Entanglement entropies for Lifshitz fermionic fields at finite density or Entanglement Hamiltonians for the massless Dirac field
by
Erik Tonni(SISSA)
→
Europe/Rome
281 (Dipartimenti di Fisica )
281
Dipartimenti di Fisica
Description
Option 1
Entanglement entropies for Lifshitz fermionic fields at finite density
The entanglement entropies of an interval for the free fermionic spinless Schroedinger field theory at finite density and zero temperature are investigated. The interval is either on the line or at the beginning of the half line, when either Neumann or Dirichlet boundary conditions are imposed at the origin. We show that the entanglement entropies are finite functions of a dimensionless parameter proportional to the area of the rectangular region in the phase space identified by the Fermi momentum and the length of the interval.
For the interval on the line, the entanglement entropy is a monotonically increasing function. Instead, for the interval on the half line, it displays an oscillatory behaviour related to the Friedel oscillations of the mean particle density at the entangling point.
By employing the properties of the prolate spheroidal wave functions or the expansions of the tau functions of the kernels occurring in the spectral problems, determined by the two point function, we find analytic expressions for the expansions of the entanglement entropies in the asymptotic regimes of small and large area of the rectangular phase space region. Extending our analysis to a class of free fermionic Lifshitz models, we find that the parity of the Lifshitz exponent determines the properties of the bipartite entanglement.
Option 2
Entanglement Hamiltonians for the massless Dirac field
The reduced density matrix of a spatial subsystem can be written as the exponential of the entanglement (modular) Hamiltonian, hence this operator contains a lot of information about the entanglement of the corresponding spatial bipartition. For some particular models, states and bipartitions, this operator is local, but in general it is expected to be non-local. We study this feature for the massless Dirac field on the half-line, by considering the bipartition given by an interval in generic position. Imposing the most general boundary conditions ensuring the global energy conservation leads to two inequivalent phases where either the vector or the axial symmetry is preserved. In these two phases, we find analytic expressions for the modular Hamiltonians of an interval on the half-line when the system is in its ground state, for the the modular flows of the Dirac field and for the modular correlators. The method can be employed to obtain analytic expressions for the modular Hamiltonians, the modular flows and the modular correlators when the bipartition is given by two disjoint equal intervals at the same distance from a point-like defect characterised by a unitary scattering matrix, that allows both reflection and transmission. We also discuss how local entanglement Hamiltonians in free fermionic and bosonic model are obtained from the corresponding lattice results.
