Relaxation in a Completely Integrable Many-Body Quantum System: AnAb InitioStudy of the Dynamics of the Highly Excited States of 1D Lattice Hard-Core Bosons

Abstract

In this Letter we pose the question of whether a many-body quantum system with a full set of conserved quantities can relax to an equilibrium state, and, if it can, what the properties of such state are. We confirm the relaxation hypothesis through a thorough ab initio numerical investigation of the dynamics of hard-core bosons on a one-dimensional lattice. Further, a natural extension of the Gibbs ensemble to integrable systems results in a theory that is able to predict the mean values of physical observables after relaxation. Finally, we show that our generalized equilibrium carries more memory of the initial conditions than the usual thermodynamic one. This effect may have many experimental consequences, some of which having already been observed in the recent experiment on the non-equilibrium dynamics of one-dimensional hard-core bosons in a harmonic potential [T. Kinoshita, T. Wenger, D. S. Weiss, Nature (London) 440, 900 (2006)].

Dynamic correlations, fluctuation-dissipation relations, and effective temperatures after a quantum quench of the transverse field Ising chain

“…(3), it is necessary to introduce a number of Lagrange multipliers (see further below), one for each "conserved quantity", which eventually turn into a set of "effective temperatures" {T k eff } determined by the condition ψ 0 |n k |ψ 0 = n k T =T k eff . These quantities prove to be particularly useful since they naturally appear in the calculation of (stationary and non-stationary) expectation values [17,18,20]. It was in fact suggested [17] that the stationary behavior of the system after quenches towards Hamiltonians of the form (7) can be described in terms of the density matrixρ GGE obtained by maximizing the von Neumann entropy S[ρ] under the constraints on the expectation values of n k .…”

“…In several studies presented in the literature it was shown that a number of quantum isolated systems with short-range interactions reach a stationary state [17,18,20,21,30], while some fully-connected models [33,35,54] and some mean-field approximations to models with short-range interactions [55][56][57] keep a nonstationary behavior. For the isolated quantum Ising chain a stationary state is indeed reached after the quench and we will therefore focus on this relatively simple case.…”

“…The discussion above implicitly assumes that the energy is the only quantity conserved by the dynamics and that in minimizing S[ρ] one has to account only for one constraint, which naturally leads to the Gibbs canonical ensemble. However, if the system is integrable, the situation turns out to be rather subtle because of the many "independent" quantities which are conserved by the dynamics in addition to the energy [17,18,20,21]. Consider, for simplicity, a non-interacting Hamiltonian that can be written in the diagonal form…”

“…In Sec. II we review several definitions of effective temperatures proposed in the context of quantum quenches [17,18,20,21,30,31,36] and classical and quantum dissipative glassy dynamics [12-14, 43, 51]. Section III summarizes those features of the static and dynamic behavior of a quantum Ising chain that are relevant to our study.…”

“…Integrable systems, instead, are not expected to thermalize. However, their asymptotic stationary state should nonetheless be described by the so-called generalized Gibbs ensemble (GGE) in which each conserved quantity is characterized by a generally different effective temperature [17][18][19][20][21][22][23]. On the other hand, other works [24][25][26][27][28][29][30][31][32][33][34][35] have shown (or at least argued) that this scenario could actually be significantly richer.…”