Real-time dynamics of string breaking in quantum spin chains

String breaking is a central dynamical process in theories featuring confinement, where a string connecting two charges decays at the expense of the creation of new particle-antiparticle pairs. Here, we show that this process can also be observed in quantum Ising chains where domain walls get confined either by a symmetry-breaking field or by long-range interactions. We find that string breaking occurs, in general, as a two-stage process: First, the initial charges remain essentially static and stable. The connecting string, however, can become a dynamical object. We develop an effective description of this motion, which we find is strongly constrained. In the second stage, which can be severely delayed due to these dynamical constraints, the string finally breaks. We observe that the associated time scale can depend crucially on the initial separation between domain walls and can grow by orders of magnitude by changing the distance by just a few lattice sites. We discuss how our results generalize to one-dimensional confining gauge theories and how they can be made accessible in quantum simulator experiments such as Rydberg atoms or trapped ions.Comment: 16 pages, 8 figures; version published in Physical Review

Variational classical networks for dynamics in interacting quantum matter

Dynamics in correlated quantum matter is a hard problem, as its exact solution generally involves a computational effort that grows exponentially with the number of constituents. While a remarkable progress has been witnessed in recent years for one-dimensional systems, much less has been achieved for interacting quantum models in higher dimensions, since they incorporate an additional layer of complexity. In this work, we employ a variational method that allows for an efficient and controlled computation of the dynamics of quantum many-body systems in one and higher dimensions. The approach presented here introduces a variational class of wavefunctions based on complex networks of classical spins akin to artificial neural networks, which can be constructed in a controlled fashion. We provide a detailed prescription for such constructions and illustrate their performance by studying quantum quenches in one- and two-dimensional models. In particular, we investigate the nonequilibrium dynamics of a genuinely interacting two-dimensional lattice gauge theory, the quantum link model, for which we have recently shown -- employing the technique discussed thoroughly in this paper -- that it features disorder-free localization dynamics [P. Karpov, et al., arXiv preprint, arXiv:2003.04901 (2020)]. The present work not only supplies a framework to address purely theoretical questions but also could be used to provide a theoretical description of experiments in quantum simulators, which have recently seen an increased effort targeting two-dimensional geometries. Importantly, our method can be applied to any quantum many-body system with a well-defined classical limit.Comment: 19 pages, 12 figure

Network science Ising states of matter

Network science provides very powerful tools for extracting information from interacting data. Although recently the unsupervised detection of phases of matter using machine learning has raised significant interest, the full prediction power of network science has not yet been systematically explored in this context. Here we fill this gap by providing an in-depth statistical, combinatorial, geometrical and topological characterization of 2D Ising snapshot networks (IsingNets) extracted from Monte Carlo simulations of the 2D Ising model at different temperatures, going across the phase transition. Our analysis reveals the complex organization properties of IsingNets in both the ferromagnetic and paramagnetic phases and demonstrates the significant deviations of the IsingNets with respect to randomized null models. In particular percolation properties of the IsingNets reflect the existence of the symmetry between configurations with opposite magnetization below the critical temperature and the very compact nature of the two emerging giant clusters revealed by our persistent homology analysis of the IsingNets. Moreover, the IsingNets display a very broad degree distribution and significant degree-degree correlations and weight-degree correlations demonstrating that they encode relevant information present in the configuration space of the 2D Ising model. The geometrical organization of the critical IsingNets is reflected in their spectral properties deviating from the one of the null model. This work reveals the important insights that network science can bring to the characterization of phases of matter. The set of tools described hereby can be applied as well to numerical and experimental data.Comment: 17 pages, 18 figure

Non-parametric learning critical behavior in Ising partition functions: PCA entropy and intrinsic dimension

We provide and critically analyze a framework to learn critical behavior in classical partition functions through the application of non-parametric methods to data sets of thermal configurations. We illustrate our approach in phase transitions in 2D and 3D Ising models. First, we extend previous studies on the intrinsic dimension of 2D partition function data sets, by exploring the effect of volume in 3D Ising data. We find that as opposed to 2D systems for which this quantity has been successfully used in unsupervised characterizations of critical phenomena, in the 3D case its estimation is far more challenging. To circumvent this limitation, we then use the principal component analysis (PCA) entropy, a "Shannon entropy" of the normalized spectrum of the covariance matrix. We find a striking qualitative similarity to the thermodynamic entropy, which the PCA entropy approaches asymptotically. The latter allows us to extract -- through a conventional finite-size scaling analysis with modest lattice sizes -- the critical temperature with less than $1\%$ error for both 2D and 3D models while being computationally efficient. The PCA entropy can readily be applied to characterize correlations and critical phenomena in a huge variety of many-body problems and suggests a (direct) link between easy-to-compute quantities and entropies.Comment: Corrected affiliation informatio

Nonequilibrium dynamics in lattice gauge theories: disorder-free localization and string breaking

Lattice gauge theories are crucial for our understanding of many physical phenomena ranging from fundamental particle interactions in high-energy physics to frustration and topological order in condensed matter. Hence, many equilibrium aspects of these theories have been studied intensively over the past decades. Recent developments, however, have shown that the study of nonequilibrium dynamics in lattice gauge theories also provides a very fertile ground for interesting phenomena. This thesis is devoted to the study of two particular dynamical processes in lattice gauge theories and related quantum spin models. First, we show that an interacting two-dimensional lattice gauge theory can exhibit disorder-free localization: a mechanism for ergodicity breaking due to local constraints imposed by gauge invariance. This result is particularly remarkable as the stability in two dimensions of the more conventional (disorder-induced) many-body localization is still debated. Concretely, we show this type of nonergodic behavior in the quantum link model. Our central result is based on a bound on the localization-delocalization transition, which is established through a concomitant classical percolation problem. Further, we develop a numerical method dubbed “variational classical networks”, to study the quantum dynamics in this system. This technique provides an efficient and perturbatively controlled representation of the wave function in terms of networks of classical spins akin to artificial neural networks. This allows us to identify distinguishing transport properties in the localized and ergodic phases, respectively. In the second problem, we study the dynamics of string breaking, a key process in confining gauge theories, where a string connecting two charges decays due to the creation of new particle-antiparticle pairs. Our main result here is that string breaking can also be observed in quantum Ising chains, in which domain walls get confined either by a symmetry-breaking field or by long-range interactions. We identify, in general, two distinct stages in this process. While at the beginning the initial charges remain stable, the string can exhibit complex dynamics with strong quantum correlations. We provide an effective description of this string motion, and find that it can be highly constrained. In the second stage, the string finally breaks at a timescale that depends sensitively on the initial separation of domain walls. We observe that the second stage can be significantly delayed as a consequence of the dynamical constraints appearing in the first stage. Finally, we discuss the generalization of our results to low-dimensional confining gauge theories. As a general aspect of this work, we discuss how the phenomena studied here could be realized experimentally with current and future technologies in quantum simulation. Furthermore, the methods developed in this thesis can also be applied to other lattice gauge theories and constrained quantum many-body models, not only to address purely theoretical questions but also to provide a theoretical description of experiments in quantum simulators.Gittereichtheorien sind ein wichtiger Bestandteil im Verständnis vieler physikalischer Phänomene und Grundlage verschiedener Theorien, welche sich von der elementaren Wechselwirkungen in der Hochenergiephysik, Frustration in Spinmodellen bis hin zu topologischer Ordnung in der Festkörperphysik erstrecken. Die Eigenschaften von Eichtheorien im Gleichgewicht waren in den letzten Jahrzehnten ein zentraler Punkt der Forschung. Obwohl sich Untersuchungen der Dynamik jenseits des Gleichgewichs als eine große Herausfordung dargestellt haben, haben kürzliche Erkenntnisse gezeigt, dass die Dynamik in Gittereichtheorien überraschende und interessante Entdeckungen bereithält. Diese Dissertation behandelt zwei zentrale dynamische Prozesse in Gittereichtheorien und verwandten Spinmodellen. Einerseits soll die Dynamik von zweidimensionalen und wechselwirkenden Gittereichtheorien untersucht werden im Falle des sogenan- nten Quanten-Link-Modells untersucht werden. Entgegen der Ergodenhypothese zeigt das System Lokalisierung ohne Unordnung aufgrund lokaler Zwangsbedingungen durch Eininvarianz. Dieses Ergebnis ist insofern bemerkenswert, als die gewöhnliche, durch Unordnung induzierte, Vielteilchenlokalisierung in zwei Dimensionen umstritten ist. Als ein Hauptergebnis finden wir einen Übergang zwischen einer lokalisierten und ergodischen Phase, dessen Existenz durch ein zugehöriges klassisches Perkolationsproblem gezeigt werden konnte. Die quantenmechanischen Transporteigenschaften, elementar verschieden in der lokalisierten und ergodischen Phase, werden charakterisiert und untersucht. Die Lösung der quantenmechanischen Zeitentwicklung wird durch eine methodische Weiterentwicklung der sogenannten „variationellen klassischen Netzwerke“ erreicht Diese Methode stellt eine perturbative, aber kontrollierte Repräsentation von zeitentwickelten quantenmechanischen Wellenfunktionen dar in Form von Netzwerken klassischer Spins, ähnlich wie bei einem künstlichen neuronalen Netz. Im zweiten Teil untersuchen wir die Dynamik eines Schlüsselprozesses in Eichtheorien mit Confinement, welcher als „String-Breaking“ bezeichnet wird In diesem Prozess zerfällt der der Strang, der zwei elementare Ladungen verbindet, durch die Bildung neuer Teilchen-Antiteilchen-Paare. Ein Hauptresultat dieser Arbeit ist die Beobachtung dieses dynamischen Phänomens in Quantum-Ising-Ketten und damit in Systemen ohne Eichinvarianz. Das Confinement entsteht dabei zwischen Domänenwänden entweder durch eine langreichweitige Wechselwirkung zwischen den beteiligten Spins oder durch symmetriebrechende Magnetfelder. Es wird gezeigt, dass während des „String-breaking“ Prozesses das Modell zwei Phasen durchläuft: Während zu Beginn die Anfangsladungen stabil bleiben, weist der Strang eine komplexe Dynamik mit starken Quantenkorrelationen auf. Für diese erste Phase wird eine effektive Beschreibung eingeführt, um die verschiedenen Aspekte zu analysieren und zu verstehen. Die Zeitskalen zur Destabilisierung des Strangs innerhalb einer zweiten Phase zeigen eine starke Abhängigkeit von der anfänglichen Trennung der Domänenwände. Es wird gezeigt, dass die zweite Phase als Konsequenz der dynamischen Beschränkungen der ersten Phase signifikant verzögert werden kann. Diese Resultate können in niedrigdimensionalen Eichtheorien verallgemeinert werden. Weiterführend sollen die Ergebnisse als Grundlage einer experimentellen Realisierung durch Quantensimulationen dienen. Die entwickelten Methoden können auf andere Eichtheorien und verwandten Vielteilchenmodellen angewendet werden und bieten eine Plattform für weitere Ansätze

Empagliflozin in Patients with Chronic Kidney Disease

Background The effects of empagliflozin in patients with chronic kidney disease who are at risk for disease progression are not well understood. The EMPA-KIDNEY trial was designed to assess the effects of treatment with empagliflozin in a broad range of such patients. Methods We enrolled patients with chronic kidney disease who had an estimated glomerular filtration rate (eGFR) of at least 20 but less than 45 ml per minute per 1.73 m(2) of body-surface area, or who had an eGFR of at least 45 but less than 90 ml per minute per 1.73 m(2) with a urinary albumin-to-creatinine ratio (with albumin measured in milligrams and creatinine measured in grams) of at least 200. Patients were randomly assigned to receive empagliflozin (10 mg once daily) or matching placebo. The primary outcome was a composite of progression of kidney disease (defined as end-stage kidney disease, a sustained decrease in eGFR to < 10 ml per minute per 1.73 m(2), a sustained decrease in eGFR of & GE;40% from baseline, or death from renal causes) or death from cardiovascular causes. Results A total of 6609 patients underwent randomization. During a median of 2.0 years of follow-up, progression of kidney disease or death from cardiovascular causes occurred in 432 of 3304 patients (13.1%) in the empagliflozin group and in 558 of 3305 patients (16.9%) in the placebo group (hazard ratio, 0.72; 95% confidence interval [CI], 0.64 to 0.82; P < 0.001). Results were consistent among patients with or without diabetes and across subgroups defined according to eGFR ranges. The rate of hospitalization from any cause was lower in the empagliflozin group than in the placebo group (hazard ratio, 0.86; 95% CI, 0.78 to 0.95; P=0.003), but there were no significant between-group differences with respect to the composite outcome of hospitalization for heart failure or death from cardiovascular causes (which occurred in 4.0% in the empagliflozin group and 4.6% in the placebo group) or death from any cause (in 4.5% and 5.1%, respectively). The rates of serious adverse events were similar in the two groups. Conclusions Among a wide range of patients with chronic kidney disease who were at risk for disease progression, empagliflozin therapy led to a lower risk of progression of kidney disease or death from cardiovascular causes than placebo

Alirocumab and Cardiovascular Outcomes after Acute Coronary Syndrome

BACKGROUN