Thermalization through unitary evolution of pure states

– The unitary time evolution of a quantum spin chain is calculated, and the entanglement evolution is shown. Moreover, we show that the reduced density matrix of a part of the chain evolves into a state in thermal equlibrium with the rest of the chain, even through unitary time evolution as opposed to the conventional picture that thermalization in quantum systems will happen during stochastic processes. Thermalization through unitary evolution of pure states Stein Olav Skrøvseth Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway PACS. 75.10.Pq – Spin chain models. PACS. 03.65.Ud – Entanglement and quantum nonlocality. PACS. 65.90.+i – Other topics in thermal properties of condensed matter. Introduction. – Systems that are left alone to evolve in interaction with an environment at a certain temperature, usually relax into a state in thermal equilibrium with its environment, and according to the ergodic hypothesis, this has been shown for numerous examples in classical systems. Feynman [1] argues that the same will happen in a quantum system that when the process is interpreted as a stochastical process. We will show that we can achieve thermal states in a quantum system that evolves unitarily, with a system that is large enough and where small amounts of noise is introduced in the form of a magnetic impurity. Quantum spin chains have been a matter of keen investigation over the latest years, and in particular their critical properties have been investigated thoroughly since it was discovered that non-classical correlations known as entanglement are characteristic in quantum phase transitions [2, 3]. This gives rise to the study of entanglement in the chains, in particular at critical points, as a part of the vast and expanding field of quantum information science [4]. Moreover, it has been identified that at critical points, conformal symmetry arises in a large class of models, and the characteristics of conformal field theory can be used to describe the universal properties of such systems [5–7]. The dynamics of quantum spin chains is therefore of great importance, and for a certain class of models the chain can be fermionized, which maskes then more accessible to analytic and numerical investigation [8]. Any existent entanglement measured by the concurrence (i.e. the entanglement of formation of two spins) in the quantum spin chain at zero temperature disappears when the temperature of the system exceeds a threshold temperature, see e.g. [9]. However, the entanglement entropy does not vanish, indeed conformal field theory predicts that an infinitely long conformally invariant state in thermal equilibrium at temperature β has entropy proportional to the temperature at high temperatures [10]. In this paper we consider a quantum spin chain as sketched in Fig. 1, i.e. a system of N spins with open boundary conditions. The chain is initially in some arbitrary excited