Quantum fronts and entanglement driven by inhomogeneities

Statistical mechanics teaches us how the equilibrium physics of many-body quantum systems emerges through the simple concept of thermal ensembles; the external world acts as a heat bath that fixes the temperature. For closed quantum systems, however, it is far from obvious how macroscopic equilibrium could emerge from the underlying microscopic dynamics. In particular, how do the mixed-state ensembles of statistical mechanics arise from the unitary time evolution of pure states? In the last two decades, such fundamental questions have been under intensive research and many aspects of the problem have been identified and understood. The notions of thermalization have even been generalized to integrable models, where ergodicity is broken due to the presence of an extensive set of conserved quantities, which have to be incorporated in the proper statistical ensembles. Although integrable systems are non-generic, it turns out that their special relaxation features survive on intermediate time scales even when the system is slightly perturbed away from integrability. Moreover, the presence of disorder can lead to a complete breakdown of ergodicity even in generic interacting systems. This phenomenon is known as many-body localization. The dynamics of integrable systems does not only give rise to generalized thermal ensembles but can support persistent currents which do not decay with time, preventing thermalization and leading to the formation of nonequilibrium steady states (NESS). This requires the presence of macroscopic inhomogeneities in the initial state, which undergo ballistic broadening under the dynamics. In the project entitled Quantum fronts and entanglement driven by inhomogeneities we shall study some key features of the far-from-equilibrium dynamics of various integrable models, which is by far less understood than the relaxation from homogenous initial states. There are many fundamental questions unanswered which we shall address. First, can the form of the NESS for integrable systems be predicted by considering only some main characteristics of the nonequilibrium initial condition? Second, can one observe universal features in the relaxation towards the NESS, i.e. in the propagation of fronts induced by the inhomogeneity? Third, what are the main characteristics of entanglement generation in the front and what is the asymptotic entanglement structure in the NESS? Fourth, how robust are all the above features against perturbing the system away from integrability? Finally, how is the transport obstructed as a consequence of many-body localization? We will attack these questions using a number of different analytical (Bethe Ansatz, conformal field theory) and numerical (matrix product states) methods, for well-known spin chain models that are most relevant for current cold-atom experiments.

Entanglement is the most genuine non-classical feature of quantum many-body systems. On one hand, it plays an important role in characterizing distinct phases of quantum matter in equilibrium at low temperatures. On the other hand, it provides invaluable information about the non-equilibrium dynamics of closed quantum systems, shedding light on the mechanisms of local relaxation and thermalization. In this project we addressed questions about the behaviour of entanglement both in and out of equilibrium. In particular, we studied the spreading of entanglement in the presence of inhomogeneities in various integrable spin chains. These models support long-lived quasiparticle excitations and their dynamics can be treated within a generalized hydrodynamic approach. Our first goal was to study simple inhomogeneous initial states, such as a domain wall or a chain with a density bias. We observed a slow increase of entanglement and uncovered an interesting relation between entropy and magnetization fluctuations. A completely different scenario is when an inhomogeneity is present in the Hamiltonian itself, in the form of a single defect. Here we identified a mechanism which leads to a rapid growth of entanglement due to the backscattering of quasiparticles from the defect. We also investigated various examples of inhomogeneities corresponding to local perturbations of the ground state. Our goal was to study how the excess entanglement produced by such local excitations is transported away in various integrable chains. We managed to find a simple interpretation of the resulting entanglement profile in terms of a simple quasiparticle picture. We also investigated the scaling of the excess entanglement as one moves from local to extended excitations in fermionic hopping chains. Finally, we also studied the so-called entanglement Hamiltonian in simple equilibrium settings. Here the reduced state of a subsystem within a larger chain is written in an exponential form and treated as a kind of statistical mechanics problem. The resulting entanglement Hamiltonian is an inherently inhomogeneous operator, markedly different from the physical Hamiltonian. We studied its structure for both critical and non-critical hopping chains, characterizing the spatial structure of entanglement within the subsystem.