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The phase diagram of order parameters for pairing and phase coherence in hole-doped cuprates is discussed. By examining carefully some recent inelastic neutron scattering data obtained on hole-doped cuprates and heavy fermion compound UPd2Al3 in which the superconductivity is mediated by spin fluctuations, we conclude that the coherent gap in hole-doped cuprates has most likely the magnetic origin and scales with Tc, on average, as 2Dc/kBTc = 5.4. We discuss a model of the superconductivity in hole-doped cuprates and the symmetries of two order parameters.

We report bulk superconductivity (SC) in Eu$_{0.2}$Sr$_{0.8}$(Fe$_{0.86}$Co$_{0.14}$)$_{2}$As$_{2}$ single crystals by means of electrical resistivity, magnetic susceptibility, and specific heat measurements with $T$$_{\mathrm{c}}$ $\simeq$ 20 K with an antiferromagnetic (AFM) ordering of Eu$^{2+}$ moments at $T$$_{\mathrm{N}}$ $\simeq$ 2.0 K in zero field. $^{75}$As NMR experiments have been performed in the two external field directions (H$\|ab$) and (H$\|c$). $^{75}$As-NMR spectra are analyzed in terms of first order quadrupolar interaction. Spin-lattice relaxation rates (1/$T_{1}$) follow a $T^{3}$ law in the temperature range 4.2-15 K. There is no signature of Hebel-Slichter coherence peak just below the SC transition indicating a non s-wave or s$_{\pm}$ type of superconductivity. The increase of 1/$T_{1}T$ with lowering the temperature in the range 160-18 K following $\frac{C}{T+\theta}$ law reflecting 2D AFM spin fluctuations.

We report Cu-NMR/NQR and F-NMR studies on the multilayered high-T_c copper oxides Ba_2Ca_{n-1}Cu_nO_{2n}F_2 with n=2,3,4, where n is the number of CuO_2 planes. It is revealed that bi-layered Ba_2CaCu_2O_4F_2 is an underdoped superconductor with hole carriers, which are introduced into CuO_2 planes by an unexpected deviation from the nominal content of apical fluorines. In a previous paper, we proposed a self-doping mechanism as the origin of carrier doping in n=3 and n=4; in the mechanism, electrons are transferred from the inner CuO_2 plane (IP) to the outer one (OP). However, since it has been found that the bi-layered compound is hole doped, we have reexamined the superconducting and magnetic properties in n=3 and n=4 by Cu-NMR/NQR and F-NMR. The extensive NMR studies have confirmed that the apical-fluorine compounds are not self-doped but hole-doped, and that antiferromagnetism (AFM) and superconductivity (SC) coexist in a single CuO_2 plane. In n=4, the AFM ordering occurs at T_N = 80 K, well above T_c=55 K, where the respective AFM moments are M_AFM=0.11 mu_B and 0.18 mu_B at the OP and the IP. In n=3, on the other hand, the underdoped single IP exhibits a spontaneous moment M_AFM=0.12 mu_B at low temperatures and a peak in the nuclear-spin-lattice relaxation rate 1/T_1 of F at T_N=23 K, much lower than T_c = 76 K. We note that the increase in the number of IPs from one to two leads to an increase in T_N due to strengthening the interlayer coupling, although the doping levels for both compounds are almost comparable. The present results strongly suggest that the uniform mixing of AFM and SC is a general property inherent to a single CuO_2 plane in the underdoped regime for hole-doping.

Using intrinsic multiple Andreev reflection effect (IMARE) spectroscopy we studied superconducting properties of nearly optimal oxygen-deficient GdFeAsO$_{0.88}$ polycrystalline samples (bulk critical temperatures $T_C^{bulk} = 49 \div 52$\,K). Temperature dependences for two superconducting gaps $\Delta_{L,S}(T)$ ($T_C^{local} = 48 \div 50$\,K) have been measured in the range from 4.2 to 50\,K. The $\Delta_{L,S}(T)$ dependences were found to deviate from the BCS-like function; this suggests an importance of the $k$-space (internal) proximity effect between the two condensates.

We study the Hubbard model on the honeycomb lattice with nearest-neighbor hopping ($t>0$) and next-nearest-neighbor one ($t'

We measured the THz reflectance properties of a high quality epitaxial thin film of the Fe-based superconductor BaFe$_{1.84}$Co$_{0.16}$As$_2$ with T$_c$=22.5 K. The film was grown by pulsed laser deposition on a DyScO$_3$ substrate with an epitaxial SrTiO$_3$ intermediate layer. The measured $R_S/R_N$ spectrum, i.e. the reflectivity ratio between the superconducting and normal state reflectance, provides clear evidence of a superconducting gap $\Delta_A$ close to 15 cm$^{-1}$. A detailed data analysis shows that a two-band, two-gap model is absolutely necessary to obtain a good description of the measured $R_S/R_N$ spectrum. The low-energy $\Delta_A$ gap results to be well determined ($\Delta_A$=15.5$\pm$0.5 cm$^{-1}$), while the value of the high-energy gap $\Delta_B$ is more uncertain ($\Delta_B$=55$\pm$7 cm$^{-1}$). Our results provide evidence of a nodeless isotropic double-gap scenario, with the presence of two optical gaps corresponding to 2$\Delta/kT_c$ values close to 2 and 7.

The presence of macroscopic phase separation between the superconducting and magnetic phases in \cfcaf is demonstrated by muon spin rotation (muSR) measurements conducted across their phase boundaries (x=0.05-0.15). The magnetic phase tends to retain the high transition temperature (T_m > T_c), while Co-doping induces strong randomness. The volumetric fraction of superconducting phase is nearly proportional to the Co content $x$ with constant superfluid density. These observations suggest the formation of superconducting "islands" (or domains) associated with Co ions in the Fe$_2$As$_2$ layers, indicating a very short coherence length.

Many copper oxide based Thallium compounds have now been discovered. In comparison to the Bi-compounds, the Tl-system shows a richer diversity; viz., High Temperature Superconductors (HTSC) can be obtained with either one or two Tl-0 layers (m = 1,2); also, the triple-digit phases are easier to synthesize. The value of d, oxygen stoichiometry, is critical to achieving superconductivity. The Tl system is robust to oxygen loss; Tl may be lost or incorporated by diffusion. A diffusion coefficient equal to 10 ms at 900 C was determined. Both ortho-rhombic and tetragonal structures are evidenced, but HTSC behavior is indifferent to the crystal symmetry. This system has the highest T(sub c) confirmed. T(sub c) generally increases with p, the number of CuO layers, but tends to saturate at p = 3. Zero resistance as high as 125K has been observed. Most of these HTSC's are hole type, but the Ce-doped specimens may be electronic. The magnetic aspects were studied; because in addition to defining the perfectly diamagnetic ground state as in the conventional superconductors, magnetism of the copper oxides show a surprising variety. This is true of both the normal and the superconducting states. Also, due to the large phonon contribution to the specific heat at the high T(sub c) accurate thermal measurement of important parameters such as the sp. heat jump, electronic density of states, D(Ef) and coherence length are uncertain, and thus, are estimated from the magnetic results. Results from the Tl-system CuO, LaBaCuO, 120 and the Bi-CuO compounds are discussed. The emphasis is on the role of magnetism in the TlCuO HTSC, but technological aspects are also pointed out.

We have studied the quasiparticle excitation spectrum of the superconductor Ba8Si46 by local tunneling spectroscopy. Using high energy resolution achieved in Superconductor-Superconductor junctions we observed tunneling conductance spectra of a non-conventional shape revealing two distinct energy gaps, DeltaL = 1.3meV and DeltaS = 0.9meV. The analysis of tunneling data evidenced that DeltaL is the principal superconducting gap while DeltaS, smaller and more dispersive, is induced into an intrinsically non-superconducting band of the material by the inter-band quasiparticle scattering.

Magnetization and resistance measurements were carried out on carbon-based multiwall nanotubes. Both magnetization and resistance data can be consistently explained in terms of bulk superconductivity above 400 K although we cannot completely rule out other possible explanations to the data.

We derive a model for the highest occupied molecular orbital band of a C60 crystal which includes on-site electron-electron interactions. The form of the interactions are based on the icosahedral symmetry of the C60 molecule together with a perturbative treatment of an isolated C60 molecule. Using this model we do a mean-field calculation in two dimensions on the [100] surface of the crystal. Due to the multi-band nature we find that electron-electron interactions can have a profound effect on the density of states as a function of doping. The doping dependence of the transition temperature can then be qualitatively different from that expected from simple BCS theory based on the density of states from band structure calculations.

We present the results of numerical studies of superconductivity and antiferromagnetism in a strongly correlated electron system. To do this we construct a Hubbard model on a lattice of self-consistently embedded multi-site clusters by means of a dynamical mean-field theory in which intra-cluster dynamics is treated essentially exactly. We show that a class of characteristic features which have been seen in the excitation spectra of high-$T_{c}$ cuprates (e.g., pseudogap and the spin-flip resonance), as well as their interplay with the onset of a pairing correlations, can be captured within a dynamical mean-field theory in which short-wavelength dynamics are rigorously treated. Thus we infer that the observation of the neutron scattering resonance in the superconducting state of the cuprate superconductors does not appear to be directly tied to their quasi-2D character. Although our approach is defined strictly in terms of fermion degrees of freedom, we show that we can readily identify the emergence of effective low energy bosonic degrees of freedom in the presence of a well-defined broken symmetry phase as long as their dynamics are dominated by short-range, short-wavelength fluctuations. Our results reveal that the dynamics of staggered spin degrees of freedom builds up coherence and a resonance-like sharp feature emerges as pairing correlations set in. Under conditions of superconducting broken symmetry our approach thus extends static BCS mean field theory to provide an exact treatment of quantum fluctuations of the BCS order parameter.

A possibility of the realization of the p-wave spin-singlet superconductivity ($p$SS), whose gap function is odd both in momentum and in frequency, is investigated by solving the gap equation with the phenomenological interaction mediated by the antiferromagnetic spin fluctuation. The $p$SS is realized prevailing over the d-wave singlet superconductivity ($d$SS) in the vicinity of antiferromagnetic quantum critical pint (QCP) both on the paramagnetic and on the antiferromagnetic sides. Off the QCP in the paramagnetic phase, however, the $d$SS with line-nodes is realized as \textit{conventional} anisotropic superconductivity. For the present $p$SS state, there is no gap in the quasiparticle spectrum everywhere on the Fermi surface due to its odd frequency. These features can give a qualitative understanding of the anomalous behaviors of NQR relaxation rate on CeCu$_2$Si$_2$ or CeRhIn$_5$ where the antiferromagnetism and superconductivity coexist on a microscopic level.

We investigate the relativistic model of superconductivity in (2+1)-dimension. We employ the massless Gross-Neveu model at finite temperature and density, to study the superconductivity and superconducting instability. Our investigation is related to the superconductivity in (2+1)-dimensional two-band systems like ${\rm MgB_{2}}$ or intercalated graphite.

In this review article we present a survey of unconventional superconductors and their coexistence behaviour with magnetism. We focus on Ce- and U-based heavy fermion superconductors, in addition we discuss the rare earth borocarbide and skutterudite superconductors. We show that heavy fermion behaviour in Ce- and U- compounds should be described differently. While the Kondo-lattice model with localised 4f-electrons is appropriate for Ce-compounds, dual models including localised and itinerant 5f-electrons are necessary for U-compounds. We demonostrate that this leads to consistent results for Fermi surface topology and mass enhancement in various U-compounds. We also show that virtual exchange of localised 5f-electron excitations provides a new mechanism for unconventional superconductivity in addition to the known itinerant spin-fluctuation exchange mechanism. We discuss symmetry properties and coexistence of unconventional superconducting and density wave (hidden) order parameters and methods for their detection, notably the new angle resolved magntetotransport methods. The latter have been very succesful for the investigtion of gap anisotropies in the nonmagnetic borocarbide superconductors. Recent results on the new multiphase Pr-skutterudite superconductor are also reviewed.

In this review article we present a survey of unconventional superconductors and their coexistence behaviour with magnetism. We focus on Ce- and U-based heavy fermion superconductors, in addition we discuss the rare earth borocarbide and skutterudite superconductors. We show that heavy fermion behaviour in Ce- and U- compounds should be described differently. While the Kondo-lattice model with localised 4f-electrons is appropriate for Ce-compounds, dual models including localised and itinerant 5f-electrons are necessary for U-compounds. We demonostrate that this leads to consistent results for Fermi surface topology and mass enhancement in various U-compounds. We also show that virtual exchange of localised 5f-electron excitations provides a new mechanism for unconventional superconductivity in addition to the known itinerant spin-fluctuation exchange mechanism. We discuss symmetry properties and coexistence of unconventional superconducting and density wave (hidden) order parameters and methods for their detection, notably the new angle resolved magntetotransport methods. The latter have been very succesful for the investigtion of gap anisotropies in the nonmagnetic borocarbide superconductors. Recent results on the new multiphase Pr-skutterudite superconductor are also reviewed.

We propose an extended Hubbard model with three molecular orbitals on a hexagonal lattice with $D_{3h}$ symmetry to study recently discovered superconductivity in A$_2$Cr$_3$As$_3$ (A=K,Rb,Cs). Effective pairing interactions from paramagnon fluctuations are derived within the random phase approximation, and are found to be most attractive in spin triplet channels. At small Hubbard $U$ and moderate Hund's coupling, the pairing arises from 3-dimensional (3D) $\gamma$ band and has a spatial symmetry $f_{y(3x^{2}-y^{2})}$, which gives line nodes in the gap function. At large $U$, a fully gapped $p$-wave state, $p_{z}\hat{z}$ dominates at the quasi-1D $\alpha $-band.

We report a high-pressure study of simultaneous low-temperature electrical resistivity and Hall effect measurements on high quality single-crystalline KFe2As2 using designer diamond anvil cell techniques with applied pressures up to 33 GPa. In the low pressure regime, we show that the superconducting transition temperature T_c finds a maximum onset value of 7 K near 2 GPa, in contrast to previous reports that find a minimum T_c and reversal of pressure dependence at this pressure. Upon applying higher pressures, this T_c is diminished until a sudden drastic enhancement occurs coincident with a first-order structural phase transition into a collapsed tetragonal phase. The appearance of a distinct superconducting phase above 13 GPa is also accompanied by a sudden reversal of dominant charge carrier sign, from hole- to electron-like, which agrees with our band calculations predicting the emergence of an electron pocket and diminishment of hole pockets upon Fermi surface reconstruction. Our results suggest the high-temperature superconducting phase in KFe2As2 is substantially enhanced by the presence of nested electron and hole pockets, providing the key ingredient of high-T_c superconductivity in iron pnictide superconductors.

The variations of in- and inter- plane London penetration depths, $\Delta\lambda(T)$, were measured using a tunnel diode resonator in single crystals of the intrinsic pnictide superconductor LiFeAs. This compound appears to be in the clean limit with a residual resistivity of 4 ($T\to0$) to 8 ($T_c$) $\mu \Omega\cdot$cm and $RRR$ of 65 to 35, respectively. The superfluid density, $\rho_s(T)=\lambda^2(0)/\lambda^2(T)$, is well described by the self-consistent two-gap $\gamma-$model. Together with the previous data, our results support the universal evolution of the superconducting gap from nodeless to nodal upon departure from optimal doping. We also conclude that pairbreaking scattering plays an important role in the deviation of the low-temperature behavior of $\lambda(T)$ from exponential in Fe-based compounds.

Superconductivity for Type II superconductors in external magnetic fields of magnitude between the second and third critical fields is known to be restricted to a narrow boundary region. The profile of the superconducting order parameter in the Ginzburg-Landau model is expected to be governed by an effective one-dimensional model. This is known to be the case for external magnetic fields sufficiently close to the third critical field. In this text we prove such a result on a larger interval of validity.

Superconductivity for Type II superconductors in external magnetic fields of magnitude between the second and third critical fields is known to be restricted to a narrow boundary region. The profile of the superconducting order parameter in the Ginzburg-Landau model is expected to be governed by an effective one-dimensional model. This is known to be the case for external magnetic fields sufficiently close to the third critical field. In this text we prove such a result on a larger interval of validity.

High quality single crystals of UPt_3 have been prepared by vertical float-zone refining in ultra-high vacuum. We find that the superconducting transition temperature can be varied systematically by annealing, revealing that the transition temperature intrinsic to UPt_3 is 563 +/- 5 mK. The suppression of the superconducting transition from defects is consistent with a modified Abrikosov-Gor'kov formula that includes anisotropic pairing, Fermi surface anisotropy, and anisotropic scattering by defects.

The phase diagram for quark matter is investigated within a simple Nambu-Jona-Lasinio model without vector correlations. It is found that the phase structure in the temperature-density plane depends sensitively on the parametrization of the model. We present two schemes of parametrization of the model where within the first one a first order phase transition from a phase with broken chiral symmetry to a color superconducting phase for temperatures below the triple point at T_t= 55 MeV occurs whereas for the second one a second order phase transition for temperatures below T_t = 7 MeV is found. In the latter case, there is also a coexistence phase of broken chiral symmetry with color superconductivity, which is a new finding within this class of models. Possible consequences for the phenomenology of the QCD phase transition at high baryon densities are discussed.

This article published in Hadronic Journal is copyrighted and Hadronic Press owns the copyright. In its first part, this article develops a variational formulation for modeling a chemical reaction suitable to represent a combustion process. The results are obtained through standard tools of calculus of variations and optimization theory in function spaces. We assume such a chemical reaction develops in a control volume which allows the entering and leaving of the concerned reacting chemical substances. We highlight the related fluid motion is addressed in an Eulerian context. Finally, in the last sections, we present variational formulations for models in superconductivity, including a magnetic field and respective magnetic potential. We highlight in such last sections it is included an internal variables approach as well. 

Based on the recent developed real-space picture of superconductivity, we study the stability of the superconducting vortex lattices in layered superconductors. It is shown that the effective c-axis lattice constant play a significant role in promoting the superconducting transition temperature in these materials. An unified expression Tc(max)=10c*-28 can be applied to estimate the highest possible Tc(max) for a given layered superconductor with an effective c-axis lattice constant c*. For the newly discovered iron-based superconductors, our results suggest that their Tc cannot exceed 60K, 50K and 40K for the 1111, 21311 and 122 series, respectively. In the case of copper-based oxide superconductors, it seems that the highest Tc can reach about 161K without applying of the external pressure. In our theoretical framework, we could interpret the experimental results of the completely different superconducting transition temperatures obtained in two very similar cuprate superconductors (La{2-x}Ba{x}CuO{4} of 40K and Sr{2-x}Ba{x}CuO{3+\delta} of 98K). In addition, the physical reason why the superconductivity does not occur in noble metals (like gold, silver and copper) is discussed. Finally, we argue that the metallic hydrogen cannot exhibit superconductivity at room temperature, it even cannot be a superconductor at any low temperature.

We have performed electrical resistivity measurements on single crystal BaFe2As2 under high pressure P up to 16 GPa with a cubic anvil apparatus, and up to 3 GPa with a modified Bridgman anvil cell. The samples were obtained from the same batch, which was grown with a self-flux method. A cubic anvil apparatus provides highly hydrostatic pressure, and a modified Bridgman anvil cell, which contains liquid pressure transmitting medium, provides quasi hydrostatic pressure. For highly hydrostatic pressure, the crystal phase and magnetic transition temperature decreases robustly with P and disappears at around 10 GPa. The superconducting phase appears adjacent to magnetic phase in narrow pressure region between 11 and 14 GPa. The tiny difference of hydrostaticity between the cubic anvil apparatus and modified Bridgman anvil cell induces a drastic effect on the phase diagram of BaFe2As2. This result indicates that small uniaxial stress along c-axis strongly suppresses the structural/antiferromagnetic ordering and stabilizes superconductivity at much lower pressure.

Various recent experimental data and especially the large Fe-isotope effect point against unconventional pairings, since the large intra-band impurity scattering is strongly pair-breaking for them. The strength of the inter-band impurity scattering in some single crystals may be strong and probably beyond the Born scattering limit. In that case the proposed s(+-) pairing (hole(h)- and electron(el)-gaps are of opposite signs) is suppressed but possibly not completely destroyed. The data imply that the intra-band pairing in the h- and in the el-band, which are inevitably due to some nonmagnetic el-boson interaction (EBI), must be taken into account. EBI is either due to phonons (EPI) or possibly due to excitons (EEI), or both are simultaneously operative. We discuss their interplay briefly. The large Fe-isotope effect favors the EPI and the s(+) pairing (the h- and el-gaps are in-phase).

The nature of the lattice instability connected to the structural transition and superconductivity of (Sr,Ca)$_3$Ir$_4$Sn$_{13}$ is not yet fully understood. In this work density functional theory (DFT) calculations of the phonon instabilities as a function of chemical and hydrostatic pressure show that the primary lattice instabilities in Sr$_3$Ir$_4$Sn$_{13}$ lie at phonon modes of wavevectors $\mathbf{q}=(0.5,0,0)$ and $\mathbf{q}=(0.5,0.5,0)$. Following these modes by calculating the energy of supercells incorporating the mode distortion results in an energy advantage of -14.1 meV and -9.0 meV per formula unit respectively. However, the application of chemical pressure to form Ca$_3$Ir$_4$Sn$_{13}$ reduces the energetic advantage of these instabilities, which is completely removed by the application of a hydrostatic pressure of 35 kbar to Ca$_3$Ir$_4$Sn$_{13}$. The evolution of these lattice instabilities is consistent with experimental phase diagram. The structural distortion associated with the mode at $\mathbf{q}=(0.5,0.5,0)$ produces a distorted cell with the same space group symmetry as the experimentally refined low temperature structure. Furthermore, calculation of the deformation potential due to these modes quantitatively demonstrates a strong electron-phonon coupling. Therefore, these modes are likely to be implicated in the structural transition and superconductivity of this system.

We measured the electronic-structure of FeSe_xTe_1-x above and below Tc. In the normal state we find multiple bands with remarkably small values for the Fermi energy, E_F. Yet,below Tc we find a superconducting gap, Delta, that is comparable in size to E_F, leading to a ratio Delta/E_F~0.5 that is much larger than found in any previously studied superconductor. We also observe an anomalous dispersion of the coherence peak which is very similar to the dispersion found in cold Fermi-gas experiments and which is consistent with the predictions of the BCS-BEC crossover theory.

Motivated by the proposed topological state in Cu$_x$Bi$_2$Se$_3$, we study the possibility of phonon-mediated odd-parity superconductivity in spin-orbit coupled systems with time-reversal and inversion symmetry. For such systems, we show that, in general, pure electron-phonon coupling can never lead to a triplet state with a higher critical temperature than the leading singlet state. The Coulomb pseudopotential, which is the repulsive part of the electron-electron interaction and is typically small in weakly correlated systems, is therefore critical to stabilizing the triplet state. We introduce a chirality quantum number, which identifies the electron-phonon vertex interactions that are most favorable to the triplet channel as those that conserve chirality. Applying these results to Cu$_x$Bi$_2$Se$_3$, we find that a phonon-mediated odd-parity state may be realized in the presence of weak electronic correlations if the chirality-preserving electron-phonon vertices are much stronger than the chirality-flipping vertices.

We have performed systematic angle-resolved photoemission spectroscopy (ARPES) of iron-chalcogenide superconductor FeTe1-xSex to elucidate the electronic states relevant to the superconductivity. While the Fermi-surface shape is nearly independent of x, we found that the ARPES spectral line shape shows prominent x dependence. A broad ARPES spectrum characterized by a small quasiparticle weight at x = 0, indicative of incoherent electronic states, becomes progressively sharper with increasing x, and a well-defined quasiparticle peak appears around x = 0.45 where bulk superconductivity is realized. The present result suggests the evolution from incoherent to coherent electronic states and its close relationship to the emergence of superconductivity.

We have performed Ce $L_3$-edge x-ray absorption spectroscopy (XAS) measurements on CeO$_{1-x}$F$_x$BiS$_2$, in which the superconductivity of the BiS$_2$ layer and the ferromagnetism of the CeO$_{1-x}$F$_x$ layer are induced by the F-doping, in order to investigate the impact of the F-doping on the local electronic and lattice structures. The Ce $L_3$-edge XAS spectrum of CeOBiS$_2$ exhibits coexistence of $4f^1$ (Ce$^{3+}$) and $4f^0$ (Ce$^{4+}$) state transitions revealing Ce mixed valency in this system. The spectral weight of the $4f^0$ state decreases with the F-doping and completely disappears for $x>0.4$ where the system shows the superconductivity and the ferromagnetism. The results suggest that suppression of Ce-S-Bi coupling channel by the F-doping appears to drive the system from the valence fluctuation regime to the Kondo-like regime, leading to the coexistence of the superconducting BiS$_2$ layer and the ferromagnetic CeO$_{1-x}$F$_x$ layer.

The recent observation of fully-gapped superconductivity in Yb doped CeCoIn5 poses a paradox, for the disappearance of nodes suggests that they are accidental, yet d-wave symmetry with protected nodes is we ll established by experiment. Here, we show that composite pairing provides a natural resolution: in this scenario, Yb doping drives a Lifshitz transition of the nodal Fermi surface, forming a fully-gapped d-wave molecular superfluid of composite pairs. The T4 dependence of the penetration depth associated with the sound mode of this condensate is in accord with observation.

The non-relativistic $G_0 G$ formalism of BCS-BEC crossover at finite temperature is extended to relativistic fermion systems. The uncondensed pairs contribute a pseudogap to the fermion excitations. The theory recovers the BCS mean field approximation at zero temperature and the non-relativistic results in a proper limit. For massive fermions, when the coupling strength increases, there exist two crossovers from the weak coupling BCS superfluid to the non-relativistic BEC state and then to the relativistic BEC state. For color superconductivity at moderate baryon density, the matter is in the BCS-BEC crossover region, and the behavior of the pseudogap is quite similar to that found in high temperature superconductors.

Recent theoretical studies of various strongly-correlated systems in condensed matter physics reveal that the lattice gauge theory(LGT) developed in high-energy physics is quite a useful tool to understand physics of these systems. Knowledges of LGT are to become a necessary item even for condensed matter physicists. In the first part of this paper, we present a concise review of LGT for the reader who wants to understand its basics for the first time. For illustration, we choose the abelian Higgs model, a typical and quite useful LGT, which is the lattice verison of the Ginzburg-Landau model interacting with a U(1) gauge field (vector potential). In the second part, we present an account of the recent progress in the study of ferromagnetic superconductivity (SC) as an example of application of LGT to topics in condensed matter physics, . As the ferromagnetism (FM) and SC are competing orders with each other, large fluctuations are expected to take place and therefore nonperturbative methods are required for theoretical investigation. After we introduce a LGT describing the FMSC, we study its phase diagram and topological excitations (vortices of Cooper pairs) by Monte-Carlo simulations.

Recently, Wang $et$ $al.$ have reported the observation of "unconventional superconductivity" in the Weyl semimetal TaAs [arXiv:1607.00513]. The authors have written "\textit{A conductance plateau and sharp double dips are observed in the point contact spectra, indicating p-wave like unconventional superconductivity. Furthermore, the zero bias conductance peak in low temperature regime is detected, suggesting potentially the existence of Majorana zero modes. The experimentally observed tunneling spectra can be interpreted with a novel mirror-symmetry protected topological superconductor induced in TaAs, which can exhibit zero bias and double finite bias peaks, and double conductance dips in the measurements}." In this comment we show that for a superconducting point contact, the features like a zero-bias conductance peak, a plateau and single or multiple conductance dips might arise due to simple contact-heating related effects. Such features are routinely observed in point contacts involving a wide variety of superconductors when the experiments are not performed in the right regime of mesoscopic transport. We also show that the data presented by Wang $et$ $al.$ in a single transport regime of point contact do not confirm tip induced superconductivity (TISC). Even if it is assumed that Wang $et$ $al.$ achieved TISC on TaAs, all the spectra that they have reported show striking similarities with the type of spectra expected in thermal regime of transport. Such data cannot be used for extracting any spectroscopic information and based on such data any discussion on "p-wave" superconductivity or the emergence of Majorana modes should be considered invalid. This version (v2) also includes a brief discussion on the response of Wang $et$ $al.$ to the first version (v1) of this comment. Correct ballistic regime data on TaAs point contacts can be found in arXiv:1607.05131 (2016).

With the purpose of investigating coexistence between magnetic order and superconductivity, we consider a model in which conduction electrons interact with each other, via an attractive Hubbard on-site coupling $U$, and with local moments on every site, via a Kondo-like coupling, $J$. The model is solved on a simple cubic lattice through a Hartree-Fock approximation, within a `semi-classical' framework which allows spiral magnetic modes to be stabilized. For a fixed electronic density, $n_c$, the small $J$ region of the ground state ($T=0$) phase diagram displays spiral antiferromagnetic (SAFM) states for small $U$. Upon increasing $U$, a state with coexistence between superconductivity (SC) and SAFM sets in; further increase in $U$ turns the spiral mode into a N\'eel antiferromagnet. The large $J$ region is a (singlet) Kondo phase. At finite temperatures, and in the region of coexistence, thermal fluctuations suppress the different ordered phases in succession: the SAFM phase at lower temperatures and SC at higher temperatures; also, reentrant behavior is found to be induced by temperature. Comparison with experimental data for the borocarbides family reveals that, when $J$ is properly renormalized by the total angular momentum, the magnetic critical temperature, $T_N$, scales with $dG$, the de Gennes factor, in line with the measured behavior. By correlating $t/U$ with the ionic radii, we obtain a very satisfactory description of the trend in magnetic and superconducting phases observed in $R$Ni$_2$B$_2$C, with $R$ running across the magnetic rare earths. And, finally, our model reproduces qualitatively some features of $R$(Ni$_{1-x}$Co$_x$)$_2$B$_2$C, with $R=$ Ho and Er, namely the suppression of superconductivity for $x\ll 1$, a slowly varying $T_N(x)$, and a region of coexistence between superconductivity and spiral phases.

Recently, it was experimentally realized that 2D superconducting transition metal dichalcogenides (TMD) such as gated MoS$_2$ and monolayer NbSe$_2$ have in-plane upper critical magnetic fields much higher than the Pauli limit. This is due to the so-called Ising spin-orbit coupling (SOC) of TMD which pins the electron spins along the out-of-plane directions and protects the Cooper pairs from in-plane magnetic fields. However, many TMD materials with extremely large Ising SOC, in the order of a few hundred meV, are not superconducting. In this work, we show that TMD materials can induce strong Ising SOC on ordinary $s$-wave superconductors through proximity effect. By solving the self-consistent gap equation of the TMD/superconductor heterostructure, we found that the $H_{c2}$ of the $s$-wave superconductor can be strongly enhanced. Importantly, when the in-plane field is larger than the Pauli limit field and weaker than $H_{c2}$, the heterostructure becomes a nodal topological superconductor which supports Majorana flat bands.

The mechanism of unconventional superconductivity in iron-based superconductors (IBSs) is one of the most intriguing questions in current materials research. Among non-oxide IBSs, (Ba$_{1-x}$K$_x$)Fe$_2$As$_2$ has been intensively studied because of its high superconducting transition temperature and fascinating evolution of the superconducting gap structure from being fully isotropic at optimal doping (\(x\approx\)0.4) to becoming nodal at $x > $0.8. Although this marked evolution was identified in several independent experiments, there are no details of the gap evolution to date because of the lack of high-quality single crystals covering the entire K-doping range of the superconducting dome. We conducted a systematic study of the London penetration depth, $\lambda (T)$, across the full phase diagram for different concentrations of point-like defects introduced by 2.5 MeV electron irradiation. Fitting the low-temperature variation with the power law, $\Delta \lambda \sim T^{n}$, we find that the exponent $n$ is the highest and $T_c$ suppression rate with disorder is the smallest at optimal doping, and they evolve with doping being away from optimal, which is consistent with increasing gap anisotropy, including an abrupt change around $x\simeq 0.8$, indicating the onset of nodal behavior. Our analysis using a self-consistent $t$-matrix approach suggests the ubiquitous and robust nature of s$_{\pm}$ pairing in IBSs and argues against a previously suggested transition to a $d-$wave state near $x=1$ in this system.

Motivated by the recent realization of the three-dimensional hyperhoneycomb and stripyhoneycomb lattices in lithium iridate (Li$_2$IrO$_3$), we study the possible spin-singlet superconducting states on the whole series of harmonic honeycomb lattices. Beginning with an isolated out-of-plane twist making the honeycomb lattice three-dimensional, we find that the chiral d$\pm$id state, well-known from the honeycomb lattice, is realized in the largest members of the series at low to intermediate doping. Along the twist, four chiral edge states form a two-dimensional dispersive band. Reducing the distance between the twists to form the smaller members of the harmonic honeycomb lattices, the degeneracy between the d-wave states is lifted, which finally destroys the chiral state. By analyzing the hyper- and stripyhoneycomb lattices and generalizing using the $D_{2h}$ point group of all the harmonic honeycomb lattices, we show that the superconducting state often belongs to the trivial irreducible representation. This state has nodal lines at low to intermediate doping, which is possible because the full lattice symmetry allows sign changes between different sets of bonds. We also find time-reversal symmetry broken states, which are either fully gapped or feature nodal points, in certain parts of the phase diagram. Finally, we draw a comparison between the states classified in terms of the $D_{2h}$ symmetries and those observed on the $D_{6h}$ honeycomb lattice.

We address the question of why strongly correlated d-wave superconductors, such as the cuprates, prove to be surprisingly robust against the introduction of non-magnetic impurities. We show that, very generally, both the pair-breaking and the normal state transport scattering rates are significantly suppressed by strong correlations effects arising in the proximity to a Mott insulating state. We also show that the correlation-renormalized scattering amplitude is generically enhanced in the forward direction, an effect which was previously often ascribed to the specific scattering by charged impurities outside the copper-oxide planes.

The $t\!\!-\!\!J$ Hamiltonian is one of the cornerstones in the theoretical study of strongly correlated copper-oxide based materials. Using the density matrix renormalization group method we calculate the phase diagram of the one-dimensional (1D) $t\!\!-\!\!J$ chain in the presence of a periodic hopping modulation, as a prototype of coupled-segment models. While in the uniform 1D $t\!\!-\!\!J$ model near half-filling superconducting (SC) state dominates only at unphysically large values of the exchange coupling constant $J/t>3$, we show that a small hopping and exchange modulation very strongly reduces the critical coupling to be as low as $J/t\sim1/3$ -- well within the physical regime. The phase diagram as a function of the electron filling also exhibits metallic, insulating line phases and regions of phase separation. We suggest that a SC state is easily stabilized if $t\!\!-\!\!J$ segments creating local spin-singlet pairing are coupled to each other -- another example is ladder system.

The thermal conductivity of the layered s-wave superconductor NbSe_2 was measured down to T_c/100 throughout the vortex state. With increasing field, we identify two regimes: one with localized states at fields very near H_c1 and one with highly delocalized quasiparticle excitations at higher fields. The two associated length scales are most naturally explained as multi-band superconductivity, with distinct small and large superconducting gaps on different sheets of the Fermi surface.

We study the temperature dependence of nuclear magnetic relaxation (NMR) rates to detect a sign of topological superconductivity in doped topological insulators, such as $M$($=$Cu,Nb,Sr)$_{x}$Bi$_{2}$Se$_{3}$ and Sn$_{1-x}$In$_{x}$Te. The Hebel-Slichter coherence effect below a critical temperature Tc depends on the superconducting states predicted by a minimal model of doped topological insulators. In a nodal anisotropic topological state similar to the ABM-phase in $^{3}$He, the NMR rate has a conventional $s$-wave like coherence peak below Tc. In contrast, in a fully-gapped isotropic topological superconducting state, this rate below Tc exhibits an anti-peak profile. Moreover, in a two-fold in-plane anisotropic topological superconducting state, there is no coherence effect, which is similar to that in a chiral $p$-wave state. Thus, we reveal that the NMR rates shed light on unconventional superconductivity in doped topological insulators.

A variational ground state of the repulsive Hubbard model on a square lattice is investigated numerically for an intermediate coupling strength (U = 8t) and for moderate sizes (from 6 x 6 to 10 x 10). Our ansatz is clearly superior to other widely used variational wave functions. The results for order parameters and correlation functions provide new insight for the antiferromagnetic state at half filling as well as strong evidence for a superconducting phase away from half filling.

This review discusses the main experiments and theoretical views related to observation of high-temperature superconductivity in intercalated FeSe compounds and single layer films of FeSe on substrates like SrTiO_3. We consider in detail the electronic structure of these systems, both theoretical calculations of this structure at hand and their correspondence with ARPES experiments. It is stressed that electronic spectrum of these systems is qualitatively different from typical picture of the spectrum in well studied FeAs superconductors and the related problems of theoretical description of spectrum formation are also discussed. We also discuss the possible mechanisms of Cooper pairing in monolayers of FeSe and problems appearing here. As single layer films of FeSe on SrTiO_3 can be represented as typical Ginzburg "sandwiches", we analyze the possibility of rising the critical temperature of superconducting transition T_c due to different variants of "excitonic" mechanism of superconductivity. It is shown, that in its classic variant (as proposed for such systems by Allender, Bray and Bardeen) this mechanism is unable to explain the observed values of T_c, but situation is different when we consider instead of "excitons" the optical phonons in SrTiO_3 (with energy of the order of 100 meV). We consider both the simplest model of T_c enhancement due to interaction with such phonons and more specific models with dominant "forward" scattering, which allow to understand the growth of T_c as compared with the case of bulk FeSe and intercalated FeSe systems. We also discuss the problems connected with antiadiabatic nature of superconductivity due to such mechanism.

In the hole-doped cuprate superconductors, the superconducting transition temperature $T_c$ exhibits a dome-like feature against the doping rate. By contrast, recent experiments reveal that $T_c$ in the electron-doped systems monotonically increases as the doping is reduced, at least up to a very small doping rate. Here we show that this asymmetry is reproduced by performing a two-particle self-consistent analysis for the three-band model of the CuO$_2$ plane. This is explained as a combined effect of the intrinsic electron-hole asymmetry in systems comprising Cu3$d$ and O2$p$ orbitals and the band-filling-dependent vertex correction.

We consider superconducting properties of a two-dimensional Dirac material such as graphene under strain that produces a flat band spectrum in the normal state. We show that in the superconducting state, such a model results in a highly increased critical temperature compared to the case without the strain, inhomogenous order parameter with two-peak shaped local density of states and yet a large and almost uniform and isotropic supercurrent. This model could be realized in strained graphene or ultracold atom systems and could be responsible for unusually strong superconductivity observed in some graphite interfaces and certain IV-VI semiconductor heterostructures.