We consider two semi-infinite quantum Ising chains initially at thermal equilibrium at two different temperatures and subsequently joined by an interaction between their end points. Transport properties such as the heat current are determined by the dynamics of the left- and right-moving fermionic quasiparticles which characterize the ensuing unitary dynamics. Within the so-called semiclassical space-time scaling limit we extend known results by determining the full space and time dependence of the density and current of energy and of fermionic quasiparticles. Upon approaching the edge of the propagating front, these quantities as well as the two-point correlation function display qualitatively different behaviors depending on the transverse field of the chain being critical or not. While in the latter case corrections to the leading behavior are described, as expected, by the Airy kernel, in the former a novel scaling form emerges with universal features.
We study the relaxation behaviour of the quantum Ising chain, focusing our attention onto the non-equilibrium dynamics of the transverse magnetization. The initial states, from which the magnetization relaxes, are product states such as those generated by setting in contact several systems, each initially equilibrated at a given temperature. Due to the free fermionic structure of the chain, the dynamics of the transverse magnetization is easily expressed in a compact form. In the completely factorized initial state, corresponding to a situation where all the spins are thermalized independently, we obtain in the scaling limit the Green function associated to the transverse magnetization. The relaxation behaviour is also considered in the system-bath case, where part of the chain called the system is thermalized at a temperature Ts and the remaining part is at a temperature Tb. The magnetization profiles show a scaling behaviour. Moreover, in the extreme case Tb=∞ and Ts=0, it is shown that the magnetization relaxes in quantized steps in the strong transverse field region.
We study the out-of-equilibrium dynamics in the XX chain with step-like magnetic field, which maps to an inhomogeneous tight-binding chain after Jordan-Wigner transformation. We obtain exact analytic expressions for the fermionic two-point correlation function after a quantum quench from several initial product states, both homogeneous and inhomogeneous ones. This is achieved by using a combination of Fourier and Laplace transforms, which al low us to map the problem to a standard Riemann-Hilbert problem on the unit circle. For arbitrary positions and times the correlators are not expressed in terms of elementary functions. However, in the hydrodynamic limit
x,y,t→∞ with fixed ratios, we provide explicit formulas that depend only on the effective transmission coefficient across the origin. We benchmark our analytic predictions against exact numerical simulations and find excellent agreement in the hydrodynamic limit, apart from finite-time corrections.
We study the quantum dynamics resulting from preparing a one-dimensional quantum system in the ground state of initially two decoupled parts which are then joined together (local quench). Specifically we focus on the transverse Ising chain and compute the time-dependence of the magnetization profile, m_l(t), and correlation functions at the critical point, in the ferromagnetically ordered phase and in the paramagnetic phase. At the critical point we find finite size scaling forms for the nonequilibrium magnetization and compare predictions of conformal field theory with our numerical results. In the ferromagnetic phase the magnetization profiles are well matched by our predictions from a quasi-classical calculation.
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 consider the non-equilibrium physics induced by joining together two tight-binding fermionic chains to form a single chain. Before being joined, each chain is in a many-fermion ground state. The fillings (densities) in the two chains might be different. We present a number of exact results, focusing on two-point correlators and the Loschmidt echo (return probability). For the non-interacting case, we identify through an exact derivation the regime in which a semiclassical ansatz is valid. We present a number of analytical results beyond semiclassics, such as the approach to the non-equilibrium steady state and the appearance of Tracy-Widom distributions at the front of the light cone. The light cone behavior is quantified through a series expansion in time, and this description is shown to be valid for interacting systems as well. Results on the Loschmidt echo, presented for finite and zero interactions, illustrate that the physics is different from both local and global quenches.