Entangled phase diagrams of the
2D Kugel-Khomskii models
Wojciech Brzezicki
Andrzej M. Oleś
M. Smoluchowski Institute of Physics, Kraków, Poland
Kugel-Khomskii model
• Spin-orbital model; Si - spins ½, τiγ (γ=a,b,c) - eg orbital operators
defined in terms of pseudospins ½ - σi.
• r1, r2 , r4 depend on Hund coupling JH and Hubbard U ratio η= JH/U:
• J is a superexchange constant: J=4t2/U
Meaning of the operator
• For all {i, γ}:
• Eigenstates:
-1/2:
1/2:
where orbitals’ orientation is given by γ. For all {i}:
Origin of the KK model
• Effective superexchange model for transition metal oxides (eg.
La2CuO4) or fluorides (eg. KCuF3) with a single hole localized at the
metal ion in d9 orbital configuration
• Only eg are active, no Jahn-Teller distortions
• Superexchange Hamiltonian contains: kinetic term Ht, Hund and
Hubbard interactions between the hole Hint and crystal-field splitting Hz
favorizing one of the eg 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 Ht as a perturbation (see: A. M. Oleś, L. F. Feiner,
and J. Zaanen, Phys. Rev. B 61, 6257 (2000))
Motivation for cluster mean-field calculations
for bilayer and single layer 2D systems
• Interlayer singlet phase found in a bilayer
fluride K3Cu2F7 by magnetic susceptibility
measurement using SQUID magnetometer
H. Manaka, et. al. J. Phys. Soc. Jpn. 76, 044710 (2007).
• Pressure induced phase transition in monolayer K2CuF4: 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 Ez in KK model can be regarded as
uniaxial pressure along c direction
• Rough estimation:
J=0.225 eV (for CuO2 planes and x orbitals)
Ez/P=4.5 meV/GPa (for t2g states of iron)
gives:
Ez=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)
• Rich phase diagram of a three-dimensional KCuF3:
L. F. Feiner, A. M. Oleś, and J. Zaanen, Phys. Rev. Lett. 78, 2799 (1997).
(cited 155 times)
• Role of spin-orbital entanglement in t2g systems;
violation of the Goodenough-Canamori rules
• In shaded area spins order AF despite Jij<0 for d1
and d2 models
• Cluster MF can capture all of these effects!
A. M. Oleś, P. Horsch, L. F. Feiner, and
G. Khaliullin, Phys. Rev. Lett. 96, 147205 (2006).
Single-site mean field for a bilayer
• The KK Hamiltonian:
• The MF decoupling:
• Because of single-site character of this approximation, no spin
flutuactions can be taken into account (total Mz is conserved)
• This means that spin and orbital degrees of freedom factorize
• We assume certain magnetic order to determine orbital order
• The order parameters are:
with γ=a,b and
γ γ
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
Single-site MF phase diagram for a bilayer
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 – <operator> bond
• We cover whole lattice with such clusters
Spin and orbital order
• Our order parameters are:
with γ=a,b
• 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: si = s
• AF phase – for all i є {1,4,5,8}: si = s and for all i є {2,3,6,7}: si = -s
+ assumption that neighboring clusters are identical
Orbital order:
• AO phase – for all i є {1,4,5,8}: tia,b = ta,b and for all i є {2,3,6,7}:
tia,b = tb,a + assumption that neighboring clusters are identical
• PVB phase – for all i є {1,...,8}: tia,b = ta,b + assumption that
neighboring clusters are rotated by π/2 in the ab plane
Spin-orbital order: we treat via,b as if via,b = s tia,b
• Example of the PVB orbital order:
Iterative procedure
• We choose some values of {s, ta,b, va,b}, desired order and (Ez, η)
• Using Lanczos algorith me find the ground state ane new {s, ta,b, va,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 va,b = s ta,b
Factorizable phase diagram for a bilayer
• Singlet phases shaded
yellow
• No perfect orderings
• 2nd order phase transition
between VBz and VBm
• High frustration for nonzero η
• No C-AF phase
Phase diagram for a bilayer
• Phases with SO entanglement
are shaded orange: ESO, EPVB
• Intermediate phase between
PVB and G-AF: continuous
transitions
• Four-critical point at
Order parameters, correlations and entanglement
• In what follows we will use the notation for the correlation functions:
where γ= direction of the bond <1,i >, i=2,3,7.
• The simplest parameters of SO entaglement are:
- on-site SO non-factorizibility
- bond SO non-factorizibility
Passage through the intermediate PVB-AF
Entangled phases
Passage through the intermediate VBm
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
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: Mz conservation and
invariance under the permutation of sites
(1→3, 2→4, 3→1, 4→2)
• Diagonal blocks of the cluster Hamiltonian:
Mz = 0
d+= 52, d– = 44,
Mz = ±1 d+= 32, d– = 32,
Mz = ±2 d+= 10, d– = 6.
Phase diagram for a monolayer in T=0
• Phases observed before
plus..
• Plaquette Liquid Phase
(orange)
PSL phase
• 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:
• the degeneracy is not lifted by the mean fields
• 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 via,b is:
Melting PSL
• Ez=-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*Ra,b
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
Thank you for your attention!
Gracias por su atención!