One Dimensional Free Entanglement Hamiltonians for Off Critical Systems and in the Presence of Defects

The reduced density matrix of a quantum system can be expressed as a "thermal" partition function of a different Hamiltonian called the entanglement (or modular) Hamiltonian. For ground states of critical one dimensional free fermion chains this Hamiltonian is well understood, both on the lattice and continuum limit. In this work, we use high precision numerics to directly calculate the entanglement Hamiltonian for the critical and off critical Ising chain in the presence of defects, for connected and disconnected subsystems. While the critical system gives rise to mostly local Entanglement Hamiltonians, the non-local couplings are bolstered in the presence of defects. In the free fermion language, the non-local couplings are described by Majorana bilinears and thus by strings of spin operators in the spin language. We also extend this analysis to off-critical phases where we reproduce know behavior for the open chain and are able to study the periodic chain. For defects with zero modes, we show that there are relatively large non-local contributions. Finally, we expand upon the idea that for the critical Ising system, the lattice entanglement Hamiltonian can be expressed as a power series of a local Hamiltonian by finding this Hamiltonian for certain defects. This Hamiltonian is the expected continuum limit entanglement Hamiltonian.