# Local Temperature Ansatz

# Abstract

The entanglement Hamiltoian, $H_E$ is related to the reduced density matrix associated with a sub-system A as: $\rho_A = e^{-H_E}$. An interesting question is (a) how local $H_E$ is, and (b) how is it related to the local Hamiltonian density of the system, $\mathcal{H}(x)$. 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 $H_E = \int_{x\in A} dx \mathcal{H}(x)/T(x)$. 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.

We use perturbation theory as well as quantum Monte Carlo calculation and discuss $H_E$ 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, we see 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.

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