Wave Order Parameter Research Articles

Quantitative determination of bond order and lattice distortions in nickel oxide heterostructures by resonant x-ray scattering

We present a combined study of Ni $K$-edge resonant x-ray scattering and density functional calculations to probe and distinguish electronically driven ordering and lattice distortions in nickelate heterostructures. We demonstrate that due to the low crystal symmetry, contributions from structural distortions can contribute significantly to the energy-dependent Bragg peak intensities of a bond-ordered NdNiO$_3$ reference film. For a LaNiO$_3$-LaAlO$_3$ superlattice that exhibits magnetic order, we establish a rigorous upper bound on the bond-order parameter. We thus conclusively confirm predictions of a dominant spin density wave order parameter in metallic nickelates with a quasi-two-dimensional electronic structure.

Multi-Component Ginzburg-Landau Theory: Microscopic Derivation and Examples

This paper consists of three parts. In part I, we microscopically derive Ginzburg--Landau (GL) theory from BCS theory for translation-invariant systems in which multiple types of superconductivity may coexist. Our motivation are unconventional superconductors. We allow the ground state of the effective gap operator $K_{T_c}+V$ to be $n$-fold degenerate and the resulting GL theory then couples $n$ order parameters. In part II, we study examples of multi-component GL theories which arise from an isotropic BCS theory. We study the cases of (a) pure $d$-wave order parameters and (b) mixed $(s+d)$-wave order parameters, in two and three dimensions. In part III, we present explicit choices of spherically symmetric interactions $V$ which produce the examples in part II. In fact, we find interactions $V$ which produce ground state sectors of $K_{T_c}+V$ of arbitrary angular momentum, for open sets of of parameter values. This is in stark contrast with Schr\"odinger operators $-\nabla^2+V$, for which the ground state is always non-degenerate. Along the way, we prove the following fact about Bessel functions: At its first maximum, a half-integer Bessel function is strictly larger than all other half-integer Bessel functions.

Phase diagram of a non-Abelian Aubry-André-Harper model withp-wave superfluidity

We theoretically study a one-dimensional quasi-periodic Fermi system with topological $p$-wave superfluidity, which can be deduced from a topologically non-trivial tight-binding model on the square lattice in a uniform magnetic field and subject to a non-Abelian gauge field. The system may be regarded a non-Abelian generalization of the well-known Aubry-Andr\'e-Harper model. We investigate its phase diagram as functions of the strength of the quasi-disorder and the amplitude of the $p$-wave order parameter, through a number of numerical investigations, including a multifractal analysis. There are four distinct phases separated by three critical lines, i.e., two phases with all extended wave-functions (I and IV), a topologically trivial phase (II) with all localized wave-functions and a critical phase (III) with all multifractal wave-functions. The phase I is related to the phase IV by duality. It also seems to be related to the phase II by duality. Our proposed phase diagram may be observable in current cold-atom experiments, in view of simulating non-Abelian gauge fields and topological insulators/superfluids with ultracold atoms.

Random field disorder and charge order driven quantum oscillations in cuprates

IIn the pseudogap regime of the cuprates, charge order breaks a $\mathbb{Z}_{2}$ symmetry. Therefore, the interaction of charge order and quenched disorder due to potential scattering, can, in principle, be treated as a random field Ising model. A numerical analysis of the ground state of such a random field Ising model reveals local, glassy dynamics in both $2D$ and $3D$. The glassy dynamics are treated as a heat bath which couple to the itinerant electrons, leading to an unusual electronic non-Fermi liquid. If the dynamics are strong enough, the electron spectral function has no quasiparticle peak and the effective mass diverges at the Fermi surface, precluding quantum oscillations. In contrast to charge density, $d$-density wave order (reflecting staggered circulating currents) does not directly couple to potential disorder, allowing it to support quantum oscillations. At fourth order in Landau theory, there is a term consisting of the square of the $d$-density wave order parameter, and the square of the charge order. This coupling could induce parasitic charge order, which may be weak enough for the Fermi liquid behavior to remain uncorrupted. Here, we argue that this distinction must be made clear, as one interprets quantum oscillations in cuprates.

Coexistence of Superconductivity and Spin Density Wave (SDW) in Ferropnictide Ba1−x K x Fe2As2

This work focuses on the theoretical investigation of the coexistence of superconductivity and spin density wave (SDW) in ferropnictide Ba1−x K x Fe2As2. By developing a model Hamiltonian for the system and by using quantum field theory Green’s function formalism, we have obtained mathematical expressions for superconducting transition temperature (T C), spin density wave transition temperature (T sdw), superconductivity order parameter (ΔSc), and spin density wave order parameter (Δsdw). By employing the experimental and theoretical values of the parameters in the obtained expressions, phase diagrams of superconducting transition temperature (T C) versus superconducting order parameter (ΔSc) and spin density wave transition temperature (T sdw), versus spin density wave order parameter (Δsdw) have been plotted. By combining the two phase diagrams, we have demonstrated the possible coexistence of superconductivity and spin density wave (SDW) in ferropnictide Ba1−x K x Fe2As2.

Temperature dependence of the penetration depth of a magnetic field in the presence of dispersion of the superconducting and charge density wave order parameters

A formula for computing the temperature dependence of the London penetration depth of a magnetic field in the regime of coexistence of charge density waves and superconductivity has been proposed taking into account the dependence of both order parameters on the wave vector. It has been shown that an anomalously high diamagnetic response of the system and a finite value of the superconducting current persist even at Tc ≤ T ≤ TCDW.

Superconductivity and Spin Density Wave (SDW) in NaFe1-xCoxAs

A model is presented utilizing a Hamiltonian with equal spin singlet and triplet pairings based on quantum field theory and green function formalism, to show the correlation between the superconducting and spin density wave (SDW) order parameters. The model exhibits a distinct possibility of the coexistence of superconductivity and long-range magnetic phase, which are two usually incompatible cooperative phenomena. The work is motivated by the recent experimental evidences on high-TC superconductivity in the FeAs-based superconductors. The theoretical results are then applied to show the coexistence of superconductivity and spin density wave (SDW) in NaFe1-xCoxAs.

Nodeless superconductivity in the presence of spin-density wave in pnictide superconductors: The case ofBaFe2−xNixAs2

The characteristics of Fe-based superconductors are manifested in their electronic, magnetic properties, and pairing symmetry of the Cooper pair, but the latter remain to be explored. Usually in these materials, superconductivity coexists and competes with magnetic order, giving unconventional pairing mechanisms. We report on the results of the bulk magnetization measurements in the superconducting state and the low-temperature specific heat down to 0.4 K for BaFe$_{2-x}$Ni$_{x}$As$_{2}$ single crystals. The {electronic} specific heat displays a pronounced anomaly at the superconducting transition temperature and a small residual part {at low temperatures in the superconducting state}. The normal-state Sommerfeld coefficient increases with Ni doping for $x$ = 0.092, 0.096, and 0.10, which illustrates the competition between magnetism and superconductivity. Our analysis of the temperature dependence of the superconducting-state specific heat and the London penetration depth provides strong evidence for a two-band $s$-wave order parameter. Further, the data of the London penetration depth calculated from the lower critical field follow an exponential temperature dependence, characteristic of a fully gapped superconductor. These observations clearly show that the superconducting gap in the nearly optimally doped compounds is nodeless.

Multigap superconductivity in locally noncentrosymmetric SrPtAs: AnAs75nuclear quadrupole resonance investigation

We report detailed $^{75}$As-NQR investigations of the locally non-centrosymmetric superconductor SrPtAs. The spin-lattice relaxation studies prove weakly coupled multi-gap superconductivity. The Hebel-Slichter peak, a hallmark of conventional superconductivity, is strongly suppressed, which points to an unconventional superconducting state. The observed behavior excludes a superconducting order parameter with line nodes and is consistent with proposed $f$-wave and chiral $d$-wave order parameters.

Observation of a Bulk Nodal-Gap in Overdoped Y $$_{0.9}$$ 0.9 Ca $$_{0.1}$$ 0.1 Ba $$_{2}$$ 2 Cu $$_{3}$$ 3 O $$_{7-\delta }$$ 7 - δ Thin Films

Complex conductivity of overdoped Y\(_{0.9}\)Ca\(_{0.1}\)Ba\(_{2}\)Cu\(_{3}\)O\(_{7-\delta }\) thin films was measured in Terahertz frequency using frequency and time domain methods. The films were measured in the frequency range of 3–100 cm\(^{-1}\) and in the temperature range of 20–300 K. Results show a possible deviation from a pure \(d_{x^{2}-y^{2}}\)-wave superconductor, indicated by the existence of an energy sub-gap in overdoped Y\(_{0.9}\)Ca\(_{0.1}\)Ba\(_{2}\)Cu\(_{3}\)O\(_{7-\delta }\) films. Evidence for this sub-gap appears as non-monotonic behavior of \(\sigma _{1}(\omega ,T)\) as a function of frequency followed by a sharp decrease at low frequencies, and a dip in the imaginary part of the optical conductivity multiplied by frequency, \(\omega {{\sigma }_{2}}(\omega ,T)\). The mentioned features were observed at energy of about 1.2 meV in 10 % Ca-doped YBCO thin films. Our complex conductivity spectra are in agreement with the theoretical prediction obtained by using a mixed symmetry order parameter within the Born limit, shown by Schurrer et al. We suggest that these observations are direct evidence for a nodal gap obtained in a \(d_{x^{2}-y^{2}}\)-wave superconductor and can be theoretically clarified by adding an imaginary component as \(is\) or \(i{{d}_{xy}}\) to the main \(d_{x^{2}-y^{2}}\)-wave order parameter.

Microscopic theory for phase-sensitive experiments to determine the symmetry of the order parameter in Fe-based superconductors

We present a microscopic theory for the dc Josephson current, based on the construction of a coherent temperature Green's function in the tight-binding approximation, in junctions with multiband superconductors. This theory is applied to junctions with a multiband Fe-based superconductor (FeBS) described by -wave and -wave order parameter symmetries, which is probably realized in FeBS. For the first time, phase dependences of the Josephson current have been calculated for different directions of an interface with respect to crystallographic axes of FeBS. The present approach is also suitable for a consistent description of the Josephson transport in structures with topological superconductors.

Specific Heat of K $$_{0.71}$$ 0.71 Na $$_{0.29}$$ 0.29 Fe $$_2$$ 2 As $$_2$$ 2 at Very Low Temperatures

A commercially available calorimeter has been used to investigate the specific heat of a high-quality \kn\ single crystal. The addenda heat capacity of the calorimeter is determined in the temperature range $0.02 \, \mathrm{K} \leq T \leq 0.54 \, \mathrm{K}$. The data of the \kn\ crystal imply the presence of a large $T^2$ contribution to the specific heat which gives evidence of $d$-wave order parameter symmetry in the superconducting state. To improve the measurements, a novel design for a calorimeter with a paramagnetic temperature sensor is presented. It promises a temperature resolution of $\Delta T \approx 0.1 \, \mathrm{\mu K}$ and an addenda heat capacity less than $200 \, \mathrm{pJ/K}$ at $ T < 100 \, \mathrm{mK}$.

Chirald-wave superconductivity in SrPtAs

Recent $\mu$SR measurements on SrPtAs revealed time-reversal-symmetry breaking with the onset of superconductivity [Biswas et al., Phys. Rev. B 87, 180503(R) (2013)], suggesting an unconventional superconducting state. We investigate this possibility via functional renormalization group and find a chiral $(d+\mathrm{i}d)$-wave order parameter favored by the multiband fermiology and hexagonal symmetry of SrPtAs. This $(d+\mathrm{i}d)$-wave state exhibits significant gap anisotropies as well as gap differences on the different bands, but only has point nodes on one of the bands at the Brillouin zone corners. We study the topological characteristics of this superconducting phase, which features Majorana-Weyl nodes in the bulk, protected surface states, and an associated thermal Hall response. The lack of extended nodes and the spontaneously broken time-reversal symmetry of the $(d+\mathrm{i}d)$-wave state are in agreement with the $\mu$SR experiments. Our theoretical findings together with the experimental evidence thus suggests that SrPtAs is the first example of chiral $d$-wave superconductivity.

Topological defects of Néel order and Kondo singlet formation for the Kondo-Heisenberg model on a honeycomb lattice

Heavy fermion systems represent a prototypical setting to study magnetic quantum phase transitions. A particular focus has been on the physics of Kondo destruction, which captures quantum criticality beyond the Landau framework of order-parameter fluctuations. In this context, we study the spin one-half Kondo-Heisenberg model on a honeycomb lattice at half filling. The problem is approached from the Kondo destroyed, antiferromagnetically ordered insulating phase. We describe the local moments in terms of a coarse grained quantum non-linear sigma model, and show that the skyrmion defects of the antiferromagnetic order parameter host a number of competing order parameters. In addition to the spin Peierls, charge and current density wave order parameters, we identify for the first time Kondo singlets as the competing orders of the antiferromagnetism. We show that the antiferromagnetism and various competing singlet orders can be related to each other via generalized chiral transformations of the underlying fermions. We also show that the conduction electrons acquire a Berry phase through their coupling to the hedgehog configurations of the N\'eel order, which cancels the Berry phase of the local moments. Our results demonstrate the competition between the Kondo-singlet formation and spin-Peierls order when the antiferromagnetic order is suppressed, thereby shedding new light on the global phase diagram of heavy fermion systems at zero temperature.

The Coexistence of Super Conductivity and Spin Density Wave (SDW) in SmO&amp;lt;sub&amp;gt;1-x&amp;lt;/sub&amp;gt;F&amp;lt;sub&amp;gt;x&amp;lt;/sub&amp;gt;FeAs

The coexistence of superconductivity and spin density wave in SmO1-xFxFeAs is theoretically studied using the model Hamiltonian which contains BCS type superconductivity and spin density wave terms. Employing green function formalism, the expression for the spin density wave order parameter (M) and expression for spin density wave transition temperature TSDW is obtained. The interplay between the superconductivity and spin density wave is examined in these parameters and the coexistence of the two states is established in the order parameter range of 0.1≤MmeV≤0.13 which is seen to be in broad experimental agreement.

London penetration depth and pair breaking

The London penetration depth is evaluated for isotropic materials for any transport and pair-breaking Born scattering rates. Besides known results, a number of features are found. The slope $|d\ensuremath{\rho}/d\ensuremath{\theta}|$ of the normalized superfluid density $\ensuremath{\rho}={\ensuremath{\lambda}}^{2}(0)/{\ensuremath{\lambda}}^{2}(\ensuremath{\theta})$ at the transition $\ensuremath{\theta}=T/{T}_{c}=1$ has a minimum near the value of the pair-breaking parameter separating gapped and gapless states. The low-$T$ exponentially flat part of $\ensuremath{\rho}$ for the $s$-wave materials is suppressed by increasing pair breaking. For strong ${T}_{c}$ suppression by magnetic impurities the ``Homes scaling'' ${\ensuremath{\lambda}}^{\ensuremath{-}2}(0)\ensuremath{\propto}\ensuremath{\sigma}{T}_{c}$ with $\ensuremath{\sigma}$ being the normal conductivity gives way to ${\ensuremath{\lambda}}^{\ensuremath{-}2}(0)\ensuremath{\propto}\ensuremath{\sigma}{T}_{c}^{2}$. For the $d$-wave order parameter, the transport and spin-flip Born scattering rates enter the theory only as a sum; in particular, they affect the ${T}_{c}$ depression in the same manner. We confirm that the linear low-temperature behavior of $\ensuremath{\rho}$ in a broad range of the combined scattering parameter turns to the ${T}^{2}$ behavior only when the critical temperature is suppressed at least by a factor of 3 relative to the clean limit ${T}_{c0}$. Moreover, in this range, $\ensuremath{\rho}(\ensuremath{\theta})$ is only weakly dependent on the scattering parameter, i.e., it is nearly universal.

Temperature dependence of lower critical fieldHc1(T)shows nodeless superconductivity in FeSe

We investigate the temperature dependence of the lower critical field ${H}_{c1}(T)$ of a high-quality FeSe single crystal under static magnetic fields $H$ parallel to the $c$ axis. The temperature dependence of the first vortex penetration field has been experimentally obtained by two independent methods and the corresponding ${H}_{c1}(T)$ was deduced by taking into account demagnetization factors. A pronounced change in the ${H}_{c1}$(T) curvature is observed, which is attributed to anisotopic s-wave or multiband superconductivity. The London penetration depth ${\ensuremath{\lambda}}_{ab}(T)$ calculated from the lower critical field does not follow an exponential behavior at low temperatures, as it would be expected for a fully gapped clean $s$-wave superconductor. Using either a two-band model with $s$-wave-like gaps of magnitudes ${\ensuremath{\Delta}}_{1}=0.41\ifmmode\pm\else\textpm\fi{}0.1$ meV and ${\ensuremath{\Delta}}_{2}=3.33\ifmmode\pm\else\textpm\fi{}0.25$ meV or a single anisotropic $s$-wave order parameter, the temperature dependence of the lower critical field ${H}_{c1}(T)$ can be well described. These observations clearly show that the superconducting energy gap in FeSe is nodeless.

Space group symmetry, spin-orbit coupling, and the low-energy effective Hamiltonian for iron-based superconductors

We construct the symmetry adapted low energy effective Hamiltonian for the electronic states in the vicinity of the Fermi level in iron based superconductors. We use Luttinger's method of invariants, expanding about Gamma and M points in the Brillouin zone corresponding to two iron unit cell, and then matching the coefficients of the expansion to the 5- and 8-band models. We then use the method of invariants to study the effects of the spin-density wave order parameters on the electronic spectrum, with and without spin-orbit coupling included. Among the results of this analysis is the finding that the nodal spin-density wave is unstable once spin-orbit coupling is included. Similar analysis is performed for the A_{1g} spin singlet superconducting state. Without spin-orbit coupling there is one pairing invariant near the Gamma point, but two near the M point. This leads to an isotropic spectral gap at the hole Fermi surface near Gamma, but anisotropic near M. The relative values of these three parameters determine whether the superconducting state is s_{++}, s_{+-}, or nodal. Inclusion of spin-orbit coupling leads to additional mixing of spin triplet pairing, with one additional pairing parameter near Gamma and one near M. This leads to an anisotropic spectral gap near both hole and electron Fermi surfaces, the latter no longer cross, but rather split.

Fully gapped superconductivity in a nanometre-size YBa2Cu3O7–δ island enhanced by a magnetic field

The symmetry of Cooper pairs is central to constructing a superconducting state. The demonstration of a d(x²-y²)-wave order parameter with nodes represented a breakthrough for high critical temperature superconductors (HTSs). However, despite this fundamental discovery, the origin of superconductivity remains elusive, raising the question of whether something is missing from the global picture. Deviations from d(x²-y²)-wave symmetry, such as an imaginary admixture d(x²-y²)+ is (or id(xy)), predict a ground state with unconventional properties exhibiting a full superconducting gap and time reversal symmetry breaking. The existence of such a state, until now highly controversial, can be proved by highly sensitive measurements of the excitation spectrum. Here, we present a spectroscopic technique based on an HTS nanoscale device that allows an unprecedented energy resolution thanks to Coulomb blockade effects, a regime practically inaccessible in these materials previously. We find that the energy required to add an extra electron depends on the parity (odd/even) of the excess electrons on the island and increases with magnetic field. This is inconsistent with a pure d(x²-y²)-wave symmetry and demonstrates a complex order parameter component that needs to be incorporated into any theoretical model of HTS.

Anomalous proximity effects at the interface ofs- ands±- superconductors

We study proximity effects close to a boundary between $s$ and ${s}_{\ifmmode\pm\else\textpm\fi{}}$ superconductors. Frustration, caused by interaction of the $s$-wave gap parameter with the opposite-sign gaps of ${s}_{\ifmmode\pm\else\textpm\fi{}}$ superconductor, leads to several anomalous features. In the case of strong frustration a nontrivial time-reversal-symmetry breaking (TRSB) state, with nonzero phase angles between all gap parameters, is possible. In a more typical state, the $s$-wave order parameter is aligned with one of the ${s}_{\ifmmode\pm\else\textpm\fi{}}$ gaps. The other (antialigned) gap induces negative feature in the $s$-wave density of states, which can serve as a fingerprint of the ${s}_{\ifmmode\pm\else\textpm\fi{}}$ state. Another consequence of the frustration is an extended region in the parameter space in which $s$-wave superconductivity is suppressed, despite being in contact with nominally stronger superconductor. This negative proximity effect is always present for the TRSB state, but extends even into the aligned states. We study these effects within a simple microscopic model assuming a dirty limit in all bands, which allows us to model the system in terms of minimum number of the most relevant parameters. The described anomalous features provide a route to establishing the possible ${s}_{\ifmmode\pm\else\textpm\fi{}}$ state in the iron-based superconductors.