Several lattice models have recently been shown to display *emergent* supersymmetry at length scales long enough that the lattice can be coarse-grained into a continuum (e.g. see here, here, and here). Could a lattice model display *exact* supersymmetry even at the lattice length scales? Clearly the answer is no, because the lattice breaks the Poincare subgroup of the supersymmetry group down to $S \times \mathbb{R}$, where $S$ is the lattice's space group and the $\mathbb{R}$ corresponds to time translational invariance.

According to this answer, the supersymmetry Lie supergroup $G$ corresponding to 3+1D SUSY with $N$ fermionic generators and no additional internal symmetries is Inonu-Wigner contracted $OSP(4/N)$. Does there exist a Hamiltonian, defined on a lattice with space group $S$, with a symmetry Lie supergroup $H < G$ such that the bosonic part of $H$ is $S \times \mathbb{R}$ and the fermionic part of $H$ is nontrivial? (In other words, I want to reduce the full supersymmetry supergroup $G$ down to a sub-supergroup $H$, such that the bosonic part of $G$ reduces from the Poincare group down to a lattice space group (times time translation), but *without* reducing the fermionic part all the way down to the identity, which would eliminate the supersymmetry entirely.) This seems to me like the natural way to restrict supersymmetry to a lattice.

This post imported from StackExchange Physics at 2017-01-11 10:27 (UTC), posted by SE-user tparker