We present a manifestly gauge-invariant description of Chern numbers associated with the Berry connection defined on a discretized Brillouin zone. It provides an efficient method of computing (spin) Hall conductances without specifying gauge-fixing conditions. We demonstrate that it correctly reproduces quantized Hall conductances even on a coarsely discretized Brillouin zone. A gauge-dependent integer-valued field, which plays a key role in the formulation, is evaluated in several gauges. An extension to the non-Abelian Berry connection is also given.

with among the highest numbers of patients with COVID-19 in the country. We selected HCWs, including physicians, nurses, laboratory medical technologists, radiological technologists, and pharmacists, who worked in departments in which they had contact with patients with COVID-19, including emergency departments, general internal medicine departments, respiratory medicine departments, infectious disease departments, general wards, and intensive care units. An explanation of the sample size calculation appears in the eAppendix in the Supplement. This study was approved by the institutional review board of St Luke's International Hospital in Tokyo, Japan. A letter of informed consent was distributed to the participants via email, and completion of the questionnaire implied their consent. This study followed the Strengthening the Reporting of Observational Studies in Epidemiology (STROBE) reporting guideline.The web-based survey was generated using SurveyMonkey, a cloud-based survey development application. The survey solicited responses regarding participants' demographic characteristics (age and gender), professional history (job category and years of experience), working environment characteristics (mean weekly working hours, days off per month, and hours of sleep per day), types of anxiety perceived, changes compared with before the pandemic, and types of support needed.The primary outcome was the prevalence of burnout among frontline HCWs in departments with direct contact with patients with COVID-19, using the validated Japanese version of the Maslach Burnout Inventory-General Survey, 3 which is currently considered the criterion standard for measuring burnout. This 16-item questionnaire contains 3 subscales that evaluate what are considered the 3 major domains of burnout, ie, emotional exhaustion, cynicism (ie, depersonalization), and professional efficacy (ie, personal accomplishment). High levels of exhaustion (>3.5) plus either high cynicism (>3.5) or low professional efficacy (<2.5) were selected as the primary criteria for burnout.We first compared the baseline characteristics of those who did and did not have burnout by using the χ 2 difference test for categorical variables and the Mann-Whitney U test for continuous variables. Given the limited information available on confounding by potential risk factors for burnout, we used logistic regression analyses to evaluate whether any factors, such as types of anxiety perceived, changes compared with the prepandemic period, and types of support needed, were significant by integrating data on the participants' backgrounds into the model. All analyses

Background and Purpose-About one half of those who develop adult-onset moyamoya disease experience intracranial hemorrhage. Despite the extremely high frequency of rebleeding attacks and poor prognosis, measures to prevent rebleeding have not been established. The purpose of this study is to determine whether extracranial-intracranial bypass can reduce incidence of rebleeding and improve patient prognosis. Methods-This study was a multicentered, prospective, randomized, controlled trial conducted by 22 institutes in Japan.Adult patients with moyamoya disease who had experienced intracranial hemorrhage within the preceding year were given either conservative care or bilateral extracranial-intracranial direct bypass and were observed for 5 years. Primary and secondary end points were defined as all adverse events and rebleeding attacks, respectively. Results-Eighty patients were enrolled (surgical, 42; nonsurgical, 38). Adverse events causing significant morbidity were observed in 6 patients in the surgical group (14.3%) and 13 patients in the nonsurgical group (34.2%). Kaplan-Meier survival analysis revealed significant differences between the 2 groups (3.2%/y versus 8.2%/y; P=0.048). The hazard ratio of the surgical group calculated by Cox regression analysis was 0.391 (95% confidence interval, 0.148-1.029).Rebleeding attacks were observed in 5 patients in the surgical group (11.9%) and 12 in the nonsurgical group (31.6%), significantly different in the Kaplan-Meier survival analysis (2.7%/y versus 7.6%/y; P=0.042). The hazard ratio of the surgical group was 0.355 (95% confidence interval, 0.125-1.009). Conclusions-Although statistically marginal, Kaplan-Meier analysis revealed the significant difference between surgical and nonsurgical group, suggesting the preventive effect of direct bypass against rebleeding. Clinical Trial Registration

Inspired by a recent discovery of a peculiar integer quantum Hall effect ͑QHE͒ in graphene, we study QHE on a honeycomb lattice in terms of the topological quantum number, with two interests. First, how the zero-mass Dirac QHE around the center of the tight-binding band crosses over to the ordinary finite-mass fermion QHE around the band edges. Second, how the bulk QHE is related with the edge QHE for the entire spectrum including Dirac and ordinary behaviors. We find the following. ͑i͒ The zero-mass Dirac QHE ͓with xy = ϯ ͑2N +1͒e 2 / h , N: integer͔ persists, surprisingly, up to the van Hove singularities, at which the ordinary fermion behavior abruptly takes over. Here a technique developed in the lattice gauge theory enabled us to calculate the behavior of the topological number over the entire spectrum. This result indicates a robustness of the topological quantum number, and should be observable if the chemical potential can be varied over a wide range in graphene. ͑ii͒ To see if the honeycomb lattice is singular in producing the anomalous QHE, we have systematically surveyed over square ↔ honeycomb ↔-flux lattices, which is scanned by introducing a diagonal transfer tЈ. We find that the massless Dirac QHE ͓ϰ͑2N +1͔͒ forms a critical line, that is, the presence of Dirac cones in the Brillouin zone is preserved by the inclusion of tЈ and the Dirac region sits side by side with ordinary one persists all through the transformation. ͑iii͒ We have compared the bulk QHE number obtained by an adiabatic continuity of the Chern number across the square ↔ honeycomb ↔-flux transformation and numerically obtained edge QHE number calculated from the whole energy spectra for sample with edges, which shows that the bulk QHE number coincides, as in ordinary lattices, with the edge QHE number throughout the lattice transformation.