Two-component Order Parameter Research Articles

The observation of π-shifts in the Little-Parks effect in 4Hb-TaS2

Finding evidence of non-trivial pairing states is one of the greatest experimental challenges in the field of unconventional superconductivity. Such evidence requires phase-sensitive probes susceptible to the internal structure of the order parameter. We report the measurement of the Little-Parks effect in the unconventional superconductor candidate 4Hb-TaS2. In half of our rings, which are fabricated from single-crystals, we find a π-shift in the transition-temperature oscillations. According to theory, such a π-shift is only possible if the order parameter is non-s-wave. In the absence of crystallographic defects, the shift provides evidence of a multi-component order parameter. Thus, this observation increases the likelihood of the two-component order parameter scenario in 4Hb-TaS2. Furthermore, we show that Tc is enhanced as a function of the out-of-plane field when a constant in-plane field is applied, which we explain using a two-component order-parameter.

Interaction between spin and Abrikosov vortices in doped topological insulators

In the topological superconductor with the nematic superconductivity in $E_u$ representation, it is possible to have different types of vortices. One is associated with the vorticity in the particle-hole space and corresponds to the Abrikosov vortex. Another type corresponds to the vorticity in the spin space and is called spin vortex. We study the interaction of the Abrikosov vortex with the spin vortices. We derive the free energy of the sample with the Abrikosov and the strain-induced spin vortices using the Ginzburg-Landau approach for the two-component superconducting order parameter. We calculate the critical strain at which the spin vortex is formed. We show that the spin vortex and the Abrikosov vortex attract to each other and, as a result, they have a common core. We show that there are no zero-energy states (Majorana fermions) localized near the common vortex core of the Abrikosov vortex and the spin vortex of any type. Possible experimental realization is discussed.

Nonabelian Ginzburg–Landau theory for ferroelectrics

The Ginzburg–Landau theory, which was introduced to phenomenologically describe the destruction of superconductivity by a magnetic field at the beginning, has brought up much more knowledge beyond the original one as a mean-field theory of thermodynamics states. There the complex order parameter plays an important role. Here we propose a macroscopic theory to describe the features of ferroelectrics by a two-component complex order parameter coupled to nonabelian gauge potentials that provide more freedom to reflect interplays between different measurables. Within this theoretical framework, some recently discovered empirical static and time-independent phenomena, such as vortex, anti-vortex, spiral orders can be obtained as solutions for different gauge potentials. It is expected to bring in a new angle of view with more elucidation than the traditional one that takes the polarization as order parameter.

Role of electromagnetic gauge-field fluctuations in the selection between chiral and nematic superconductivity

Motivated by the observation of nematic superconductivity in several systems, we revisit the problem of the leading pairing instability of two-component unconventional superconductors on the triangular lattice -- such as $(p_{x},\,p_{y})$-wave and $(d_{x^{2}-y^{2}},\,d_{xy})$-wave. Such a system has two possible superconducting states: the chiral state (e.g. $p+ip$ or $d+id$), which breaks time-reversal symmetry, and the nematic state (e.g. $p+p$ or $d+d$), which breaks the threefold rotational symmetry of the lattice. Weak-coupling calculations generally favor the chiral over the nematic superconducting state, raising the question of what mechanism can stabilize the latter. Here, we show that the electromagnetic field fluctuations can play a crucial role in selecting between these two states. Specifically, we derive and analyze the effective free energy for the two-component superconducting order parameter after integrating out the gauge-field fluctuations, which is formally justified if the spatial order parameter fluctuations can be neglected. A non-analytic cubic term arises, as in the case of a conventional $s$-wave superconductor. However, unlike the latter, the cubic term depends on the relative phase and on the relative amplitudes between the two order parameter components, in such a way that it generally favors the nematic state. This result is a direct consequence of the fact that the stiffness of the superconducting order parameter is not isotropic. Competition with the quartic term, which favors the chiral state, leads to a renormalized phase diagram in which the nematic state displaces the chiral state over a wide region in the parameter space. We analyze the stability of the fluctuation-induced nematic phase, generalize our results to tetragonal lattices, and discuss their applicability to candidate nematic superconductors, including twisted bilayer graphene.

Strong increase in ultrasound attenuation below Tc in Sr2RuO4 : Possible evidence for domains

Recent experiments suggest that the superconducting order parameter of Sr$_2$RuO$_4$ has two components. A two-component order parameter has multiple degrees of freedom in the superconducting state that can result in low-energy collective modes or the formation of domain walls -- a possibility that would explain a number of experimental observations including the smallness of the time reversal symmetry breaking signal at T$_\mathrm{c}$ and telegraph noise in critical current experiments. We perform ultrasound attenuation measurements across the superconducting transition of Sr$_2$RuO$_4$ using resonant ultrasound spectroscopy (RUS). We find that the attenuation for compressional sound increases by a factor of seven immediately below T$_\mathrm{c}$, in sharp contrast with what is found in both conventional ($s$-wave) and high-T$_\mathrm{c}$ ($d$-wave) superconductors. We find our observations to be most consistent with the presence of domain walls between different configurations of the superconducting state. The fact that we observe an increase in sound attenuation for compressional strains, and not for shear strains, suggests an inhomogeneous superconducting state formed of two distinct, accidentally-degenerate superconducting order parameters that are not related to each other by symmetry. Whatever the mechanism, a factor of seven increase in sound attenuation is a singular characteristic with which any potential theory of the superconductivity in Sr$_2$RuO$_4$ must be reconciled.

Interorbital singlet pairing in Sr2RuO4 : A Hund's superconductor

We study the superconducting gap function of Sr$_2$RuO$_4$. By solving the linearized Eliashberg equation with a correlated pairing vertex extracted from a dynamical mean-field calculation we identify the dominant pairing channels. An analysis of the candidate gap functions in orbital and quasiparticle band basis reveals that an inter-orbital singlet pairing of even parity is in agreement with experimental observations. It reconciles in particular the occurrence of a two-component order parameter with the presence of line-nodes of quasiparticles along the c-axis in the superconducting phase. The strong angular dependence of the gap along the Fermi surface is in stark contrast to its quasi-locality when expressed in the orbital basis. We identify local inter-orbital spin correlations as the driving force for the pairing and thus reveal the continuation of Hund's physics into the superconducting phase.

Shear modulus anomaly of unconventional superconductors in a symmetry breaking field

Using Ginzburg-Landau formalism, we theoretically study the isothermal shear modulus anomaly of an unconventional superconductor with a two-component order parameter, where the superconducting transition is split by a symmetry-breaking field. Experimental signatures are proposed for both chiral and nematic superconductors. Particularly striking is the vanishing of ${C}_{66}$ across the lower transition to a nematic superconducting state. Our findings can guide future experiments and shed new light on materials such as ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$ and ${M}_{x}{\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$.

Ultrasound evidence for a two-component superconducting order parameter in Sr2RuO4

The quasi-two-dimensional metal Sr2RuO4 is one of the best characterized unconventional superconductors, yet the nature of its superconducting order parameter is still under debate1,2,3. This information is crucial to determine the pairing mechanism of Cooper pairs. Here we use ultrasound velocity to probe the superconducting state of Sr2RuO4. This thermodynamic probe is sensitive to the symmetry of the material, and therefore, it can help in identifying the symmetry of the superconducting order parameter4,5. Indeed, we observe a sharp jump in the shear elastic constant c66 as the temperature is increased across the superconducting transition. This directly implies that the superconducting order parameter is of a two-component nature. On the basis of symmetry arguments and given the other known properties of Sr2RuO4 (refs. 6,7,8), we discuss which states are compatible with this requirement and propose that the two-component order parameter {dxz; dyz} is the most likely candidate.

Thermodynamic evidence for a two-component superconducting order parameter in Sr2RuO4

Sr$_2$RuO$_4$ has stood as the leading candidate for a spin-triplet superconductor for 26 years. Recent NMR experiments have cast doubt on this candidacy, however, and it is difficult to find a theory of superconductivity that is consistent with all experiments. What is needed are symmetry-based experiments that can rule out broad classes of possible superconducting order parameters. Here we use resonant ultrasound spectroscopy to measure the entire symmetry-resolved elastic tensor of Sr$_2$RuO$_4$ through the superconducting transition. We observe a thermodynamic discontinuity in the shear elastic modulus $c_{66}$, requiring that the superconducting order parameter is two-component. A two-component $p$-wave order parameter, such as $p_x+i p_y$, naturally satisfies this requirement. As this order parameter appears to be precluded by recent NMR experiments, we suggest that two other two-component order parameters, namely $\left\{d_{xz},d_{yz}\right\}$ or $\left\{d_{x^2-y^2},g_{xy(x^2-y^2)}\right\}$, are now the prime candidates for the order parameter of Sr$_2$RuO$_4$.

Exotic phases of frustrated antiferromagnet LiCu2O2

7Li NMR spectra were measured in a magnetic field up to 17 T at temperatures 5-30 K on single crystalline LiCu2O2. Earlier reported anomalies on magnetization curves correspond to magnetic field values where we observe changes of the NMR spectral shape. For the interpretation of the field and temperature evolutions of our NMR spectra, the magnetic structures were analyzed in the frame of the phenomenological theoretical approach of the Dzyaloshinskii-Landau theory. A set of possible planar and collinear structures was obtained. Most of these structures have an unusual configuration; they are characterized by a two-component order parameter and their magnetic moments vary harmonically not only in direction, but also in size. From the modeling of the observed spectra, a possible scenario of magnetic structure transformations is obtained.

Topological phase transitions in small mesoscopic chiral p -wave superconductors

Spin-triplet chiral p-wave superconductivity is typically described by a two-component order parameter, and as such is prone to unique emergent effects when compared to the standard single-component superconductors. Here we present the equilibrium phase diagram for small mesoscopic chiral p-wave superconducting disks in the presence of magnetic field, obtained by solving the microscopic Bogoliubov-de Gennes equations self-consistently. In the ultra-small limit, the cylindrically-symmetric giant-vortex states are the ground state of the system. However, with increasing sample size, the cylindrical symmetry is broken as the two components of the order parameter segregate into domains, and the number of fragmented domain walls between them characterizes the resulting states. Such domain walls are topological defects unique for the p-wave order, and constitute a dominant phase in the mesoscopic regime. Moreover, we find two possible types of domain walls, identified by their chirality-dependent interaction with the edge states.

Majorana Kramers pair in a nematic vortex

A time-reversal (TR) invariant topological superconductor is characterized by a Kramers pair of Majorana zero-energy modes on boundaries and in a core of a TR invariant vortex. A vortex defect that preserves TR symmetry has remained primarily of theoretical interest, since typically a magnetic field, which explicitly breaks TR, needs to be applied to create vortices in superconductors. In this work, we show that an odd-parity topological superconductor with a nematic pairing order parameter can host a nematic vortex that preserves TR symmetry and binds a Majorana Kramers pair. Such a nematic superconductor could be realized in metal-doped Bi$_2$Se$_3$, as suggested by recent experiments. We provide an analytic solution for the zero modes in a continuous nematic vortex. In lattice, crystalline anisotropy can pin the two-component order parameter along high-symmetry directions. We show that a discrete nematic vortex, which forms when three nematic domains meet, also supports a TR pair of Majorana modes. Finally, we discuss possible experiments to probe the zero modes.

Publisher's Note: Odd-parity superconductors with two-component order parameters: Nematic and chiral, full gap, and Majorana node [Phys. Rev. B 94 , 180504(R) (2016)

Received 22 May 2017DOI:https://doi.org/10.1103/PhysRevB.95.179902©2017 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasMultiband superconductivityOdd-frequency superconductivitySuperconducting gapSuperconducting order parameterPhysical SystemsUnconventional superconductorsTechniquesMethods in superconductivityCondensed Matter, Materials & Applied Physics

Odd-parity superconductors with two-component order parameters: Nematic and chiral, full gap, and Majorana node

Motivated by the recent experiment indicating that superconductivity in the doped topological insulator Cu$_x$Bi$_2$Se$_3$ has an odd-parity pairing symmetry with rotational symmetry breaking, we study the general class of odd-parity superconductors with two-component order parameters in trigonal and hexagonal crystal systems. In the presence of strong spin-orbit interaction, we find two possible superconducting phases below $T_c$, a time-reversal-breaking (i.e., chiral) phase and an anisotropic (i.e., nematic) phase, and determine their relative energetics from the gap function in momentum space. The nematic superconductor generally has a full quasi-particle gap, whereas the chiral superconductor with a three-dimensional (3D) Fermi surface has point nodes with lifted spin degeneracy, resulting in itinerant Majorana fermions in the bulk and topological Majorana arcs on the surface.

Zero Helicity States in the LaAlO3-SrTiO3 Interface

The transverse magnetic moment and torque observed by Li et al. (Nat. Phys. 7, 762 (6)) in the LaAlO3/SrTiO3 interface is claimed to have a purely kinetic origin. To explain the data, we describe the electronic state by a two-component order parameter with zero helicity. The zero helicity state breaks the spatial reflection symmetry of the kinetic energy according to the mechanism described here. The resulting state is a lattice of vortex-skyrmions that result in a transverse magnetization due to the asymmetry in the mass and thickness anisotropy between the two sides of the interface.

Classification of phenomenological phase transition models by the methods of equivariant catastrophe theory: Models with L = C nv (n = 3, 4, 6)

A classification of phenomenological phase-transition models for the two-component order parameter by the methods of equivariant catastrophe theory is proposed. The models are constructed on the basis of the number of thermodynamic parameters that are varied in an experiment (temperature, pressure, chemical potentials of admixtures, etc.) instead of the simple expansion of the thermodynamic potential in an order parameter power series. The classification of models with the order parameter symmetry groups L = Cnv (n = 3, 4, 6) with the number of varied parameters from 1 to 5 is performed.

Electronic properties of emergent topological defects in chiralp-wave superconductivity

Chiral $p$-wave superconductors in applied magnetic field can exhibit more complex topological defects than just conventional superconducting vortices, due to the two-component order parameter (OP) and the broken time-reversal symmetry. We investigate the electronic properties of those exotic states, some of which contain clusters of one-component vortices in chiral components of the OP and/or exhibit skyrmionic character in the \textit{relative} OP space, all obtained as a self-consistent solution of the microscopic Bogoliubov-de Gennes equations. We reveal the link between the local density of states (LDOS) of the novel topological states and the behavior of the chiral domain wall between the OP components, enabling direct identification of those states in scanning tunneling microscopy. For example, a skyrmion always contains a closed chiral domain wall, which is found to be mapped exactly by zero-bias peaks in LDOS. Moreover, the LDOS exhibits electron-hole asymmetry, which is different from the LDOS of conventional vortex states with the same vorticity. Finally, we present the magnetic field and temperature dependence of the properties of a skyrmion, indicating that this topological defect can be surprisingly large in size, and can be pinned by an artificially indented non-superconducting closed path in the sample. These features are expected to facilitate the experimental observation of skyrmionic states, thereby enabling experimental verification of chirality in emerging superconducting materials.

Monte Carlo simulations of vector pseudospins for strains: Microstructures and martensitic conversion times

We present systematic temperature-quench Monte Carlo simulations on discrete-strain pseudospin model Hamiltonians to study microstructural evolutions in ferroelastic transitions with two-component vector order parameters (NOP=2) in 2-spatial dimensions. The zero value pseudospin is the single high-temperature phase while the low-temperature phase has Nv variants. Thus the number of nonzero values of pseudospin are triangle-to-centered rectangle (Nv=3), square-to-oblique (Nv=4) and triangle-to-oblique (Nv=6). The model Hamiltonians contain a transition-specific Landau energy term, a domain wall cost or Ginzburg term, and power-law anisotropic interaction potential, induced from a strain compatibility condition. On quenching below a transition temperature, we find behaviour similar to the previously studied square-to-rectangle transition (NOP=1,Nv=2), showing that the rich behaviour found, is generic. Thus we find for two-component order parameters that the same Hamiltonian can describe both athermal and isothermal martensite regimes for different material parameters. The athermal/isothermal/austenite parameter regimes and temperature–time–transformation diagrams are understood, as previously, through parametrization of effective-droplet energies. In the athermal regime, we find rapid conversions below a spinodal-like temperature and austenite-martensite conversion delays above it, as in the experiment. The delays show early incubation behaviour, and at the transition to austenite the delay times have Vogel–Fulcher divergences and are insensitive to Hamiltonian energy scales, suggesting that entropy barriers are dominant. Systematic temperature quench experiments can look for martensite formation and growth during conversion-incubations, divergences, and distributions close to the transition.

The construction of phenomenological models of phase transitions using equivariant catastrophe theory methods

A method for the construction of phenomenological models of phase transitions using the equivariant theory of singularities of differentiable maps with the symmetry of the order parameter (the equivariant catastrophe theory) is considered. This method differs from the traditional series expansion in powers of the order parameter in that, firstly, there is no requirement that the order parameters be small, and only the number of thermodynamic parameters that are varied in the experiment (temperature, pressure, etc.) is given; secondly, derived models have structural stability for an adequate description of the anomalies of the physical properties near the phase transitions. As an example to illustrate the method, a model of phase transitions with a two-component order parameter is constructed.

Critical Exponents Can Be Different on the Two Sides of a Transition: A Generic Mechanism.

We present models where γ(+) and γ(-), the exponents of the susceptibility in the high- and low-temperature phases, are generically different. In these models, continuous symmetries are explicitly broken down by discrete anisotropies that are irrelevant in the renormalization-group sense. The Z(q)-invariant models are the simplest examples for two-component order parameters (N=2) and the model with icosahedral symmetry for N=3. We accurately compute γ(+)-γ(-) as well as the ratio ν/ν' of the exponents of the two correlation lengths present for T<T(c).