1. Entangled phase diagrams of the 2D Kugel-Khomskii models Wojciech Brzezicki Andrzej M. Oleś M. Smoluchowski Institute of Physics, Kraków, Poland

3. Kugel-Khomskii model Spin-orbital model; S i - spins ½, τ i γ (γ=a,b,c) - e g orbital operators defined in terms of pseudospins ½ - σ i. r 1, r 2, r 4 depend on Hund coupling J H and Hubbard U ratio η= J H /U: J is a superexchange constant: J=4t 2 /U

4. Meaning of the operator For all {i, γ}: Eigenstates: 1/2:-1/2: where orbitals’ orientation is given by γ. For all {i}:

5. Origin of the KK model Effective superexchange model for transition metal oxides (eg. La 2 CuO 4 ) or fluorides (eg. KCuF 3 ) with a single hole localized at the metal ion in d 9 orbital configuration Only e g are active, no Jahn-Teller distortions Superexchange Hamiltonian contains: kinetic term H t, Hund and Hubbard interactions between the hole H int and crystal-field splitting H z favorizing one of the e g orbitals: Effective hopping amplitudes between two sites depend on the shape of occupied orbitals – only cigar-cigar amplitudes are non-zero. Only two orthogonal orbitals for three space directions! Effective (Kugel-Khomskii) Hamiltonian can be derived from the atomic limit treating H t as a perturbation (see: A. M. Oleś, L. F. Feiner, and J. Zaanen, Phys. Rev. B 61, 6257 (2000))

6. Motivation for cluster mean-field calculations for bilayer and single layer 2D systems Interlayer singlet phase found in a bilayer fluride K 3 Cu 2 F 7 by magnetic susceptibility measurement using SQUID magnetometer H. Manaka, et. al. J. Phys. Soc. Jpn. 76, 044710 (2007).

7. Pressure induced phase transition in monolayer K 2 CuF 4 : AO order with FM spin configuration changes into FOx with AF order at 9-10 GPa Lattice structure changes: orthorombic Bbcm → Ammm Crystal-field splitting energy E z in KK model can be regarded as uniaxial pressure along c direction Rough estimation: J=0.225 eV (for CuO 2 planes and x orbitals) E z /P=4.5 meV/GPa (for t 2g states of iron) gives: E z =0.18-0.2 J for P=9-10 GPa M. Ishizuka, I. Yamada, K. Amaya and S. Endo, J. Phys. Soc. Jpn. 65, 1927 (1996)

8. Rich phase diagram of a three-dimensional KCuF 3 : L. F. Feiner, A. M. Oleś, and J. Zaanen, Phys. Rev. Lett. 78, 2799 (1997). (cited 155 times) A. M. Oleś, P. Horsch, L. F. Feiner, and G. Khaliullin, Phys. Rev. Lett. 96, 147205 (2006). Role of spin-orbital entanglement in t 2g systems; violation of the Goodenough-Canamori rules In shaded area spins order AF despite J ij <0 for d 1 and d 2 models Cluster MF can capture all of these effects!

9. Single-site mean field for a bilayer The KK Hamiltonian: This means that spin and orbital degrees of freedom factorize Because of single-site character of this approximation, no spin flutuactions can be taken into account (total M z is conserved) The MF decoupling:

10. We assume certain magnetic order to determine orbital order The order parameters are: where we assume the AO order as the most general in a single site approach Self consistency equations can be solved exactly (see W. Brzezicki and A. M. Oleś, Phys. Rev. B 83, 214408 (2011)) Comparing the ground-state energies of different phases we obtain phase diagram with γ=a,b and

11. Single-site MF phase diagram for a bilayer

12. Cluster MF for a bilayer Cluster apprach gives the possibility of spin fluctuations inside the cluster – we can get singlet phases and spin-orbital entanglement Our cluster: Solid lines = operator – operator bonds Dashed lines = operator – bond We cover whole lattice with such clusters

13. Spin and orbital order Our order parameters are: Configuration of site 1 determine all other sites inside and outside of the cluster depending on assumed spin and orbital order Spin order: FM phase – for all i: s i = s AF phase – for all i є {1,4,5,8}: s i = s and for all i є {2,3,6,7}: s i = -s + assumption that neighboring clusters are identical Orbital order: AO phase – for all i є {1,4,5,8}: t i a,b = t a,b and for all i є {2,3,6,7}: t i a,b = t b,a + assumption that neighboring clusters are identical PVB phase – for all i є {1,...,8} : t i a,b = t a,b + assumption that neighboring clusters are rotated by π /2 in the ab plane with γ=a,b

14. Spin-orbital order: we treat v i a,b as if v i a,b = s t i a,b Example of the PVB orbital order:

15. Iterative procedure We choose some values of {s, t a,b, v a,b }, desired order and (E z, η) Using Lanczos algorith me find the ground state ane new {s, t a,b, v a,b } We continue until the convergence is reached For AF phases we work in zero magnetization sector, d = 17920 For FM phases we work in maximal magnetization sector, d = 256 To determine the relevance of the non-factorizable SO field we first assume that v a,b = s t a,b

16. Factorizable phase diagram for a bilayer Singlet phases shaded yellow No perfect orderings 2 nd order phase transition between VBz and VBm High frustration for non- zero η No C-AF phase

17. Phases with SO entanglement are shaded orange: ESO, EPVB Intermediate phase between PVB and G-AF: continuous transitions Four-critical point at Phase diagram for a bilayer

18. Order parameters, correlations and entanglement In what follows we will use the notation for the correlation functions: where γ= direction of the bond, i=2,3,7. The simplest parameters of SO entaglement are: - on-site SO non-factorizibility - bond SO non-factorizibility

19. Passage through the intermediate PVB-AF

20. Entangled phases

21. Passage through the intermediate VBm

22. Bilayer KK model: conclusions Including quantum fluctuations within the cluster gives new phases comparing to the single-site MF phase diagram: PVB and VBz singlet phases and destroys perfect order Independent SO order parameter stabilizes new phases in the most frustrated part of the phase diagram Spin-orbital order can exist independently from magnetic and orbital orders Spin-orbital entaglement can exist in some phases away from the phase transitions, as a permanent effect

23. Cluster MF for a monolayer Cluster = four-site square Calculations in T>0 – thermal averages We consider phases: AO-FM and PVB-AF Symmetries used: M z conservation and invariance under the permutation of sites (1→3, 2→4, 3→1, 4→2) Diagonal blocks of the cluster Hamiltonian: M z = 0 d + = 52, d – = 44, M z = ±1 d + = 32, d – = 32, M z = ±2 d + = 10, d – = 6.

24. Phase diagram for a monolayer in T=0 Phases observed before plus.. Plaquette Liquid Phase (orange)

25. all inter-cluster spin-spin correlations are zero short-range FM order inside the cluster orbital and spin-orbital order each cluster is degenerate between two configurations: PSL phase the degeneracy is not lifted by the mean fields

26. average cluster configuration (not superposition) is characterized by: a) zero magnetization b) no orbital alternation (FO order) c) spin-orbital order such that for all i: and the global pattern for v i a,b is:

27. Melting PSL E z =-0.7J, η =0.17 thermal fluctuations lift the degeneracy global FM order appears, entropy S drops, orbitals start to alternate we get on-site non-factorizibility high but not necessarily entanglement orbital order survives longer than magnetization maximal entropy = 0.554 bond non-factorizibility low – plot shows 10*R a,b

28. Monolayer KK model: conclusions Two-fold degeneracy of the cluster’s ground state in certain parameter range for T=0 (and T>0) not lifed by the mean fields This stabilizes a spin liquid phase with uniform orbital order and SO order doubling the unit cell PSL phase exhibits SO non-factorizibility which is not an effect of SO entanglement even in T=0 Thermal decay of the PSL phase is due to the order by disorder mechanism and leads to the FM phase

29. Thank you for your attention! Gracias por su atención!