Inhomogeneous quantum critical systems in one spatial dimension have beenstudied by using conformal field theory in static curved backgrounds. Twointeresting examples are the free fermion gas in the harmonic trap and theinhomogeneous XX spin chain called rainbow chain. For conformal field theoriesdefined on static curved spacetimes characterised by a metric which is Weylequivalent to the flat metric, with the Weyl factor depending only on thespatial coordinate, we study the entanglement hamiltonian and the entanglementspectrum of an interval adjacent to the boundary of a segment where the sameboundary condition is imposed at the endpoints. A contour function for theentanglement entropies corresponding to this configuration is also considered,being closely related to the entanglement hamiltonian. The analytic expressionsobtained by considering the curved spacetime which characterises the rainbowmodel have been checked against numerical data for the rainbow chain, findingan excellent agreement.
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