We show there are analogues to the Unruh temperature that can be defined for any quantum field theory and region of the space. These local temperatures are defined using relative entropy with localized excitations. We show important restrictions arise from relative entropy inequalities and causal propagation between Cauchy surfaces. These suggest a large amount of universality for local temperatures, specially the ones affecting null directions. For regions with any number of intervals in two space-time dimensions the local temperatures might arise from a term in the modular Hamiltonian proportional to the stress tensor. We argue this term might be universal, with a coefficient that is the same for any theory, and check analytically and numerically this is the case for free massive scalar and Dirac fields. In dimensions $d\ge 3$ the local terms in the modular Hamiltonian producing these local temperatures cannot be formed exclusively from the stress tensor. For a free scalar field we classify the structure of the local terms.Comment: 35 pages, 12 figure

We calculate the analytic form of the vacuum modular Hamiltonian for a two interval region and the algebra of a current j(x) = ∂φ(x) corresponding to a chiral free scalar φ in d = 2. We also compute explicitly the mutual information between the intervals. This model shows a failure of Haag duality for two intervals that translates into a loss of a symmetry property for the mutual information usually associated with modular invariance. Contrary to the case of a free massless fermion, the modular Hamiltonian turns out to be completely non local. The calculation is done diagonalizing the density matrix by computing the eigensystem of a correlator kernel operator. These eigenvectors are obtained by a novel method that involves solving an equivalent problem for an holomorphic function in the complex plane where multiplicative boundary conditions are imposed on the intervals. Using the same technique we also re-derive the free fermion modular Hamiltonian in a more transparent way. arXiv:1809.00026v2 [hep-th]

We study the temporal evolution of the entanglement Hamiltonian of an interval after a global quantum quench in free lattice models in one spatial dimension. In a harmonic chain we explore a quench of the frequency parameter. In a chain of free fermions at half filling we consider the evolution of the ground state of a fully dimerised chain through the homogeneous Hamiltonian. We focus on critical evolution Hamiltonians. The temporal evolutions of the gaps in the entanglement spectrum are analysed. The entanglement Hamiltonians in these models are characterised by matrices that provide also contours for the entanglement entropies. The temporal evolution of these contours for the entanglement entropy is studied, also by employing existing conformal field theory results for the semi-infinite line and the quasi-particle picture for the global quench.