Modular Hamiltonian of holographic time band states

A holographic time band is a causal incomplete boundary spacetime subregion whose causal wedge is a causal complete bulk spacetime subregion. In an AdS$_3$ spacetime with a specifically modified IR geometry, its causal wedge coincides with its entanglement wedge, which suggests the existence of a local modular Hamiltonian for the holographic time band state. In this work, we construct the local modular Hamiltonian for holographic time bands using two independent methods: from the quantum information properties of the time band state and from the construction of consistent geometric modular flows. Both methods lead to the same unique result of the local modular Hamiltonian, reflecting the intrinsic property of the time band state. The entanglement first law has also been checked to hold for the simplest time band state. This is a substantial addition to the known holographic subsystems with a local modular Hamiltonian, beyond the few cases previously identified.