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I review the quantum theory of the electron moving in a random environment. First, the quantum mechanics of individual particles scattered on a random potential is discussed. The quantum-mechanical description is extended to many-body systems by using many-body Green functions. The many-body approach is used to derive the coherent-potential approximation and to show how it fits into the dynamical mean-field theory. The generating functional of the coherent-potential approximation is obtained in an analytic form from the limit to infinite dimensions of the general many-body description of non-interacting electrons in random lattices. The analytic generating functional of the mean-field description of random systems is extended to the Falicov-Kimball model with thermally equilibrated scattering lattice potential. The many-body Green functions are then used to calculate transport properties. The electrical conductivity of the coherent-potential approximation is derived from the two-particle Green function calculated in infinite spatial dimensions. Finally, a perturbation theory for the vertex corrections to the mean-field conductivity is introduced. In particular, it is shown how to make the expansion beyond the local approximations conserving., 30 pages, Lecture Notes for the Autumn School on Correlated Electrons: The Physics of Correlated Insulators, Metals, and Superconductors, 25-29 September 2017, Forschungszentrum J\"ulich (https://www.cond-mat.de/events/correl17/)

We develop the general many-body perturbation theory for a superconducting quantum dot represented by a single-impurity Anderson model attached to superconducting leads. We build our approach on a thermodynamically consistent mean-field approximation with a two-particle self-consistency of the parquet type. The two-particle self-consistency leading to a screening of the bare interaction proves substantial for suppressing the spurious transitions of the Hartree-Fock solution. We demonstrate that the magnetic field plays a fundamental role in the extension of the perturbation theory beyond the weakly correlated $0$-phase. It controls the critical behavior of the $0-��$ quantum transition, lifts the degeneracy in the $��$-phase, where the limits to zero temperature and zero magnetic field do not commute. The response to the magnetic field is quite different in $0$- and $��$-phases. While the magnetic susceptibility vanishes in the $0$-phase it becomes of the Curie type and diverges in the $��$-phase at zero temperature., RevTex 4-2, 19 pages, 12 figures

We apply a two-particle semi-analytic approach to a single Anderson impurity attached to two biased metallic leads. The theory is based on reduced parquet equations justified in critical regions of singularities in the Bethe-Salpeter equations. It offers a way to treat one-particle and two-particle thermodynamic and spectral quantities on the same footing. The two-particle vertices are appropriately renormalized so that spurious transitions into the magnetic state of the weak-coupling approximations are suppressed. The unphysical hysteresis loop in the current-voltage characteristics is thereby eliminated. Furthermore, in the linear response regime, we qualitatively reproduce the three transport regimes with the increasing temperature: from the Kondo resonant tunneling through the Coulomb blockade regime up to a sequential tunneling regime. Far from equilibrium, we find that the bias plays a similar role as the temperature in destroying the Kondo resonant peak when the corresponding energy scale is comparable with the Kondo temperature. Besides that, the applied voltage in low bias is shown to develop spectral peaks around the lead chemical potentials as observed in previous theoretical and experimental studies., 15 pages, 10 figures

I review the way the many-body Green functions are used to renormalize the perturbation theory of correlated fermions. The Green functions are introduced to implement systematically dynamical corrections to the static mean-field theory. The renormalizations enter the perturbation theory via self-consistent evaluations of one-particle and two-particle Green functions. The one-particle self-consistency is discussed within the Baym-Kadanoff construction. We point to an inherent dichotomy of the Baym-Kadanoff approach in the relation between one and two-particle Green functions. They are the Schwinger-Dyson equation, directly derived from the generating Luttinger-Ward functional, and the Ward identity. The latter imposes a severe restriction on the two-particle irreducible vertex and is not guaranteed in the Baym-Kadanoff approach. No approximate theory is capable to obey both relations. I further discuss the two-particle approach with two-particle renormalizations that offers a way to reconcile the Schwinger-Dyson equation and the Ward identity. The two-particle irreducible vertex from the critical Bethe-Salpeter equation becomes a generating functional in the two-particle renormalization of the perturbation theory. The irreducible vertex is used in the Ward identity to determine the order parameter beyond the critical point and in the Bethe-Salpeter equation representing the vertex of the Schwinger-Dyson equation to determine the spectral self-energy in such a way that the thermodynamic critical behavior from both equations is qualitatively the same. Finally, I discuss a static mean-field-like approximation with a renormalized effective interaction determined self-consistently that leads to a qualitatively correct description of strong electron correlations in impurity and lattice models., Lecture notes to Online Autumn School on Correlated Electrons: Simulating Correlations with Computers, J\"ulich, 20-24 September 2021, (https://www.cond-mat.de/events/correl21/)

The zero-temperature physics of interacting quantum dots attached to superconducting leads is now well understood. The overall qualitative picture is obtained from the static mean-field approximation. The situation drastically changes at non-zero temperatures. No reliable solutions apart from numerical simulations exist there. We show that any static mean-field approximation fails at non-zero temperatures since magnetic fluctuations induce dynamical corrections that lead to broadening of the in-gap state energies to energy bands. Spin-symmetric equilibrium state at non-zero temperatures is unstable with respect to magnetic fluctuations and the zero magnetic field can be reached only as a weak limit of the spin-polarized solution like in a magnetically ordered phase., 6 pages, 3 figures, accepted in AIP Advances

Kondo temperature is standardly defined from the local zero-temperature susceptibility in the regime of strong electron correlations as a new scale controlling the low-temperature asymptotics of thermodynamic quantities. We show by using a two-particle self-consistent theory that the Kondo temperature can be identified as a crossover temperature at which the zero-temperature quantum fluctuations equal the thermal ones in the electron-hole correlation function. The high-temperature Curie-Weiss susceptibility is shown to go over to the Pauli one below the Kondo temperature., 6 pages, 4 figures, submitted to SCES 2019

We discuss the mean-field theory of spin-glass models with frustrated long-range random spin exchange. We analyze the reasons for breakdown of the simple mean-field theory of Sherrington and Kirkpatrick. We relate the replica-symmetry breaking to ergodicity breaking and use the concept of real replicas to restore thermodynamic homogeneity of the equilibrium free energy in a replicated phase space. Embedded replications of the spin variables result in a set of hierarchical free energies and overlap susceptibilities between replica hierarchies as order parameters. The limit to infinite number of replica hierarchies leads to the Parisi solution with a continuous replica-symmetry breaking. We present a closed-form representation of the Parisi mean-field theory that is independent from stability of solutions with finite-many replica hierarchies. Hence, solutions with continuous and discrete replica-symmetry breaking can coexist. We demonstrate the construction of the spin-glass mean-field solutions via real replicas on three spin-glass models: Ising, Potts and p-spin. An asymptotic expansion in each model is used to demonstrate various types of the transition from the paramagnetic to the glassy phase., 28 pages, Lecture Notes for the Autumn School on Correlated Electrons, J\"ulich, 2015 (http://www.cond-mat.de/events/correl15). arXiv admin note: text overlap with arXiv:1501.01653

A single-level quantum dot with Coulomb repulsion attached to two superconducting leads is studied via the perturbation expansion in the interaction strength. We use the Nambu formalism and the standard many-body diagrammatic representation of the impurity Green functions to formulate the Matsubara self-consistent perturbation expansion. We show that at zero temperature second order of the expansion in its $\it{spin-symmetric}$ version yields a nearly perfect agreement with the numerically exact calculations for the position of the $0-��$ phase boundary at which the Andreev bound states reach the Fermi energy as well as for the values of single-particle quantities in the 0-phase. We present results for phase diagrams, level occupation, induced local superconducting gap, Josephson current, and energy of the Andreev bound states with the precision surpassing any (semi)analytical approaches employed thus far., 7 pages, 4 figures; v2: published version, minor updates

Self-consistent perturbation expansion up to the second order in the interaction strength is used to study a single-level quantum dot with local Coulomb repulsion attached asymmetrically to two generally different superconducting leads. At zero temperature and wide range of other parameters the spin-symmetric version of the expansion yields excellent results for the position of the $0-��$ impurity quantum phase transition boundary and Josephson current together with the energy of Andreev bound states in the $0$-phase as confirmed by numerical calculations using the Numerical Renormalisation Group method. We analytically prove that the method is charge-conserving as well as thermodynamically consistent. Explicit formulas for the position of the $0-��$ phase boundary are presented for the Hartree-Fock approximation as well as for its variant called Generalized Atomic Limit. It is shown that the Generalized Atomic Limit can be used as a quick estimate for the position of the phase boundary at half-filling in a broad range of parameters. We apply our second order perturbation method to the interpretation of the existing experimental data on the phase boundary with very satisfactory outcome suggesting that the so far employed heavy numerical tools such as Numerical Renormalization Group and/or Quantum Monte Carlo are not necessary in a class of generic situations and can be safely replaced by a perturbative approach., 19 pages, 11 figures

Behavior of Andreev gap states in a quantum dot with Coulomb repulsion symmetrically attached to superconducting leads is studied via the perturbation expansion in the interaction strength. We find the exact asymptotic form of the spin-symmetric solution for the Andreev states continuously approaching the Fermi level. We thereby derive a critical interaction at which the Andreev states at zero temperature merge at the Fermi energy, being the upper bound for the $0-��$ transition. We show that the spin-symmetric solution becomes degenerate beyond this interaction, in the $��$ phase, and the Andreev states do not split unless the degeneracy is lifted. We further demonstrate that the degeneracy of the spin-symmetric state extends also into the $0$ phase in which the solutions with zero and non-zero frequencies of the Andreev states may coexist., 12 pages, 4 figures

Background: Applications of atrial speckle tracking echocardiography (STE) strain (ε) analysis in pediatric cardiac surgery have been limited. This study aims to evaluate the feasibility of atrial STE ε analysis and the progression of atrial ε values as a function of post-operative time in children after pediatric cardiac surgery. Methods: 131 children (mean 1.69 ± 2.98; range 0.01–15.16 years) undergoing cardiac surgery were prospectively enrolled. Echocardiographic examinations were performed pre-operatively and at 3 different post-operative intervals: Time 1 (24–36 h), Time 2 (3–5 days), Time 3 (>5 days, before discharging). The right and left atrium longitudinal systolic contractile (Ct), Conduit (Cd), and Reservoir (R) ε were evaluated with a novel atrial specific software with both P- and R-Gating methods. One hundred and thirty-one age-matched normal subjects (mean 1.7 ± 3.2 years) were included as controls. Results: In all, 309 examinations were performed over the post-operative times. For each post-operative interval, all STE atrial ε parameters assessed were significantly lower compared to controls (all p < 0.0001). The lowest atrial ε values were found at Time 1, with only partial recovery thereafter (p from 0.02 to 0.04). All atrial ε values at discharge were decreased compared to the controls (all p < 0.0001). Significant correlations of the atrial ε values with cardio-pulmonary-bypass time, left and right ventricular ε values (p < 0.05), and ejection fraction (p < 0.05) were demonstrated. Conclusions: Atrial ε is highly reduced after surgery with only partial post-operative recovery in the near term. Our study additionally demonstrates that post-surgical atrial and ventricular ε responses correlated with each other.

The aim of this review is to highlight the strengths and limitations of major echocardiographic biventricular repair (BVR) prediction models for borderline left ventricle (LV) in complex congenital heart disease (CHD). A systematic search in the National Library of Medicine for Medical Subject Headings and free text terms including echocardiography, CHD, and scores, was performed. The search was refined by adding keywords for critical aortic stenosis (AS), borderline LV, complex left ventricular outflow tract (LVOT) obstruction, hypoplastic left heart syndrome/complex (HLHS/HLHC), and unbalanced atrio-ventricular septal defects (uAVSD). Fifteen studies were selected for the final analysis. We outlined what echocardiographic scores for different types of complex CHD with diminutive LV are available. Scores for CHD with LVOT obstruction including critical AS, HLHS/HLHC, and aortic arch hypoplasia have been validated and implemented by several studies. Scores for uAVSD with right ventricle (RV) dominance have also been established and implemented, the first being the atrioventricular valve index (AVVI). In addition to AVII, both LV/RV inflow angle and LV inflow index have all been validated for the prediction of BVR. We conclude with a discussion of limitations in the development and validation of each of these scores, including retrospective design during score development, heterogeneity in echocardiographic parameters evaluated, variability in the definition of outcomes, differences in adopted surgical and Interventional strategies, and institutional differences. Furthermore, scores developed in the past two decades may have little clinical relevance now. In summary, we provide a review of echocardiographic scores for BVR in complex CHD with a diminutive LV that may serve as a guide for use in modern clinical practice.

Background: Speckle-tracking echocardiography (STE) has gained increasing value in the evaluation of congenital heart diseases (CHD), however, its use in pediatric cardiac surgery is limited. Aim: To evaluate left ventricular (LV) systolic impairment after biventricular pediatric cardiac surgery by STE strain (ε) analysis. Methods: We prospectively enrolled 117 children undergoing cardiac surgery for CHD. Echocardiography was performed at four different times: pre-operatively, 12–36 h (Time 1), 3–5 days (Time 2), and 6–8 days (Time 3). Images were obtained in the 4-2-and 3 apical chamber’s views to derive LV global and regional (basal/mid/apical) ε values. Results: At different postoperative times, we performed 320 examinations in 117 children (mean age: 2.4 ± 3.9, range: 0–16 years), 117 age-matched healthy children served as controls. All global, basal, and mid LVε values decreased after surgery, the lowest values being at Time 1 (p &lt, 0.0001), which increased thereafter. At discharge, all global, basal, and mid LVε values remained lower than in pre-operative and healthy children (p &lt, 0.05). Instead, apical segments (lowest at baseline) increased after surgery (p &lt, 0.0001) but remained lower compared to controls. LV ejection fraction (LVEF) decreased at Time 1 (p = 0.0004) but promptly recovered to Time 2 and normalized at Time 3. Conclusions: STE ε analysis revealed a significant LV systolic impairment after surgery with amelioration thereafter but incomplete normalization at discharge. Base-apex differences emerged with apical segments that, contrary to all the other regions, showed relative hypercontractility after surgery. The slower recovery of LVε values compared to LVEF suggests that STE ε analysis may be more accurate for the follow-up of mild LV post-surgical impairment.

Lung ultrasound (LUS) is increasing in its popularity for the diagnosis of pulmonary complications in acute pediatric care settings. Despite the high incidence of pulmonary complications for patients with pediatric cardiovascular and congenital heart disease, especially in children undergoing cardiac surgery, the use of LUS remains quite limited in these patients. The aim of this review is to provide a comprehensive overview and list of current potential applications for LUS in children with congenital heart disease, post-surgery. We herein describe protocols for LUS examinations in children, discuss diagnostic criteria, and introduce methods for the diagnosis and classification of pulmonary disease commonly encountered in pediatric cardiology (e.g., pleural effusion, atelectasis, interstitial edema, pneumothorax, pneumonia, and diaphragmatic motion analysis). Furthermore, applications of chest ultrasounds for the evaluation of the retrosternal area, and in particular, systematic search criteria for retrosternal clots, are illustrated. We also discussed the potential applications of LUS, including the guidance of interventional procedures, namely lung recruitment and drainage insertion. Lastly, we analyzed current gaps in knowledge, including the difficulty of the quantification of pleural effusion and atelectasis, and the need to differentiate different etiologies of B-lines. We concluded with future applications of LUS, including strain analysis and advanced analysis of diaphragmatic mechanics. In summary, US is an easy, accurate, fast, cheap, and radiation-free tool for the diagnosis and follow-up of major pulmonary complications in pediatric cardiac surgery, and we strongly encourage its use in routine practice.

BACKGROUND: Small rodent models are routinely used to evaluate the safety and efficacy of blood transfusions. Limited comprehensive literature exists about effect of different storage solutions in rat red blood cells (RBCs) characteristics. RBCs undergo time dependent biochemical and biophysical changes during storage known as hypothermic storage lesions (HSLs). OBJECTIVE: This study evaluates the effects of RBC additive solutions (AS) during storage of rat RBCs. METHODS: Blood was leukoreduced and stored as per manufacturer instructions at 4 degrees C up to 42-days. Three solutions, CPDA-1; AS-1; and AS-7 (SOLX), were evaluated. Biochemical parameters measured included extracellular K+, pH, hemolysis, 2,3-diphosphoglycerate (2,3-DPG), oxygen affinity, ATP, and lactate. Mechanical properties measured included RBC deformability, elongation index (EI), RBC membrane shear elastic modulus (SEM), mean corpuscular volume (MCV), viscosity, and aggregability. RESULTS: There were no differences in biochemical or mechanical parameters at baseline or after one week of storage. However, after two weeks, AS-7 preserved biochemical and mechanical properties as compared to CPDA-1 and AS-1. Changes were observed to be significant after 14-days of storage. AS-7 prevented extracellular K+ increase, reduced acidosis, showed lower hemolysis, preserved ATP and 2,3-DPG levels (consequently oxygen affinity), and reduced lactate. AS-7, when compared to CPDA-1 and AS-1, prevented the reduction in RBC deformability and was found to preserve the EI at multiple shear stresses, the membrane SEM, the aggregability and viscosity. DISCUSSION: Rat RBCs stored with AS-7 presented reduced changes in biochemical and mechanical parameters, when compared with rat RBCs stored in CPDA-1 and AS-1, after as early as two weeks of storage., NIH grants from the Heart Lung and Blood Institute