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 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.

Entanglement entropies of an interval for the massless scalar field in the presence of a boundary

“…Other possible extensions to consider are non-vanishing temperature for the entire system, or a non-vanishing mass [105], or a subsystem made by the union of a generic number of disjoint intervals [51,56,65], or spatially inhomogeneous backgrounds [66,67,106], or defects [107][108][109][110].…”

“…However, for n > 2 very few results are available in the literature. The case of an interval for the free massless Dirac field on the half line, which is the prototypical fermionic BCFT with c = 1, has been analyzed in [63] (see [64] for a lattice computation) and later extended to an arbitrary number of intervals [65].…”

Exact Renormalization of Wave Functionals yields continuous MERA

“…Other choices correspond to Lifshitz-like dispersions ω ∼ p 1+α . We leave the exploration of such networks, their entanglement properties, whether the operator algebra is of type I or type III, and the connection to recent work in [12] to future work. Note that the ERG formalism discussed here can accommodate any Gaussian theory, including theories with no local actions such as generalized free fields (GFF).…”

“…Here we will take K 0 to be a function of zp/M and p/µ for some new intermediate scale µ. The transformation in (12) then remains a symmetry of the action and we can repeat the previous ERG construction. To summarize, we consider…”

Entanglement Hamiltonian in the non-Hermitian SSH model

“…In [94][95][96] it was shown that in order to recover the CFT entanglement temperature β(x) in equation ( 5), it is necessary to perform a careful continuum limit which takes into account all of these higher contributions. 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.…”