Globally 118 319 confirmed (4620 new) 4292 deaths (280 new) China 80 955 confirmed (31 new) 3162 deaths (22 new) Outside of China 37 364 confirmed (4589 new) 1130 deaths (258 new) 113 countries/territories/ areas (4 new)

No abstract

We present a truly canonical theory of superconductivity in ultrasmall metallic grains by variationally optimizing fixed-N projected BCS wave-functions, which yields the first full description of the entire crossover from the bulk BCS regime (mean level spacing d ≪ bulk gap∆) to the "fluctuation-dominated" few-electron regime (d ≫∆). A wave-function analysis shows in detail how the BCS limit is recovered for d ≪∆, and how for d ≫∆ pairing correlations become delocalized in energy space. An earlier grand-canonical prediction for an observable parity effect in the spectral gaps is found to survive the fixed-N projection.PACS numbers: 74.20.Fg, 74.25.Ha, 74.80.Fp In the early days of BCS theory, its use of essentially grand-canonical (g.c.) wave-functions was viewed as one of its most innovative, if not perplexing features: the variational BCS ansatz for the ground state is a superposition of states with different electron numbers, although BCS [1] themselves had emphasized that the true ground state of an isolated superconductor must be a state of definite electron number. That this ansatz was nevertheless rapidly accepted and tremendously successful had two reasons: Firstly, calculational convenience -determining the variational parameters is incomparably much simpler in a g.c. framework, where the particle number is fixed only on the average, than in a canonical one, where a further projection to fixed electron number is required; and secondly, it becomes exact in the thermodynamic limitfixed-N projections yield corrections to the BCS ground state energy per electron that are only of order N −1 , as shown e.g. by Anderson [2] and Mühlschlegel [3].Recently, however, a more detailed examination of the range of validity of BCS's g.c. treatment has become necessary, in the light of measurements by Ralph, Black and Tinkham (RBT) [4] of the discrete electronic spectrum of an individual ultrasmall superconducting grain: it had a charging energy so large (E C ≫∆) that electron number fluctuations are strongly suppressed, calling for a canonical description, and the number of electrons N within the Debye frequency cutoff ω D from the Fermi energy ε F was only of order 10 2 , hence differences between canonical and g.c. treatments might become important. Moreover, its mean level spacing d ∝ N −1 was comparable to the bulk gap∆, hence it lies right in the crossover regime between the "fluctuation-dominated" few-electron regime (d ≫∆) and the bulk BCS regime (d ≪∆), which could not be treated satisfactorily in any of the recent theoretical papers inspired by these experiments: the results of [5][6][7][8][9], including the predictions of parity effects, were obtained in a g.c. framework; and Mastellone, Falci and Fazio's (MFF) [10] fixed-N exact numerical diagonalization study, the first detailed analysis of the f.d. regime, was limited to N ≤ 25.In this Letter we achieve the first canonical description of the full crossover. We explicitly project the BCS ansatz to fixed N (for N ≤ 600) before variationally optimizing it...

Several recent papers have predicted parity effects, based on even-odd ground state energy differences, in ultrasmall ͑nm scale͒ superconductors having a discrete electronic eigenspectrum with mean level spacing d Ӎ⌬ ͑bulk gap͒. The motivation for the present paper is to analyze the measurability of these and related parity effects in the present generation of experiments ͓e.g., those of Ralph, Black, and Tinkham ͑RBT͔͒. To this end we develop a general theory of superconductivity in ultrasmall metallic grains, based on calculating the eigenspectrum using a generalized BCS variational approach. We discuss how conventional mean field theory breaks down with decreasing sample size, how the so-called blocking effect weakens pairing correlations in states with nonzero total spin, and how this affects the discrete eigenspectrum's behavior in a magnetic field, which favors nonzero total spin. Our calculations qualitatively reproduce the magnetic-field-dependent tunneling spectra for individual aluminum grains measured by RBT. Our main results regarding parity effects are ͑i͒ the conclusion that those based on even-odd ground state energy differences are currently not measurable and ͑ii͒ the proposal of a parity effect for the pair-breaking energy, which should be measurable provided that the grain size can be controlled sufficiently well.

We study the magnetic-field-induced breakdown of superconductivity in nm-scale metal grains having a mean electron level spacing d ≃∆ (bulk gap). Using a generalized variational BCS approach that yields good qualitative agreement with measured spectra, we argue that Pauli paramagnetism dominates orbital diamagnetism, as in the case of thin films in a parallel magnetic field. However, the first-order transition observed for the latter can be made continuous by finite size effects. The meanfield procedure of describing the system by a single pairing parameter ∆ breaks down for d ≃∆.PACS numbers: 74.20.Fg, 74.25.Ha, 74.80.Fp When a system of (correlated) electrons is sufficiently small, the electronic spectrum becomes discrete. This allows one to study the nature of electron correlations in unprecedented detail by analyzing the details of the spectrum. It has recently become possible to measure such discrete spectra directly by studying electron transport through nm-scale metallic grains (radius r ≈ 5nm), for which the mean spacing d = 1/N (ε F ) is ≃ 0.1 − 1 meV [1,2]. For Al grains the effects on the spectrum of spinorbit interactions [1], non-equilibrium excitations [3] and superconductivity [1,4,5] have been investigated.Studying the latter is particularly interesting in grains with d ≃∆ (bulk gap), near the lower size limit [6] of observable superconductivity. The number of free-electron states that pair-correlate (those within∆ of ε F ) is then of order one. Thus, even in grains in which a gap can still be observed [1], pairing correlations are expected to become so weak that they might be destroyed by the presence of a single unpaired electron [4]. A direct way to probe this is to turn on a magnetic field, whose Zeeman energy favors paramagnetic states with non-zero total spin.In this Letter, we develop a theory for the paramagnetic breakdown of superconductivity in nm-scale grains. We exploit analogies to thin films in a parallel magnetic field [7], but explicitly take account of the discreteness of the spectrum. To calculate the eigenenergies E n of the grain's lowest-lying eigenstates |n , we adopt a generalized variational BCS approach that goes beyond standard mean-field theory by using a different pairing parameter ∆ n for each |n . Using the E n to reconstruct the tunneling spectra, we find qualitative agreement with measured spectra [1], and show that the H-induced first-order transition to the paramagnetic normal state observed for thin films can be softened in ultrasmall grains.Experimental Results.-Our goal is to understand in detail the H-dependence of the measured discrete tunneling spectrum (see Fig. 3 of [2]) of an ultrasmall Al grain, coupled via tunnel barriers to one gate and two lead electrodes to form a nm-scale transistor. Each line in the spectrum corresponds to the H-dependent energy) needed for some rate-limiting electron tunneling process |n i N ±1 → |n f N off or onto the grain, where |n N is an eigenstate (with eigenenergy E N n + E N C ) of the N -electron island with charging e...

The reduced BCS model that is commonly used for ultrasmall superconducting grains has an exact solution worked out long ago by Richardson in the context of nuclear physics. We use it to check the quality of previous treatments of this model, and to investigate the effect of level statistics on pairing correlations. We find that the ground state energies are on average somewhat lower for systems with non-uniform than uniform level spacings, but both have an equally smooth crossover from the bulk to the few-electron regime. In the latter, statistical fluctuations in ground state energies strongly depend on the grain's electron number parity. . Nevertheless, these two regimes are qualitatively very different [9,10]: the condensation energy, e.g., is an extensive function of volume in the former and almost intensive in the latter, and pairing correlations are quite strongly localized around the Fermi energy ε F , or more spread out in energy, respectively.After the appearance of all these works, we became aware that the reduced BCS Hamiltonian on which they are based actually has an exact solution. It was published by Richardson in the context of nuclear physics (where it is known as the "picket-fence model"), in a series of papers between 1963 and 1977 [12,13] which seem to have completely escaped the attention of the condensed matter community. The beauty of this solution, besides its mathematical elegance, is that it also works for the case of randomly-spaced levels. It thus presents us with two rare opportunities, which are the subject of this Letter: (i) to compare the results of various previously-used approximations against the benchmark set by the exact solution, in order to gauge their reliability for related problems for which no exact solutions exist; and very interestingly, (ii) to study the interplay of randomness and interactions in a non-trivial model exactly, by examining the effect of level statistics on the SC/FD crossover.There is a previous study of the latter question by Smith and Ambegaokar using the g.c. mean-field BCS approach [5], who concluded, interestingly, that randomness enhances pairing correlations: compared to the case of uniform spacings [2], they found that a random spacing of levels (distributed according to the gaussian orthogonal ensemble) on average lowers the condensation energy E C to more negative values and increases the critical level spacings at which E C vanishes abruptly, but these still are parity dependent ( d c 1 = 1.8∆, d c 0 ≃ 14∆). However, the abrupt vanishing of E C found by SA can be suspected to be an artifact of their g.c. mean-field treatment, as was the case in [2][3][4]. Indeed, our exact results for random levels show (1) that the SC/FD crossover is as smooth as for the case of uniformly-spaced levels; this means, remarkably, that (2) even in the presence of randomness pairing correlations never vanish, no matter how large d/∆ becomes; quite the opposite, (3) the randomness-induced lowering of E C is strongest in the FD regime; in the latter, moreover, (...

Extrapulmonary manifestations of COVID-19 have gained attention due to their links to clinical outcomes and their potential long-term sequelae1. Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) displays tropism towards several organs, including the heart and kidney. Whether it also directly affects the liver has been debated2,3. Here we provide clinical, histopathological, molecular and bioinformatic evidence for the hepatic tropism of SARS-CoV-2. We find that liver injury, indicated by a high frequency of abnormal liver function tests, is a common clinical feature of COVID-19 in two independent cohorts of patients with COVID-19 requiring hospitalization. Using autopsy samples obtained from a third patient cohort, we provide multiple levels of evidence for SARS-CoV-2 liver tropism, including viral RNA detection in 69% of autopsy liver specimens, and successful isolation of infectious SARS-CoV-2 from liver tissue postmortem. Furthermore, we identify transcription-, proteomic- and transcription factor-based activity profiles in hepatic autopsy samples, revealing similarities to the signatures associated with multiple other viral infections of the human liver. Together, we provide a comprehensive multimodal analysis of SARS-CoV-2 liver tropism, which increases our understanding of the molecular consequences of severe COVID-19 and could be useful for the identification of organ-specific pharmacological targets.

The transcription factor ∆Np63 is a master regulator of epithelial cell identity and essential for the survival of squamous cell carcinoma (SCC) of lung, head and neck, oesophagus, cervix and skin. Here, we report that the deubiquitylase USP28 stabilizes ∆Np63 and maintains elevated ∆NP63 levels in SCC by counteracting its proteasome‐mediated degradation. Impaired USP28 activity, either genetically or pharmacologically, abrogates the transcriptional identity and suppresses growth and survival of human SCC cells. CRISPR/Cas9‐engineered in vivo mouse models establish that endogenous USP28 is strictly required for both induction and maintenance of lung SCC. Our data strongly suggest that targeting ∆Np63 abundance via inhibition of USP28 is a promising strategy for the treatment of SCC tumours.