Invariants of spin chains

Question

I consider modelling a particular physical phenomenon using a spin chain (Ising, XYZ, Potts, etc.). Once I establish the mapping from experimental data to the states of spins for, I get the values $\{s_i(t)\}$.

In order to constrain the possible forms of the Hamiltonian, I would like to check if the system respects any symmetries. To do so, I would like to calculate some functionals $I_\alpha[\{s_i(t)]$ which may (potentially) be the invariants of the system. If $I_\alpha[\{s_i(t)\}]\approx I_\alpha[\{s_i(t_2)\}] \approx I_\alpha[\{s_i(t_3)\}]$, I would conclude that the system respects a certain symmetry, and, therefore, the Hamiltonian contains a smaller number of free parameters.

What are the typical physically motivated symmetries of spin chains and their corresponding invariant quantities?

Answer 1

It really depends on your Hamiltonian... E.g., the free energy for the Heisenberg model has symmetry in respect ti flipping all spins, in respect to reversing the magnetic field, and sometimes also in respect to exchanging sublattices.

In general, thermodynamic functions are good candidates for being symmetry invariant functionals.