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.

The aromaticity argument to me seems to be the precise cause of the effect. The point is that the bond strength alternates in the direction transverse to the 4n+2 direction (i.e. along the long axis of their cylinders).

ReplyDeleteIf the circumference of their cylinders are spanned by 4n+2 sites, there is an extra aromatic resonance that prefers strong bonds around the circumference.