I am studying entanglement and its measurements in the context of a lattice model of the Dirac theory. The idea is that one has two bands, symmetric with respect to $E=0$, and the groundstate is obtained by filling the lower band, $$|GS\rangle=\prod_k\hat{\gamma}^\dagger_{k,-}|0\rangle,$$ where $\hat\gamma^\dagger_{k,\pm}$ annihilates a particle with momentum $k$ in the lower band. Later, a particle-hole transformation can be applied to interpret the filled lower band as a vacuum of antiparticles, $\hat{\gamma}_{k,+}\rightarrow\hat{b}_k$ (particles), $\hat{\gamma}_{k,-}\rightarrow\hat{d}_{-k}^\dagger$ (antiparticles), recovering thus the typical picture from QED.
If one chooses a partition and calculates the Entanglement Entropy of it, one obtains different results depending on the mass of the field (e.g. near the critical point, for $m\approx0$, one recovers predictions of logarithmic scaling from the underlying CFT, while for $m\gg 1$ a strict area law is satisfied).
However, I am now interested in excited states, i.e., states of the form $$|\psi\rangle\propto\prod_kf(k)\hat\gamma_{k,+}^\dagger\hat{\gamma}_{k,-}|GS\rangle,$$ with $f(k)$ a certain distribution of excitations in momentum space. After the particle-hole transformation, this state is understood as containing pairs of particles-antiparticles with opposite momentum. For example, if $f(k)=\delta_{k,q}$, one obtains a state with only one pair of particle-antiparticle with momentum $q$. This state is still an eigenstate of the Hamiltonian, $$\hat{H}=\sum_k\omega_k\left(\hat{\gamma}_{k,+}^\dagger\hat{\gamma}_{k,+}-\hat{\gamma}_{k,-}^\dagger\hat{\gamma}_{k,-}\right),$$ with energy $2\omega_q$, and therefore is an eigenstate of the evolution operator.
Here is where my question appears: the time-evolution of this state seems trivial, since the state is an eigenstate. Therefore, the associated covariance matrix (which is used in lattice models to calculate the Entanglement Entropy) has a trivial evolution in time, and therefore the Entanglement Entropy does not evolve in time. This goes against my intuition, which tells me that, since this state contains pairs of particles-antiparticles with opposite momentum, the entanglement between a partition and its complementary partition should grow as time evolves, in a way somehow related to the group velocity of these particles. Why isn't this the case?
