Seeking a specific quantum spin system of interacting spin 1/2 particles

Question

Is there a system of interacting quantum spin 1/2 particles (of any topology) whose the states where all spins are up or down are eigenstates of its Hamiltonian and yet does not conserve the total spin polarization (in the z-direction) ?

Answer 1

There are plenty, the condition is very weak. For a simple example, with N spin 1/2, all spins up and all spins down both have total angular momentum N/2. You can make a Hamiltonian be zero on the N/2 total angular momentum states, so that all N/2 states are eigenstates, but it can be anything at all to the states of lower total angular momentum, without violating your condition. So, for example letting $\vec\sigma_j$ be the Pauli matrix for the n-th spin, and

$$ P = ({N/2(N/2+1)\over 2} - |\sum_j \vec\sigma_j|^2)$$

be the projection operator which zeros out the total angular momentum N/2 state, then

$$ H = P A P $$

will work, where A is any Hermitian operator at all (except for specially chosen ones). The projection on both sides guarantees that the N/2 total angular momentum states are all eigenvectors with eigenvalue zero.

But I assume you are more interested in a local spin-coupling. The construction above is global--- it requires you to decompose the total angular momentum. In this case, it's still easy to do. Make a matrix element which flips spins only when they are different. It will take adjacent |-+-> to |+-+> and adjacent |+-+> to |-+->. Both of these operations do not conserve total z spin.

The Pauli matrices suffice to expand any operator. In Pauli form $(\sigma^z_i + 1)$, and $(\sigma^z_i-1)$ project out the upper and lower component of a spin, and $\sigma^x_i$ flips a spin. So the Hamiltonian above is:

$$H = \sum_i \sigma^x_{i-1}\sigma^x_{i}\sigma^x_{i+1}( (\sigma^z_{i-1} +1)(\sigma^z_i-1)(\sigma^z_{i+1} +1) + (\sigma^z_{i+1} -1)(\sigma^z_i +1)(\sigma^z_i -1) ) $$

Which is Hermitian (the second term in the sum gives the Hermitian conjugate of the first when multiplied by all the sigma-x's).

The reason you were having trouble is probably because you wanted a two-spin nearest neighbor coupling. To preserve all up and all down, you can't change aligned-spins. This means that the only nonzero matrix elements of H are between antialigned spins, so you only have matrix elements between |+-> and |-+> at every site, and each of these two allowed transitions preserves total spin.

So only for the special case of two-spin interactions, the condition of having all up and all down preserved implies that the total z spin is conserved.