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# Mott Insulators in the Strong Spin-Orbit Coupling Limit: From Heisenberg to a Quantum Compass and Kitaev Models

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### Abstract

We study the magnetic interactions in Mott-Hubbard systems with partially filled $$t_{2g}$$-levels and with strong spin-orbit coupling. The latter entangles the spin and orbital spaces, and leads to a rich variety of the low energy Hamiltonians that extrapolate from the Heisenberg to a quantum compass model depending on the lattice geometry. This gives way to "engineer" in such Mott insulators an exactly solvable spin model by Kitaev relevant for quantum computation. We, finally, explain "weak" ferromagnetism, with an anomalously large ferromagnetic moment, in Sr$$_2$$IrO$$_4$$.

### Most cited references10

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### Spin current and magneto-electric effect in non-collinear magnets

(2005)
A new microscopic mechanism of the magneto-electric (ME) effect based on the spin supercurrent is theoretically presented for non-collinear magnets. The close analogy between the superconductors (charge current) and magnets (spin current) is drawn to derive the distribution of the spin supercurrent and the resultant electric polarization. Application to the spiral spin structure is discussed.
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### Complexity in Strongly Correlated Electronic Systems

(2005)
A wide variety of experimental results and theoretical investigations in recent years have convincingly demonstrated that several transition metal oxides and other materials, have dominant states that are not spatially homogeneous. This occurs in cases in which several physical interactions -- spin, charge, lattice, and/or orbital -- are simultaneously active. This phenomenon causes interesting effects, such as colossal magnetoresistance, and it also appears crucial to understand the high temperature superconductors. The spontaneous emergence of electronic nanometer-scale structures in transition metal oxides, and the existence of many competing states, are properties often associated with complex matter where nonlinearities dominate, such as soft materials and biological systems. This electronic complexity could have potential consequences for applications of correlated electronic materials, because not only charge (semiconducting electronic), or charge and spin (spintronics) are of relevance, but in addition the lattice and orbital degrees of freedom are active, leading to giant responses to small perturbations. Moreover, several metallic and insulating phases compete, increasing the potential for novel behavior.
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### Orbital liquid in three dimensional Mott insulator: $$LaTiO_3$$

(2000)
We present a theory of spin and orbital states in Mott insulator $$LaTiO_3$$. The spin-orbital superexchange interaction between $$d^1(t_{2g})$$ ions in cubic crystal suffers from a pathological degeneracy of orbital states at classical level. Quantum effects remove this degeneracy and result in the formation of the coherent ground state, in which the orbital moment of $$t_{2g}$$ level is fully quenched. We find a finite gap for orbital excitations. Such a disordered state of local degrees of freedom on unfrustrated, simple cubic lattice is highly unusual. Orbital liquid state naturally explains observed anomalies of $$LaTiO_3$$.
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### Author and article information

###### Journal
26 September 2008
2008-12-04
###### Article
10.1103/PhysRevLett.102.017205
0809.4658