Natalia Perkins (Wisconsin) - Spin-orbital physics in frustrated vanadium oxides

It's about ten past five on Tuesday evening, Natasha kindly moved her talk to this late slot to allow some of us to go snorkling by Miramare castle this afternoon - thank you Natasha!!


Natahsa starts by saying that she has a very general title, but we will quickly go into some specific aspects, after a general introduction.

Motivations to study vanadates and other TMO's:
--understanding of reality!! many compunds, many experiments, novel phases and possible applications
--Intellectual adventure: TMO's are complex systems with number of different degrees of freedom (charge, spin, orbitals...)

Let's look at more detail about the orbital degrees of freedom:
--electrons surrounding atoms fall into different orbitals
--much degeneracy, but partially split by crystal field

For example, the octahedral ligand field in LaMnO3: spins align, t2g levels filled, eg levels have only one electron in them, giving a degeneracy of 2, or in other words, a pseudo-spin of 1/2.

In MnV2O4, MgTi2O4, find pseudo-spin 1, etc...

Question from Andriy - can you clarify what psuedo-spin means?
Answer: Psuedo-spin is simply counting the degeneracy of the orbital manifold, i.e. the number of (low-energy) ways of filling the orbitals with the required number of electrons.

As it will be important, now lets speak a bit more about the single ion properties of V^3+ (d^2, S=1)
--the 3d level splits into 2g (very high in energy, forget about them), and t2g manifold - effective orbital angular momentum L'=1. There will also be some spin-orbit coupling \lambda (S.L) splitting this t2g manifold.

So for a description of physics of Vanadates, need a multi-orbital description
.. here we see a generic Hamiltonian of multi-orbital Hubbard model ..

Then make a second-order perturbation theory in t (the materials are insulators) - giving two degrees of freedom per site: spin and pseudo-spin (orbital) - known as the Kugel-Khomskii model.

This model is still very complicated due to the presence of the two different degrees of freedom. However, if these happen at sufficiently different energy scales, then simplifications are possible.

So let's get some inspiration from the materials: normally, orbital ordering happens first - the pseudo-spin variables gain an expectation value - mapping to an anisotropic spin model is possible, which will then order at a lower temperature order.

For example: magnetic ordering on pyrochlore, ZnV2O4
--See two transitions: First structural (Ts=52K, cubic->tetragonal), then magnetic (AFM at Tn=44K)
--This gives us AFM 1D order, consistent with a set of spin chains
--This is a temperature-induced change in dimensionality
[see S.H. Lee et al. PRL 93 (2004)]

By 1D magnetic order, means you see different regimes - a temperature regime where 1D correlations are clearly seen with no magnetic order, then at a lower temperature, get full magnetic order.

Question (Paul C.) : So we need to remove this orbital degeneracy to get to the transition leading to AFM?
Answer: the structural change is very important to lead to the low-T structure of the magnetic Hamiltonian.
P: So the transitions could in principle happen at the same place?
N: Nothing prevents it, but I don't know of any examples when this happens.

Aside: there is big development in this field by group of Bela Lake, with a related Mg compound where they have very good (neutron) measurments of 1D fluctuations.


However, we can't always get rid of orbital degrees of freedom. For example, spin-order may come first (or even spin disorder - correlated paramagnet so know correlation function), leading to a pure orbital model.


Super-exchange in vanadates (AV2O4)
--Octahedra are edge sharing - VOV angles are about 90 degrees so V-O-V superexchange is weak. Small V-V distance means direct overlap of t2g orbitals. Structural distortions life orbital degeneracies... A static Potts-like orbital interaction happens if only dd\sigma overlap is taken into account. In this case, the spin exchange is AF if the bond is occupied by the `good' orbitals, FM for bonds with one `good' and one `bad' orbital, and no exchange otherwise.

Now that the generic introduction is over, we have the plan for:

Rest of talk:
--Spin-Orbital model for MnV2O4
--Spin-Orbital model for CaV2O4
--Open questions

Collaboration with:
Gia-Wei Chern (Wisconsin)
Zhihao Hao (John Hopkins)
Gia Japaridze (Tbilisi)
Shura Nersesyan (Tbilisi/ICTP)

Spin-Orbital model for MnV2O4

Two sub-lattices: A diamond lattice interweaving with B pyrochlore lattice
On A: Mn^2+, on B V^3+

Experiments tell us: Two phase transitions in all insulating AV2O4. Specifically for MnV2O4, find
--magnetic (para-ferrimangetic) at Tn=56K
--tetragonal+ collinear-noncollinear (orthogonal in xy plane) magnetic at Ts=53K
[see Garlea et al, PRL 100, 066404 (2008) ]

Suggestions that magnetic order is stabalized by some form of orbital order (generally believed to be A type - but we will see that this is not correct).

Also find in this material a trigonal distortion and modulation of electronic density, which may be very important. Will also find at low-T a tetragonal distortion - but dealing with both will be non-fun, so lets assume that the trigonal distortion is the largest one.

The trigonal distortion gives a splitting \Delta between the a1g and the eg1/eg2 doublet, which after careful examination will lead to an appropriate super-exchange Hamiltonian, with various projectors whose form comes directly from the structure of the trigonal distortion.

At temperatures higher than the structural transitional, can reduce to an effective orbital Hamiltonian, known as the 120 degrees orbital model (see e.g. Khomskii and Mostovoy; Nussinov and Fradkin; etc...) Note that this is related to the eg-orbital model in manganites with a cubic-lattice.

This model on the cubic lattice is highly frustrated (due to various rotations), however in our case this frustration is not so bad... we'll see why in a moment.

Classical Orbital Order:
Coupling constant is positive - AFM order in pseudospins - but frustrated lattice so can't satisfy them all.
Find a discrete degeneracy of the ground state (total degeneracy=6, all of which are collinear), and can write staggered orbital order parameters, which are the difference in orbital occupation on bonds of the same type.

Let's look at one of these possible GS. Then find that all three order parameters are the same (2/3), which is a different state than that previously proposed (which had 2 out of 3 orbitals occupied, rather than all occupied by 2/3).

Of course, orbital ordering will give some modulation of the electron density - basically from the staggering of the trigonal axes. We then see some nice pictures of this.

What are the effects of orbital-ordering on the magnetic state?
Note that the magnetic frustration is relieved by:
--interaction with magnetic A-sites
--an anisotropic spin exchange in the presence of orbital ordering
--a single-ion anisotropy in the presence of orbital ordering due to spin orbit coupling

Competition between orbital exchange and spin-orbit coupling: both are of the same order or magnitude -> need to go to second order in \lambda -> gives staggered spin anisotropy, which is consistent with the proposed magnetic state found from experiment for MnV2O4.

No time for CaV2O4 - ... blogger sees slides flashing before his eyes...
... but one point: here, the one-dimensionality comes from the beginning, due to the structure - zig-zag spin-chains. Find a toy model with symmetry U(1)xZ2xZ2 (first U(1)xZ2 is from a reduction of SU(2) spin (with S=1), second Z2 is from orbital Ising-like chain)... shame there was no time for this as it sounds very interesting. Another few words: Can de-couple spin/orbitals by mean-field -- see G.-W Chern, N.Perkins, G.Japarize, arXiv:1007.3472. Find two phase transitions, both Ising-like - but in another limit there is only one Gaussian transition - but this is work in progress.

Open Questions:
--Reduction of dimensionality due to orbital anisotropy...
--Orbital excitations: both experiment and theory...
--Entanglement of orbital and spin degrees...
--New models...


Questions from the audience

Ordering in Haldane-like phase? Well, true magnetic ordering is always 3D.

A discussion has now broken out between people in the front row, relating to mutli-ferroics in these materials, which the blogger is not really able to hear clearly.

No more questions? I think we are all tired. Good evening all.