Bundled matrix product states represent low-energy excitations faithfully

Finding the ground-state energy of many-body lattice systems is exponentially costly due to the size of the Hilbert space. Ground-state wave functions satisfying the area law of entanglement entropy can be efficiently expressed as a matrix product states (MPS) for local, gapped Hamiltonians. The extension to a bundled MPS describes excitations, and we provide a formal proof of its efficiency. We define a bundled density matrix as a set of independent density matrices which are all written in a common (truncated) basis. We demonstrate that the truncation error is a practical metric that determines how well an excitation is described. We also show that states with volume law entanglement are not necessarily more costly to include in the bundle. The same is true for gapless systems if sufficient lower energy solutions are already present. This result implies that bundled MPSs can describe low-energy excitations without significantly increasing the bond dimension over the cost of the ground-state calculation with caveats we explain.