There are many ways to characterize quantum systems, for example, by measuring system-specific observables or probing their response to external perturbations. A complementary approach is based on quantum information measures. Independent of the microscopic details, it offers a universal perspective on quantum systems and enables a unified characterization of diverse quantum systems. Central, in this regard, is the determination of the quantum entanglement within a quantum system. Recently, in the interest of studying genuine nonequilibrium properties of a quantum system, the dynamics of the quantum entanglement have been established as a powerful approach to gain deep insights into the structure of quantum matter. In this Thesis, we investigate the entanglement dynamics in one-dimensional non-interacting fermionic chains following a quantum quench, i.e. after a sudden change of the system parameters. We propose a new and generic setting for the analysis of entanglement spreading in complex inhomogeneous systems, which we call a hybrid quench-probe setup. The key feature of this new approach is a separation between the quenched part of the system and a static probe region, where the quench induced entanglement spreading is measured. This separation allows a spatial and energy-resolved analysis of the entanglement dynamics. We demonstrate the capability of our protocol by identifying fractionally quantized entanglement jumps of log(2)/2 as a signature of fractionalized topological quasi-particles, such as isolated Majorana modes, that appear at the edges of an open-boundary Kitaev chain in the topological phase. In addition, we observe in this setup an interesting nonequilibrium effect: a static and a propagating contribution to the entanglement dynamics. To understand the origin of this effect, a local perspective on the entanglement structure is necessary. For this purpose, we employ the information lattice, which offers a local scale-resolved decomposition of the correlations present in a quantum state. In nonequilibrium systems with unitary dynamics, this framework enables a hydrodynamic description of the information flow through local densities and currents. Using the information lattice, we analyze three quantum quenches in one-dimensional fermionic systems and identify spatiotemporal features of the local information flow, such as information interfaces and long-range transport from topological edge modes. This analysis enables us to derive an analytical description of the nonequilibrium effect observed in the quenched Kitaev chain.
There are many ways to characterize quantum systems, for example, by measuring system-specific observables or probing their response to external perturbations. A complementary approach is based on quantum information measures. Independent of the microscopic details, it offers a universal perspective on quantum systems and enables a unified characterization of diverse quantum systems. Central, in this regard, is the determination of the quantum entanglement within a quantum system. Recently, in the interest of studying genuine nonequilibrium properties of a quantum system, the dynamics of the quantum entanglement have been established as a powerful approach to gain deep insights into the structure of quantum matter. In this Thesis, we investigate the entanglement dynamics in one-dimensional non-interacting fermionic chains following a quantum quench, i.e. after a sudden change of the system parameters. We propose a new and generic setting for the analysis of entanglement spreading in complex inhomogeneous systems, which we call a hybrid quench-probe setup. The key feature of this new approach is a separation between the quenched part of the system and a static probe region, where the quench induced entanglement spreading is measured. This separation allows a spatial and energy-resolved analysis of the entanglement dynamics. We demonstrate the capability of our protocol by identifying fractionally quantized entanglement jumps of log(2)/2 as a signature of fractionalized topological quasi-particles, such as isolated Majorana modes, that appear at the edges of an open-boundary Kitaev chain in the topological phase. In addition, we observe in this setup an interesting nonequilibrium effect: a static and a propagating contribution to the entanglement dynamics. To understand the origin of this effect, a local perspective on the entanglement structure is necessary. For this purpose, we employ the information lattice, which offers a local scale-resolved decomposition of the correlations present in a quantum state. In nonequilibrium systems with unitary dynamics, this framework enables a hydrodynamic description of the information flow through local densities and currents. Using the information lattice, we analyze three quantum quenches in one-dimensional fermionic systems and identify spatiotemporal features of the local information flow, such as information interfaces and long-range transport from topological edge modes. This analysis enables us to derive an analytical description of the nonequilibrium effect observed in the quenched Kitaev chain.
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.
The crossover between topology and nonequilibrium dynamics has emerged as a rich frontier, in which quantum systems can exhibit unique dynamical phenomena that lie beyond the reach of equilibrium. Of particular interest are quench dynamics across topological phase, as it may reveal the information about the underlying Majorana zero-energy states. Here, we investigate the fate of Majorana boundary modes in a quenched fermionic superfluid using self-consistent time-dependent Bogoliubov-de Gennes theory. For quantum quenches within the topological regime, Majorana boundary modes survive but undergo coherent boundary oscillations arising from the nonadiabatic deformation of their wave functions. Furthermore, a pronounced ripple pattern appears in the post-quench density distribution following a sudden topology-changing quench. Here we identify that these ripple structures originates from the coherent hybridization and interference of the initially separated Majorana boundary states. Our findings establish nonequilibrium boundary dynamics as a new approach for probing Majorana physics, which is complementary to conventional equilibrium measurements.
With the discovery of the integer quantum Hall effect by von Klitzing and collaborators in 1980, the mathematical field of topology entered the world of condensed matter physics. Almost three decades later, this eventually led to the theoretical prediction and the experimental realization of many intriguing topological materials and topology-based devices. In this Ph.D. thesis, we will study the interplay between topology and another key topic in condensed matter physics, namely the study of inter-particle interactions in many-body systems. This interplay is analyzed from two different perspectives. Firstly, we studied how the presence of electron-electron interactions affects single-electron injection into a couple of counter-propagating one-dimensional edge channels. The latter appear at the edges of topologically non-trivial systems in the quantum spin Hall regime and they can also be engineered by exploiting the integer quantum Hall effect. Because of inter-channel interactions, the injected electron splits up into a couple of counter-propagating fractional excitations. Here, we carefully study and discuss their properties by means of an analytical approach based on the Luttinger liquid theory and the bosonization method. Our results are quite relevant in the context of the so-called electron quantum optics, a fast developing field which deeply exploits the topological protection of one-dimensional edge states to study the coherent propagation of electrons in solid-state devices. As an aside, we also showed that similar analytical techniques can also be used to study the time-resolved dynamics of a Luttinger liquid subject to a sudden change of the interaction strength, a protocol known as quantum quench which is gaining more and more attention, especially within the cold-atoms community. Secondly, we study how inter-particle interactions can enhance the topological properties of strictly one-dimensional fermionic systems. More precisely, the starting point is the seminal Kitaev chain, a free-fermionic lattice model which hosts exotic Majorana zero-energy modes at its ends. The latter are extremely relevant in the context of topological quantum computation because of their non-Abelian anyonic exchange statistics. Here we show that, by properly adding electron-electron interactions to the Kitaev chain, it is possible to obtain lattice models which feature zero-energy parafermionic modes, an even more intriguing generalization of Majoranas. To this end, we develop at first an exact mapping between Z4 parafermions and ordinary fermions on a lattice. We subsequently exploit this mapping to analytically obtain an exactly solvable fermionic model hosting zero-energy parafermions. We study their properties and numerically investigate their signatures and robustness even when parameters are tuned away from the exactly solvable point.
Quantum systems display counterintuitive features, escaping our everyday-life intuition, whose origin is related to entanglement. This phenomenon arises when different parts of the same systems display correlations, which cannot be understood via classical mechanics, and they are dubbed as quantum. To characterize the correlations above, it is necessary to find good entanglement measures which are able to capture quantitatively the quantum features of a given system. This motivates the investigation and the computation of quantities such as the von Neumann entropy of regions, which provides information about how a system is entangled. Our focus is quantum one-dimensional systems close to their critical point, where quantum correlations are enhanced, and a characterization via Quantum Field Theory (QFT) can be provided. In particular, we aim to understand the role of symmetries and localized defects on the entanglement properties. Those aspects display an intimate connection. First, via replica trick, the entropy is expressed via a replica permutation symmetry. Then, single defects break symmetries that are present in the corresponding homogeneous systems. Due to these reasons, the characterization of symmetry actions on subsystems can be viewed as a unifying framework for quantifying entanglement. We split this thesis into two parts. The first part is devoted to the symmetry-resolved entanglement. We first compute it in massive integrable field theories, giving analytical predictions for free fermions/bosons and the 3-state Potts model. For instance, we characterize the universal corrections to area law via form factor bootstrap. We also study excited states of field theories, and we characterize both low-energy states of Conformal Field Theory (CFT) and quasi-particle states. The second part deals with quantum correlations across defects. We provide a CFT framework to compute Rényi entropy and negativity in the ground state of conformal junctions, giving analytical predictions for free fermions and bosons. We find logarithmic growth of the entanglement measures above, and we relate the universal prefactor to the boundary conditions, encoded in a boundary scattering matrix. Then, we focus on the entanglement dynamics, and we study the time evolution of correlations for two paradigmatic classes of quench protocols. For a global quench, we find a linear growth of entropy across the defect whose slope is computed both in CFT, for thermal states, and for free fermions, via a modification of the quasi-particle picture. For a domain wall melting protocol, also known as inhomogeneous quench, we find that, while local observables are easily understood via hydrodynamics and local generalized Gibbs ensembles, the same is not true for the entropy. In contrast, by employing the quasi-particle picture, we provide precise predictions for the dynamics of entanglement, revealing the influence of long-range correlations between points that are specular with respect to the defect.
We study the quench dynamics of a topologically trivial one-dimensional gapless wire following its sudden coupling to topological bound states. We find that as the bound states leak into and propagate through the wire, signatures of their topological nature survive and remain measurable over a long lifetime. Thus, the quench dynamically induces topological properties in the gapless wire. Specifically, we study a gapless wire coupled to fractionally charged solitons or Majorana fermions, and we characterize the dynamically induced topology in the wire, in the presence of coupling to the bath, disorder, and short-range interactions, by analytical and numerical calculations of the dynamics of fractional charge, fermion parity, entanglement entropy, and fractional exchange statistics. In a dual effective description, this phenomenon is described by correlators of boundary changing operators, which, remarkably, generate topologically nontrivial monodromies in the gapless wire, both for Abelian and non-Abelian quantum statistics of the bound states.