The simplest of these problems involve only the spin operators S

_{i}of electrons residing on the sites, i, of a regular lattice. Each electron can have its spin oriented either up or down, leading to a Hilbert space of 2^{N}states, on a lattice of N sites. On this space acts the Heisenberg HamiltonianH=∑_{i<j }J_{ij}S_{i}⋅S_{j}, (1)where the J

_{ij}are a set of short-range exchange interactions, the strongest of which have J_{ij}>0, i.e., are antiferromagnetic. We would like to map the ground-state phase diagram of H as a function of the J_{ij}for a variety of lattices in the limit of N→∞. Note that we are not interested in obtaining the exact wave function of the ground state: this is a hopeless task in dimensions greater than one. Rather, we would be satisfied in a qualitative characterization of each phase in the space of the J_{ij}. Among the possible phases are(i) a Néel phase, in which the spins have a definite orientation just as in the classical antiferromagnet, with the spin expectation values

all parallel or antiparallel to each other; (ii) a spiral antiferromagnet, which is magnetically ordered like the Néel phase, but the

spins are not collinear; (iii) a valence bond solid (VBS), with the spins paired into S=0 valence bonds, which then crystallize into a preferred arrangement that breaks the lattice symmetry; and

(iv) a spin liquid, with no broken symmetries, neutral S=1/2 elementary excitations, and varieties of a subtle “topological” order.

Specific examples occur for Heisenberg models on the (i) square lattice, (ii), triangular lattice, (iii) kagome lattice (probably).

An example of a model which contains all three is here.

We are still searching for an example of (iv).

## No comments:

## Post a Comment