The question of how systems respond to perturbations is ubiquitous in physics. Predicting this response for large classes of systems becomes particularly challenging if many degrees of freedom are involved and linear response theory cannot be applied. Here, we consider isolated many-body quantum systems which either start out far from equilibrium and then thermalize, or find themselves near thermal equilibrium from the outset. We show that time-periodic perturbations of moderate strength, in the sense that they do not heat up the system too quickly, give rise to the following phenomenon of stalled response: While the driving usually causes quite considerable reactions as long as the unperturbed system is far from equilibrium, the driving effects are strongly suppressed when the unperturbed system approaches thermal equilibrium. Likewise, for systems prepared near thermal equilibrium, the response to the driving is barely noticeable right from the beginning. Numerical results are complemented by a quantitatively accurate analytical description and by simple qualitative arguments.
The question of how systems respond to perturbations is ubiquitous in physics. Predicting this response for large classes of systems becomes particularly challenging if many degrees of freedom are involved and linear response theory cannot be applied. Here, we consider isolated many-body quantum systems which either start out far from equilibrium and then thermalize, or find themselves near thermal equilibrium from the outset. We show that time-periodic perturbations of moderate strength, in the sense that they do not heat up the system too quickly, give rise to the following phenomenon of stalled response: While the driving usually causes quite considerable reactions as long as the unperturbed system is far from equilibrium, the driving effects are strongly suppressed when the unperturbed system approaches thermal equilibrium. Likewise, for systems prepared near thermal equilibrium, the response to the driving is barely noticeable right from the beginning. Numerical results are complemented by a quantitatively accurate analytical description and by simple qualitative arguments.
The eigenstate thermalization hypothesis (ETH) and the theory of linear response (LRT) are celebrated cornerstones of our understanding of the physics of many-body quantum systems out of equilibrium. While the ETH provides a generic mechanism of thermalization for states arbitrarily far from equilibrium, LRT extends the successful concepts of statistical mechanics to situations close to equilibrium. In our work, we connect these cornerstones to shed light on the route to equilibrium for a class of properly prepared states. We unveil that, if the off-diagonal part of the ETH applies, then the relaxation process can become independent of whether or not a state is close to equilibrium. Moreover, in this case, the dynamics is generated by a single correlation function, i.e., the relaxation function in the context of LRT. Our analytical arguments are illustrated by numerical results for idealized models of random-matrix type and more realistic models of interacting spins on a lattice. Remarkably, our arguments also apply to integrable quantum systems where the diagonal part of the ETH may break down.
Heating under periodic driving is a generic nonequilibrium phenomenon, and it is a challenging problem in nonequilibrium statistical physics to derive a quantitatively accurate heating rate. In this work, we provide a simple formula on the heating rate under fast and strong periodic driving in classical and quantum many-body systems. The key idea behind the formula is constructing a time-dependent dressed Hamiltonian by moving to a rotating frame, which is found by a truncation of the high-frequency expansion of the micromotion operator, and applying the linear-response theory. It is confirmed for specific classical and quantum models that the second-order truncation of the high-frequency expansion yields quantitatively accurate heating rates beyond the linear-response regime. Our result implies that the information on heating dynamics is encoded in the first few terms of the high-frequency expansion, although heating is often associated with an asymptotically divergent behavior of the high-frequency expansion.
The nature of the behavior of an isolated many-body quantum system periodically driven in time has been an open question since the beginning of quantum mechanics. After an initial transient period, such a system is known to synchronize with the driving; in contrast to the nondriven case, no fundamental principle has been proposed for constructing the resulting nonequilibrium state. Here, we analytically show that, for a class of integrable systems, the relevant ensemble is constructed by maximizing an appropriately defined entropy subject to constraints, which we explicitly identify. This result constitutes a generalization of the concepts of equilibrium statistical mechanics to a class of far-from-equilibrium systems, up to now mainly accessible using ad hoc methods.
We show that when a quantum many-body system is subjected to coherent periodic driving, the response may exhibit exotic freezing behavior in high driving frequency $(\ensuremath{\omega})$ regime. In a periodically driven classical thermodynamic system, freezing at high $\ensuremath{\omega}$ occurs when $1/\ensuremath{\omega}$ is much smaller than the characteristic relaxation time of the system and hence the freezing always increases there as $\ensuremath{\omega}$ is increased. Here, in the contrary, we see surprising nonmonotonic freezing behavior of the response with $\ensuremath{\omega}$, showing curious peak-valley structure. Quite interestingly, the entire system tends to freeze almost absolutely (the freezing peaks) when driven with a certain combination of driving parameters values (amplitude and $\ensuremath{\omega}$) due to coherent suppression of dynamics of the quantum many-body modes, which has no classical analog. We demonstrate this new freezing phenomenon analytically (supported by large-scale numerics) for a general class of integrable quantum spin systems.
This Ph.D. thesis concerns a very active area of current research in theoretical physics, which is non-equilibrium statistical mechanics. More specifically, the focus will be on the study of the non-equilibrium dynamics and fluctuations in isolated many body systems driven out of equilibrium. We shall consider primarily quantum systems, albeit many of the theoretical techniques that will be used can be applied to both classical and quantum statistical systems. The novel contribution of this work is two-fold. First, we have developed new analytical techniques to study exactly the large-deviation statistics of physical quantities relevant for the description of systems driven out of equilibrium, like the work done in a quantum quench or the time-integrated current characterizing transport of matter, energy or charge etc, discussed in Part I of the thesis. Second, we have singled out a class of experimentally relevant nondisordered quantum systems showing a dramatically slow approach to thermal equilibrium and we have provided quantitative predictions for this behavior, which is discussed in Part II.
Despite the exponentially large amount of information required in the quantum description of many-body systems, finite size numerical simulations have paved the way for recent progress in the understanding of quantum phases of matter both in and out-of-equilibrium. \n \nIn this spirit, using exact diagonalization for a small number of spins ~ 20, first, we investigate the fate of both ergodic quantum many-body systems that thermalize at long times and systems with quenched disorder that exhibit many-body localization (MBL) and fail to thermalize (due to an extensive set of local conserved degrees of freedom) under periodic driving. Ergodic systems always delocalize in energy space and heat up to infinite temperature, for both local driving, in which the time dependence of the Hamiltonian is restricted to a few contiguous degrees of freedom, and global driving. On the other hand, MBL systems remain localized at finite energy density for local driving, while in the case of global driving these systems can either heat up to infinite temperature or remain localized depending on the parameters of the drive. Underlying the latter is the emergence of an effective time independent MBL Hamiltonian describing the long time dynamics of the system. These numerical predictions have recently been verified experimentally in a one dimensional quasi-disordered optical lattice of interacting fermions wherein unitary time evolution can be experimentally probed. \n \nSubsequently, we study the stability of MBL systems with respect to another perturbation, in this case by thermal inclusions, which remains poorly understood. We consider a simple model to address this question: a two level system interacting with 𝑁 localized bits subject to random fields. In the large 𝑁 limit, the localized bits are thermalized for any nonzero ratio of interactions to disorder strength. Moreover, at finite 𝑁, in the transition region, the single-site eigenstate entanglement entropies exhibit bimodal distributions, so that localized bits are either "on" (strongly entangled) or "off" (weakly entangled) in eigenstates. The clusters of "on" bits vary significantly between eigenstates of the same sample, which provides evidence for a heterogenous discontinuous transition out of the localized phase in single site observables. These results also imply the MBL phase is unstable in systems with short-range interactions and quenched randomness in dimensions 𝑑 that are high but finite. \n \nThe second part of this thesis is motivated by the recent breakthroughs of artificial intelligence in domains such as computer vision involving highly dimensional data of complexity not unlike the one found in the description of many-body systems. Using Monte Carlo simulations to generate synthetic data, we find that kernel methods, a class of machine learning algorithms, can learn the mathematical \nform of physical discriminators associated with phase transitions, such as order parameters and Hamiltonian constraints, for a set of two dimensional spin models: the ferromagnetic Ising model, a conserved-order-parameter Ising model, and the Ising gauge theory. \n \nOur work shows that the combination of numerical simulations with the ability of machine learning algorithms for automatic feature detection hold promise to become standard tools in the exploration and characterization of different phases of matter in condensed matter and other many-body systems.