The transition metal oxides have been the source of many fascinating physical phenomena such as high T

_{c}superconductivity, colossal magnetoresistance, and orbiton physics [ 1 ]. These surprising and diverse physical properties arise from strong correlation effects in the 3dbands. Most theoretical attempts to understand such systems are based on the Hubbard model. Here we report that for high symmetry transition metal oxides with threefoldtbands, this model possesses several novel hidden symmetries with many surprising consequences on the ground state properties [ 2, 3 ]._{2g}We consider cubic 3

d^{1}perovskites (i.e., ABO_{3}) where the five initially degenerate 3dstates are split into a two-foldemanifold and a lower-energy threefold_{g}tmanifold with wavefunctions d_{2g}_{yz}X, d_{xz}Y, and d_{xy }Z ( see Figure 1 ). Keeping only thetstates, we base our discussion on the following generic three-band Hubbard Hamiltonian:_{2g}

Here creates a

telectron ( or hole ) on the i_{2g}^{th}ion in the spatial orbital (i.e., = X, Y or Z ) with spin , and is the on-site energy of the orbital at siteiof a simple cubic lattice. is the on-site Coulomb repulsion between orbitals and at sitei. is the effective hopping parameter from the orbital at siteito the orbital at its nearest neighbour sitej.As shown in Figure 2, for cubic perovskites where the metal-oxygen-metal ( M-O-M ) bond is linear, the hopping parameter is diagonal in the orbital indices and . This suggests that the total number of electrons in each orbital is a good quantum number. Furthermore is zero along the “inactive” axis perpendicular to the orbital plane , due to symmetry ( Figure 2 ). In other words an -electron can only hop in -plane. Thus, for the n

^{th }plane perpendicular to the -axis, the total number N_{n alpha}of electrons in the -orbital is conserved,i.e., it is a good quantum number. Hence, one can consider the three dimensional lattice as a superposition of interpenetrating planes perpendicular to thex, y, andz-directions, each having a constant number of X, Y, and Z electrons, respectively, which are good quantum numbers ( see Figure 3 ( a ) ).In Reference [ 3 ] we also show that the global rotation of the spins of -orbital electrons

in any given planeperpendicular to the -axis leaves the Hubbard Hamiltonian given in Equation ( 1 ) invariant. As a consequence of this rotational symmetry, one may conclude that both the Hubbard model and its 2^{nd}order perturbation at order of t^{2}/U, the Kugel- Khomskii ( KK) Hamiltonian,cannotsupport any long range spin order: if oneassumeslong-range spin order, the spins associated with -orbitals within any given plane can be rotated at zero cost in energy, thereby destroying the supposed correlations among planes and/or among orbitals, and therefore the long-range order [ 2, 3 ]. The crucial conclusion here is that any credible theory of spin-ordering in these systemscannotbe based solely on the KK Hamiltonian, as currently done in the literature.The hidden symmetries discussed here are very useful in simplifying the exact numerical studies of small clusters. For example, to treat the simpler KK Hamiltonian for a cube of eight sites even using the conservation of the total spin ( a widely used symmetry ) requires the diagonalization of a matrix of dimensionality on the order of ˝ million. Using the conservation laws applied to each face of the cube (see Figure 3 ( a ) ), there is an astonishing numeri- cal simplification. The ground state can be found from a Hamiltonian matrix within a manifold of just 16 states!! As seen in Figure 3 ( b - d ), these states are all products of four dimer states in each of which two electrons are paired into a spin zero singlet state. In this model a very unusual phenomenon occurs: when the electrons hop from site to site along active axes, they retain their membership in the singlet they started in. An example of this transformation is seen by comparing Figures 3 ( c ) and 3 ( d ): pairs of electrons are tied together, as if by quantum mechanical rubber bands!

In conclusion, we uncovered several novel symmetries of the Hubbard model for orthogonal

tsystems. Using these symmetries, we rigorously showed that both the original Hubbard Hamiltonian [ 2 ] and the KK effective Hamiltonian [ 1 ] ( without spin-orbit interactions ) do not permit the development of long-range spin order in a three dimensional orthogonal lattice at nonzero temperature. It is important to take proper account of the symmetries identified here and to recognize that the observed longrange spin order can only be explained with the Hubbard or KK Hamiltonian providing suitable symmetry breaking terms are included. Such perturbations therefore play a crucial role in determining the observed behavior of these transition metal oxides. We hope that these results will inspire experimentalists to synthesize new_{2g}ttransition metal oxides with tetragonal or higher symmetry. Such systems would have quite striking and anomalous properties._{2g}## References:

[1] Y. Tokura and N. Nagaosa, Science

288, 462 (2000) and references therein.[2] A. B. Harris, T. Yildirim, A. Aharony, O. Entin-Wohlman, and I. Ya. Korenblit, Phys. Rev. Lett. (in press, 2003)(cond-mat/0303219).

[3] A. B. Harris, T. Yildirim, O. Entin-Wohlman, and A. Aharony, Phys. Rev. B (cond-mat/0307515).

T. Yildirim

NIST Center for Neutron Research

National Institute of Standards and Technology

Gaithersburg, MD 20899

A. B. Harris

University of Pennsylvania

Philadelphia, PA 19104

A. Aharony, O. Entin-Wohlman, and I. Ya. Korenblit

Tel Aviv University

Tel Aviv 69978, Israel

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