We study the time evolution of the R\'enyi entanglement entropies following a quantum quench in a two-dimensional (2D) free fermion system. By employing dimensional reduction, we effectively transform the 2D problem into decoupled chains, a technique applicable when the system exhibits translational invariance in one direction. Various initial configurations are examined, revealing that the behavior of entanglement entropies can often be explained by adapting the one-dimensional quasiparticle picture. However, intriguingly, for specific initial states the entanglement entropy saturates to a finite value without the reduced density matrix converging to a stationary state. We discuss the conditions necessary for a stationary state to exist and delve into the necessary modifications to the quasiparticle picture when such a state is absent.
The time evolution of a quantum state with short-range correlations after a quench to a one-dimensional critical Hamiltonian can be understood using the quasiparticle picture, which states that local entanglement spreads as if it was carried by quasiparticles which separate at a fixed speed. We extend the quasiparticle picture using the recently introduced link representation of entanglement, allowing us to apply it to initial states presenting long-range correlations. The entanglement links are current correlators, and therefore follow a wave equation on the appropriate configurational space which allows us to predict the time evolution of the entanglement entropies. Our results are checked numerically for free fermionic chains with different initial entanglement patterns.
In recent years the out-of-equilibrium dynamics of many-body quantum systems has attracted growing interest, both for its fundamental implications in the understanding of quantum thermalization and for potential applications in quantum information and quantum computation. In particular, quantum quenches provide a natural setting to investigate the mechanisms of correlation propagation and entanglement growth far from equilibrium. Interest in these phenomena stems not only from the possibility of testing theoretical predictions coming from emergent hydrodynamic descriptions, but also from understanding whether, and to what extent, such descriptions remain valid in the presence of spatial gradients or interfaces. Two paradigmatic frameworks in this context are the Generalized Hydrodynamics (GHD) and quasi-particle picture. The former is an effective theory devised to describe the long-time evolution after a quench in the thermodynamic limit. Within this perspective, inhomogeneous quenches are a crucial testbed for extending hydrodynamic theories to more realistic situations where, for example, scattering at an interface can occur. This is the main goal of the present thesis. The reference physical model is the one-dimensional Kitaev chain, an integrable spinless p-wave superconductor that exhibits both a topological and a trivial phase. The bilinear structure of the Hamiltonian in the fermionic creation and annihilation operators admits an exact rewriting in terms of Majorana fermions, in which the Hamiltonian remains quadratic. This representation allows a compact description of the ground-state properties via the Majorana correlation matrix (Gamma) at time t=0: the entries of Gamma completely encode the quantum information of the Gaussian ground state. The time evolution of Gamma is studied in the Heisenberg picture, where unitary dynamics correspond to an orthogonal rotation of Gamma implemented by an orthogonal matrix determined by the Kitaev Hamiltonian in the Majorana basis. Consequently, the entanglement entropy dynamics following a quench, both homogeneous and inhomogeneous, can be simulated through simple matrix products, making the approach numerically efficient even for sizable systems. The evolution code for Gamma was validated on a homogeneous quench with periodic boundary conditions: subsystem-size (l) curves exhibit the expected finite-size scaling, (Δ \propto 1/l), and numerical growth rates in the linear regime (t < v/2l), extracted by linear fits at fixed l, match the quasi-particle predictions for various values of the new chemical potential. The rate discrepancy behaves asymptotically as (Δ_{rate}} \propto 1/t_{\min}\) for large (t_{\min}\), up to oscillations from sub-extensive terms. The physical problem of interest is then an inhomogeneous quench of the chemical potential on the infinite Kitaev chain ground state with open boundaries, (μ_0 to μ_L, μ_R), producing a central interface. Initial and final Hamiltonians are free in the quasiparticle sense (ballistic propagation), but the presence of two distinct energy bands implies interface scattering. Simulations varying only μ_R (within the same trivial phase) show a nonmonotonic dependence of the entanglement growth rate, with a maximum at a particular μ_R. To explain this, an analytic ansatz for the stationary Majorana wavefunction was constructed: plane waves multiplied by a Majorana spinor, with bulk spinors obtained from the homogeneous PBC problem via the time-independent Schrödinger equation. Imposing matching at the interface yields four nontrivial equations for two unknowns (overdetermined in general). In the special cases (Δ ̸= t) two equations become trivial, the system is determined, and one obtains an explicit transmission coefficient. Combining this result with known quasi-particle formulas for two semi-infinite chains yields a hydrodynamic prediction for entanglement growth with interface scattering. Energy conservation across the interface selects the momenta that can be transmitted and thus contribute to inter-region correlations. Numerical tests for several total lengths (L) confirm the finite-size scaling (Δ_{\mathrm{e.e.}} \propto 1/L\), and show excellent agreement between numerical and theoretical growth rates for both quenches restricted to the right half and quenches applied to both halves. For the right-only case the rate discrepancy again follows (Δ_{rate} \propto 1/t_{min}\) as
tmin→∞. Possible future directions include: the study of the general inhomogeneous quench with (Δ ̸= t) via a direct extension of the analytic ansatz; the analysis of quenches starting from the topological ground state; the investigation of universal effects near the quantum critical point (μ=2t); the introduction of multiple interfaces or regions with spatially varying parameters to explore interference phenomena between quasiparticle fronts (e.g. periodic alternation (μ_1,μ_2,μ_1,μ_2,....) and, finally, the extension to interacting models where the Gaussian formalism is no longer applicable.
We investigate the dynamics of Entanglement Hamiltonians (EHs) in dissipative free-fermionic systems using a recent operator-based formulation of the quasiparticle picture. Focusing on gain and loss dissipation, we study the post-quench evolution and derive explicit expressions for the EH at the ballistic scale. In the long-time and weak-dissipation regime, the EH is shown to take the form of a time-dependent Generalized Gibbs Ensemble (t-GGE), with a structure that is universal across different initial states of the quench protocol. Within this framework, the emergence of the t-GGE is fully accounted for by the quasiparticle picture, and we argue that this description remains valid whenever the Lindbladian admits an appropriate coarse-grained representation.
The large-scale behaviour of entanglement entropy in finite-density states, in and out of equilibrium, can be understood using the physical picture of particle pairs. However, the full theoretical origin of this picture is not fully established yet. In this work, we clarify this picture by investigating entanglement entropy using its connection with the large-deviation theory for thermodynamic and hydrodynamic fluctuations. We apply the universal framework of Ballistic Fluctuation Theory (BFT), based the Euler hydrodynamics of the model, to correlation functions of branch-point twist fields, the starting point for computing Rényi entanglement entropies within the replica approach. Focusing on free fermionic systems in order to illustrate the ideas, we show that both the equilibrium behavior and the dynamics of Rényi entanglement entropies can be fully derived from the BFT. In particular, we emphasise that long-range correlations develop after quantum quenches, and accounting for these explain the structure of the entanglement growth. We further show that this growth is related to fluctuations of charge transport, generalising to quantum quenches the relation between charge fluctuations and entanglement observed earlier. The general ideas we introduce suggest that the large-scale behaviour of entanglement has its origin within hydrodynamic fluctuations.
Abstract We investigate the dynamics of entanglement Hamiltonians (EHs) in dissipative free-fermionic systems using a recent operator-based formulation of the quasiparticle picture. Focusing on gain and loss dissipation, we study the post-quench evolution and derive explicit expressions for the EH at the ballistic scale. In the long-time and weak-dissipation regime, the EH is shown to take the form of a time-dependent generalized Gibbs ensemble (t-GGE), with a structure that is universal across different initial states of the quench protocol. Within this framework, the emergence of the t-GGE is fully accounted for by the quasiparticle picture, and we argue that this description remains valid whenever the Lindbladian admits an appropriate coarse-grained representation.