Condensed Matter

We study disorder-induced ergodicity breaking transition in high-energy eigenstates of interacting spin-1/2 chains. Using exact diagonalization we introduce a cost function approach to quantitatively compare different scenarios for the eigenstate transition. We study ergodicity indicators such as the eigenstate entanglement entropy and the spectral level spacing ratio, and we consistently find that an (infinite-order) Kosterlitz-Thouless transition yields a lower cost function when compared to a finite-order transition. Interestingly, we observe that the transition point in finite systems exhibits nearly thermal properties, i.e., ergodicity indicators at the transition are close to the random matrix theory predictions.

We performed a series of aging experiments of an inorganic glass (As 2 Se 3 ) at a temperature T 2 near the glass transition point T g by first relaxing it at T 1 . The relaxation of Young's modulus was monitored, which was(almost if not ideally) exponential with a T 1 -dependent relaxation time τ( T 1 ) . We demostrate the Kovacs' paradox for the first time in an inorganic glasses. Associated with the divergence of τ , the quasi-equilibrated Young's modulus E ∞ does not converge either. An elastic model of relaxation time and a Mori-Tanaka analysis of E ∞ lead to a similar estimate of the persistent memory of the history, ergodicity breaking within the accessible experimental time. Experiments with different T 2 exhibits a critical temperature T p ∼ T g , i.e., when T 2 > T p , both τ and E ∞ converge.

Many-body localization (MBL) behavior is analyzed {in an extended Bose-Hubbard model with quasiperiodic infinite-range interactions. No additional disorder is present. Examining level statistics and entanglement entropy of eigenstates we show that a significant fraction of eigenstates of the system is localized in the presence of strong interactions. In spite of this, our results suggest that the system becomes ergodic in the standard thermodynamic limit in which the energy of the system is extensive. At the same time, the MBL regime seems to be stable if one allows for a super-extensive scaling of the energy. We show that our findings can be experimentally verified by studies of time dynamics in many-body cavity quantum electrodynamics setups. The "quench spectroscopy" is a particularly effective tool that allows us to systematically study energy dependence of time dynamics and to investigate a mobility edge in our system.

In this thesis I discuss combinatorial optimization problems, from the statistical physics perspective. The starting point are the motivations which brought physicists together with computer scientists and mathematicians to work on this beautiful and deep topic. I give some elements of complexity theory, and I motivate why the point of view of statistical physics leads to many interesting results, as well as new questions. I discuss the connection between combinatorial optimization problems and spin glasses. Finally, I briefly review some topics of large deviation theory, as a way to go beyond average quantities. As a concrete example of this, I show how the replica method can be used to explore the large deviations of a well-known toy model of spin glasses, the p-spin spherical model. In the second chapter I specialize in Euclidean combinatorial optimization problems. In particular, I explain why these problems, when embedded in a finite dimensional Euclidean space, are difficult to deal with. I analyze several specific problems in one dimension to explain a quite general technique to deal with one dimensional Euclidean combinatorial optimization problems. Whenever possible, and in a detailed way for the traveling-salesman-problem case, I also discuss how to proceed in two (and also more) dimensions. In the last chapter I outline a promising approach to tackle hard combinatorial optimization problems: quantum computing. After giving a quick overview of the paradigm of quantum computation, I discuss in detail the application of the so-called quantum annealing algorithm to a specific case of the matching problem, also by providing a comparison between the performance of a recent quantum annealer machine and a classical super-computer equipped with an heuristic algorithm. Finally, I draw the conclusions of my work and I suggest some interesting directions for future studies.

We investigate the number entropy S N ---which characterizes particle-number fluctuations between subsystems---following a quench in one-dimensional interacting many-body systems with potential disorder. We find evidence that in the regime which is expected to show many-body localization (MBL) and where the entanglement entropy grows as S∼lnt as function of time t , the number entropy grows as S N ∼lnlnt , indicating continuing particle transport at a very slow rate. We demonstrate that this growth is consistent with a relation between entanglement and number entropy recently established for non-interacting systems.

We present a large-scale simulation of the three-dimensional Ising spin glass with Gaussian disorder to low temperatures and large sizes using optimized population annealing Monte Carlo. Our primary focus is investigating the number of pure states regarding a controversial statistic, characterizing the fraction of centrally peaked disorder instances, of the overlap function order parameter. We observe that this statistic is subtly and sensitively influenced by the slight fluctuations of the integrated central weight of the disorder-averaged overlap function, making the asymptotic growth behaviour very difficult to identify. Modified statistics effectively reducing this correlation are studied and essentially monotonic growth trends are obtained. The effect of temperature is also studied, finding a larger growth rate at a higher temperature. Our state-of-the-art simulation and variance reduction data analysis suggest that the many pure state picture is most likely and coherent.

The nature of the amorphous state has been notably difficult to ascertain at the microscopic level. In addition to the fundamental importance of understanding the amorphous state, potential changes to amorphous structures as a result of radiation damage have direct implications for the pressing problem of nuclear waste encapsulation. Here, we develop new methods to identify and quantify the damage produced by high-energy collision cascades that are applicable to amorphous structures and perform large-scale molecular dynamics simulations of high-energy collision cascades in a model zircon system. We find that, whereas the averaged probes of order such as pair distribution function do not indicate structural changes, local coordination analysis shows that the amorphous structure substantially evolves due to radiation damage. Our analysis shows a correlation between the local structural changes and enthalpy. Important implications for the long-term storage of nuclear waste follow from our detection of significant local density inhomogeneities. Although we do not reach the point of convergence where the changes of the amorphous structure saturate, our results imply that the nature of this new converged amorphous state will be of substantial interest in future experimental and modelling work.

We propose a family of one-dimensional mosaic models inlaid with a slowly varying potential V n =λcos(πα n ν ) , where n is the lattice site index and 0<ν<1 . Combinating the asymptotic heuristic argument with the theory of trace map of transfer matrix, mobility edges (MEs) and pseudo-mobility edges (PMEs) in their energy spectra are solved semi-analytically, where ME separates extended states from weakly localized ones and PME separates weakly localized states from strongly localized ones. The nature of eigenstates in extended, critical, weakly localized and strongly localized is diagnosed by the local density of states, the Lyapunov exponent, and the localization tensor. Numerical calculation results are in excellent quantitative agreement with theoretical predictions.

Complex systems are often non-stationary, typical indicators are continuously changing statistical properties of time series. In particular, the correlations between different time series fluctuate. Models that describe the multivariate amplitude distributions of such systems are of considerable interest. Extending previous work, we view a set of measured, non-stationary correlation matrices as an ensemble for which we set up a random matrix model. We use this ensemble to average the stationary multivariate amplitude distributions measured on short time scales and thus obtain for large time scales multivariate amplitude distributions which feature heavy tails. We explicitly work out four cases, combining Gaussian and algebraic distributions. The results are either of closed forms or single integrals. We thus provide, first, explicit multivariate distributions for such non-stationary systems and, second, a tool that quantitatively captures the degree of non-stationarity in the correlations.

We propose a general analytic method to study the localization transition in one-dimensional quasicrystals with parity-time ( PT ) symmetry, described by complex quasiperiodic mosaic lattice models. By applying Avila's global theory of quasiperiodic Schrödinger operators, we obtain exact mobility edges and prove that the mobility edge is identical to the boundary of PT -symmetry breaking, which also proves the existence of correspondence between extended (localized) states and PT -symmetry ( PT -symmetry-broken) states. Furthermore, we generalize the models to more general cases with non-reciprocal hopping, which breaks PT symmetry and generally induces skin effect, and obtain a general and analytical expression of mobility edges. While the localized states are not sensitive to the boundary conditions, the extended states become skin states when the periodic boundary condition is changed to open boundary condition. This indicates that the skin states and localized states can coexist with their boundary determined by the mobility edges.