In this thesis we consider isolated quantum systems influenced by disorder and external driving, and explore aspects of localisation and ergodicity-breaking in three different models. We first consider the XYZ spin chain in a disordered magnetic field; this system breaks the U(1) symmetry of the XXZ model and therefore violates the conservation of the z component of the total magnetisation. We identify a quantum phase transition between an ergodic phase and a many-body localised phase at a finite disorder strength that is strongly dependent on the size of the U(1) symmetry-breaking term. We also investigate the breakdown of thermalisation by studying the matrix elements of local operators in the basis of the system's eigenstates, and we find suggestive evidence of subdiffusive energy transport at disorder strengths preceding the localisation transition. We then consider a two-level time-dependent quantum system that is influenced by a fluctuating classical noise term. We show that the noise does not fully randomise the quantum state if the fluctuations of every noise parameter coupled to the system are perfectly correlated; the state of the system instead becomes distributed over a finite subset of the full state space. Lastly we consider a one-dimensional Anderson-localised system that is affected by randomly fluctuating onsite potentials. We explore how the spatiotemporal correlations between these onsite potentials affect the delocalisation of a particle initially localised on a single site.
In this thesis we consider isolated quantum systems influenced by disorder and external driving, and explore aspects of localisation and ergodicity-breaking in three different models. We first consider the XYZ spin chain in a disordered magnetic field; this system breaks the U(1) symmetry of the XXZ model and therefore violates the conservation of the z component of the total magnetisation. We identify a quantum phase transition between an ergodic phase and a many-body localised phase at a finite disorder strength that is strongly dependent on the size of the U(1) symmetry-breaking term. We also investigate the breakdown of thermalisation by studying the matrix elements of local operators in the basis of the system's eigenstates, and we find suggestive evidence of subdiffusive energy transport at disorder strengths preceding the localisation transition. We then consider a two-level time-dependent quantum system that is influenced by a fluctuating classical noise term. We show that the noise does not fully randomise the quantum state if the fluctuations of every noise parameter coupled to the system are perfectly correlated; the state of the system instead becomes distributed over a finite subset of the full state space. Lastly we consider a one-dimensional Anderson-localised system that is affected by randomly fluctuating onsite potentials. We explore how the spatiotemporal correlations between these onsite potentials affect the delocalisation of a particle initially localised on a single site.
The study of collective behaviour in many-body systems often explores fundamentally new ideas absent from the mere constituents of such a system. A paradigmatic model for these studies is the spin-1/2 XXZ chain and its fermionic equivalent. This thesis can be broadly divided into the study of two fundamental aspects of this model. Firstly, we discuss localisation phenomena in one dimensional lattices as often experimentally realised in cold atom systems. Secondly, we investigate how disorder and symmetry influence heat transport in spin chains. More specifically, in the first part we consider a system of non-interacting fermions in one dimension subject to a single-particle potential consisting of a strong optical lattice, a harmonic trap, and uncorrelated on-site disorder. We investigate a global inhomogeneous quantum quench and present numerical and analytical results for static and dynamical properties. We show that the approach to the non-thermal equilibrium state is extremely slow and that it implies a sensitivity to disorder parametrically stronger than that expected from Anderson localisation. We also consider the above system in a strong non-uniform electric field. In the non-interacting case, due to Wannier-Stark localisation, the single-particle wave functions are exponentially localised without quenched disorder. We show that this system remains localised in the presence of nearest-neighbour interactions and exhibits physics analogous to models of conventional many-body localisation. The second part explores the hydrodynamics of the disordered XYZ spin chain. Using time-evolving block decimation on open chains of up to 400 spins attached to thermal baths, we probe the energy transport of this system. Our principal findings are as follows. For weak disorder there is a stable diffusive region that persists up to a critical disorder strength that depends on the XY anisotropy. Then, for disorder strengths above this critical value energy transport becomes increasingly subdiffusive.
The interplay between quenched disorder and interaction effects opens the possibility in a closed quantum many-body system of a phase transition at finite energy density between an ergodic phase, which is governed by the laws of statistical physics, and a localized one, in which the degrees of freedom are frozen and ergodicity breaks down. The possible existence of a quantum phase transition at finite energy density is strongly questioning our understanding of the fundamental laws of nature and has generated an active field of research called many-body localization. This thesis consists of three parts and is dedicated to the understanding and characterization of the phenomenon of many-body localization, approaching it from complementary facets. In particular, borrowing methods and tools from different fields, we analyze timely problems. The first part of the thesis is devoted to detecting the many-body localization transition and to characterize both the ergodic and the localized phase it separates. Here we provide a characterization from two different perspectives: the first one is based on the study of local entanglement properties. In the second one, using tools from quantum-chaos theory, we attempt to answer the question of understanding time-irreversibility, and thus probing the breaking of ergodicity. We analyze experimentally viable observables. Moreover, we propose two different quantities to distinguish an Anderson insulating phase from a many-body localized one, which is one of the issues in experiments. The second part focuses on understanding the existence of a putative subdiffusive multifractal phase. Analyzing the quantum dynamics of the system in this region of the phase diagram, we point out the importance of finite-size effects, questioning the existence of this multifractal phase. We speculate with a possible scenario in which the diffusivity and thus ergodicity could be restored in the thermodynamic limit. Furthermore, we find that the propagation is highly non-Gaussian, which could have an important effect on understanding the critical point of the according transition. We tackle this problem also from a different angle. A possible toy-model to understand many-body localization entails the Anderson model on a random-regular graph. Also in the latter model the possible existence of an intermediate multifractal phase has been conjectured. There, studying the survival return probability of a particle with time, we give a new characterization of multifractal phases and give indication of the possible existence of this phase. Nevertheless, we also outline possible caveats. In the last part of this thesis we study the interplay between symmetry and correlated disorder in a non-interacting fermionic system. We show another possible mechanism for breaking localization. In particular, we focus on studying information and particle transport, emphasizing how the two types of propagation can be different.
A central goal in the study of modern condensed matter physics is the characterization of the dynamical properties of quantum systems. Many decades of effort towards this goal, studying a diverse range of (near-equilibrium) quantum matter, from Fermi liquids, to quantum two-level systems, to interacting spin models, and more, has revealed a remarkable pervasiveness of the simple dynamical description of these complex systems in terms of quasi-particles that carry spin, charge, and heat, and that are generally able to equilibrate systems. \n\nThis thesis is an examination of some exceptions to this rule. Specifically, we study a number of instances of quantum matter where equilibration phenomena happens at rather long time scales, or does not occur at all. Particular emphasis is laid on the role of dimensionality, disorder, and dissipation in engendering such novel dynamical behavior.\n\nFirst, we consider non-equilibrium dynamics in one-dimensional quasi-condensates. Low dimensionality inhibits scattering in these systems, and low-energy excitations are long-lived phase fluctuations that exhibit an enriched conformal symmetry. Utilizing this symmetry, we generalize sudden quenches typically used to study non-equilibrium dynamics to quenches along general relativistic and conformal trajectories. Gases never truly equilibrate after such a quench; instead, they evolve into a `prethermal' state with thermal-looking correlations and a chiral asymmetry. \n\nWe then study the problem of the dynamical transition driven by disorder, from an ergodic to a non-ergodic phase, in one-dimensional quantum spin chains. In particular, in XXZ chains with on-site disorder, we find a unique intermediate phase straddling the boundary of the dynamical phase transition, wherein rare-region effects lead to long-time tails in equilibration and vanishing DC conduction before the onset of non-ergodicity. We propose generalizations of such `Griffiths' behavior to arbitrary dimensions. We also study the dynamics of random-bond Heisenberg chains by developing a strong-disorder renormalization group protocol for these systems. We discuss how magnetic noise from such disordered systems contains signatures of their anomalous dynamical properties. \n\nNext, we re-examine the phenomenological theory of two-level systems in amorphous materials in the light of new experimental evidence that these states have large electric/magnetic dipole moments. We propose and justify an interpretation of the model as one of tunneling electrons slowed down by a large phonon drag and discuss the dynamical consequences of such polaronic effects. \n\nFinally, we discuss how magnetic noise measurements can be used to non-invasively access the anomalous properties of systems such as those discussed above. In particular, we examine how scattering properties of isolated magnetic impurities and non-local transport in a variety of two-dimensional materials can be probed experimentally using NV centers as noise magnetometers.
The work presented in this thesis is divided into two main parts: strong zero modes in quantum spin chains with open boundaries and large deviations in classical spin systems. The results concerning the strong zero modes are presented in chapters 2 and 3, while chapter 4 contains the results about large deviations in systems of many, but independent, classical degrees of freedom. These two distinct classes of problems in spin chains are both about aspects of slow relaxation to equilibrium in many-body systems. Nowadays slow dynamics and non-ergodicity in many-body quantum systems are receiving much attention due to their notable technological potential for storing and processing, for very long times, the quantum information encoded in boundary degrees of freedom. In the first part of the work presented in this thesis we mainly focus on quantum spin chains with open boundaries, which can display coherence times for edge spins, that diverge with the system size, as a consequence of the emergence, in the thermodynamic limit, of conserved local quantities, called strong zero modes. First, we discuss the fate of these coherence times when dynamics becomes dissipative. Understanding the behavior of these long correlation times in open quantum systems, undergoing Markovian (memoryless) dynamics, would allow for the possibility of exploiting the protected information encoded in the boundary spins for applications in quantum devices in realistic non-equilibrium settings. Second, we study one possible simple generalization of the transverse field Ising model to a class of ladder spin systems with plaquette interactions in the presence of a transverse field. Also for these systems one can explicitly construct the strong zero mode operators and understand the behavior of the consequential long coherence times for the edge degrees of freedom under isolated (unitary) dynamics. This is possible because the models we consider, while appearing complex superficially, can be brought to simple forms by exploiting an extensive number of conserved quantities they possess. The results presented in chapters 2 and 3 are closely related, even if they focus on different models and investigate different aspects of their dynamics. Nevertheless, the underlying concept they are based on is the same one, that is, an eigenstate phase transition for the strong zero modes that has interesting consequences for the full energy spectrum. In the second part of the thesis we focus on a different type of phase transitions. Here we address the issue of dynamical large deviation phase transitions from a disordered phase to an ordered one, in classical systems which consist in a collection of independent Ising spins evolving stochastically. Surprisingly, by reweighing trajectories accordingly to a certain type of time-integrated observable, even a system of independently evolving spins can display some critical behavior, at the fluctuation level, in trajectory space in the large size and long time limit. By making use of the tilted generator techniques we study the statistics of two particular time-integrated observables. Their associated tilted generators have an exact Kramers-Wannier duality and they look like, up to a sign and an additive constant, some specific well known quantum Hamiltonians. In this sense, the dynamical large deviation phase transitions we observe in these classical stochastic systems translate in ground state quantum phase transitions taking place at the self dual point of the tilted generators we study.
Understanding the effects of disorder in quantum many-body systems is one of the central challenges of contemporary statistical physics. Its interplay with interactions and quantum fluctuations profoundly modifies the nature of quantum states and their dynamics, giving rise to phenomena such as localization, ergodicity breaking, and the emergence of complex collective behavior. This thesis presents new results on two classes of disordered quantum systems. The first part is devoted to the study of the many-body localization transition, a paradigmatic example of disorder-induced ergodicity breaking. The critical properties of the transition are investigated using a numerical renormalization-group approach, establishing a connection between its scaling behavior and that of the Anderson localization transition on tree-like graphs. The second part focuses on the zero-temperature quantum spin-glass phase of the two-dimensional bond-disordered Heisenberg model. Combining large-scale numerical simulations with a semiclassical expansion, evidence is provided for the existence of the spin-glass phase. The effective model governing the semiclassical fluctuations is further analyzed to characterize the low-energy excitation spectrum and its localization properties. Finally, the weak-disorder regime is investigated and its implications for the ferromagnetic phase are discussed. Overall, the results presented in this thesis contribute to the understanding of the effects of disorder in interacting quantum many-body systems, encompassing both localization phenomena and the collective behavior of disordered quantum magnets.
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.