Local temperature approximation of entanglement Hamiltonian

The entanglement Hamiltoian, H_E is related to the reduced density matrix associated with a sub-system A as: ρ_A = e^-H_E. An interesting question is (a) how local H_E is, and (b) how is it related to the local Hamiltonian density of the system, 𝒽(x).It has been mathematically shown in the context of black-hole physics that for quantum systems with conformal symmetry, or those with Lorentz symmetry but at zero temperature, H_E is local and can be written as H_E = ∫_{x ∈ A} (𝒽(x)/T(x)) dx, when entangling surfaces (boundaries between A and its complement) are flat. In this expression T(x) is a local temperature that diverges as 1/x near the entangling surfaces. This particular form of H_E is known as the Rindler Hamiltonian.

In this talk, I use perturbation theory as well as quantum Monte Carlo calculation and discussthe the form of H_E in the following situations: (a) massive theories with neither conformal nor Lorentz symmetry, (b) low temperature thermal states, and (c) non-smooth or singular entangling surfaces. In all of these cases, I show that H_E can still be expressed in the Rindler Hamiltonian form, albeit by choosing a proper profile for T(x). Our findings can help us to understand quantum entanglement better and develop more efficient numerical algorithms accordingly.