Orbital Fluctuations Research Articles

Raman scattering as a probe of nematic correlations

We use the symmetry constrained low energy effective Hamiltonian of iron based superconductors to study the Raman scattering in the normal state of underdoped iron-based superconductors. The incoming and scattered Raman photons couple directly to orbital fluctuations and indirectly to the spin fluctuations. We computed both couplings within the same low energy model. The symmetry constrained Hamiltonian yields the coupling between the orbital and spin fluctuations of only the same symmetry type. Attraction in B2g symmetry channel was assumed for the system to develop the subleading instability towards the discrete in-plane rotational symmetry breaking, referred to as Ising nematic transition. We find that upon approaching this instability, the Raman spectral function develops a quasi-elastic peak as a function of energy transferred by photons to the crystal. We attribute this low-energy B2g scattering to the critical slow-down associated with the build up of nematic correlations.

Two kinds of in-plane resistivity anisotropy inFe1+δTe(δ=0.09)as seen via synchrotron radiation x-ray diffraction andin situresistivity measurements

We have investigated correlation between structural and electronic anisotropies in a parent compound of Fe-chalcogenide superconductor ${\mathrm{Fe}}_{1+\ensuremath{\delta}}\mathrm{Te}$ with $\ensuremath{\delta}=0.09$ by means of synchrotron x-ray diffraction and in situ in-plane resistivity anisotropy measurements with uniaxial stress applied along a tetragonal $a$ axis. This system is known to exhibit a tetragonal-to-monoclinic structural transition at ${T}_{\mathrm{S}}\ensuremath{\sim}60$ K. We have confirmed that the in-plane resistivity anisotropy in the low-temperature monoclinic phase is attributed to the asymmetry in volume fractions of the monoclinic domains, as was suggested in a previous study [Jiang et al., Phys. Rev. B 88 115130 (2013)]. On the other hand, we found another in-plane resistivity anisotropy above ${T}_{\mathrm{S}}$. The present x-ray diffraction and resistivity anisotropy measurements have revealed that this anisotropy is not due to an onset of the low-temperature monoclinic phase but to the lattice softening enhanced toward ${T}_{\mathrm{S}}$. As one of the possibilities, we suggest that the orbital fluctuation contributes to the lattice softening and the resistivity anisotropy above ${T}_{\mathrm{S}}$.

Spin-orbital liquid and quantum critical point inY1−xLaxTiO3

The specific heat, the susceptibility under pressure, and the dielectric constant were measured for single crystals ${\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{La}}_{x}{\mathrm{TiO}}_{3}$. The observed ${T}^{2}$-dependent specific heat at low temperatures for $0.17\ensuremath{\le}x\ensuremath{\le}0.3$ samples shows a spin-orbital liquid state between the ferromagnetic/orbital ordering $(xl0.17)$ and antiferromagnetic/possible orbital liquid phase $(xg0.3)$. The nonmonotonous pressure dependence of ${T}_{\text{C}}$ and the glassy behavior of the dielectric loss for the $x=0.23$ sample suggest that it is approaching a possible quantum critical point. All these properties result from the coupling between the strong spin and orbital fluctuations while approaching the phase boundary.

Emergence of Orbital Nematicity in the Tetragonal Phase of BaFe2(As1−xPx)2

We report on $^{75}$As-NMR measurements in single crystalline BaFe$_{2}$(As$_{0.96}$P$_{0.04}$)$_{2}$ for magnetic fields parallel to the orthorhombic $[110]_{\rm o}$ and $[100]_{\rm o}$ directions above the structural transition temperature $T_{\rm S}\simeq121$ K. A large difference in the linewidth between the two field directions reveals in-plane anisotropy of the electric field gradient, even in the tetragonal phase. This provides microscopic evidence of population imbalance between As-$4p_{x}$ and $4p_{y}$ orbitals, which reaches $|n_x-n_y|/|n_x+n_y| \sim 15$\% at $T\rightarrow T_{\rm S}$ and is a natural consequence of the orbital ordering of Fe-3$d_{xz}$ and $d_{yz}$ electrons. Surprisingly, this orbital polarization is found to be already static near room temperature, suggesting that it arises from the pinning of anisotropic orbital fluctuations by disorder. The effect is found to be stronger below $\sim 160$ K, which coincides with the appearance of nematicity in previous torque and photoemission measurements. These results impose strong constraints on microscopic models of the nematic state.

Spin-triplet superconductivity inSr2RuO4due to orbital and spin fluctuations: Analyses by two-dimensional renormalization group theory and self-consistent vertex-correction method

We study the mechanism of the triplet superconductivity (TSC) in Sr$_2$RuO$_4$ based on the multiorbital Hubbard model. The electronic states are studied using the recently developed renormalization group method combined with the constrained random-phase-approximation, called the RG+cRPA method. Thanks to the vertex correction (VC) for the susceptibility, which is dropped in the mean-field-level approximations, strong orbital and spin fluctuations at $Q \approx(2\pi/3,2\pi/3)$ emerge in the quasi one-dimensional Fermi surfaces (FSs) composed of $d_{xz}+d_{yz}$ orbitals. Due to the cooperation of both fluctuations, we obtain the triplet superconductivity in the $E_u$ representation, in which the superconducting gap is given by the linear combination of $(\Delta_x(k),\Delta_y(k))\sim (\sin 3k_x,\sin 3k_y)$. Very similar results are obtained by applying the diagrammatic calculation called the self-consistent VC method. Thus, the idea of "orbital+spin fluctuation mediated TSC" is confirmed by both RG+cRPA method and the self-consistent VC method. We also reveal that a substantial superconducting gap on the $d_{xy}$-orbital FS is induced from the gaps on the quasi one-dimensional FSs, in consequence of the large orbital-mixture due to the 4$d$ spin-orbit interaction.

Competition between Quadrupole and Magnetic Kondo Effects in Non-Kramers Doublet Systems

We discuss possible competition between magnetic and quadrupole Kondo effects in non-Kramers doublet systems in cubic symmetry. The quadrupole Kondo effect leads to non-Fermi-liquid (NFL) ground state, while the magnetic one favors ordinary Fermi-liquid (FL) ground state. In terms of the j-j coupling scheme, we argue that the orbital fluctuation must develop in the vicinity of the NFL-FL boundary. A change of temperature dependence of the f-electron entropy in both the FL and NFL regimes is demonstrated by the Wilson's numerical renormalization-group (NRG) method on the basis of the extended two-channel Kondo exchange model. We present implications to PrT2X20 (T=Ti, V, Ir; X=Al, Zn) systems which exhibit both quadrupole ordering and peculiar superconductivity. We discuss how the magnetic field lifts the non-Kramers degeneracy. Our model also represents the alternative FL state accompanied by a free magnetic spin, as a consequence of stronger competition between the magnetic and the quadrupole Kondo effects.

Ultrasound velocity measurements in orbital-degenerate frustrated spinel MgV2O4

Ultrasound velocity measurements of the orbital-degenerate frustrated spinel MgV2O4 are performed in the disorder-free high-purity single crystal which exhibits successive structural and antiferromagnetic phase transitions, and in the disorder-introduced single crystal which exhibits spin-glass-like behavior. The measurements reveal coexisting two types of anomalous temperature dependence of the elastic moduli in the cubic paramagnetic phase: Curie-type softening with decreasing temperature, and softening with a characteristic minimum with decreasing temperature. These elastic anomalies should respectively originate from the coexisting orbital fluctuations and spin-cluster excitations.

Spin-orbital exchange of strongly interacting fermions in the p band of a two-dimensional optical lattice.

Mott insulators with both spin and orbital degeneracy are pertinent to a large number of transition metal oxides. The intertwined spin and orbital fluctuations can lead to rather exotic phases such as quantum spin-orbital liquids. Here, we consider two-component (spin 1/2) fermionic atoms with strong repulsive interactions on the p band of the optical square lattice. We derive the spin-orbital exchange for quarter filling of the p band when the density fluctuations are suppressed, and show that it frustrates the development of long-range spin order. Exact diagonalization indicates a spin-disordered ground state with ferro-orbital order. The system dynamically decouples into individual Heisenberg spin chains, each realizing a Luttinger liquid accessible at higher temperatures compared to atoms confined to the s band.

Noise in oscillators: a review of state space decomposition approaches

We review the state space decomposition techniques for the assessment of the noise properties of autonomous oscillators, a topic of great practical and theoretical importance for many applications in many different fields, from electronics, to optics, to biology. After presenting a rigorous definition of phase, given in terms of the autonomous system isochrons, we provide a generalized projection technique that allows to decompose the oscillator fluctuations in terms of phase and amplitude noise, pointing out that the very definition of phase (and orbital) deviations depends of the base chosen to define the aforementioned projection. After reviewing the most advanced theories for phase noise, based on the use of the Floquet basis and of the reduction of the projected model by neglecting the orbital fluctuations, we discuss the intricacies of the phase reduction process pointing out the presence of possible variations of the noisy oscillator frequency due to amplitude-related effects.

Comparative studies of the thermal conductivity of spinel oxides with orbital degrees of freedom

We studied the thermal conductivity of various transition-metal oxides with the spinel structure $({\mathrm{MnV}}_{2}{\mathrm{O}}_{4}$, ${\mathrm{FeV}}_{2}{\mathrm{O}}_{4}$, ${\mathrm{CoV}}_{2}{\mathrm{O}}_{4}$, and ${\mathrm{Mn}}_{3}{\mathrm{O}}_{4})$ upon varying the temperature and magnetic field. We found that for the spinel oxides having ${\mathrm{V}}^{3+}$ ions (two electrons in the ${t}_{2g}$ states) at the octahedral site, the orbital ordering suppresses the thermal resistivity (inverse thermal conductivity) in contrast to purely magnetic ordering, indicating that the orbital fluctuation of the V ${t}_{2g}$ states is the main factor affecting the thermal conductivity. On the other hand, for ${\mathrm{Mn}}_{3}{\mathrm{O}}_{4}$, which has ${\mathrm{Mn}}^{3+}$ ions (one electron in the ${e}_{g}$ states) at the octahedral site, the thermal resistivity is suppressed in association with the successive magnetic phase transitions with decreasing temperature and increasing magnetic field. This can be explained by the frustration of ${\mathrm{Mn}}^{3+}$ spins and the fluctuation of ${\mathrm{Mn}}^{3+}$ ${e}_{g}$ orbitals.

Anisotropy of the superconducting gap in the iron-based superconductor BaFe2(As(1-x)P(x))2.

We report peculiar momentum-dependent anisotropy in the superconducting gap observed by angle-resolved photoemission spectroscopy in BaFe2(As1-xPx)2 (x = 0.30, Tc = 30 K). Strongly anisotropic gap has been found only in the electron Fermi surface while the gap on the entire hole Fermi surfaces are nearly isotropic. These results are inconsistent with horizontal nodes but are consistent with modified s± gap with nodal loops. We have shown that the complicated gap modulation can be theoretically reproduced by considering both spin and orbital fluctuations.

Multiple lattice instabilities resolved by magnetic-field and disorder sensitivities inMgV2O4

Ultrasound velocity measurements of the orbital-degenerate frustrated spinel MgV$_2$O$_4$ are performed in the high-purity single crystal which exhibits successive structural and antiferromagnetic phase transitions, and in the disorder-introduced single crystal which exhibits spin-glass-like behavior. The measurements reveal that two-types of unusual temperature dependence of the elastic moduli coexist in the cubic paramagnetic phase, which are resolved by magnetic-field and disorder sensitivities: huge Curie-type softening with decreasing temperature, and concave temperature dependence with a characteristic minimum. These elastic anomalies suggest the coupling of lattice to coexisting orbital fluctuations and orbital-spin-coupled excitations.

Anomalous Superconducting-Gap Structure of Slightly Overdoped Ba(Fe1−xCox)2As2

We observed the anisotropic superconducting-gap (SC-gap) structure of a slightly overdoped superconductor, Ba(Fe$_{1-x}$Co$_{x}$)$_{2}$As$_{2}$ ($x=0.1$), using three-dimensional (3D) angle-resolved photoemission spectroscopy. Two hole Fermi surfaces (FSs) observed at the Brillouin zone center and an inner electron FS at the zone corner showed a nearly isotropic SC gap in 3D momentum space. However, the outer electron FS showed an anisotropic SC gap with nodes or gap minima around the M and A points. The different anisotropies obtained the SC gap between the outer and inner electron FSs cannot be expected from all theoretical predictions with spin fluctuation, orbital fluctuation, and both competition. Our results provide a new insight into the SC mechanisms of iron pnictide superconductors.

Linear response theory for shear modulus C66 and Raman quadrupole susceptibility: evidence for nematic orbital fluctuations in Fe-based superconductors.

The emergence of the nematic order and fluctuations has been discussed as a central issue in Fe-based superconductors. To clarify the origin of the nematicity, we focus on the shear modulus C(66) and the Raman quadrupole susceptibility χ(x)(2)-y(2))(Raman). Because of the Aslamazov-Larkin vertex correction, the nematic-type orbital fluctuations are induced, and they enhance both 1/C(66) and χ(x(2)-y(2))(Raman) strongly. However, χ(x)(2)-y(2))(Raman) remains finite even at the structure transition temperature T(S), because of the absence of the band Jahn-Teller effect and the Pauli (intraband) contribution, as proved in terms of the linear response theory. The present study clarifies that the origin of the nematicity in Fe-based superconductors is the nematic orbital order and fluctuations.

Exotic Spin Order due to Orbital Fluctuations

We investigate the phase diagrams of the spin-orbital $d^9$ Kugel-Khomskii model for increasing system dimensionality: from the square lattice monolayer, via the bilayer to the cubic lattice. In each case we find strong competition between different types of spin and orbital order, with entangled spin-orbital phases at the crossover from antiferromagnetic to ferromagnetic correlations in the intermediate regime of Hund's exchange. These phases have various types of exotic spin order and are stabilized by effective interactions of longer range which follow from enhanced spin-orbital fluctuations. We find that orbital order is in general more robust and spin order melts first under increasing temperature, as observed in several experiments for spin-orbital systems.

Reproduction of experimental gap structure in LiFeAs based on orbital-spin fluctuation theory:s++-wave,s±-wave, and hole-s±-wave states

The absence of nesting between electron and hole pockets in LiFeAs with ${T}_{\mathrm{c}}=18$ K attracts great attention, as an important hint to understand the pairing mechanism of Fe-based superconductors. Here, we study the five-orbital model of LiFeAs based on the recently developed orbital-spin fluctuation theories. It is found that the experimentally observed gap structure of LiFeAs, which is a ``fingerprint'' of the pairing mechanism, is quantitatively reproduced in terms of the orbital-fluctuation-mediated ${s}_{++}$-wave state. Specifically, the largest gap observed on the two small hole pockets composed of (${d}_{xz},{d}_{yz}$) orbitals can be explained, and this is a hallmark of the orbital-fluctuation-mediated superconductivity. The ${s}_{++}$-wave gap structure becomes more anisotropic in the presence of weak spin fluctuations. As the spin fluctuations increase, we obtain the ``hole-${s}_{\ifmmode\pm\else\textpm\fi{}}$-wave state,'' in which only the gap of the large hole pocket made of the ${d}_{xy}$ orbital is sign reversed, due to the cooperation of orbital and spin fluctuations. This gap structure with ``sign reversal between hole pockets'' is similar to that recently reported in $(\mathrm{Ba},\mathrm{K}){\mathrm{Fe}}_{2}{\mathrm{As}}_{2}$.

Phase diagram and optical conductivity ofLa1.8−xEu0.2SrxCuO4

The absence of nesting between electron and hole-pockets in LiFeAs with $T_c = 18$K attracts great attention, as an important hint to understand the pairing mechanism of Fe-based superconductors. Here, we study the five-orbital model of LiFeAs based on the recently-developed orbital-spin fluctuation theories. It is found that the experimentally observed gap structure of LiFeAs, which is a "fingerprint" of the pairing mechanism, is quantitatively reproduced in terms of the orbital-fluctuation-mediated $s_{++}$-wave state. Especially, the largest gap observed on the small two hole-pockets composed of ($d_{xz}, d_{yz}$) orbitals can be explained, and this is a hallmark of the orbital-fluctuation-mediated superconductivity. The $s_{++}$-wave gap structure becomes more anisotropic in the presence of weak spin fluctuations. As the spin fluctuations increase, we obtain the "hole-$s_\pm$-wave state", in which only the gap of the large hole-pocket made of $d_{xy}$-orbital is sign-reversed, due to the cooperation of orbital and spin fluctuations. %out of the five pockets. This gap structure with "sign-reversal between hole-pockets" is similar to that recently reported in (Ba,K)Fe$_2$As$_2$.

LaSrVO4: A candidate for the spin-orbital liquid state

A layered perovskite ${\mathrm{LaSrVO}}_{4}$ was studied by neutron diffraction, pair distribution function measurement using synchrotron x-ray, susceptibility, and specific heat measurements, and first-principles calculation. The results show (i) a weak structural distortion around 100 K with the existence of orbital fluctuations both above and below it; (ii) the absence of the long range magnetic ordering down to 0.35 K but the appearance of a short range magnetic ordering around 11 K with a ${T}^{2}$ behavior of the specific heat below it. Meanwhile, the calculation based on the density functional theory predicts a magnetic ordered ground state. All facts indicate a melting of the magnetic ordering due to the orbital fluctuations in ${\mathrm{LaSrVO}}_{4}$, which makes it a rare candidate for the spin-orbital liquid state related to ${t}_{2g}$ orbitals.

Critical behavior of spinel MnV2O4 investigated by dc-magnetization

Critical behavior of spinel vanadate oxide MnV2O4 has been investigated by the dc-magnetization study. Critical exponents β = 0.349 ± 0.009, γ = 0.909 ± 0.002, and δ = 2.913 ± 0.003 are obtained in the critical asymptotic region around the critical temperature TC. With these critical exponents, experimental M–T–H curves collapse onto two independent universal branches below and above TC, respectively, indicating the reliability of the obtained results. The critical exponents γ and δ are close to the mean-field model with long-range interaction, while the value of β deviates to the 3D-Heisenberg model with short-range coupling. It is suggested that a long-range ferromagnetic coupling exists for Mn2+-Mn2+ interaction, which competes with the short-range interaction of V3+ sublattice. Moreover, the precursor phenomenon of the orbital ordering could be observed from the critical behavior, which indicates orbital fluctuation in MnV2O4.

Coexistence of orbital degeneracy lifting and superconductivity in iron-based superconductors

In contrast to conventional superconducting (SC) materials, superconductivity in high-temperature superconductors (HTCs) usually emerges in the presence of other fluctuating orders with similar or higher energy scales, thus instigating debates over their relevance for the SC pairing mechanism. In iron-based superconductors (IBSCs), local orbital fluctuations have been proposed to be directly responsible for the structural phase transition and closely related to the observed giant magnetic anisotropy and electronic nematicity. However, whether superconductivity can emerge from, or even coexist with orbital fluctuations, remains unclear. Here we report the angle-resolved photoemission spectroscopy (ARPES) observation of the lifting of symmetry-protected band degeneracy, and consequently the breakdown of local tetragonal symmetry in the SC state of Li(Fe1-xCox)As. Supported by theoretical simulations, we analyse the doping and temperature dependences of this band-splitting and demonstrate an intimate connection between ferro-orbital correlations and superconductivity.