Frustrated quantum spin models in a nutshell

Subir Sachdev has a Physics article which provides a background to recent work using tensor networks (inspired by quantum information theory) to find the ground state of quantum spin lattice models. I really like the following succinct summary of the problem:

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 Hamiltonian

H=∑_i<j J_ijS_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).