Solving gapped Hamiltonians locally

We show that any short-range Hamiltonian with a gap between the ground and excited states can be written as a sum of local operators, such that the ground state is an approximate eigenvector of each operator separately. We then show that the ground state of any such Hamiltonian is close to a generalized matrix product state. The range of the given operators needed to obtain a good approximation to the ground state is proportional to the square of the logarithm of the system size times a characteristic “factorization length.” Applications to many-body quantum simulation are discussed. We also consider density matrices of systems at non zero temperature.

Figure 1

Illustration of iterative and hierarchical procedures. (a) Iterative procedure. Filled circles represent sites, ovals surrounding sites representing grouping of sites into a super site. The first stage groups sites 1 and 2. The second groups the combined site with site 3, and so on. (b) Hierarchical procedure. The first stage groups pairs of sites, the second stages groups four sites into a single site, and so on.

Figure 2

Example of a set of active bonds shown as solid lines; dashed lines represent bonds which are not active while circles represent sites. The Hamiltonian is a sum of terms which act on pairs of neighboring sites, so that the bonds connect only neighboring sites. Using a Manhattan metric for the lattice, the set of active bonds shown here is a term in $\rho (\beta ,l_{\mathrm{proj}}=2)$. There are four distinct clusters, three with diameter two and one with diameter one.