Entanglement and negativity Hamiltonians for the massless Dirac field on the half line

Abstract

We study the ground-state entanglement Hamiltonian of several disjoint intervals for the massless Dirac fermion on the half-line. Its structure consists of a local part and a bi-local term that couples each point to another one in each other interval. The bi-local operator can be either diagonal or mixed in the fermionic chiralities and it is sensitive to the boundary conditions. The knowledge of such entanglement Hamiltonian is the starting point to evaluate the negativity Hamiltonian, i.e. the logarithm of the partially transposed reduced density matrix, which is an operatorial characterisation of entanglement of subsystems in a mixed states. We find that the negativity Hamiltonian inherits the structure of the corresponding entanglement Hamiltonian. We finally show how the continuum expressions for both these operators can be recovered from exact numerical computations in free-fermion chains.

40 pages, 9 figures

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• Entanglement entropies of an interval for the massless scalar field in the presence of a boundary
• Modular Hamiltonian for de Sitter diamonds
• Symmetry resolution of the computable cross-norm negativity of two disjoint intervals in the massless Dirac field theory
• Breakdown of Measurement-Induced Phase Transitions Under Information Loss
• Exact Renormalization of Wave Functionals yields Continuous MERA
• Quench dynamics of negativity Hamiltonians
• A contour for the entanglement negativity of bosonic Gaussian states