How is geometric frustration different from (not) being frustration free?

Question

In the context of Ising models, some Hamiltonians can be described as geometrically frustrated - such as, I think, the antiferromagnetic kagome lattice, as well as a one-dimensional anisotropic, next-nearest neighbor Ising chain. There appears to be a competition between spin-up and spin-down qubits, influenced by the geometry/topology of the lattice.

But we can also describe certain local Hamiltonians as being "frustration free", where, in the ground state of the global Hamiltonian, each local term also obtains its own lowest energy configuration.

Apparently there is a difference between the two terms - but I'm not entirely sure why?

Certainly a Hamiltonian can be "computationally local" with each term acting only on a subset of $n$ qubits, without being "geometrically local" where each of the $k$ qubits are near to each other. But if a lattice is geometrically frustrated, then surely its natural description as a Hamiltonian is not frustration free?

Answer 1

These two concepts are mostly unrelated:

1. A generic nearest-neighbor antiferromagnetic Hamiltonian on a non-bipartite lattice (e.g., triangular lattice Heisenberg) is geometrically frustrated and not frustration-free.

2. A generic nearest-neighbor antiferromagnetic Hamiltonian on a bipartite (e.g., square lattice Heisenberg) lattice is neither geometrically frustrated nor frustration-free.

3. The AKLT model (which favors some antiferromagnetic order) on the square lattice is frustration-free and has no geometric frustration.

4. The last case -- being frustration-free and having geometric frustration -- is the more tricky one, as it depends on your precise definition both concepts:

   • A classical (i.e., Ising) antiferromagnet on a frustrated lattice is geometrically frustrated, but is in some sense frustration free -- the ground state is an eigenstate of every interaction term, though not a ground state.
   • The AKLT model on the kagome lattice is frustration-free, but has has some kind of geometric frustration, as the interaction favors antiferromagnetic order (again, one can argue about this, as it is not strictly antiferromagnetic).
   • Finally, the Majumdar-Ghosh chain (a chain with NN and NNN Heisenberg interactions) is geometrically frustrated, as well as frustration free (however, one can voice the critique that the frustration free Hamiltonian is obtained by taking the NN and NNN term together, so one might start to argue how frustrated the frustration-free Hamiltonian is).