# Presentation: 108857

## 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).**

## Outline

- Motivations: *High T**c** 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

## 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.

## d-electrons in transition metal oxides

-  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.

- = negative

- =  positive

## Single-band Hubbard model

- U,  is  very  large  and  there  is  one  electron  per  site.

- Heisenberg Hamiltonian

- Perturbation  processes

## -

*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).

- +

- +

## -

*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).

- +

- +

## Hubbard Hamiltonian

**Hubbard Hamiltonian:**

- creates an electron at site *i* in orbital with spin

## KUGEL – KHOMSKII  (1975)

## ``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!!

## 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.

## Hubbard Hamiltonian

- So since N is a rotational invariant, *H****C*** is
- invariant against rotation of the spin of
- -flavor electrons.

- In fancier language:  the operator which rotates
- the spin of -flavor electrons commutes with *H****C***.

## This is also a rotational invariant PROVIDING

- we rotate -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).

## MERMIN-WAGNER  PROOF

- In  d=2  dimensions  thermal  excitations  of
- spin-waves  destroy  long-range order.
- The number of thermal spin excitations is

- This  diverges  for d=2:  the ground
-  state is unstable to thermal excitation

- Can  we trust  this heuristic argument??

## 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.

## MEAN-FIELD  THEORY

- THERE   IS  NO  WAVEVECTOR  SELECTION

- The  spin  susceptibility  for  z-flavor orbitals  is

- This  has  an  instability  to  order  in
- antiferromagnetic  planes  which are  active:

- kz = anything

## 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).

## Consequences of the Novel Hidden Symmetries 

**of t****2g****  Hubbard Hamiltonian**

- The  exact  ground state  is
- a  sum  of  16  dimer states,
- which  can be obtained from
- the dominant  configurations
-   and  (b)  by  hopping  as
- in  panel  (d).

- Symmetry  reduces 1.7 million  to  16!!

## J = 15.5 meV

- Gap = 3.3 meV   D = 1.1 meV

- Reduced moment = 0.45 B             ( =  B  for S = 1⁄2)

- La Ti O3

- B.  Keimer *et al*

## ``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!!

## 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.

## 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.

## SPIN-ORBIT  INTERACTIONS

- With  spin-orbit  interactions  one  would  think
- that  the  spin  knows  where  the  crystal
- axes  are  ->  we  expect  anisotropy

- WRONG!!

- with

## Consequences of the Novel Hidden Symmetries

** ****of t****2g****  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

## CONCLUSIONS

**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.