We introduce a one-dimensional model which interpolates between the Ising model and the quantum compass model with frustrated pseudospin interactions σ z i σ z i+1 and σ x i σ x i+1 , alternating between even/odd bonds, and present its exact solution by mapping to quantum Ising models. We show that the nearest neighbor pseudospin correlations change discontinuosly and indicate divergent correlation length at the first order quantum phase transition. At this transition one finds the disordered ground state of the compass model with high degeneracy 2 × 2 N/2 in the limit of N → ∞.

Quantum phase transitions in the two-dimensional Kugel-Khomskii model on a square lattice are studied using the plaquette mean field theory and the entanglement renormalization Ansatz. When 3z(2)-r(2) orbitals are favored by the crystal field and Hund's exchange is finite, both methods give a noncollinear exotic magnetic order that consists of four sublattices with mutually orthogonal nearest-neighbor and antiferromagnetic second-neighbor spins. We derive an effective frustrated spin model with second- and third-neighbor spin interactions which stabilize this phase and follow from spin-orbital quantum fluctuations involving spin singlets entangled with orbital excitations.

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We investigate the changes in spin and orbital patterns induced by magnetic transition metal ions without an orbital degree of freedom doped in a strongly correlated insulator with spin-orbital order. In this context we study the 3d ion substitution in 4d transition metal oxides in the case of 3d 3 doping at either 3d 2 or 4d 4 sites which realizes orbital dilution in a Mott insulator. Although we concentrate on this doping case as it is known experimentally and more challenging than other oxides due to finite spin-orbit coupling, the conclusions are more general. We derive the effective 3d−4d (or 3d − 3d) superexchange in a Mott insulator with different ionic valencies, underlining the emerging structure of the spin-orbital coupling between the impurity and the host sites and demonstrate that it is qualitatively different from that encountered in the host itself. This derivation shows that the interaction between the host and the impurity depends in a crucial way on the type of doubly occupied t2g orbital. One finds that in some cases, due to the quench of the orbital degree of freedom at the 3d impurity, the spin and orbital order within the host is drastically modified by doping. The impurity acts either as a spin defect accompanied by an orbital vacancy in the spinorbital structure when the host-impurity coupling is weak, or it favors doubly occupied active orbitals (orbital polarons) along the 3d−4d bond leading to antiferromagnetic or ferromagnetic spin coupling. This competition between different magnetic couplings leads to quite different ground states. In particular, for the case of a finite and periodic 3d atom substitution, it leads to striped patterns either with alternating ferromagnetic/antiferromagnetic domains or with islands of saturated ferromagnetic order. We find that magnetic frustration and spin degeneracy can be lifted by the quantum orbital flips of the host but they are robust in special regions of the incommensurate phase diagram. Orbital quantum fluctuations modify quantitatively spin-orbital order imposed by superexchange. In contrast, the spin-orbit coupling can lead to anisotropic spin and orbital patterns along the symmetry directions and cause a radical modification of the order imposed by the spin-orbital superexchange. Our findings are expected to be of importance for future theoretical understanding of experimental results for 4d transition metal oxides doped with 3d 3 ions. We suggest how the local or global changes of the spin-orbital order induced by such impurities could be detected experimentally.

We derive the Kugel-Khomskii spin-orbital (SO) model for a bilayer and investigate its phase diagram depending on Hund's exchange $J_H$ and the $e_g$ orbital splitting $E_z$. In the (classical) mean-field approach with on-site spin $$ and orbital $<\tau_i^z>$ order parameters and factorized spin-and-orbital degrees of freedom, we demonstrate a competition between the phases with either $G$-type or $A$-type antiferromagnetic (AF) or ferromagnetic long-range order. Next we develop a Bethe-Peierls-Weiss method with a Lanczos exact diagonalization of a cube coupled to its neighbors in $ab$ planes by the mean-field terms --- this approach captures quantum fluctuations on the bonds which decide about the nature of disordered phases in the highly frustrated regime near the orbital degeneracy. We show that the long-range spin order is unstable in a large part of the phase diagram which contains then six phases, including also the valence-bond phase with interlayer spin singlets stabilized by holes in $3z^2-r^2$ orbitals (VB$z$ phase), a disordered plaquette valence-bond (PVB) phase and a crossover phase between the VB$z$ and the $A$-type AF phase. When on-site SO coupling is also included by the $$ order parameter, we discover in addition two entangled phases which compete with $A$-type AF phase and another crossover phase in between the $G$-AF phase with occupied $x^2-y^2$ orbitals and the PVB phase. Thus, the present bilayer model provides an example of SO entanglement which generates novel disordered phases. We analyze the order parameters in all phases and identify situations where SO entanglement is crucial and factorization of the spins and orbitals leads to qualitatively incorrect results. We point out that SO entanglement may play a role in a bilayer fluoride K$_3$Cu$_2$F$_7$ which is a realization of the VB$z$ phase.Comment: 22 pages, 21 figure

We study zero temperature phase diagram of the three-dimensional Kugel-Khomskii model on a cubic lattice using the cluster mean field theory and different perturbative expansions in the orbital sector. The phase diagram is rich, goes beyond the single-site mean field theory due to spin-orbital entanglement. In addition to the antiferromagnetic (AF) and ferromagnetic (FM) phases, one finds also a plaquette valence-bond phase with singlets ordered either on horizontal or vertical bonds. More importantly, for increasing Hund's exchange we identify three phases with exotic magnetic order stabilized by orbital fluctuations in between the AF and FM order: (i) an AF phase with two mutually orthogonal antiferromagnets on two sublattices in each ab plane and AF order along the c axis (ortho-G-type phase), (ii) a canted-A-type AF phase with a non-trivial canting angle between nearest neighbor FM layers along the c axis, and (iii) a striped-AF phase with anisotropic AF order in the ab planes. We elucidate the mechanism responsible for each of the above phases by deriving effective spin models which involve second and third neighbor Heisenberg interactions as well as four-site spin interactions going beyond Heisenberg physics, and explain how the entangled nearest neighbor spin-orbital superexchange generates spin interactions between more distant spins. Published in Physical Review B 87, 064407 (2013).

We introduce a spin ladder with antiferromagnetic Ising ZZ interactions along the legs, and interactions on the rungs which interpolate between the Ising ladder and the quantum compass ladder. We show that the entire energy spectrum of the ladder may be determined exactly for finite number of spins 2N by mapping to the quantum Ising chain and using Jordan-Wigner transformation in invariant subspaces. We also demonstrate that subspaces with spin defects lead to excited states using finite size scaling, and the ground state corresponds to the quantum Ising model without defects. At the quantum phase transition to maximally frustrated interactions of the compass ladder, the ZZ spin correlation function on the rungs collapses to zero and the ground state degeneracy increases by 2. We formulate a systematic method to calculate the partition function for a mesoscopic system, and employ it to demonstrate that fragmentation of the compass ladder by kink defects increases with increasing temperature. The obtained heat capacity of a large compass ladder consisting of 2N=104 spins reveals two relevant energy scales and has a broad maximum due to dense energy spectrum. The present exact results elucidate the nature of the quantum phase transition from ordered to disordered ground state found in the compass model in two dimensions.Comment: 15 pages, 11 figures, to appear in Phys. Rev.

We present rigorous topological order which emerges in a one-dimensional spin-orbital model due to the ring topology. Although an exact solution of a spin-orbital ring with SU(2) spin and XY orbital interactions separates spins from orbitals by means of a unitary transformation, the spins are not independent when the ring is closed, but form two half-rings carrying opposite pseudomomenta. We show that an inverse transformation back to the physical degrees of freedom entangles the spin half-rings with the orbitals once again. This surprising correlation arises on changing the topology from an open to a closed chain, which reduces the degeneracy of the ground-state manifold, leaving in it only the states in which pseudomomenta compensate each other. Spin-orbital physics is one of the foundation stones in the theory of frustrated magnetism [1][2][3][4][5]. When degenerate 3d orbitals in a transition-metal oxide are partly filled, electrons localize due to large on-site Coulomb interaction and superexchange between magnetic ions includes both spin and orbital degrees of freedom that are strongly interrelated [6]. The orbital degeneracy leads in many cases to a dramatic increase of quantum fluctuation [7], which may trigger exotic order [8] or may stabilize a spin-liquid [9, 10] when different states compete near a quantum critical point. While spin-orbital separation is possible in one-dimensional (1D) systems [11], as observed recently in Sr 2 CuO 3 [12], spins and orbitals are usually entangled strongly, as in the archetypal KugelKhomskii model [13]. In the S = 1/2 SU(2)⊗SU(2) chain [14,15], both ground state [16] and excited states [17] are entangled, similar to the S = 1 SU(2)⊗SU(2) chain which plays a prominent role in perovskite vanadates [3][4][5]18]. Only in exceptional cases can such 1D models be solved exactly, for example at the SU(4) point [19] or for a valence-bond state [20] of alternating spin and orbital singlets [21], but even in these situations the spins and orbitals cannot be separated from each other.In real materials the symmetry between spin and orbital interactions is absent. Orbital interactions generically have lower symmetry than spin ones [22], being usually Ising-or XY-like [23]. The XY case is quantum and in general the orbitals cannot be separated from the spins [24]. In this context the 1D spin-orbital SU(2)⊗XY model introduced by Kumar [25] is surprising -by a change of basis, the S = 1/2 spins decouple from the orbitals in an open chain. The orbital interactions remain formally unchanged but the spin ones are gauged away. The spins then appear free and the ground state has large degeneracy (2 L for chain length L) [25].Frustrated spin systems are at the forefront of modern condensed matter theory and experiment [26][27][28], in large part for the investigation of topology in manybody physics. A particular manifestation is the topological spin liquid (TSL) [29], a category including resonating valence-bond (RVB) states [30] and states hosting excitations with non-Abelian fractional ...