Finite temperature negativity Hamiltonians of the massless Dirac fermion

Abstract

The negativity Hamiltonian, defined as the logarithm of a partially transposed density matrix, provides an operatorial characterisation of mixed-state entanglement. However, so far, it has only been studied for the mixed-state density matrices corresponding to subsystems of globally pure states. Here, we consider as a genuine example of a mixed state the one-dimensional massless Dirac fermions in a system at finite temperature and size. As subsystems, we consider an arbitrary set of disjoint intervals. The structure of the corresponding negativity Hamiltonian resembles the one for the entanglement Hamiltonian in the same geometry: in addition to a local term proportional to the stress-energy tensor, each point is non-locally coupled to an infinite but discrete set of other points. However, when the lengths of the transposed and non-transposed intervals coincide, the structure remarkably simplifies and we retrieve the mild non-locality of the ground state negativity Hamiltonian. We also conjecture an exact expression for the negativity Hamiltonian associated to the twisted partial transpose, which is a Hermitian fermionic matrix. We finally obtain the continuum limit of both the local and bi-local operators from exact numerical computations in free-fermionic chains.

Entanglement Hamiltonian in the non-Hermitian SSH model

“…This limiting procedure has allowed to reconstruct the CFT EH in many systems at criticality, both at finite temperature and in the ground state [94][95][96] and also in the presence of boundaries [95][96][97], in inhomogeneous and out-of-equilibrium systems [98,99] and in higher dimensions [100]. In [97,101] it has also been extended to the recently introduced negativity Hamiltonian [102], i.e. the logarithm of the partial transposed density matrix.…”