The Role of Rotational Invariance in the Properties of Hamiltonians

TL;DR

This paper investigates whether rotational invariance simplifies the computation of Hamiltonian properties, concluding it does not reduce complexity for ground state energies and discussing limitations at finite temperatures.

Contribution

It demonstrates that rotational invariance does not ease the calculation of ground state energies and preserves translational invariance, highlighting limitations at finite temperatures.

Findings

Rotational invariance does not reduce computational difficulty for ground state energies.

The construction preserves translational invariance.

Limitations are discussed for thermal states at finite temperatures.

Abstract

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 it preserves the translational invariance of a Hamiltonian. The failure of the construction for the properties of thermal states at finite temperatures is discussed.