A spin liquid in my favourite frustrated spin model

At last weeks cake meeting (condensed matter group meeting/journal club) I gave a talk about an interesting recent paper Incommensurate correlations in the anisotropic triangular Heisenberg lattice by Andreas Weichselbaum and Steve White.

The model is the spin-1/2 Heisenberg model on the anisotropic triangular lattice with antiferromagnetic interactions. By varying the relative strength (or spatial anisotropy) of the interactions the model can interpolate between the square lattice, triangular lattice, and weakly coupled chains (with a frustrated interchain interaction J'). Back in 1998 I argued that this is the minimal model for the spin excitations in the Mott insulating phase of a family of organic superconductors. I have since written 7 papers on the model. A recent review article looks at the model in light of theoretical and experimental studies, which reveal its richness including the possibility of spin liquid ground states.

Weichselbaum and White perform extensive DMRG (density matrix renormalisation group) studies considering lattices as large as 64 x 8. This is arguably the highest power numerical study of the model to date. They mostly focus on the very specific question of whether in the coupled chain limit (J' much less than J, the intrachain coupling) the spin correlations ever become commensurate. [An Sp(N), large N study (not referenced) suggested this was the case]. This limit is particularly relevant to the material Cs2CuCl4, which has J'/J ~0.3, and is a candidate material to have deconfined spinon excitations, a connection that is clearly mentioned in the paper.

Here are a few of the interesting results in the paper. First, the results are quite sensitive to the boundary conditions and to the system size, even for these relatively large systems.

For all system sizes the authors see a spin gap in the parameter range of J'/J ~ 1-1.2. [see the yellow curve in the figure above which is for 64 x 6.] This range and the magnitude of the gap do vary significantly with the size of the system. The authors claim this gap will vanish in the thermodynamic limit. However, I am not convinced, partly because series expansions do produce a gap in this parameter range (see Figure 10 in this PRB).

For this same parameter regime J'/J ~ 1-1.2 a ground state which breaks translation symmetry is seen. The figure below shows the magnitude of the spin correlations on different links in the lattice. A similar ground state was found in a recent DMRG study of a four leg ladder.

The results vary significantly depending on whether the system size in the vertical direction is 4n or 4n+2. Periodic boundary conditions are applied in this direction. The authors state:

Overall, the dimerization seen here suggests a qualitative difference of the systems of width 4n+2 (symmetry-broken systems), with n an integer, to systems of width 4n (uniform systems), while nevertheless, a two-chain periodicity perpendicular to the chains is maintained in either case. Equivalently, this translates into an even-odd effect in the number of laterally coupled zigzag chains. 

No mention is made of a very famous phenomena in organic chemistry, which must be related. But the precise connection is not completely clear to me, because in that case 4n+2 tends to be uniform and 4n tends to break symmetry.

Huckel's rule states that rings with 4n+2 pi electrons (e.g. benzene) are stable and do not dimerise. They are aromatic. The electrons are delocalised.
In contrast, rings with 4n electrons (e.g. cyclobutadiene) are unstable to dimerisation (i.e., they have alternating single and double bonds). They are anti-aromatic.
This 4n/4n+2 dichotomy can also be formulated in valence bond theory, as emphasized passionately by Shaik and Hiberty. 

Finding the lost twin

This beautiful picture is on the cover of A Chemist's guide to Valence Bond Theory by Shaik and Hiberty. It summarises the main idea in a paper, The Twin-Excited State as a Probe for the Transition State in Concerted Unimolecular Reactions: The Semibullvalene Rearrangement.
It illustrates how the use of diabatic states (K1 and K2) based on chemical intuition can lead to adiabatic potential energy surfaces with complex structure. Furthermore, it illustrates the notion of an excited state (K1 - K2) which is a "twin state" to the ground state, K1+K2. The relevant vibrational frequency is higher in the excited state than in the ground state.
An earlier post discussed the analogous picture for benzene.

What did Escher know?

In the interesting physics colloquium that Marcelo Gleiser gave on friday at UQ he used the Escher print above to illustrate CP symmetry in a system which violates both C and P symmetry.
[C is charge conjugation and P is parity].
I found the image here.
Another example of breaking of a symmetry while conserving a combined symmetry concerns vibronic transitions in molecules. For molecules which have inversion symmetry selection rules suggest that electronic transitions that involve no change in parity should not be observed. However, they sometimes are because they are combined with a vibrational transition. Hence, vibronic [= vib(rational)-(elect)ronic] transitions. I think the first observed case of this may have been in benzene.

Desperately seeking spin liquids V

I just read a very interesting Nature paper, which appeared last month.

Quantum spin liquid emerging in two-dimensional correlated Dirac fermions,

by Z. Y. Meng, T. C. Lang, S. Wessel, F. F. Assaad & A. Muramatsu

The authors perform Quantum Monte Carlo simulations on the Hubbard model at half-filling on the honeycomb lattice. [This is the relevant lattice for graphene].

As U/t increases there is a phase transition from a semi-metal (SM) (which has gapless excitations at corners of the Brillouin zone, Dirac fermions) to a Mott insulating phase. But, they also find that there is a spin liquid (SL) phase with a spin gap before entering a phase with antiferromagnetic order (AFMI). The latter what one expects from a strong coupling expansion, i.e U >>t), which is described by an unfrustrated Heisenberg model. This is summarised in the figure below.

A few random observations:

The spin gap is very small Delta_{s ~ t/40~J/40.}

The single-particle charge gap Delta_sp(K) is quite small in the spin liquid state (about t/10 ~ U/40).

Although the honeycomb lattice is bi-partite and so not frustrated the authors, suggest that near the Mott transition effective frustrating interactions occur.

The spin liquid state has dimer-dimer correlations similar to that in a single hexagon which can be described by the RVB states of benzene. See the figure below.

A great quantum many-body theorist

Today is Ada Lovelace day [I thank Ben Powell for bringing this to my attention] and so I asked myself, "Which woman has made the greatest contribution to quantum many-body theory?"

Maria Goeppert-Mayer is best known for receiving the Nobel Prize in Physics for her role in developing the nuclear shell model.

However, she did more. In 1938, together with Sklar, she did one of the first non -empirical quantum chemistry calculations: the excited states of benzene. This is in a widely cited Journal of Chemical Physics paper.

A signature of resonating valence bonds

In most molecules when you different vibrational modes have the highest frequency in the electronic ground state. Another way of looking at this is that it is chemical bonding which stabilises the ground state. These bonds will be strongest and stiffest in the ground state. Excited states are associated with less bonding and so most are associated with a reduction in vibrational frequencies.

An important except is benzene. In the lowest excited state the b2u vibrational mode increases in frequency by about 20 per cent. This has a natural explanation in terms of valence bond theory (see figure below) and is discussed in a Accounts of Chemical Research paper by Shaik, Zilberg, and Haas.

The ground state (which has A1g symmetry) can be described as a linear superposition of two valence bond structures, Kr + Kl.

The lowest excited state (which has B2u symmetry) can be described as the antisymmetric superposition, Kr - Kl.

The b2u distortion couples linearly to the energy of Kr and Kl near the degeneracy point. This leads to the curves shown above.

It is clear that the b2u vibrational mode will have a higher frequency in the excited state than the ground state.

High-level quantum chemistry calculations support this simple picture, which underscores the importance of using the appropriate Hilbert space to describe strongly correlated systems.

How big a Hilbert space do you need?

How big a Hilbert space do I need to describe the electronic properties of a molecule?

Specifically, suppose in the molecule there are N valence electrons.

One must decide then how many spatial orbitals are required and how many Slater determinants? This issue is brought out in this review paper. Benzene is an illuminating case. McWeeny shows that a "brute force" approach based on molecular orbital theory requires hundreds and sometimes thousands of Slater determinants to obtain results of comparable accuracy to that obtained with a valence bond description.The latter uses just six localised orbitals and two Slater determinants (corresponding to the two Kekule structures).

Why does this matter? First, the priority of chemical insight favours the valence bond description over the "black box" approach embodied in the molecular orbital theory approach. The issues are described nicely in this Nature paper, which emphasises that the delocalised molecular orbitals are physically misleading. Second, computational efficiency may be more likely to be obtained with the simpler description.

A key question is whether one can codify these issues in a systematic way (perhaps with ideas from quantum information theory) to develop quantitative criteria to decide what is the minimal number of orbitals and Slater determinants.

Thanks to Anthony Jacko for providing the cartoon.

A great way to start learning (and understanding) quantum physics?

At UQ the Faculty of Science has an Advanced Study Program, which gets gifted (and hopefully highly motivated!) undergraduates involved in research early. I have a second year student, Michael Horn, who is doing a project with me on optically excited states of biomolecules. However, he hasn't taken any courses in quantum physics yet.

So what should he read? Volume III of The Feynman lectures turns out to be ideal. I never cease to be amazed at Feynman's originality and profound insight. He certainly does not treat topics in the conventional order! Just using a Hamiltonian matrix he describes the ammonia maser, the hydrogen molecule, benzene, dyes, the neutral K-meson, semiconductor devices, ... Only after all this in chapter (lecture) 16 does he introduce the Schrodinger equation (i.e., the differential equation) and the hydrogen atom. Michael only needs to read 12 lectures to learn what he needs to know to understand the essential quantum physics associated with a lot of the photochemical properties of organic molecules.

Quantum dancing monkeys? Valence bond theory of benzene

Hopefully this will be the first of several posts related to valence bond theory. I attach a nice treatment of a simple Heisenberg model for benzene, copied from a book, Electronic Properites of Conjugated Polymers by William Barford. The book also shows how strong electronic correlations lead to a significant different ordering of excited electronic states than predicted by molecular orbital theory and density functional theory based approximations. So what does this have to do with dancing monkeys and quantum entanglement?