Topological insulators are materials that behave like an insulator in the bulk but have conducting edge states. These edge states are topologically protected and exhibit unique properties. Studying the dynamics of these systems far-from equilibrium can provide information on the stability of these edge states and provide insight on the propagation of entanglement which is of interest in quantum information. In this thesis, I study the far-from-equilibrium dynamics of the Haldane model. Various quenches between different topological phases are performed, and the dynamics of both the entanglement and particle numbers are analyzed. Using a correlation matrix, the time evolution of the entanglement and charge fluctuations are calculated. The dynamics of the entanglement and charge fluctuations provides insight on the evolution of this system following a quench. The charge fluctuations allow us to probe the entanglement of these systems using a quantity that can be measured experimentally. I showed that for the various quenches, the charge fluctuations mirrors the entanglement entropy providing examples for which the second moment provides the majority of the weight. Additionally, I showed that the entanglement entropy drastically increases immediately following a quench until the system reaches a steady state. By analyzing the entanglement spectrum I showed that the bulk states contribute to the entanglement following a quench. This thesis establishes a framework that can be applied to other topological systems.
Topological insulators are materials that behave like an insulator in the bulk but have conducting edge states. These edge states are topologically protected and exhibit unique properties. Studying the dynamics of these systems far-from equilibrium can provide information on the stability of these edge states and provide insight on the propagation of entanglement which is of interest in quantum information. In this thesis, I study the far-from-equilibrium dynamics of the Haldane model. Various quenches between different topological phases are performed, and the dynamics of both the entanglement and particle numbers are analyzed. Using a correlation matrix, the time evolution of the entanglement and charge fluctuations are calculated. The dynamics of the entanglement and charge fluctuations provides insight on the evolution of this system following a quench. The charge fluctuations allow us to probe the entanglement of these systems using a quantity that can be measured experimentally. I showed that for the various quenches, the charge fluctuations mirrors the entanglement entropy providing examples for which the second moment provides the majority of the weight. Additionally, I showed that the entanglement entropy drastically increases immediately following a quench until the system reaches a steady state. By analyzing the entanglement spectrum I showed that the bulk states contribute to the entanglement following a quench. This thesis establishes a framework that can be applied to other topological systems.
One of the appealing features of topological systems is the presence of robust edge modes. Under a sudden quantum quench, the edge modes survive for a characteristic time that scales with the system size, during which the nontrivial topology continues to manifest in entanglement properties, even though the post-quench Hamiltonian belongs to a trivial phase. We exemplify this in the quench dynamics of a two-dimensional Haldane model with the help of one-particle entanglement spectrum and the probability density of its mid-states. We find that, beyond our knowledge in one-dimensional models, the momentum dependence of the transverse velocity plays a crucial role in the out-of-equilibrium evolution of the entanglement properties.
Quantum entanglement plays an important role in characterizing the property of many-body systems and quantum field theories. In this thesis, we study the quantum entanglement in (1+1)-dimensional critical systems and (2+1)-dimensional topologically ordered phases. For (1+1)-d critical systems, we mainly study the non-equilibrium property of quantum entanglement, by focusing on three interesting cases: (i) the time evolution of entanglement hamiltonian during thermalization of a subsystem after a global quantum quench; (ii) time evolution of entanglement entropy after an inhomogeneous quantum quench; (iii) entanglement negativity evolution after a local quantum quench. For (2+1)-d topologically ordered phases, we use edge theory approach to study the topological entanglement entropy, mutual information and entnanglement negativity of Chern-Simons theories, which are further confirmed based on the surgey approach. In addition, we study the entanglement renormalization of topological insulators, and investigate the geometric and topological properties in the bulk of entanglement renormalization.
Higher-order topological insulators are materials with a bulk energy gap featuring gapless modes at (d−n)-dimensional edges, where n>1. Despite this modified bulk-boundary correspondence, their nontrivial topology can nevertheless be observed through the bipartite entanglement spectrum. Inspired by works suggesting fluctuations in conserved quantities could be used to study topology in one-dimensional systems, which likewise feature zero-dimensional edge modes, we extend this treatment to a two-dimensional second-order topological insulator where corner states play an analogous role. We show that the standard bipartite entanglement entropy and particle number fluctuation lack striking signals of topological phase transitions, both in the static case and in the time evolution after a quench, obscured by an area-law term from the subsystem edges. By introducing a quadripartite construction of the subsystem, we show that it is possible to isolate the topological contributions, producing sharp transition peaks in both quantities, observable by studying subsystems much smaller than the full system. As particle number fluctuations are accessible in existing experiments, these results establish a concrete and scalable path to experimental detection of the entanglement in higher-order topological insulators.
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.
We explore the nonequilibrium response of Chern insulators. Focusing on the Haldane model, we study the dynamics induced by quantum quenches between topological and nontopological phases. A notable feature is that the Chern number, calculated for an infinite system, is unchanged under the dynamics following such a quench. However, in finite geometries, the initial and final Hamiltonians are distinguished by the presence or absence of edge modes. We study the edge excitations and describe their impact on the experimentally observable edge currents and magnetization. We show that, following a quantum quench, the edge currents relax towards new equilibrium values, and that there is light-cone spreading of the currents into the interior of the sample.
We investigate evolution of entanglement spectra of the Haldane model for Chern insulators upon a sudden quench within the same topological phase. In particular, we focus on the location of the entanglement spectrum crossing, which signifies the bulk topology. It is shown that the coplanarity condition for the pseudomagnetic field of the model, which can be used to determine the crossing in the equilibrium case, needs to be relaxed. We analytically derive the non-equilibrium condition with the help of an edge-state wave function ansatz and a dynamically induced length-scale cutoff. With spectral analyses, it is realized that the oscillatory behavior of the crossing is dominated by the interband excitations at the van Hove singularities.