Numerical Reconstruction of Entaglement Hamiltonians using Matrix Product States

The area law of entanglement for ground states of 1+1D local Hamiltonian enables us to study the reduced density operator (RDO) of large scale bi-partitions efficiently using Density Matrix Renormalization Group (DMRG) and Matrix Product States (MPS). This helped us to understand that quantum entanglement is not only governing the complexity of simulations on classical computers but shows signatures for multiple of interesting many body phenomena like phase transitions. But an intuitive and or analytical understanding of the RDO is still incomplete. We will use the fact that any RDO can be represented as a “thermal” state of an effective Hamiltonian. This is called the modular or entanglement Hamiltonian (EH). The main contribution of this work is to introduce a way to find nice easy to understand EHs numerically. This calculation is efficient and feasible for system sizes of hundreds of lattice sites, given that the state is available in form of an MPS. We will show results for lattice chains like the Quantum Ising Model in different phases with and without the influence of defects and show agreement to analytic results as well as further extend to cases where non are available.