Stability of the Enhanced Area Law of the Entanglement Entropy

We consider a multi-dimensional continuum Schr\"odinger operator which is given by a perturbation of the negative Laplacian by a compactly supported potential. We establish both an upper and a lower bound on the bipartite entanglement entropy of the ground state of the corresponding quasi-free Fermi gas. The bounds prove that the scaling behaviour of the entanglement entropy remains a logarithmically enhanced area law as in the unperturbed case of the free Fermi gas. The central idea for the upper bound is to use a limiting absorption principle for such kinds of Schr\"odinger operators.Comment: Changes in v2: result extended from cubes to Lipschitz domains with piecewise smooth boundar

On enhanced area laws of the entanglement entropy

In many-body systems the extent and range of spatial quantum correlations induced by entanglement provide a great deal of information about several qualitative physical properties. One way of studying this information is to examine the scaling behaviour of the ground state entanglement entropy with respect to a scaled version of a distinguished spatial subregion. In various systems the entanglement entropy grows proportionally to the surface area of the subregion which is referred to as an area law. In this thesis we examine the connection between the scaling behaviour of the entanglement entropy and many-body localisation. In recent years it was show that a number of systems, which are known to be in the localised phase, exhibit area laws of the entanglement entropy. It is commonly expected that the entanglement entropies of delocalised ground states do not satisfy area laws, though not many examples of different scaling behaviours have been shown, yet. The aim of this thesis is to provide further examples of violations of area laws in the context of delocalised systems. In three different models we show that the entanglement entropy of the ground states grows at least like a logarithmically enhanced area law. The first part of this thesis, based on joint work with P. Müller and L. Pastur [MPS20], considers the random dimer model. Even though this non-interacting, one-dimensional model is spectrally localised, there exist critical points in its spectrum at which the localisation length diverges. We consider the ground state corresponding to a Fermi energy positioned at one of these critical energies. In the case of small disorder we show a logarithmic lower bound to the expectation of the entanglement entropy. Moreover, we proof a logarithmic lower bound to the finite-volume entanglement en- tropy at these critical points for any disorder strength. In the second part of this thesis, which is based on joint work with P. Müller [MS20], we consider a multi-dimensional continuum Schrödinger operator, which is given by a perturbation of a negative Laplacian by a compactly supported, bounded potential. We establish both an upper and a lower bound to the entanglement entropy corresponding to a positive Fermi energy. These bounds prove that the scaling behaviour of the entanglement entropy is a logarithmically enhanced area law. This is the same scaling behaviour as the one occurring in the case of free fermions, one of the few delocalised systems for which an asymptotic expansion of the entanglement entropy is known. Finally, in the third and last part, based on joint work with C. Fischbacher [FS20], we consider the finite XXZ spin chain with periodic boundary conditions in the Ising phase. We show that for each eigenvalue in the droplet band there exists at least one eigenvector such that the corresponding entanglement entropy grows at least logarithmically, provided the anisotropy parameter ∆ is sufficiently large. In addition, we show a Combes–Thomas estimate for this model, which may be of independent interest.In Vielteilchensystemen liefert die Reichweite der durch Verschränkung induzierten räumlichen Quantenkorrelationen eine Vielzahl von Informationen über verschiedene physikalische Eigenschaften. Eine Möglichkeit, diese Informationen zu untersuchen, ist die Betrachtung des Skalierungsverhaltens der Verschränkungsentropie des Grundzustandes in Bezug auf eine skalierte Version eines räumlichen Gebietes. In vielen Systemen wächst die Verschränkungsentropie proportional zur Oberflä̈che des Gebietes, was als Oberflächengesetz bezeichnet wird. In dieser Arbeit untersuchen wir den Zusammenhang zwischen dem Skalierungsverhalten der Verschränkungsentropie und Vielteilchenlokalisierung. In den letzten Jahren konnte gezeigt werden, dass eine Reihe von Systemen, von denen bekannt ist, dass sich ihr Grundzustand in der lokalisierten Phase befindet, Oberflächengesetze der Verschränkungsentropie aufweisen. Auf der anderen Seite wird allgemein angenommen, dass die Verschra ̈nkungsentropie von delokalisierten Grundzuständen nicht einem Oberflächengesetz genügt. Allerdings gibt es nur wenige Beispiele, für die ein abweichendes Skalierungsverhalten bereits gezeigt wurde. Ziel dieser Arbeit ist es, weitere Beispiele für solche Abweichungen von Oberflächengesetzen der Verschränkungsentropie im Zusammenhang mit delokalisierten Systemen zu liefern. In drei verschiedenen Modellen zeigen wir, dass die Verschränkungsentropie des Grundzustandes zumindest ein logarithmisch erweitertes Oberflächengesetz aufweist. Der erste Teil dieser Dissertation, welcher auf einer gemeinsamen Arbeit mit L. Pastur und P. Müller [MPS20] basiert, befasst sich mit dem zufälligen Dimer-Modell. Obwohl dieses nicht-interagierende, eindimensionale Modell spektral lokalisiert ist, gibt es kritische Punkte in dem Spektrum, an denen die Lokalisierungslänge divergiert. Im Falle von geringer Unordnung wird in dieser Arbeit eine logarithmische Untergrenze für den Erwartungswert der Verschränkungsentropie gezeigt. Darüber hinaus wird für eine beliebige Unordnungsstärke eine logarithmische Untergrenze an die Verschränkungsen- tropie für endliche Volumen an diesen kritischen Punkten bewiesen. Im zweiten Teil dieser Arbeit, welcher auf einer gemeinsamen Arbeit mit P. Müller [MS20] basiert, betrachten wir einen mehrdimensionalen, kontinuierlichen Schrödinger-Operator, der durch die Störung eines negativen Laplace-Operators durch ein kompakt getragenes, beschränktes Potential gegeben ist. Sowohl eine obere als auch eine untere Grenze für die Verschränkungsentropie zu einer positiven Fermi-Energie wird gezeigt. Diese Schranken beweisen, dass das Skalierungsverhalten der Verschra ̈nkungsentropie einem logarithmisch erweiterten Oberfl ̈achengesetz entspricht. Dies ist das gleiche Skalierungsverhalten, das auch bei freien Fermionen auftritt. Das Modell der freien Fermionen ist eines der wenigen delokalisierten Systeme, für die eine asymptotische Entwicklung der Verschränkungsentropie bekannt ist. Im dritten und letzten Teil wird, basierend auf einer gemeinsamen Arbeit mit C. Fischbacher [FS20], die endliche XXZ-Spinkette mit periodischen Randbedingungen in der Ising-Phase betrachtet. Dieses Modell hat aufgrund seiner Translationsinvarianz delokalisierte Eigenzustände. Wir zeigen, dass für jeden Eigenwert im Droplet-Band mindestens ein Eigenvektor existiert, sodass die zugehörige Verschränkungsentropie mindestens logarithmisch anwächst. Für dieses Resultat setzen wir voraus, dass der Anisotropie-Parameter ∆ ausreichend groß ist. Zusätzlich dazu zeigen wir eine Combes–Thomas-Abschätzung für dieses Modell, was für sich genommen ebenfalls von Interesse ist

Isotopic systematics of ultramafic and mafic rocks of the Taitao ophiolite, southern Chile

A variety of ultramafic and mafic rocks from a well-mapped sampling of the ~6 Ma Taitao Ophiolite, Chile, have been examined. Calculated initial 187Os/188Os ratios of the peridotitic rocks range from 0.1168 to 0.1282. Similar ranges of Os isotopic compositions have been reported for abyssal peridotites, peridotites, and chromites from other ophiolites. A correlation between Mg # of primary olivine and Os isotopic compositions of whole rock peridotites suggests that the isotopic variability is due to variable extents of partial melting at ~1.5 Ga. This conclusion is also supported by a linear correlation between 187Re/188Os and 187Os/188Os. The ancient melting event requires that this km-scale block of the mantle remained isolated and unmixed within the convecting upper mantle for a period of ~1.5 Ga. The results are comparable to global observations of Os isotopic compositions in mid-ocean ridge basalts versus abyssal peridotites, but the geologic relations here are well-defined

Comorbidities Between Specific Learning Disorders and Psychopathology in Elementary School Children in Germany

Children with reading and/or spelling disorders have increased rates of behavioral and emotional problems and combinations of these. Some studies also find increased rates of attention-deficit/hyperactivity disorder (ADHD), conduct disorder, anxiety disorder, and depression. However, the comorbidities of, e.g., arithmetic disorders with ADHD, anxiety disorder, and depression have been addressed only rarely. The current study explored the probability of children with specific learning disorders (SLD) in reading, spelling, and/or arithmetic to also have anxiety disorder, depression, ADHD, and/or conduct disorder. The sample consisted of 3,014 German children from grades 3 and 4 (mean age 9;9 years) who completed tests assessing reading, spelling as well as arithmetic achievement and intelligence via a web-based application. Psychopathology was assessed using questionnaires filled in by the parents. In children with a SLD we found high rates of anxiety disorder (21%), depression (28%), ADHD (28%), and conduct disorder (22%). Children with SLD in multiple learning domains had a higher risk for psychopathology and had a broader spectrum of psychopathology than children with an isolated SLD. The results highlight the importance of screening for and diagnosing psychiatric comorbidities in children with SLD

Psychological interventions for adults with bipolar disorder: a systematic review and meta-analysis

Background Psychological interventions may be beneficial for bipolar disorder. Aims Efficacy evaluation of psychological interventions for adults with bipolar disorder. Methods A systematic review of randomised controlled trials.. Outcomes were meta-analysed using RevMan and confidence assessed using the GRADE-method. Results We included 55 trials with 6010 participants. Moderate quality evidence associated individual psychological interventions with reduced relapses at post-treatment and follow-up, and collaborative care with a reduction in hospitalisations. . Low quality evidence associated group interventions with fewer depression relapses at post-treatment and follow-up, and family psycho-education with reduced symptoms of depression and mania at post-treatment. Conclusions There is evidence that psychological interventions are effective for people with bipolar disorder. Limits were the very low quality of much of the evidence and therefore inconclusive. Further research should identify the most (cost)-effective interventions for each phase of this disorder

Estragole: DNA adduct formation in primary rat hepatocytes and genotoxic potential in HepG2-CYP1A2 cells

Estragole is a natural constituent in herbs and spices and in products thereof such as essential oils or herbal teas. After cytochrome P450-catalyzed hydroxylation and subsequent sulfation, estragole acts as a genotoxic hepatocarcinogen forming DNA adducts in rodent liver. Because of the genotoxic mode of action and the widespread occurrence in food and phytomedicines a refined risk assessment for estragole is needed. We analyzed the time- and concentration-dependent levels of the DNA adducts N2-(isoestragole-3‘-yl)-2‘-desoxyguanosine (E3′N2dG) and N6-(isoestragole-3‘-yl)-desoxyadenosine (E3′N6dA), reported to be the major adducts formed in rat liver, in rat hepatocytes (pRH) in primary culture after incubation with estragole. DNA adduct levels were measured via UHPLC-ESI-MS/MS using stable isotope dilution analysis. Both adducts were formed in pRH and could already be quantified after an incubation time of 1 h (E3′N6dA at 10 μM, E3′N2dG at 1μM estragole). E3′N2dG, the main adduct at all incubation times and concentrations, could be detected at estragole concentrations < 0.1 μM after 24 h and < 0.5 μM after 48 h. Adduct levels were highest after 6 h and showed a downward trend at later time-points, possibly due to DNA repair and/or apoptosis. While the concentration-response characteristics of adduct formation were apparently linear over the whole concentration range, strong indication for marked hypo-linearity was obtained when the modeling was based on concentrations < 1 μM only. In the micronucleus assay no mutagenic potential of estragole was found in HepG2 cells whereas in HepG2-CYP1A2 cells 1 μM estragole led to a 3.2 fold and 300 μM to a 7.1 fold increase in micronuclei counts. Our findings suggest the existence of a ‘practical threshold’ dose for DNA adduct formation as an initiating key event of the carcinogenicity of estragole indicating that the default assumption of concentration-response-linearity is questionable, at least for the two major adducts studied here

A new method of RF power generation for two-beam linear colliders

In this paper we discuss a new approach to two-beam acceleration. The energy for RF production is initially stored in a long-pulse electron beam which is efficiently accelerated to about 1.2 GeV by a fully loaded, conventional, low frequency (~1 GHz) linac. The beam pulse length is twice the length of the high-gradient linac. Segments of this long pulse beam are compressed using combiner rings to create a sequence of higher peak power drive beams with gaps in between. This train of drive beams is distributed from the end of the linac against the main beam direction down a common transport line so that each drive beam can power a section of the main linac. After a 180-degree turn, each high-current, low-energy drive beam is decelerated in low-impedance decelerator structures, and the resulti ng power is used to accelerate the low-current, high-energy beam in the main linac. The method discussed here seems relatively inexpensive is very flexible and can be used to accelerate beams for lin ear colliders over the entire frequency and energy range

Hydration versus N-acetylcysteine for protection of renal function in patients with renal insufficiency undergoing percutaneous coronary intervention

A Clarion Call for Large-Scale Collaborative Studies of Pediatric Proton Therap