Description
The last two decades have witnessed an increasingly growing interest in the study and characterisation of the entanglement structure of many-body quantum systems, also due to the development of related experiments. In this framework, a central object is the so-called entanglement Hamiltonian (EH), defined as the logarithm of the reduced density matrix, that provides a full description of the entanglement of a quantum state. In this seminar, I will present some recent results on the EH of inhomogeneous free fermionic systems. First, I will consider the two paradigmatic cases of the hopping chain with a linear potential, and the Fermi gas with a quadratic chemical potential. For both systems, we find that the EH is given by a deformed version of the physical Hamiltonian, with local inverse temperatures increasing linearly from the entanglement cut, in agreement with CFT predictions. Moreover, we show that the structure predicted by CFT is inherited by a tridiagonal matrix and a differential operator that commute exactly with the EH in the lattice and continuum cases, respectively. This result allows to obtain a very good approximation of the entanglement spectrum and entropy, using the properly rescaled eigenvalues of the commuting operator. Second, I will present the results for the EH of free-fermion chains with a particular form of inhomogeneity, namely the hopping amplitudes and chemical potentials are chosen such that the single particle eigenstates are related to discrete orthogonal polynomials of the Askey scheme. The bispectral properties of these functions allows the construction of an operator which commutes exactly with the EH. We show that also for these systems the commuting operators have the form of a spatial deformation of the physical Hamiltonian, with a deformation term identified as the local inverse temperature derived from a CFT treatment of the problem in the continuum limit. As in the previous cases, rescaling the eigenvalues of the commuting operators provides an excellent approximation of the entanglement spectrum.
