Nonequilibrium Quantum Dynamics of Many-Body Systems

Abstract

We review our results for the dynamics of isolated many-body quantum systems described by onedimensional spin-1/2 models. We explain how the evolution of these systems depends on the initial state and the strength of the perturbation that takes them out of equilibrium; on the Hamiltonian, whether it is integrable or chaotic; and on the onset of multifractal eigenstates that occurs in the vicinity of the transition to a many-body localized phase. We unveil different behaviors at different time scales. We also discuss how information about the spectrum of a many-body quantum system can be extracted by the sole analysis of its time evolution, giving particular attention to the so-called correlation hole. This approach is useful for experiments that routinely study dynamics, but have limited or no direct access to spectroscopy, as experiments with cold atoms and trapped ions.3. The speed of the decay depends on the energy of the initial state. The decay is faster for initial states with energy close to the middle of the spectrum, where there is a large concentration of eigenstates, than for states with energies near the border of the spectrum [26-28].4. Decays faster than Gaussian occur when the energy distribution of the initial state is bimodal [28], in which case the quantum speed limit can be reached. Moving away from realistic systems, fast decays can be obtained by increasing the number of particles that interact simultaneously [26][27][28].5. After the initial fast (often Gaussian) decay, the dynamics slows down and becomes power-law. The power-law exponent depends on how the spectrum approaches its energy bounds [24,25] and on the level of delocalization of the eigenstates [31-33].6. In interacting systems with onsite disorder, the value of the power-law decay exponent detects the transition from chaos to many-body localization. This exponent coincides with the fractal dimension of the system [31][32][33] and with the slope of the logarithmic growth of the Shannon and entanglement entropies [33].7. At long times, after the power-law behavior and before saturation, the survival probability shows a dip below its infinite time average [35,36]. This is known as correlation hole and appears only in systems with level repulsion (that is, not in integrable models). The correlation hole provides a way to detect level repulsion from the dynamics, instead of having to resort to the eigenvalues. This is useful for the experiments mentioned above, which have limited access to the spectra of their systems. Since the correlation hole is a general indicator of the integrable-chaos transition, it serves also as a detector of the metal-insulator transition in interacting systems [33,35].Additional highlights of our research, which are not described in this chapter, but may be found in our references, include the following topics.1. The dynamical behavior of the Shannon entropy and entanglement entropy is equivalent [33,34]. The first is easier to compute numerically and is potentially accessible experimentally, although it is...

Equilibration time scales in closed many-body quantum systems

“…In this case, then, both the energy fluctuations of the initial state and the bandwidth of the observable written in the energy basis are σ E . Hence, our considerations within the dephasing picture predict, for local interacting lattice systems, an equilibration time determined by σ E , in agreement with the results of [19].…”

“…We can also understand some of the results obtained in another approach to the problem of relaxation of many-body systems, namely the study of the survival probability given by the quantum fidelity t t 0 2  y y á ñ ( ) ≔ | ( )| ( ) | , often in situations where the initial state 0 y ñ | ( ) is generated after a sudden displacement ('quantum quench') that brings the system out of equilibrium (see [19] and references therein for a review). Note that the quantum fidelity is, up to an additive constant, equivalent to the time evolution of the observable A 0 0 y y = ñ á | ( ) ( )|.…”

“…In contrast, an out-of-equilibrium initial state, for a distribution of gaps with a non-zero dispersion and a large enough system, will evolve to an equilibrium state, represented by an isotropic cloud of points centred at the origin of the complex plane as shown in figure 1 (right). This mechanism for equilibration, or more generally the vanishing amplitude of a signal over time, is usually called dephasing and is a well-known feature in many different fields of physics [19,[26][27][28][29][30][31][32],…”

Equilibration towards generalized Gibbs ensembles in non-interacting theories

“…In particular, studying quenches has been particularly rewarding [39]. In this context, numerical studies often provide useful insights [1][2][3][4][40][41][42][43][44][45][46][47][48].…”

Analytical expressions for the evolution of many-body quantum systems quenched far from equilibrium

“…Since 2014, we have been searching for expressions that could describe the dynamics of many-body quantum systems out of equilibrium [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. We started our studies with a very simple quantity, referred to as the survival probability.…”