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Lattice instability in frustrated systems Maxim Mostovoy MPI, Stuttgart Groningen, April 22, 2004 D. Khomskii, Cologne J. Knoester, Groningen R. Moessner, Paris N.V. Prokof´ev, Amherst

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Outline Geometrical frustration Magnetoelastic transitions in frustrated spin systems Orbital interactions Frustration of orbital ordering and its lifting

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Geometrical frustration

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AFM Ising on triangular lattice Ground state entropy Spin correlations at T=0 (Wannier, Houtappel, 1950) (Stephenson, 1970)

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Ordered state

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Hard sphere liquid Alder & Wainwright (1957) Hoover & Ree (1968) S solid > S liquid

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Kinks Kink energy Interactions between kinks in neighboring chains

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Kink crystal 1 kink / 3 sites M.M., N.Prokof’ev, D.Khomskii & J.Knoester, PRL 90 (2003)

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Frustrated spins coupled to lattice Strains:

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Free energy

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Frustration-induced transition Free energy First order transition Ordered state Disordered state

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Cr 3+ S = 3/2 CW = -390K T N = 12.5K first order ZnCr 2 O 4 S.-H. Lee et al., PRL 84, 3718 (2000) O. Tchernyshyov et al., PRL 88, 067203 (2001)

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AV 2 O 4 V 3+ S = 1 first order cubic to tetragonal c < a T N (K) 40 45 35 T st (K) 50 65 97 Zn Mg Cd H. Mamiya et al., J. Appl. Phys. 81, 5289 (1997)

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Interactions between t 2g -orbitals

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Tetrahedron states

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Frustration and its lifting Jahn-Teller S.-H. Lee et al., cond-mat/0312558

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Orbital interactions Jahn-Teller interaction Coupled orbital & spin exchange (Kugel-Khomskii) `Orbital Casimir´ effect Peierls-like interaction

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e g -orbitals

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Jahn-Teller interaction

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Lattice-mediated interaction

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Kugel-Khomskii model (a): (b)(c)(d) (b)+(c): (a)

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Orbital Casimir effect (U p,U d = ) HolesElectrons M.M. & D. Khomskii, Phys.Rev.Lett 89, 227203 (2002), cond-mat 0304494

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180 0 -exchange in 2D K 2 CuF 4 1e g hole/Cu canted antiferroorbital ferromagnetic

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Frustrated orbital models xy yz zx xy yzzx triangular cubic pyrochlore 180 o -exchange 90 o -exchange LiNiO 2 ZnMn 2 O 4 xx yy zz KCuF 3

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Mean field Triangular, pyrochlore ferro-obital Cubic antiferro-orbital G-type Disordered ground states arbitrary superposition

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Order from disorder Quantum orbital fluctuations chose six unform states: G.Khaliullin, PR B 64, 212405 (2001) M.M. & D.Khomskii,PRL 89, 227203 (2002)

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Strains

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Orbital-strain interaction electron hole Jahn-Teller coupling Peierls coupling Strain energy

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Ordered state layered canted antiferroorbital

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Ordered state electrons holes LaMnO 3 KCuF 3

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Conclusions Frustration of spin and orbital ordering makes the crystal lattice instable JT distortion occurs together with lattice distortion lifting the frustration of orbital ordering

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