Hidden Symmetries and Their Consequences in the Hubbard Model of t2g Electrons
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(1) The Hubbard model for t2g electrons in transition metal oxides possesses novel hidden symmetries that have significant consequences. (2) These symmetries prevent long-range spin order at non-zero temperatures and lead to an extraordinary simplification in exact diagonalization studies. (3) Even with spin-orbit interactions included, the excitation spectrum remains gapless due to a continuous symmetry arising from the hidden symmetries.
![MEAN-FIELD THEORY
THERE IS NO WAVEVECTOR SELECTION
)()()(
2
1 1
kkk
k
−=
−
∑ xxF χ
)(
1
k
−
χ )]cos()[cos(412 akakkT yx ++= εz
The spin susceptibility for z-flavor orbitals is
This has an instability to order in
antiferromagnetic planes which are active:
U
t2
=ε
akk yx /π== kz = anything](https://image.slidesharecdn.com/acns04brooksharris-170126074008/85/Hidden-Symmetries-and-Their-Consequences-in-the-Hubbard-Model-of-t2g-Electrons-16-320.jpg)
![Consequences of the Novel Hidden Symmetries
of t2g Hubbard Hamiltonian
(a) (b)
(c) (d)
n2z
n2y
n1z
n1y
x
y
z
n2x
n1x
>−
>=>≡
+
↑
+
↓
+
↓
+
↑
−
vac][2
)(|)(|
2/1
jxixjxix
xx
cccc
jiij
An i-j singlet wavefunction:
The exact ground state is
a sum of 16 dimer states,
which can be obtained from
the dominant configurations
(c) and (b) by hopping as
in panel (d).
Symmetry reduces 1.7 million to 16!!](https://image.slidesharecdn.com/acns04brooksharris-170126074008/85/Hidden-Symmetries-and-Their-Consequences-in-the-Hubbard-Model-of-t2g-Electrons-18-320.jpg)
![SPIN-ORBIT INTERACTIONS
With spin-orbit interactions one would think
that the spin knows where the crystal
axes are -> we expect anisotropy
WRONG!!
βναµ
αβγ
νµγγ σβλ ii
i
OS ccLaV +
− ∑∑ ><= ,][||
αττσασ i
t
a
i cVc
~)(
,∑= αα
α
σσ UV
)(
=with](https://image.slidesharecdn.com/acns04brooksharris-170126074008/85/Hidden-Symmetries-and-Their-Consequences-in-the-Hubbard-Model-of-t2g-Electrons-23-320.jpg)
Hidden Symmetries and Their Consequences in the Hubbard Model of t2g Electrons
- 1. Hidden Symmetries and Their Consequences in the Hubbard Model of t2g Electrons* A. B. HARRIS In collaboration with Dr. T. Yildirim, (NIST, Gaithersburg, MD) Profs. A. Aharony, Ora Entin and I. Kornblit (Tel Aviv University, Israel) Department of Physics and Astronomy ∗ Phys. Rev. B 69, 094409 (2004) and 69, 035107 (2004).
- 2. Outline • Motivations: High Tc oxides, CMR materials… • d-electrons in transition metal oxides • Hubbard Model of the t2g electrons • Novel Hidden Symmetries • Consequences 1. Absence of long-range spin order in the Kugel-Khomskii (KK) Hamiltonian 2. Extraordinary simplifications in numerical exact diagonalization studies 3. Gapless excitation spectrum even with spin- orbit interactions
- 3. Motivations High temperature superconductivity and colossal magnetoresistance sparked much recent interest in the magnetic properties of strongly correlated systems (i.e. transition metals), particularly those with orbital degeneracy. In most of the transition metal oxides (such as LaTiO3), one has to deal with not only the spin degrees of freedom but also the orbital degrees of freedom of the strongly correlated electrons. Recent studies indicate that superexchange interactions between ions with spin and orbital degrees of freedom is a fascinating problem and suggest the possibility of exotic ground states with strong interplay between spin and orbital sectors.
- 4. For an ion with a single 3d electron the cubic crystal field gives rise to a two-fold degenerate eg and a three-fold degenerate t2g manifold. d-electrons in transition metal oxides = negative = positive z yx z y x t2g e g Cubic crystal field 3d1 dxz=Y dyz=X dxy=Z
- 5. We first review the case when the on-site Coulomb interaction, U, is very large and there is one electron per site. Single-band Hubbard model Heisenberg Hamiltonian ↓↑∑∑ += + ii i ji ji ji nnUcctH , , −⋅= →→ 14 2 SS U t H To leading order in Ut / Perturbation processes
- 6. x y z t=0 t≠ 0 t≠ 0 Z (which is an xy wavefunction) can hop (via oxygen ions) to neighbors along either the x-axis or the y-axis, but NOT along the z-axis. The z-axis is called the ``inactive axis’’ for orbital ``flavor’’ Z (xy). + - +-
- 7. x y z t=0 t=0 t=0 A Z-flavor orbital can not hop into a different flavor (here we show Z trying to hop into Y). + - + -
- 8. Hubbard Hamiltonian Chop HHH +=Hubbard Hamiltonian: ασασ ασ α ji ij ijhop cctH + >< ∑∑= ),()( 2 1 iNiNUH i C βα αβ αβ∑∑= ασ σ ασα ii cciN ∑ + =)( + ασic creates an electron at site i in orbital α with spin σ
- 9. KUGEL – KHOMSKII (1975) H = ∑ ∑ <ij> ασ tα ij c† iασ cjασ ∑ iαβση + U c† iασ ciασ c† iβη ciβη J =x Σ α β =/ x Σση c† iα ciβ c† jβ cjα H= Hx HzH y + + .( )∑< −⋅= ji jixx SSJH 14
- 10. ``Inconsistencies between experimental data and theoretical models need to be resolved.’’ Paraphrasing a reminiscence of R. Shull concerning the philosophy of his Nobel Laureate father. If experiments agree with the theoretical analysis of the model, DO WE DECLARE VICTORY? Papers appear in PRL that explain properties Of LaTiO using exactly this model. Only if both are correct!!
- 11. ROTATIONAL INVARIANCE If terms are invariant when the coordinate axes of the spin are rotated, then they are invariant when the spin is rotated. (We will obtain a remarkable spin symmetry.) Simple example: the number of electrons of a given flavor, say X, is the sum of the number of ``up’’ spins of that flavor plus the number of ``down’’ spins of the flavor. But the directions of ``up’’ and ``down’’ don’t matter. So NX is a rotational invariant.
- 12. Hubbard Hamiltonian ),()( 2 1 iNiNUH i C βα αβ αβ∑∑= ασ σ ασα ii cciN ∑ + =)( So since N is a rotational invariant, HC is invariant against rotation of the spin of α-flavor electrons. In fancier language: the operator which rotates the spin of α-flavor electrons commutes with HC.
- 13. ασασ αβσ α ji ij ijhop cctH + >< ∑∑= This is also a rotational invariant PROVIDING we rotate α-flavor electron spins on site i and those on site j IN THE SAME WAY. Now consider Normally, all sites are coupled, so this means that rotational invariance only happens when ALL spins are rotated the same way. Here due to the inactive axes, dxy electrons can only hop within an x-y plane, so we get rotational invariance when we rotate together all the dxy spins in any single x-y plane (and similarly for other flavors).
- 14. MERMIN-WAGNER PROOF In d=2 dimensions thermal excitations of spin-waves destroy long-range order. The number of thermal spin excitations is ∑∑ − =><= kkk k 1 1 magnon e T e nN β ∫∫ ≈ − = − 3 1 1 2 k dkk e dkk d kc d β This diverges for d=2: the ground state is unstable to thermal excitation Can we trust this heuristic argument??
- 15. MERMIN-WAGNER PROOF We have constructed the analogous rigorous proof that there is no long range spin order in any flavor orbital at nonzero temperature. (PRB 69, 035107) Although each orbital flavor has two dimensional dynamics, this system is really three dimensional. It is very unusual for thermal fluctuations to destroy long-range order in a three dimensional system. But it happens for this special model. Any small deviations from cubic symmetry invalidate this analysis.
- 16. MEAN-FIELD THEORY THERE IS NO WAVEVECTOR SELECTION )()()( 2 1 1 kkk k −= − ∑ xxF χ )( 1 k − χ )]cos()[cos(412 akakkT yx ++= εz The spin susceptibility for z-flavor orbitals is This has an instability to order in antiferromagnetic planes which are active: U t2 =ε akk yx /π== kz = anything
- 17. ROTATIONAL SYMMETRY Because we have rotational invariance FOR EACH FLAVOR separately THE TOTAL SPIN, AS WELL AS ITS z-COMPONENT, FOR EACH FLAVOR, SUMMED OVER ALL SPINS IN THE ACTIVE PLANE OF THIS FLAVOR, ARE BOTH GOOD QUANTUM NUMBERS. The spin at any given site is the sum over the spins of each orbital flavor (X, Y, and Z).
- 18. Consequences of the Novel Hidden Symmetries of t2g Hubbard Hamiltonian (a) (b) (c) (d) n2z n2y n1z n1y x y z n2x n1x >− >=>≡ + ↑ + ↓ + ↓ + ↑ − vac][2 )(|)(| 2/1 jxixjxix xx cccc jiij An i-j singlet wavefunction: The exact ground state is a sum of 16 dimer states, which can be obtained from the dominant configurations (c) and (b) by hopping as in panel (d). Symmetry reduces 1.7 million to 16!!
- 19. J = 15.5 meV Gap = 3.3 meV D = 1.1 meV Reduced moment = 0.45 µB ( = µB for S = ½) La Ti O3 B. Keimer et al
- 20. ``Inconsistencies between experimental data and theoretical models need to be resolved.’’ Paraphrasing a reminiscence of R. Shull concerning the philosophy of his Nobel Laureate father. But suppose the experimental measurements are correct and the theory analysis of the model is accurate, but they disagree, THEN WHAT? THE MODEL IS WRONG!!
- 21. In LaTiO the octahedra are rotated, so there are no inactive axes and this system is much like any other antiferromagnet. But if a system more closely approximating the KK model could be fabricated, it would have very unusual properties: quasi 2d spin fluctuations, for example.
- 22. Almost any coupling will give rise to wave vector selection: if different flavors interact, they will all want to condense simultaneously, which can only happen at k = (pi, pi ,pi)/a. For instance, if one adds spin-orbit Interactions, then the spins of all three orbital flavors simultaneously condense, but the spins of the different flavors are NOT parallel. this might explain large zero-point spin deviation.
- 23. SPIN-ORBIT INTERACTIONS With spin-orbit interactions one would think that the spin knows where the crystal axes are -> we expect anisotropy WRONG!! βναµ αβγ νµγγ σβλ ii i OS ccLaV + − ∑∑ ><= ,][|| αττσασ i t a i cVc ~)( ,∑= αα α σσ UV )( =with
- 24. Consequences of the Novel Hidden Symmetries of t2g Hubbard Hamiltonian A global rotation of spins in a plane means that we CAN NOT have a long- range magnetic ordering at non-zero temperature (see Mermin and Wagner (PRL 13,1133,1966)). This conclusion also applies to the Kugel-Khomskii Hamiltonian which is obtained at second order perturbation (i.e. t2 /U): KK Hamiltonian – contrary to the general belief in the literature -does not support 3D magnetic ordering without additional terms such as spin-orbit interaction or allowing octahedral rotation! Inclusion of spin-orbit coupling allows such ordering but even then the excitation spectrum is gapless due to a continuous symmetry. We hope that these results will inspire experimentalists to synthesize new t2g oxides with tetragonal or higher symmetry. Such systems would have quite striking and anomalous properties. Therefore a consistent theoretical explanation of a real transition metal oxides must include other terms to the Hubbard or KK model
- 25. It is surprising that the Hubbard model has been widely used in the study of transition metal oxides for a long time but yet its remarkable symmetry properties were missed until now! Using these symmetries, we rigorously showed that the Hubbard model (and also KK model) without spin-orbit interaction does not permit the development of long- range spin order in three dimensional cubic lattice at non zero temperature. Finally,these symmetries reduce 1.7 million by 1.7 million matrix to 16x16 matrix for a cube of Ti cluster! For the first-time we uncovered several novel symmetries of the Hubbard Hamiltonian for a cubic t2g system. CONCLUSIONS