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In this paper we calculate quantum corrections to the effective potential in different models of inflationary cosmology. We show that quantum corrections lead to a modification of the initial potential uplifting its value at the minimum, which can be interpreted as a cosmological constant/dark energy. We concentrate on the models of $\alpha$-attractors and show that one can naturally get small values of the cosmological constant. In the model of quintessence, the same mechanism modifies the value of dark energy at the present time., Comment: 15 pages, 5 figures

The dodecaboride LuB$_{12}$ with cage-glass state and rattling modes has been studied to clarify the nature of the large amplitude vibrations of Lu ions. Discovered anisotropy of charge transport in conjunction with distortions of the conventional \textit{fcc} symmetry of the crystal lattice may be attributed to coherent motion of Lu ions along singular direction in the lattice. Arguments are presented in favor of cooperative dynamic Jahn-Teller effect in the boron sublattice to be the reason of the rattling mode, lattice distortion and formation of the filamentary structure of the conductive channels., Comment: 5 pages, 4 figures, 1SI

We consider the current-phase relation (CPR) in the Josephson junctions with complex insulator-superconductor-ferromagnetic interlayers in the vicinity of 0-$\pi$ transition. We find a strong impact of the second harmonic on CPR of the junctions. It is shown that the critical current can be kept constant in the region of 0-pi transition, while the CPR transforms through multi-valued hysteretic states depending on the relative values of tunnel transparency and magnetic thickness. Moreover, CPR in the transition region has multiple branches with distinct ground states., Comment: Submitted in Phys. Rev. B

The superconducting and normal state characteristics of yttrium hexaboride (YB$_6$) have been investigated for the single crystals with a transition temperatures $T_c$ ranging between 6 K and 7.6 K. The extracted set of microscopic parameters [the coherence length $\xi$(0) $\sim$ 320$\div$340 ${\AA}$, the penetration depth $\lambda$(0) $\sim$ 1100$\div$1600 ${\AA}$ and the mean free path of charge carriers $l$ = 31$\div$58 ${\AA}$, the Ginzburg-Landau-Maki parameters $\kappa$$_{1,2}$(0) $\sim$ 3.3$\div$4.8 and the superconducting gap $\Delta$(0) $\sim$ 10.3$\div$14.8 K] confirms the type II superconductivity in "dirty limit" ($\xi$$\gg$ $l$) with a medium to strong electron-phonon interaction (the electron-phonon interaction constant $\lambda_{e-ph}$ = 0.93$\div$0.96) and $s$-type pairing of charge carriers in this compound [2$\Delta$(0)$/k_BT_c$ $\approx$ 4]. The comparative analysis of charge transport (resistivity, Hall and Seebeck coefficients) and thermodynamic (heat capacity, magnetization) properties in the normal state in YB$_6$ allowed to detect a transition into the cage-glass state at $T^*$ $\sim$ 50 K with a static disorder in the arrangement of the Y$^{3+}$ ions. We argue that the significant $T_c$ variations in the YB$_6$ single crystals are determined by two main factors: (i) the superconductivity enhancement is related with the increase of the number of isolated vacancies, both at yttrium and boron sites, which leads to the development of an instability in the hexaboride lattice; (ii) the $T_c$ depression is additionally stimulated by the spin polarization of conduction electrons emerged and enhanced by the magnetic field in the vicinity of defect complexes in the YB$_6$ matrix., Comment: 17 pages, 15 figures, 5 tables

Based on low temperature resistivity, heat capacity and magnetization investigations we show that the unusually strong suppression of superconductivity in Lu$_x$Zr$_{1-x}$B$_{12}$ BSC-type superconductors in the range $x$$<$0.08 is caused by the emergence of static spin polarization in the vicinity of non-magnetic lutetium impurities. The analysis of received results points to a formation of static magnetic moments with $\mu_{eff}$$\approx$$3\mu_B$ per Lu-ion. The size of these spin polarized nanodomains was estimated to be about 5 ${\AA}$., Comment: 4 pages, 6 figures

The magnetoresistance (MR) $\Delta \rho/\rho$ of cage-glass compound Ho$_x$Lu$_{1-x}$B$_{12}$ with various concentration of magnetic holmium ions ($x$$\leq$0.5) has been studied in detail concurrently with magnetization M(T) and Hall effect investigations on high quality single crystals at temperatures 1.9-120 K and in magnetic field up to 80 kOe. The undertaken analysis of $\Delta\rho/\rho$ allows us to conclude that the large negative magnetoresistance (nMR) observed in vicinity of Neel temperature is caused by scattering of charge carriers on magnetic clusters of Ho$^{3+}$ ions, and that these nanosize regions with AF exchange inside may be considered as short range order AF domains. It was shown that the Yosida relation $-\Delta \rho/\rho$$\sim$$M^2$ provides an adequate description of the nMR effect for the case of Langevin type behavior of magnetization. Moreover, a reduction of Ho-ion effective magnetic moments in the range 3-9$\mu_B$ was found to develop both with temperature lowering and under the increase of holmium content. A phenomenological description of the large positive quadratic contribution $\Delta \rho/\rho$$\sim$$\mu_D^2 H^2$ which dominates in Ho$_x$Lu$_{1-x}$B$_{12}$ in the intermediate temperature range 20-120 K allows to estimate the drift mobility exponential changes $\mu_D$$\sim$$T^{-a}$ with $a$=1.3-1.6 depending on Ho concentration. An even more comprehensive behavior of magnetoresistance has been found in the AF state of Ho$_x$Lu$_{1-x}$B$_{12}$ where an additional linear positive component was observed and attributed to charge carriers scattering on the spin density wave (SDW). High precision measurements of $\Delta\rho/\rho=f(H,T)$ have allowed us also to reconstruct the magnetic H-T phase diagram of Ho$_{0.5}$Lu$_{0.5}$B$_{12}$ and to resolve its magnetic structure as a superposition of 4f (based on localized moments) and 5d (based on SDW) components., Comment: 25 pages, 19 figures

Most of the assertions in the theory of well ordered sets are quite simple. However, one of its central statements, Zermelo's theorem, stands out of this rule, for its well-known proofs are rather complicated. The aim of the current paper is to propose a simple proof of this theorem. Please inform us if you ever encountered such a proof., Comment: 2 pages

We report a detailed study of specific heat, electrical resistivity and optical spectroscopy in the superconducting boride LuB12 (Tc = 0.4 K) and compare it to the higher Tc compound ZrB12 (Tc = 6 K). Both compounds have the same structure based on enclosed metallic Lu or Zr ions in oversized boron cages. The infrared reflectivity and ellipsometry in the visible range allow us to extract the optical conductivity from 6 meV to 4 eV in the normal state from 20 to 280 K. By extracting the superconducting properties, phonon density of states and electron-phonon coupling function from these measurements we discuss the important factors governing Tc and explain the difference between the two compounds. The phonon density of states seems to be insignificantly modified by substitution of Zr with Lu. However, the soft vibrations of the metal ions in boron cages, responsible for the relatively high Tc in ZrB12, have almost no contribution to the electron-phonon coupling in LuB12.

We report optical (6 meV - 4 eV) properties of a boride superconductor ZrB$_{12}$ ($T_c$ = 6 K) in the normal state from 20 to 300 K measured on high-quality single crystals by a combination of reflectivity and ellipsometry. The Drude plasma frequency and interband optical conductivity calculated by self-consistent full-potential LMTO method agree well with experimental data. The Eliashberg function $\alpha_{tr}^2F(\omega)$ extracted from optical spectra features two peaks at about 25 and 80 meV, in agreement with specific heat data. The total coupling constant is $\lambda_{tr}=1.0\pm0.35$. The low energy peak presumably corresponds to the displacement mode of Zr inside $B_{24}$ cages, while the second one involves largely boron atoms. In addition to the usual narrowing of the Drude peak with cooling down, we observe an unexpected removal of about 10 % of the Drude spectral weight which is partially transferred to the region of the lowest-energy interband transition ($\sim$ 1 eV). This effect may be caused by the delocalization of the metal ion from the center of the $B_{24}$ cluster.

Magnetization measurements under hydrostatic pressure up to 10.5 kbar in zirconium dodecaboride ZrB$_{12}$ superconductor ($T_c\simeq6.0$ K at $p=0$) were carried out. A negative pressure effect on $T_c$ with ${\rm d} T_c/{\rm d} p =-0.0225(3)$ K/kbar was observed. The electron-phonon coupling constant $\lambda_{el-ph}$ decreases with increasing pressure with ${\rm d}\ln\lambda_{el-ph}/{\rm d}p\simeq-0.20$%/kbar. The magnetic field penetration depth $\lambda$ was studied in the Meissner state and, therefore, probes only the surface of the sample. The absolute values of $\lambda$ and the superconducting energy gap at ambient pressure and zero temperature were found to be $\lambda(0)=$140(30) nm and $\Delta_0=$1.251(9) meV, respectively. $\Delta_0$ scales linearly with $T_c$ as $2\Delta_0/k_BT_c=4.79(1)$. The studies of the pressure effect on $\lambda$ reveal that $\lambda^{-2}$ increases with pressure with ${\rm d}\ln\lambda^{-2}(0)/{\rm d}p=0.60(23)$ %/kbar. This effect can not be explained within the framework of conventional adiabatic electron-phonon pairing, suggesting that close to the surface, an unconventional {\it non-adiabatic} character of the electron-phonon coupling takes place., Comment: 5 pages, 3 figures