, the notion of ergodicity adopted for classical systems has failed in leading to similar conclusion in the quan- tum regime despite numerous attempts[1, 6, 7]. Most of the works have emphasized the need of coupling with an external heat bath[8], which is being done tradition- ally, in order to obtain statistical mechanics. Later, it has been shown that a finite but very small perturba- tion may lead to such a temporal evolution of the sys- tem that time average of observables are in agreement with the microcanonical ensemble, commonly known as eigenstate thermalization hypothesis (ETH)[3, 4]. The name itself signifies that thermalization happens at the level of individual eigenstates. In the last decade, experimental developments[9– 15] have made precise simulation of unitary evolution of many-body quantum systems and important exper- imental studies of thermalization possible, stimulating * Electronic address: devprasen@gmail.com † Electronic address: ghoshal.pratik00@gmail.com ‡ Electronic address: pratap6906@gmail.com § Electronic address: abhishekbanerjee2001@gmail.com theoretical interest. Discussion about those theoretical works is beyond the scope of this paper, however, one can find those in [5, 16–18] and the references therein. A generic isolated many-body quantum system thermal- ize to a microcanonical distribution consistent with their energy density[16], and the experimental results are con- sistent with this fact. The mechanism behind this is the eigenstate thermalization, as prescribed by eigenstate thermalization hypothesis. Though ETH successfully de- scribes the thermalization of a generic isolated system, in- tegrable systems possessing extensive sets of non-trivial conserved quantities do not follow it. As a result, in gen- eral, integrable system do not thermalize[17], rather they do equilibrate. To describe such intergrable systems af- ter equilibration generalized Gibbs ensembles (GGEs) are used[18]. In the last few years, a lot of research has been carried out to understand the thermalization of both in- tegrable and non-integrable systems. However, what fea- ture of many-body quantum system helps in quantum thermalization is not clear yet. Recently, experimen- tal studies[19] with ultra-cold atoms have confirmed that entanglement[20–22] acts as a thermalizing agent in iso- lated quantum many-body systems. The confirmation comes from the simultaneous measurement of entangle- ment entropy[23, 24] and thermal averages of observables of the sub-systems. As the system’s state, initialized in a pure one, moves towards thermal equilibrium, its en- tanglement entropy starts to grow. The growth of en- tanglement entropy with respect to time and size of the subsystems has been studied in [19]. Later, using stan- dard quasiparticle picture the entanglement dynamics in the space-time scaling limit has been studied[25]. In this article, we focus on two facts; firstly, a generic isolated many-body quantum system thermalizes accord- ing to ETH, and secondly, entanglement can facilitate thermalization in such a system. We show that the knowledge of a single entangled state of the global sys- tem is sufficient to compute two thermal averages, micro- canonical and canonical. Our result depicts that entan- glement not only drives an isolated many-body quantum system towards thermal equilibrium, it also helps in ther- malization at the level of individual eigenstate. 1