Stationary entropies after a quench from excited states in the Ising chain

Abstract

We consider the asymptotic state after a sudden quench of the magnetic field in the transverse field quantum Ising chain starting from excited states of the pre-quench Hamiltonian. We compute the thermodynamic entropies of the generalised Gibbs and the diagonal ensembles and we find that the generalised Gibbs entropy is always twice the diagonal one. We show that particular care should be taken in extracting the thermodynamic limit since different averages of equivalent microstates give different results for the entropies.Entropy is a fundamental concept of statistical mechanics and represents the main bridge between the microscopic description of nature and thermodynamics. Indeed, a generic isolated classical system evolves in a way to maximise its entropy reaching the microcanonical ensemble after a long time. In the quantum world the situation is more complicated: an isolated system evolves unitarily, so if the system is initially prepared in a pure state it will always remain pure with strictly zero entropy, and cannot be described asymptotically by a statistical ensemble with positive entropy. The definition of a stationary entropy for non-equilibrium quantum systems is then a complicated matter which attracted renewed interest after the cold atom experimental realisation [1,2] of isolated out of equilibrium quantum systems, in particular of the so called quantum quenches [3,4], in which a parameter of the system is changed abruptly.Two main roads have been followed to define a nonequilibrium stationary entropy after a quench. The first is to look at the system in its entirety and define the entropy in a specific basis, such as for the so-called diagonal entropy [5]. The second road is to consider subsystems of the whole system which are not isolated and therefore are described by a reduced density matrix that may be equivalent to a statistical ensemble. The two roads have both their own advantages and disadvantages. Indeed, considering only a subsystem is probably more appropriate from a fundamental perspective because taking first the thermodynamic (TD) and then the large time limit (see e.g. [6-10]), it is possible that the reduced density matrix exhibits truly stationary behaviour which is impossible for the entire system. Conversely, a global definition of entropy is surely more suitable and manageable for finite systems and numerical simulations [11][12][13].The possible connections and relations between these two apparently unrelated sets of (stationary) entropies are then very important. Some explicit calculations for integrable systems in quenches from the ground state of a pre-quench Hamiltonian show that the diagonal entropy is exactly half of the subystem entropy [14,15], reflecting the fact that the diagonal ensemble contains much more information than the one needed to describe the local observables.However, the previous studies focused on the evolution starting from the ground state of a given pre-quench Hamiltonian. Starting from an excited state makes the situation more complicated. Indeed, ...

Full counting statistics in the transverse field Ising chain

“…• We initialize the system in the ground state of H(h 0 ) and time evolve with H(h). Such transverse field quenches have been studied in detail in the literature [40][41][42][43][44][45][46][47][55][56][57][58][59][60][61][62][63][64][65].…”

Rényi entropies after releasing the Néel state in theXXZspin-chain

“…where we used that the first two orders in (65) and some parts in the third order vanish. The leading orders of S Y Y and E are then given by different powers of z being z 4 ln z and z 2 respectively.…”

“…the entropy of the GGE) which is the double of the diagonal entropy [68]. This equivalence has been investigated in several studies for the von Neumann entropy, both for free systems [60,61,63,65] and for interacting ones [62,[66][67][68], but not for the Rényi entropies with α = 1. Here we then perform extensive tDMRG simulations [105][106][107][108][109], to provide numerical evidences that in the limit of large chains and large subsystems, i.e., L, → ∞ with L, one has…”

Light-cone velocities after a global quench in a noninteracting model

“…In the simpler setting of a non-interacting model, where quasi-particles are characterized by a dispersion relation ε k there is a maximum group velocity v g allowed by the dispersion. To our best knowledge, in all the examples considered in the literature so-far (for example [21][22][23]59,60 ) correlations were found to spread with a light-cone velocity given by v g independently of the initial state. In this paper we point out for the first time that even if the dispersion of the quasi-particles after the quench is not affected by the initial state, its symmetries and in particular translation invariance can introduce additional selection rules on the momenta of the quasi-particles.…”