Role of rotational invariance in the properties of Hamiltonians

Is it possible to prove that the properties of Hamiltonians, such as the ground-state energy, results of dynamical evolution, or thermal state expectation values, can be efficiently calculated when the Hamiltonians have physically motivated constraints such as translational or rotational invariance? We report that rotational invariance does not reduce the difficulty of finding the ground-state energy of the system. Crucially, the construction preserves the translational invariance of a Hamiltonian. The failure of the construction for the properties of thermal states at finite temperatures is discussed.

Figure 1

The flag projector $P^F$. There is a duality between the projectors in the Hamiltonian and the ground state of the Hamiltonian; one selects any of the states that the flag projector projects onto. (a) A general construction that works for any values of $r$ and $j$ for sufficiently large $m$. (b) Special case construction to give $F\sim O\left(1\right)$ for $j\approx r/2$.