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The simplified scaling of the orbital upper critical field, $H_{c2}(T)$, for the isotropic case is discussed. To facilitate the analysis of the experimental data, we suggest a simple but accurate approximation in the entire temperature range of the scaled upper critical field valid for any transport scattering rate $H_{c2}/H_{c2}(0) \approx (1-t^2)/(1+0.42 \,t^{1.47})$.

This is Center on Reinventing Public Education (CRPE)'s second consensus panel report on the pandemic and delays in students' learning. Based on the evidence available a year ago, CRPE reported the average K-12 student in America mastered less academic content due to the pandemic and associated disruptions to schooling. Negative impacts were generally larger in math than in reading, substantial across all grade levels, larger where students had less access to in-person instruction, and larger for Black, Hispanic, and low-income students. This year's report, based on 23 studies published since last July, offers even stronger evidence of the pandemic's negative consequences for student growth and development. The pauses to in-person schooling and related changes in family and community life over the past year continued to interrupt most children's academic growth, but the consequences are not evenly distributed. Children whose schools were closed the longest had their learning interrupted the most--and most often, that affected low-income children and children of color in big cities. In 2022 as in 2021, the authors aim to present a coherent baseline of what is known, unknown, and needs to be known at this stage of the pandemic--and the beginning of a recovery from the pandemic. These reports are designed to help system leaders, community leaders, policymakers, researchers, philanthropies, the media, and others to define ambitious goals and clear metrics that ensure our education system meets every student's needs over the coming years. [This report is an update to "How Much Have Students Missed Academically Because of the Pandemic? A Review of the Evidence to Date" (ED614130).]

Political scientists have largely overlooked the democratic challenges inherent in the governance of U.S. public education--despite profound implications for educational delivery and, ultimately, social mobility and economic growth. In this study, we consider whether the interests of adult voters who elect school boards in each community are likely to be aligned with the educational needs of local students. Specifically, we compare voters and students in four states on several policy-relevant dimensions. Using official voter turnout records and rich microtargeting data, we document considerable demographic differences between voters who participate in school board elections and the students attending the schools that boards oversee. These gaps are most pronounced in majority nonwhite jurisdictions and school districts with the largest racial achievement gaps. Our novel analysis provides important context for understanding the political pressures facing school boards and their likely role in perpetuating educational and, ultimately, societal inequality.

The magnetic response of CaK(Fe$_{0.949}$Ni$_{0.051}$)$_4$As$_4$ was investigated by means of the muon-spin rotation/relaxation. The long-range commensurate magnetic order sets in below the N\'{e}el temperature $T_{\rm N}= 50.0(5)$~K. The density-functional theory calculations have identified three possible muon stopping sites. The experimental data were found to be consistent with only one type of magnetic structure, namely, the long-range magnetic spin-vortex-crystal order with the hedgehog motif within the $ab-$plane and the antiferromagnetic stacking along the $c-$direction. The value of the ordered magnetic moment at $T\approx3$ K was estimated to be $m_{\rm Fe}=0.38(11)$ $\mu_{\rm B}$ ($\mu_{\rm B}$ is the Bohr magneton). A microscopic coexistence of magnetic and superconducting phases accompanied by a reduction of the magnetic order parameter below the superconducting transition temperature $T_{\rm c}\simeq 9$ K is observed. Comparison with 11, 122, and 1144 families of Fe-based pnictides points to existence of correlation between the reduction of the magnetic order parameter at $T\rightarrow 0$ and the ratio of the transition temperatures $T_{\rm c}/T_{\rm N}$. Such correlations were found to be described by Machida's model for coexistence of itinerant spin-density wave magnetism and superconductivity [Machida, J. Phys. Soc. Jpn. 50, 2195 (1981) and Bud'ko et al., Phys. Rev. B 98, 144520 (2018)]., Comment: 15 pages, 14 figures

Temperature dependent $^{57}$Fe M\"ossbauer spectroscopy and specific heat measurements for CaK(Fe$_{1-x}$Ni$_x$)$_4$As$_4$ with $x$ = 0, 0.017, 0.033, and 0.049 are presented. No magnetic hyperfine field (e.g. no static magnetic order) down to 5.5 K was detected for $x$ = 0 and 0.017 in agreement with the absence of any additional feature below superconducting transition temperature, $T_c$, in the specific heat data. The evolution of magnetic hyperfine field with temperature was studied for $x$ = 0.033 and 0.049. The long-range magnetic order in these two compounds coexists with superconductivity. The magnetic hyperfine field, $B_{hf}$, (ordered magnetic moment) below $T_c$ in CaK(Fe$_{0.967}$Ni$_{0.033}$)$_4$As$_4$ is continuously suppressed with the developing superconducting order parameter. The $B_{hf}(T)$ data for CaK(Fe$_{0.967}$Ni$_{0.033}$)$_4$As$_4$, and CaK(Fe$_{0.951}$Ni$_{0.049}$)$_4$As$_4$ can be described reasonably well by Machida's model for coexistence of itinerant spin density wave magnetism and superconductivity [K. Machida, J. Phys. Soc. Jpn. {\bf 50}, 2195 (1981)]. We demonstrate directly that superconductivity suppresses the spin density wave order parameter if the conditions are right, in agreement with the theoretical analysis.

We revisit the problem of the surface superconductivity nucleation focusing on the detailed study of the critical field $H_{c3}$ as a function of temperature and disorder. Using the semiclassical Eilenberger formalism we find that away from the Ginzburg-Landau region the ratio between the nucleation critical field $H_{c3}$ and the upper critical field $H_{c2}$ deviates strongly from the Saint-James-de Gennes limit. In particular, the $H_{c3}/H_{c2}$ is found to be a nonmonotonic function of temperature, which reaches the maximum for a set of parameters corresponding to a crossover region from ballistic to diffusive scattering, when the mean free path in a bulk of a superconductor is of the same order as zero-temperature superconducting coherence length. We also analyze the robustness of the nucleated phases with respect to diffusive scattering off the sample boundary by solving exactly corresponding eigenvalue problem of an integral equation for the critical field. The implications of these results for the transport in superconductors of various geometries near $H_{c3}$ are briefly discussed. In particular, we present results for the mechanism of magnetoconductivity oscillations due to surface superconductivity effects., Comment: 14 pages, 6 figures

In superconductors with strong coupling between superconductivity and elasticity manifested in a strong dependence of transition temperature on pressure, there is an additional contribution to inter-vortex interactions due to the strain field generated by vortices. When vortex lines are along the $c$ axis of a tetragonal crystal, a square vortex lattice (VL) is favored at low vortex densities, because the vortex-induced strains contribution to the inter-vortex interactions is long range. At intermediate magnetic fields, the triangular lattice is stabilized. The triangular lattice evolves to the square lattice upon increasing magnetic field, and eventually the system locks to the square structure. We argue, however, that as $B$ approaches $H_{c2}$ the elastic inter-vortex interactions disappear faster than the standard London interactions, so that VL should return to the triangular structure. Our results are compared to VLs observed in the heavy fermion superconductor $\mathrm{CeCoIn_5}$., Comment: 6 pages and 3 figures

We consider the phase diagram of a ferromagnetic system driven to a quantum phase transition with a tuning parameter $p$. Before being suppressed, the transition becomes of the first order at a tricritical point, from which wings emerge under application of the magnetic field $H$ in the $T$-$p$-$H$ phase diagram. We show that the edge of the wings merge with tangent slopes at the tricritical point.

The pressure dependence of the upper critical field, $H_\textrm{c2,c}$, of single crystalline FeSe was studied using measurements of the inter-plane resistivity, $\rho_{\textrm{c}}$ in magnetic fields parallel to tetragonal $c$-axis. $H_\textrm{c2,c}(T)$ curves obtained under hydrostatic pressures up to $1.56$ GPa, the range over which the superconducting transition temperature, $T_\textrm{c}$, of FeSe exhibits a non-monotonic dependence with local maximum at $p_1\approx$ 0.8 GPa and local minimum at $p_2\approx$ 1.2 GPa. The slope of the upper critical field at $T_\textrm{c}$, $\left(\textrm{d}H_\text{c2,c}/\textrm{d}T\right)_{T_\textrm{c}}$, also exhibits a non-monotonic pressure dependence with distinct changes at $p_1$ and $p_2$. For $pp_2$ the slope is in good quantitative agreement with a single band, orbital Helfand-Werthamer theory with Fermi velocities determined from Shubnikov-de Haas measurements. This finding indicates that Fermi surface changes are responsible for the local minimum of $T_\textrm{c}(p)$ at $p_2\approx$ 1.2 GPa., Comment: 5 pages, 4 figures

We report measurements of electrical resistivity under pressure to 5.8 GPa, magnetization to 6.7 GPa,and ac susceptibility to 7.1 GPa in KFe2As2. The previously reported change of slope in the pressure dependence of the superconducting transition temperature Tc(p) at a pressure p0=1.8 GPa is confirmed, and Tc(p) is found to be nearly constant above p0 up to 7.1 GPa. The T-p phase diagram is very sensitive to the pressure conditions as a consequence of the anisotropic uniaxial pressure dependence of Tc. Across p0, a change in the behavior of the upper critical field is revealed through a scaling analysis of the slope of Hc2 with the effective mass as determined from the A coefficient of the T2 term of the temperature-dependent resistivity. We show that this scaling provides a quantitative test for the changes of the superconducting gap structure and suggests the development of a kz modulation of the superconducting gap above p0 as a most likely explanation.

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Magnetic flux dynamics in type-II superconductors is studied within the model of a viscous nonlinear diffusion of vortices for various sample geometries. We find that time dependence of magnetic moment relaxation after the field is switched off can be accurately approximated by $m(t)\propto 1-\sqrt{t/\tilde\tau}$ in the narrow initial time interval and by $m(t)\propto (1+t/\tau)^{-1}$ at later times before the flux creep sets in. The characteristic times $\tilde\tau$ and $\tau$ are proportional to the viscous drag coefficient $\eta$. Quantitative agreement with available experimental data is obtained for both conventional and high-temperature superconductors with $\eta$ exceeding by many orders of magnitude the Bardeen-Stephen coefficient for free vortices. Huge enhancement of the drag, as well as its exponential temperature dependence, indicates a strong influence of pinning centers on the flux diffusion. Notwithstanding complexity of the vortex motion in the presence of pinning and thermal agitation, we argue that the initial relaxation of magnetization can still be considered as a viscous flux flow with an effective drag coefficient., Comment: 5 pages, 7 figures

We argue that photon counts in a superconducting nanowire single-photon detector (SNSPD) are caused by the transition from a current-biased metastable superconducting state to the normal state. Such a transition is triggered by vortices crossing the thin film superconducting strip from one edge to another due to the Lorentz force. Detector counts in SNSPDs may be caused by three processes: (a) a single incident photon with energy sufficient to break enough Cooper pairs to create a normal-state belt across the entire width of the strip (direct photon count), (b) thermally induced single-vortex crossing in the absence of photons (dark count), which at high bias currents releases the energy sufficient to trigger the transition to the normal state in a belt across the whole width of the strip, and (c) a single incident photon with insufficient energy to create a normal-state belt but initiating a subsequent single-vortex crossing, which provides the rest of the energy needed to create the normal-state belt (vortex-assisted single photon count). We derive the current dependence of the rate of vortex-assisted photon counts. The resulting photon count rate has a plateau at high currents close to the critical current and drops as a power-law with high exponent at lower currents. While the magnetic field perpendicular to the film plane does not affect the formation of hot spots by photons, it causes the rate of vortex crossings (with or without photons) to increase. We show that by applying a magnetic field one may characterize the energy barrier for vortex crossings and identify the origin of dark counts and vortex-assisted photon counts., Comment: 9 pages, 8 figures [v3: added extensive discussion of boundary condition of Fokker-Planck equation and magnitude of vortex crossing rate]

Measurements of London penetration depth are a sensitive tool to study multi-band superconductivity and it has provided several important insights to the behavior of Fe-based superconductors. We first briefly review the "experimentalist - friendly" self-consistent Eilenberger model that relates the measurable superfluid density and structure of superconducting gaps. Then we focus on the \BaFe -derived materials, for which the results are consistent with 1) two distinct superconducting gaps; 2) development of strong in-plane gap anisotropy with the departure from the optimal doping; 3) appearance of gap nodes along the $c-$direction in a highly overdoped regime; 4) significant pair-breaking, presumably due to charge doping; 5) fully gapped (exponential) intrinsic behavior at the optimal doping if scattering is removed (probed in the "self-doped" stoichiometric LiFeAs); 6) competition between magnetically ordered state and superconductivity, which do coexist in the underdoped compounds. Overall, it appears that while there are common trends in the behavior of Fe-based superconductors, the gap structure is non-universal and is sensitive to the doping level. It is plausible that the rich variety of possible gap structures within the general $s_{\pm}$ framework is responsible for the observed behavior., Comment: review article, 39 pages

We apply the time dependent Ginzburg-Landau equations (TDGL) to study small ac currents of frequency $\omega$ in superconducting channels narrow on the scale of London penetration depth. We show that TDGL have $t$-dependent and spatially uniform solutions that describe the order parameter with an oscillating part of the double frequency coexisting with an ac electric field. We evaluate the Ohmic losses (related neither to the flux flow nor to the phase slips) and show that the resistivity reduction on cooling through the critical temperature $T_c$ should behave as $(T_c-T)^2/\omega^2$. If the channel is cut out of an anisotropic material in a direction other than the principal axes, the transverse phase difference and the Josephson voltage between the channel sides are generated., Comment: 5 pages, 1 figures, Accepted for publication in Phys. Rev. B

Pancake vortices in stacks of thin superconducting films or layers are considered. It is stressed that in the absence of Josephson coupling topological restrictions upon possible configurations of vortices are removed and various examples of structures forbidden in bulk superconductors are given. In particular, it is shown that vortices may skip surface layers in samples of less than a certain size R_c which might be macroscopic. The Josephson coupling suppresses R_c estimates.

The sin-Gordon equation for Josephson junctions with arbitrary misaligned anisotropic banks is derived. As an application, the problem of Josephson vortices at twin planes of a YBCO-like material is considered. It is shown that for an arbitrary orientation of these vortices relative to the crystal axes of the banks, the junctions should experience a mechanical torque which is evaluated. This torque and its angular dependence may, in principle, be measured in small fields, since the flux penetration into twinned crystals begins with nucleation of Josephson vortices at twin planes., Comment: 6 pages