Abstract
Physical properties of the ground and excited states of a -local Hamiltonian are largely determined by the -particle reduced density matrices (-RDMs), or simply the -matrix for fermionic systems—they are at least enough for the calculation of the ground-state and excited-state energies. Moreover, for a nondegenerate ground state of a -local Hamiltonian, even the state itself is completely determined by its -RDMs, and therefore contains no genuine -particle correlations, as they can be inferred from -particle correlation functions. It is natural to ask whether a similar result holds for nondegenerate excited states. In fact, for fermionic systems, it has been conjectured that any nondegenerate excited state of a 2-local Hamiltonian is simultaneously a unique ground state of another 2-local Hamiltonian, hence is uniquely determined by its 2-matrix. And a weaker version of this conjecture states that any nondegenerate excited state of a 2-local Hamiltonian is uniquely determined by its 2-matrix among all the pure -particle states. We construct explicit counterexamples to show that both conjectures are false. We further show that any nondegenerate excited state of a -local Hamiltonian is a unique ground state of another -local Hamiltonian, hence is uniquely determined by its -RDMs (or -matrix). These results set up a solid framework for the study of excited-state properties of many-body systems.
- Received 4 December 2011
DOI:https://doi.org/10.1103/PhysRevA.85.040303
©2012 American Physical Society