Fourfold Symmetry Breaking Research Articles

Observation of a Ubiquitous (π, π)-Type Nematic Superconducting Order in the Whole Superconducting Dome of Ultra-Thin BaFe2–x Ni x As2 Single CrystalsSupported by the National Natural Science Foundation of China (Grant Nos. 61771234, 61727805, 11674157, 11674158, 11774152, 11822405, 61521001, 61571219, and 61501222), the National Key Projects for Research and Development of China (Grant Nos. 2016YFA0300401, 2017YFB0503302, 2017YFA0304002, and 2017YFB0503300), the

In iron-based superconductors, the (0, π) or (π, 0) nematicity, which describes an electronic anisotropy with a four-fold symmetry breaking, is well established and believed to be important for understanding the superconducting mechanism. However, how exactly such a nematic order observed in the normal state can be related to the superconducting pairing is still elusive. Here, by performing angular-dependent in-plane magnetoresistivity using ultra-thin flakes in the steep superconducting transition region, we unveil a nematic superconducting order along the (π, π) direction in electron-doped BaFe2 – x Ni x As2 from under-doped to heavily overdoped regimes with x = 0.065–0.18. It shows superconducting gap maxima along the (π, π) direction rotated by 45° from the nematicity along (0, π) or (π, 0) direction observed in the normal state. A similar (π, π)-type nematicity is also observed in the under-doped and optimally doped hole-type Ba1 – y K y Fe2As2, with y = 0.2–0.5. These results suggest that the (π, π) nematic superconducting order is a universal feature that needs to be taken into account in the superconducting pairing mechanism in iron-based superconductors.

Decoupling of lattice and orbital degrees of freedom in an iron-pnictide superconductor

The interplay of structural and electronic phases in iron-based superconductors is a central theme in the search for the superconducting pairing mechanism. While electronic nematicity, defined as the breaking of four-fold symmetry triggered by electronic degrees of freedom, is competing with superconductivity, the effect of purely structural orthorhombic order is unexplored. Here, using x-ray diffraction (XRD), we reveal a new structural orthorhombic phase with an exceptionally high onset temperature ($T_\mathrm{ort} \sim 250$ K), which coexists with superconductivity ($T_\mathrm{c} = 25$ K), in an electron-doped iron-pnictide superconductor far from the underdoped region. Furthermore, our angle-resolved photoemission spectroscopy (ARPES) measurements demonstrate the absence of electronic nematic order as the driving mechanism, in contrast to other underdoped iron pnictides where nematicity is commonly found. Our results establish a new, high temperature phase in the phase diagram of iron-pnictide superconductors and impose strong constraints for the modeling of their superconducting pairing mechanism.

EXAFS investigation of the local structure in URu2−xFexSi2 : Evidence for distortions below 100 K

X-ray absorption measurements at the U ${L}_{\mathrm{III}}$, Ru $K$, and Fe $K$ edges are reported for the hidden order (HO) material ${\mathrm{URu}}_{2\ensuremath{-}x}{\mathrm{Fe}}_{x}{\mathrm{Si}}_{2}$ ($x=0$, 0.05, 0.08, 0.10, 0.12, 0.15, and 0.20) as a function of $x$ and temperature $T$. When Fe is substituted for Ru, the local structure about Fe shrinks slightly and the first neighbor Fe-Si bond length decreases by $\ensuremath{\approx}0.05\phantom{\rule{0.16em}{0ex}}\AA{}$. More importantly excess disorder is observed below 80--100 K (the coherence temperature ${T}^{*}$) in plots of the Debye-Waller factor ${\ensuremath{\sigma}}^{2}$ ($\ensuremath{\sigma}$ is the width of the pair distribution function); at low $T$ the data deviate from the usual Einstein or correlated-Debye model plots. This excess disorder is most prominent for the Ru-Si bond, and ${\ensuremath{\sigma}}^{2}$ actually increases below 80 K. These results suggest a local orthorhombic distortion with ${B}_{1g}$-like symmetry that develops below 80--100 K. A model that describes these local distortions is presented, and discussed in terms of other measurements that indicate a breaking of fourfold symmetry at low $T$. In addition, the square root of the difference between ${\ensuremath{\sigma}}^{2}(T)$ for the Ru-Si pair and a Debye fit to these data serves as an order parameter for this orthorhombic distortion, in the temperature range below 100 K. This quantity is a length related to $a\text{\ensuremath{-}}b$, the difference between the $a$ and $b$ lattice constants in the orthorhombic phase, and provides a connection between this distortion and ${T}^{*}$. X-ray absorption near edge structure (XANES) measurements also show that there are no changes in the edge positions down to 0.1 eV for any edge as a function of $x$, for $T$ in the HO regime.

Quasi-one-dimensional nanoscale modulation as sign of nematicity in iron pnictides and chalcogenides

Impurity scattering is found to lead to quasi-one dimensional nanoscale modulation of the local density of states in the iron pnictides and chalcogenides. This `quasiparticle interference' feature is remarkably similar across a wide variety of pnictide and chalcogenide phases, suggesting a common origin. We show that a unified understanding of the experiments can be obtained by simply invoking a four-fold symmetry breaking $d_{xz}-d_{yz}$ orbital splitting, of a magnitude already suggested by the experiments. This can explain the one-dimensional characteristics in the local density of states observed in the orthorhombic nematic, tetragonal paramagnetic, as well as the spin-density wave and superconducting states in these materials.

Unconventional quantum antiferromagnetism with a fourfold symmetry breaking in a spin-12Ising-Heisenberg pentagonal chain

The spin-$\frac{1}{2}$ Ising-Heisenberg pentagonal chain is investigated with use of the star-triangle transformation, which establishes a rigorous mapping equivalence with the effective spin-$\frac{1}{2}$ Ising zigzag ladder. The investigated model has a rich ground-state phase diagram including two spectacular quantum antiferromagnetic ground states with a fourfold broken symmetry. It is demonstrated that these long-period quantum ground states arise due to a competition between the effective next-nearest-neighbor and nearest-neighbor interactions of the corresponding spin-$\frac{1}{2}$ Ising zigzag ladder. The concurrence is used to quantify the bipartite entanglement between the nearest-neighbor Heisenberg spin pairs, which are quantum-mechanically entangled in two quantum ground states with or without spontaneously broken symmetry. The pair correlation functions between the nearest-neighbor Heisenberg spins as well as the next-nearest-neighbor and nearest-neighbor Ising spins were investigated with the aim to bring insight into how a relevant short-range order manifests itself at low enough temperatures. It is shown that the specific heat displays temperature dependencies with either one or two separate round maxima.

Hidden-order symmetry and superconductivity inURu2Si2investigated by quasiparticle interference

The hidden order (HO) in URu2Si2 has been determined as a high rank multipole formed by itinerant 5f-electrons with distinct orbital structure imposed by the crystalline electric field. Because this can lead to a considerable number of different multipoles it is of great importance to use microscopic techniques that are sensitive to their subtle physical differences. Here we investigate whether quasiparticle interference (QPI) method can distinguish between the two most frequently proposed HO parameter models: the even rank-4 hexadecapole and the odd-rank-5 dotriacontapole model. We obtain the quasiparticle dispersion and reconstructed Fermi surface in each HO phase adapting an effective two-orbital model of 5f bands that reproduces the main Fermi surface sheets of the para phase. We show that the resulting QPI spectrum reflects directly the effect of fourfold symmetry breaking in the rank-5 model which is absent in the rank-4 model. Therefore we suggest that QPI method should give a possibility of direct discrimination between the two most investigated models of HO in URu2Si2. Furthermore the signature of proposed chiral d-wave superconducting (SC) order parameter in QPI of the coexisting HO+SC phase is investigated.

Successive Magnetic Transitions of Ca2CoSi2O7in High Magnetic Fields

Magnetic and dielectric properties of \aa kermanite Ca$_2$CoSi$_2$O$_7$ single crystals were investigated in pulsed high magnetic fields. In magnetic fields along the $c$ axis, this material shows a magnetization plateau in a wide range of field below the saturation. Magnetization processes for fields along the $a$ and $b$ axes show multiple anomalies but different traces to each other, indicating the breaking of four-fold symmetry. Measurements of the magnetoelectric effects exhibit the changes in electric polarization according to the changes in the spin system. The experimentally determined quadratic magnetoelectric tensor is consistent with that expected in the crystal symmetry of orthorhombic $P2_12_12$.

Polarization conversion in U-shaped chiral metamaterial with four-fold symmetry breaking

Despite chiral metamaterials being widely appreciated for their giant optical activity and negative refractive index with respect to the transmitted electromagnetic waves, little research efforts are devoted to effects occurring upon optical reflection from such metamaterials. Here, we theoretically demonstrate highly energy-efficient cross-polarization conversion for a normally incident wave reflecting from metamaterial with a broken four-fold symmetry of chiral structure. We do this by designing metamaterial that makes the polarization plane of a linearly polarized electromagnetic wave reflected from its surface almost perpendicular to the polarization plane of the incident wave. Using reflection from this metamaterial, one can also freely convert between left-handed and right-handed circular polarizations without significant energy loss. The proposed chiral metamaterial may prove useful in electromagnetic communication systems, polarization controllable antennas, and on-chip biomedical sensors.

Liquid-crystal patterns of rectangular particles in a square nanocavity

Using density-functional theory in the restricted-orientation approximation, we analyze the liquid-crystal patterns and phase behavior of a fluid of hard rectangular particles confined in a two-dimensional square nanocavity of side length H composed of hard inner walls. Patterning in the cavity is governed by surface-induced order as well as capillary and frustration effects and depends on the relative values of the particle aspect ratio κ≡L/σ, with L the length and σ the width of the rectangles (L≥σ), and cavity size H. Ordering may be very different from bulk (H→∞) behavior when H is a few times the particle length L (nanocavity). Bulk and confinement properties are obtained for the cases κ=1, 3, and 6. In bulk the isotropic phase is always stable at low packing fractions η=Lσρ_{0} (with ρ_{0} the average density) and nematic, smectic, columnar, and crystal phases can be stabilized at higher η depending on κ: For increasing η the sequence of isotropic to columnar is obtained for κ=1 and 3, whereas for κ=6 we obtain isotropic to nematic to smectic (the crystal being unstable in all three cases for the density range explored). In the confined fluid surface-induced frustration leads to fourfold symmetry breaking in all phases (which become twofold symmetric). Since no director distortion can arise in our model by construction, frustration in the director orientation is relaxed by the creation of domain walls (where the director changes by 90^{∘}); this configuration is necessary to stabilize periodic phases. For κ=1 the crystal becomes stable with commensurate transitions taking place as H is varied. These transitions involve structures with different number of peaks in the local density. In the case κ=3 the commensurate transitions involve columnar phases with different number of columns. In the case κ=6 the high-density region of the phase diagram is dominated by commensurate transitions between smectic structures; at lower densities there is a symmetry-breaking isotropic to nematic transition exhibiting nonmonotonic behavior with cavity size. Apart from the present application in a confinement setup, our model could be used to explore the bulk region near close packing in order to elucidate the possible existence of disordered phases at close packing.

S-wave superconductivity in superconducting BaTi2Sb2O revealed by121/123Sb-NMR/nuclear quadrupole resonance measurements

We report the ${}^{121/123}$Sb-NMR/nuclear quadrupole resonance (NQR) measurements on the superconductor BaTi${}_{2}$Sb${}_{2}$O with a two-dimensional Ti${}_{2}$O square-net layer formed with Ti${}^{3+}$ (3${d}^{1}$). NQR measurements revealed that the in-plane four-fold symmetry is broken at the Sb site below ${T}_{\mathrm{A}}\ensuremath{\sim}40$ K, without an internal field appearing at the Sb site. These exclude a spin-density wave (SDW)/ charge density wave (CDW) ordering with incommensurate correlations, but can be understood with the commensurate CDW ordering at ${T}_{\mathrm{A}}$. The spin-lattice relaxation rate $1/{T}_{1}$, measured at the four-fold symmetry breaking site, decreases below superconducting (SC) transition temperature ${T}_{\mathrm{c}}$, indicative of the microscopic coexistence of superconductivity and the CDW/SDW phase below ${T}_{\mathrm{A}}$. Furthermore, $1/{T}_{1}$ of ${}^{121}$Sb-NQR shows a coherence peak just below ${T}_{\mathrm{c}}$ and decreases exponentially at low temperatures. These results are in sharp contrast with those in cuprate and iron-based superconductors, and strongly suggest that its SC symmetry is classified to an ordinary $s$-wave state.

Spontaneous fourfold-symmetry breaking driven by electron-lattice coupling and strong correlations in high-Tccuprates

Using dynamical-mean-field theory for clusters, we study the two-dimensional Hubbard model in which electrons are coupled with the orthorhombic lattice distortions through the modulation in the hopping matrix. Instability towards spontaneous symmetry breaking from a tetragonal symmetric phase to an orthorhombic distorted phase is examined as a function of doping and interaction strength. A very strong instability is found in the underdoped pseudogap regime when the interaction strength is large enough to yield the Mott insulating phase at half filling. The symmetry breaking accompanies the recovery of quasiparticle weights along one of the two antinodal directions, leading to the characteristic Fermi arc reconnection. We discuss the implications of our results to the fourfold symmetry breaking reported in systems where the underlying crystal does not have any structural anisotropy.

Superconductivity and electronic liquid-crystal states in twin-free YBa2Cu3O6+x studied by neutron scattering

In this paper, we first give a concise overview of recent experimental and theoretical work dealing with “electronic liquid-crystal states” which spontaneously break different symmetries of the CuO2 layers of high-T c cuprates, with an emphasis on evidence in the spin excitation spectrum. Then we describe the importance of using twin-free samples to look for evidence for fourfold symmetry breaking in the spectrum and explain the preparation procedure to obtain such samples. We present inelastic neutron scattering results for moderately underdoped YBa2Cu3O6.6(T c = 61 K) and nearly optimally doped YBa2Cu3O6.85(T c = 89 K). In YBa2Cu3O6.6, the dispersion topology changes when heating above T c from an hourglass shape with constricted, commensurate resonance peak to a “Y”-shape without resonance anomaly. This change, and the fact that the low-energy signal above T c can be described by an incommensurate, quasi-one-dimensional distribution, indicates a competition of superconductivity with an electronic liquid-crystal state. We then show a striking analogy between the difference signal I(5 K) − I(70 K) and the downward dispersing resonance mode in YBa2Cu3O6.85. We therefore argue that a resonance mode only emerges below T c, irrespective of the doping level. We finally discuss the implications of our results for the different scenarios invoked to explain the electronic liquid-crystal state in cuprates.

Breakdown of fourfold symmetry of Fermi surface and magnetic excitation spectrum

Performing the slave-boson mean-field approximation to the two-dimensional t– J model under the assumption of the uniform charge density, we find two mechanisms that lead to the breaking of fourfold symmetry of the Fermi surface (FS) and the magnetic excitation spectrum. One mechanism originates from the J-term, which has an intrinsic instability toward the formation of a quasi-one-dimensional FS. The other mechanism comes from the inclusion of the small interlayer hopping t ⊥, which can lead to the drastic breaking of fourfold symmetry in the diagonal incommensurate magnetic peaks around (π, π). These findings are discussed with emphasis on possible relevance to La 2− x Sr x CuO 4 systems.

Square vortex lattice in p-wave superconductors

Making use of the Ginzburg Landau equation for isotropic p-wave superconductors, we construct the single vortex solution in part analytically. The fourfold symmetry breaking term arising from the tetragonal symmetry distortion of the Fermi surface is crucial, since this term indicates a fourfold distortion of the vortex core somewhat similar to the one found in d-wave superconductors. This fourfold distortion of the vortex core in turn favors the square vortex lattice as observed recently by small angle neutron scattering (SANS) experiment from Sr 2 RuO 4 . We find that the hexagonal vortex lattice at H = R c1 transforms into the square one for H = H cr = 0.26H c2 . On the other hand the SANS data does not reveal such transition. The square vortex covers everywhere studied by the SANS implying R cr is very close to H c1 . Therefore some improvement in the present model is certainly desirable.

Continuously varying exponents and the value of the central charge

The author shows that the conformal field theories describing two-dimensional critical behaviour with continuously varying exponents must have central charge c=1 if (a) there are no conserved spin-2 currents other than the stress tensor and (b) the marginal operator responsible for the line of fixed points does not mix with other operators. In such cases one may show the existence of a fixed line knowing only the four-point function of the marginal operator. The author applies this to the Ashkin-Teller model and the XY model with fourfold symmetry breaking.