If several orbitals contribute to low-energy states of a transition metal oxide and if Coulomb repulsion is strong, charge fluctuations are suppressed and the Mott insulator resulting for one electron per site be described by a Kugel-Khomskii-type model , where spin spin and orbital degrees of freedom become formally almost equivalent. Excitations give information about the quantum dynamics, e.g., magnons in ferromagnetic manganites show that the main impact of alternating orbital order is a reduced magnon band width . Apart from this reduced band width, the orbital background does not show up in the magnon dispersion, which implies that one can describe this situation by decoupling spins and orbitals in a mean-field approximation.
The analogous situation to a ferromagnet with alternating orbitals, but with reversed roles for spins and orbitals, is a ferro-orbitally ordered antiferromagnet, see the first row of the figure to the left. Here, the orbital degree of freedom is "polarized" like the spin the ferromagnet and the spins alternate. The analogon to a spin flip is then an orbital excitation. The excitation can move (third and fourth rows) via virtual excitations with doubly occupied sites (second row). In the fourth row, "magnetic" and "orbital" parts of the original excitations have moved apart , in a similar manner to spin-charge separation undergone by a hole in an antiferromagnetic (AF) chain.
Indeed, it turns out that the situation of the orbital excitation can be mapped onto the case of a hole moving in an AF background, at least for small Hund's rule coupling. As a consequence, this reversed situation leads to fundamentally different excitation spectra than the magnon case mentioned above, as can be seen in the figure to the right, where the spectrum obtained via the mapping is compared to the the line one would obtain in the mean-field decoupling. There are signature of spin-"charge" separation in one dimension, the periodicity of the dispersion clearly reflects the doubled unit cell of the AF background and additional incoherent features arise [3,4]. The decisive impact of the lowered symmetries in the orbital sector is thus revealed in the differences between magnons and orbital excitations. Recently, spin-orbit separation has been observed in resonant inelastic x-ray scattering (RIXS) for precisely this case, ferro-orbital and antiferromagnetic one-dimensional chains in Sr2CuO3 .
The same mapping and analysis have also been applied to an orbital excitation in the two-dimensional iridate Sr2IrO4. In this system, spin-orbit coupling is very strong, so that the half-filled orbital consists of eigenstates of the total angular momentum with j=1/2 rather than of spins in an orbital. Nevertheless, the orbital excitation into the higher levels with j=3/2 can again be mapped onto a hole moving in an AF background, here with an additional orbital flavor. The mapping and the evaluation of the spectral density using the self-consistent Born approximation give excellent agreement with RIXS data .
We also investigate the impact of core-hole properties on the spectra, using numerical techniques . Open questions in this context are the impact of stronger Hund's rule which couples the spin of the excited electron to the background.
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Emergent frustration and Nematic states in double-exchange models
Localized spins in strongly correlated materials, e.g. transition metal oxides, can lead to particularly fascinating properties when they form a frustrated lattice, where not all interactions can be optimized. Examples include magnetic monopoles in spin ice or large thermopower due to the presence of many nearly degenerate states.
In our work , we find that on the unfrustrated honeycomb lattice, geometric frustration can spontaneously emerge as a consequence of the frustration between antiferromagnetic (AF) magnetic interactions and ferromagnetism driven by itinerant electrons. For relatively weak AF exchange, hexagons of almost ferromagnetic (FM) spins form (there is a small canting between them). Interactions between hexagons are AF and since the hexagons form a triangular lattice, their spins order in the manner expected for individual spins on a frustrated triangular lattice , namely with a 120-degree angle between NN hexagons. The resulting state is shown to the left.
For somewhat stronger AF interactions, smaller FM building blocks arise, namely dimers. Order between the dimers is AF, however, spins only order along one direction and remain perfectly uncorrelated along the other direction. This can be seen by noticing that the two states shown to the right are both valid dimer coverings and yet have different spin-spin correlation along the x-axis. Such a nematic order implies a ground-state degeneracy that is proportional to the number of columns, i.e., to the square root of the system size. It is thus intermediate between 0 and macroscopic and is related to a symmetry intermediate between global and local (gauge-like). Such symmetries are realized in Hamiltonians like the compass model. In the present case, in contrast, the peculiar symmetry is here not a property of the Hamiltonian, but emerges spontaneously as a property of the ground-state manifold.
A spontaneous decoupling into almost independent stripes was also found in a more complex two-orbital double exchange model including phonons . The model is motivated by narrow-band manganites, at hole doping 1/n, stripes of collinear spins and width n can be found. Spins in adjacent stripes are at right angles, but switching the relative orientation of stripes at larger distances only changes the total energy by a negligible amount. The spins thus form nematic order again and decompose into 1D stripes, so it is somewhat surprising that the kinetic energy of the electrons is fully two-dimensional and also depends on the long-range spin order. The reason for the nematic decoupling of the spin stripes is found in the directionality of the eg orbitals: at the border of the stripes, dispersionless states emerge that suppress electronic hoppings that would hybridize occupied and unoccupied states.
We thus find that frustration between competing charge, spin, and orbital degrees of freedom can lead to exotic magnetic phases reminiscent of geometrically frustrated lattices.
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Fractional Chern insulators in strongly correlated multi-orbital systems
The oldest known "topologically nontrivial" phase are integer quantum Hall states, which are characterized by a quantized invariant and thus robust against distortions of the system. As a consequence, the quantization of the Hall conductance is extremely precise and the von-Klitzing constant is thus used as a standard for electrical resistance. Analogous states can also arise by breaking of time-reversal symmetry by other means that by a magnetic field , e.g. via spin-orbit coupling, as in topological insulators .
Recently, it was proposed that nearly flat and topologically nontrivial bands in lattice models might allow an analogous generalization of fractional quantum-Hall (FQH) states . If the Coulomb repulsion is both large compared to the band width and small compared to the gap separating the band from its neighbors, it can stabilize FQH-like states.
We were able to show that an orbital degree of freedom can substantially flatten bands, which have a topological character due to an "effective magnetic flux" induced by non-coplanar "chiral" magnetic order . In the figure to the right, the spin pattern of the chiral phase on the triangular lattice is schematically shown: The four spin of the unit cell, in (a), point to the corners of a tetrahedron, see (b); the chirality Si(SjxSk) enclosed by three spins around a triangular plaquette is non-zero.
We showed this effect for a chiral state on both the triangular and the kagome lattices, and find that is work for either eg or t2g orbitals. The eg orbitals are also shown in panel (d), where we illustrate how octahedra in perovskites build a triangular lattice.
In our first study, we had built on the fact that frustrated lattices are known to be able to support chiral magnetic phases. We were then able to show that a chiral phase with topologically nontrivial and very flat bands arises self-consistently in a strongly correlated three-orbital model for t2g orbitals on a triangular lattice . The figure to the right shows an example for the one-particle bands obtained in mean field. Here, periodic boundary conditions were used along one directions, and open along the other, i.e., we use a tube. Momentum is thus only conserved along the first direction and gives the x-axis of the plot. For each kx, one finds several bands, their width is determined by the dispersion along the y-direction. One clearly sees some very flat bands of mostly a1g character, especially the one directly above the chemical potential. The red and blue dashes and dots decorate edge states living on the top and bottom edges of the tube, which are absent for fully periodic boundary conditions. They connect the bands above and below the chemical potential and are an indication of the topologically nontrivial character of the bands.
The band-flattening effect is very robust in this model, both the chiral phase and the nearly flat bands arise for large parameter regions. As the flat band is well separated from other bands, one can map it onto an effective one-band model with a two-site unit cell. The two-site unit cell is given by the chiral magnetic order and allows the nontrivial topological character. The flatness of the band is caused by effective longer-range hopping due to virtual excitations.
The much simpler one-band model can be treated on finite clusters with exact diagonalization and we were indeed able to establish signatures of FQH-like states for a large number of filling fractions [5,6]. These Fractional Chern insulators reveal themselves through a variety of signatures, both in the energy levels and in a fractional value of the Hall conductance. Of course, FCI states compete with other phases; as they rely on long-range Coulomb repulsion, charge-density waves are obvious competitors. We study the stability of both phases and find that the charge-density wave wins if it is favored by Fermi-surface nesting. If the Fermi surface is not well nested, even quite dispersive bands can host a fractional Chern insulator .
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As several (at the very least two) orbitals contribute weight to states near the Fermi level of iron-based superconductors, the question of orbital physics naturally arises . Recently, we looked at one particular issue, namely the rotational symmetry breaking associated with a structural phase transition, which occurs at temperatures slightly above the onset of a spin-density wave in several compounds . It could be due to the lattice distortion, but the differences in bond lengths seem too small to explain a spectral density A(k, ω) whose broad features look closer to the spin-density wave than to the symmetric high-temperature state. Spontaneous orbital order could explain this , however, tendencies to orbital polarization in the spin-density wave itself are very weak . Another possible explanation is based on breaking the rotational symmetry through spin fluctuations, which select ordering vector (π,0) over the equivalent (0,π) without establishing long-range magnetic order right way . A(k, ω) can readily be calculated in the first case, at least on a mean-filed level, but the same straight-forward approach does not work in the spin-nematic scenario: we need short-range spin correlations to break rotational symmetry, but mean-field would then immediately stabilize long-range order. We worked around this issue by coupling small real-space clusters (where short-range correlations break rotational symmetry) in momentum space (without long-range magnetic order) in cluster-perturbation theory. This allows us to establish that a spin-nematic state agrees better with experiment than brute-force orbital order . Results become a bit more complicated once onsite Coulomb correlations are included, as orbital and nematic order then lead to more similar predictions for A(k, ω). However, we found that correlations do not make it easier to induce orbital order, making spontaneous orbital order a less likely scenario . Moreover, total orbital polarization is not a good predictor of features near the Fermi surface, as states with and without orbital order may differ mostly in spectral weight at higher excitation energies . Orbital character of low-energy states and total orbital occupation numbers, orbital polatization as in the orbital-order scenario, can thus be quite different. Finally, we also found that pure lattice distortions cannot give the experimentally observed A(k, ω), with or without onsite Coulomb correlations .
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