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We present the results of a study of the vortex lattice in the heavy fermion superconductor ${\mathrm{CeCu}}_{2}{\mathrm{Si}}_{2}$, using small-angle neutron scattering (SANS). In this material at temperatures well below ${T}_{\mathrm{c}}\ensuremath{\sim}0.6$ K, the value of the upper critical field ${B}_{\mathrm{c}2}\ensuremath{\sim}2.2$ T is strongly limited by the Pauli paramagnetism of the heavy fermions. In this temperature region, our SANS data show an increase in the magnetization of the flux line cores with field, followed by a rapid fall near ${B}_{\mathrm{c}2}$. This behavior is the effect of Pauli paramagnetism and we present a theory-based model, which can be used to describe this effect in a range of materials. The pairing in ${\mathrm{CeCu}}_{2}{\mathrm{Si}}_{2}$ appears to arise from the effect of magnetic fluctuations, but the evidence for a $d$-wave order parameter is rather weak. We find that the vortex lattice structure in ${\mathrm{CeCu}}_{2}{\mathrm{Si}}_{2}$ is close to regular hexagonal. There are no phase transitions to square or rhombic structures; such transitions are expected for $d$-wave superconductors and observed in ${\mathrm{CeCoIn}}_{5}$; however, the temperature dependence of the SANS intensity indicates that both large and small gap values are present, most likely due to multiband $s$-wave superconductivity, rather than a nodal gap structure.

The theoretical description and experimental study of a reversible vibrational vortex motion induced in a type-II superconductor by a weak low-frequency ac magnetic field are discussed. The case where the external magnetic field and vortex lines are parallel to the surface of the superconductor is considered. The equilibrium positions of vortices near a surface, the amplitude of their vibrations, and the contribution of this process to the dynamic permeability ${\ensuremath{\mu}}_{v}$ are calculated. In these calculations, the interactions of vortices with each other, with the surface, and with pinning centers are considered within the discrete vortex lattice model. The calculations in the London approximation are compared with the results obtained within a more accurate model, including the spatial variation of the order parameter in the vortex core by means of a variational function. It is shown that the structural parameters of the vortex lattice near the surface depend on the accuracy of the description of the intervortex interactions, but the contribution ${\ensuremath{\mu}}_{v}$ appears to be model independent. It is important that this conclusion is valid in the case of the exponential approximation for the interaction between vortex rows. In this approximation, the problem under consideration is analytically examined, and formulas describing the dependences of the parameters of the vortex lattice and the contribution ${\ensuremath{\mu}}_{v}$ on the external dc magnetic field and the characteristics of a superconductor are derived. The theoretical results are compared with experimental data for a $\mathrm{Y}{\mathrm{Ba}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{y}$ high-${T}_{c}$ superconducting single-crystal plate. The magnetic field is parallel to the plate, i.e., to the crystallographic $ab$ plane. A strong anisotropy of the properties and shape of the crystal is considered when analyzing the experimental data. A procedure for the determination of the experimental dependence ${\ensuremath{\mu}}_{v}(H)$ is described. It is shown that the reversible vibrations of vortices occur near the planes of the single-crystal plate in the direction of the crystallographic $c$ axis, whereas vortices enter its lateral ends and leave them (along the structural layers) even at a small magnetic field amplitude of about 1 Oe. It is found that the developed theoretical description reproduces the experimental data ${\ensuremath{\mu}}_{v}(H)$ in the entire studied range of temperatures and magnetic fields under the assumptions that pinning is isotropic and the length of interaction of vortices with pinning centers is close to the coherence length.

We use large-scale Monte-Carlo simulations to study thermal fluctuations in chiral $p$-wave superconductors in an applied magnetic field in three dimensions. We consider the thermal stability of previously predicted unusual double-quanta flux-line lattice ground states in such superconductors. In previous works it was shown that, neglecting thermal fluctuations, a chiral $p$-wave superconductor forms an hexagonal lattice of doubly-quantized vortices, except extremely close to the vicinity of $H_{c2}$ where double-quanta vortices split apart. We find dissociation of double-quanta vortices driven by thermal fluctuations. However, our calculations also show that the previous predictions of hexagonal doubly-quantized vortices, where thermal fluctuations were ignored, are very robust in the considered model., Comment: 21 pages, 12 figures, submitted to Physical Review B

Chiral topological superconductors are expected to appear as intermediate states when a quantum anomalous Hall system is proximity coupled to an s-wave superconductor and the magnetization direction is reversed. In this paper we address the edge state properties of ordinary quantum Hall systems proximity coupled to s-wave superconductors, accounting explicitly for Landau quantization. We find that the appearance of topological superconducting phases with an odd number of Majorana edge modes is dependent on the structure of the system's vortex lattice. More precisely, vortex lattices containing odd number of superconducting flux quanta per unit cell, always support an even number of chiral edge channels and are therefore adiabatically connected to normal quantum Hall insulators. We discuss strategies to engineer chiral topological superconductivity in proximity-coupled quantum Hall systems by manipulating vortex lattice structure.

Thermal fluctuations always melt the vortex lattice in type-II superconductors at temperatures below the mean-field transition. The extent of the intermediate vortex-liquid state varies widely among different materials. Exploring the vortex-lattice melting in the magnetic iron arsenide RbEuFe${}_{4}$As${}_{4}$ via calorimetry and transport, the authors demonstrate here that the vortex liquid occupies a significant portion of the phase diagram and that the location of the melting line agrees with theoretical predictions without fitting parameters. Comparison with other materials reveals the universality of melting behavior.

We observe a hysteretic reorientation of the magnetic vortex lattice in the noncentrosymmetric superconductor ${\mathrm{Ru}}_{7}{\mathrm{B}}_{3}$, with the change in orientation driven by altering the magnetic field below ${T}_{\mathrm{c}}$. Normally a vortex lattice chooses either a single or degenerate set of orientations with respect to a crystal lattice at any given field or temperature, a behavior well described by prevailing phenomenological and microscopic theories. Here, in the absence of any typical VL structural transition, we observe a continuous rotation of the vortex lattice which exhibits a pronounced hysteresis and is driven by a change in magnetic field. We propose that this rotation is related to the spontaneous magnetic fields present in the superconducting phase, which are evidenced by the observation of time-reversal symmetry breaking, and the physics of broken inversion symmetry. Finally, we develop a model from the Ginzburg-Landau approach which shows that the coupling of these to the vortex lattice orientation can result in the rotation we observe.

We utilize a combination of vector magnetic field and scanning tunneling microscopy to elucidate the three-dimensional field based electronic phase diagram of a correlated iron-based superconductor, LiFeAs. We observe, under a zero-field-cooled method, an ordered hexagonal vortex lattice ground state in contrast to the disordered lattice observed under a field-cooled method. It transforms to a fourfold-symmetric state by increasing the c-axis field and distorts elliptically upon tilting the field in-plane. The vortex lattice transformations correlate with the field-dependent superconducting gap that characterizes the Cooper pairing strength. The anisotropy of the vortex lattice agrees with the field-enhanced Bogoliubov quasiparticle scattering channel that is determined by the pairing symmetry in respect to its Fermi surface structure. Our systematic tuning of the vortex lattice symmetry and study of its correlation with Cooper pairing demonstrates the many-body interplay between the superconducting order parameter and emergent vortex matter.

The magnetic penetration depth anisotropy $\gamma_\lambda=\lambda_{c}/\lambda_{ab}$ ($\lambda_{ab}$ and $\lambda_{c}$ are the in-plane and the out-of-plane components of the magnetic penetration depth) in a CaKFe$_4$As$_4$ single crystal sample (the critical temperature $T_{\rm c}\simeq 35$ K) was studied by means of muon-spin rotation ($\mu$SR). $\gamma_\lambda$ is almost temperature independent for $T\lesssim 20$ K ($\gamma_\lambda\simeq 1.9$) and it reaches $\simeq 3.0$ by approaching $T_{\rm c}$. The change of $\gamma_\lambda$ induces the corresponding rearrangement of the flux line lattice (FLL), which is clearly detected via enhanced distortions of the FLL $\mu$SR response. Comparison of $\gamma_\lambda$ with the anisotropy of the upper critical field ($\gamma_{H_{\rm c2}}$) studied in Phys. Rev B {\bf 94}, 064501 (2016), reveals that $\gamma_\lambda$ is systematically higher than $\gamma_{H_{\rm c2}}$ at low-temperatures and approaches $\gamma_{H_{\rm c2}}$ for $T \rightarrow T_{\rm c}$. The anisotropic properties of $\lambda$ are explained by the multi-gap nature of superconductivity in CaKFe$_4$As$_4$ and are caused by anisotropic contributions of various bands to the in-plane and the out-of-plane components of the superfluid density., Comment: 6 pages, 4 figures

We investigate the vortex lattice and vortex bound states in ${\mathrm{CsFe}}_{2}{\mathrm{As}}_{2}$ single crystals by scanning tunneling microscopy/spectroscopy (STM/STS) under various magnetic fields. A possible structural transition or crossover of vortex lattice is observed with the increase of magnetic field, i.e., the vortex lattice changes from a distorted hexagonal lattice to a distorted tetragonal one at the magnetic field near 0.5 T. It is found that a mixture of stripelike hexagonal and square vortex lattices emerges in the crossover region. The vortex bound state is also observed in the vortex center. The tunneling spectra crossing a vortex show that the bound-state peak position holds near zero bias with the STM tip moving away from the vortex core center. The Fermi energy estimated from the vortex bound state energy is very small. Our investigations provide experimental information to both the vortex lattice and the vortex bound states in this iron-based superconductor.

The topologically nontrivial magnetic phase with a ${\mathbb{Z}}_{2}$-vortex (${\mathbb{Z}}_{2}\mathrm{V}$) lattice is investigated by simulation in the Kitaev-Heisenberg honeycomb model expanded by considering the second- and third-nearest-neighbor Heisenberg interactions (${J}_{H2}$ and ${J}_{H3}$). On the parameter region of the ${\mathbb{Z}}_{2}\mathrm{V}$ phase, a gradual modulation of vortex density is observed, together with a transition from single-${\mathbb{Z}}_{2}\mathrm{V}$ to triple-${\mathbb{Z}}_{2}\mathrm{V}$ state driven by the variation of frustration. Additionally, ${\mathbb{Z}}_{2}$ vortices are arranged in different manners on the whole honeycomb structure for these two types of ${\mathbb{Z}}_{2}\mathrm{V}$ states. Moreover, topologically equivalent states are revealed to exist in single-${\mathbb{Z}}_{2}\mathrm{V}$ dominant and triple-${\mathbb{Z}}_{2}\mathrm{V}$ dominant styles on different parameter points, which can be controlled to switch between each other without energy consumption. It is worth noting that ${J}_{H3}$ plays a key role in expanding the ${\mathbb{Z}}_{2}\mathrm{V}$ phase, and also in stabilizing the single-${\mathbb{Z}}_{2}\mathrm{V}$ state.

Despite numerous studies the exact nature of the order parameter in superconducting Sr2RuO4 remains unresolved. We have extended previous small-angle neutron scattering studies of the vortex lattice in this material to a wider field range, higher temperatures, and with the field applied close to both the and basal plane directions. Measurements at high field were made possible by the use of both spin polarization and analysis to improve the signal-to-noise ratio. Rotating the field towards the basal plane causes a distortion of the square vortex lattice observed for H // , and also a symmetry change to a distorted triangular symmetry for fields close to . The vortex lattice distortion allows us to determine the intrinsic superconducting anisotropy between the c-axis and the Ru-O basal plane, yielding a value of ~60 at low temperature and low to intermediate fields. This greatly exceeds the upper critical field anisotropy of ~20 at low temperature, reminiscent of Pauli limiting. Indirect evidence for Pauli paramagnetic effects on the unpaired quasiparticles in the vortex cores are observed, but a direct detection lies below the measurement sensitivity. The superconducting anisotropy is found to be independent of temperature but increases for fields > 1 T, indicating multiband superconductvity in Sr2RuO4. Finally, the temperature dependence of the scattered intensity provides further support for gap nodes or deep minima in the superconducting gap., 14 pages, 8 figures

We suggest a simple model describing the temperature-driven crossover between Abrikosov vortex lattice and superconducting droplet state in dirty superconductors with fluctuations either in the impurity concentration or in the crystal axes orientation. Our analysis is based on the Usadel-type theory with a spatially modulated diffusion coefficient. This modulation appears to break a regular vortex lattice into a random set of weakly coupled superconducting droplets emerging below the fluctuating upper critical field ${H}_{c2}(T)$. These droplets cause the resistivity drop at the onset of superconducting transition, being responsible for the increasing broadening of the resistive transition in the increasing magnetic field. The above crossover reveals itself in a positive curvature of the ${H}_{c2}(T)$ curves, allowing us, thus, to explain the phase diagrams observed in a wide class of disordered superconducting materials.

Despite numerous studies the exact nature of the order parameter in superconducting ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$ remains unresolved. We have extended previous small-angle neutron scattering studies of the vortex lattice in this material to a wider field range, higher temperatures, and with the field applied close to both the $\ensuremath{\langle}100\ensuremath{\rangle}$ and $\ensuremath{\langle}110\ensuremath{\rangle}$ basal plane directions. Measurements at high field were made possible by the use of both spin polarization and analysis to improve the signal-to-noise ratio. Rotating the field towards the basal plane causes a distortion of the square vortex lattice observed for $\mathbit{H}\ensuremath{\parallel}\ensuremath{\langle}001\ensuremath{\rangle}$ and also a symmetry change to a distorted triangular symmetry for fields close to $\ensuremath{\langle}100\ensuremath{\rangle}$.The vortex lattice distortion allows us to determine the intrinsic superconducting anisotropy between the $c$ axis and the Ru-O basal plane, yielding a value of $\ensuremath{\sim}60$ at low temperature and low to intermediate fields. This greatly exceeds the upper critical field anisotropy of $\ensuremath{\sim}20$ at low temperature, reminiscent of Pauli limiting. Indirect evidence for Pauli paramagnetic effects on the unpaired quasiparticles in the vortex cores are observed, but a direct detection lies below the measurement sensitivity. The superconducting anisotropy is found to be independent of temperature but increases for fields $\ensuremath{\gtrsim}1$ T, indicating multiband superconductvity in ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$. Finally, the temperature dependence of the scattered intensity provides further support for gap nodes or deep minima in the superconducting gap.

Measurements of the $^{17}\mathrm{O}$ nuclear magnetic resonance (NMR) quadrupolar spectrum of apical oxygen in ${\mathrm{HgBa}}_{2}{\mathrm{CuO}}_{4+\ensuremath{\delta}}$ were performed over a range of magnetic fields from 6.4--30 T in the superconducting state. Oxygen-isotope-exchanged single crystals were investigated with doping corresponding to superconducting transition temperatures from 74 K underdoped, to 78 K overdoped. The apical oxygen site was chosen since its NMR spectrum has narrow quadrupolar satellites that are well separated from any other resonance. Nonvortex contributions to the spectra can be deconvolved in the time domain to determine the local magnetic field distribution from the vortices. Numerical analysis using Brandt's Ginzburg-Landau theory was used to find structural parameters of the vortex lattice, penetration depth, and coherence length as a function of magnetic field in the vortex solid phase. From this analysis we report a vortex structural transition near 15 T from an oblique lattice with an opening angle of ${73}^{\ensuremath{\circ}}$ at low magnetic fields to a triangular lattice with ${60}^{\ensuremath{\circ}}$ stabilized at high field. The temperature for onset of vortex dynamics has been identified from spin-spin relaxation. This is independent of the magnetic field at sufficiently high magnetic field similar to that reported for ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7}$ and ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{CaCu}}_{2}{\mathrm{O}}_{8+\ensuremath{\delta}}$ and is correlated with mass anisotropy of the material. This behavior is accounted for theoretically only in the limit of very high anisotropy.

The order parameter and pairing mechanism for superconductivity in heavy-fermion compounds are still poorly understood. Scanning tunneling microscopy and spectroscopy at ultralow temperatures can yield important information about the superconducting order parameter and the gap structure. Here, we study the first heavy-fermion superconductor, ${\mathrm{CeCu}}_{2}{\mathrm{Si}}_{2}$. Our data show the superconducting gap which is not fully formed and exhibits features that point to a multigap order parameter. Spatial mapping of the zero-bias conductance in magnetic field reveals the vortex lattice, which allows us to unequivocally link the observed conductance gap to superconductivity in ${\mathrm{CeCu}}_{2}{\mathrm{Si}}_{2}$. The vortex lattice is found to be predominantly triangular with distortions at fields close to $\ensuremath{\sim}0.7{H}_{c2}$.