Understanding the structure of entanglement in extended quantum systems is a fundamental problem in theoretical physics. In this framework, a central object is the so-called entanglement (or modular) Hamiltonian (EH), defined as the logarithm of the reduced density matrix, that encodes the full structure of bipartite entanglement. In general the EH  is very hard to compute and is not even expected to be a local operator.  However, in relativistic quantum field theory the locality of the modular Hamiltonian for half-space bipartitions is ensured by the Bisognano–Wichmann theorem, which expresses it as an integral of the energy density with a linear spatial weight. In the presence of conformal symmetry, this result can be extended to other geometries and to some non-equilibrium settings. Since the Bisognano–Wichmann theorem is formulated within relativistic quantum field theory, a natural question concerns its applicability to lattice many-body systems whose low-energy properties are described by a conformal field theory, but which explicitly break Lorentz invariance. Several results addressing this question exist in equilibrium situations, while a lattice realisation of the time-dependent EH in out-of-equilibrium dynamics is still missing.

In this talk, I will present the study of the dynamics of the EH in a system of one-dimensional free fermions, following a local joining quench of two initially disconnected half-chains in their ground states. Applying techniques of conformal field theory, a local expression of the EH is derived, where the left- and right-moving components of the energy density are associated with different weight functions.

The results are then compared to numerical calculations for the hopping chain, which requires to consider a proper continuum limit of the lattice EH, obtaining a good agreement with the field-theory prediction.