Quantum quenches across continuous and first-order quantum transitions in one-dimensional quantum Ising models

We investigate the quantum dynamics generated by quantum quenches (QQs) of the Hamiltonian parameters in many-body systems, focusing on protocols that cross first-order and continuous quantum transitions, both in finite-size systems and in the thermodynamic limit. As a paradigmatic example, we consider the quantum Ising chain in the presence of homogeneous transverse (g) and longitudinal (h) magnetic fields. This model exhibits a continuous quantum transition (CQT) at g=g_c and h=0, and first-order quantum transitions (FOQTs) driven by h along the line h=0 $g<g_c$. In the integrable limit h=0, the system can be mapped onto a quadratic fermionic theory; however, any nonvanishing longitudinal field breaks integrability and the spectrum of the resulting Hamiltonian is generally expected to enter a chaotic regime. We analyze QQs in which the longitudinal field is suddenly changed from a negative value h_i < 0 to a positive value h_f>0. We focus on values of h_f such that the spectrum of the post-QQ Hamiltonian H\(g,h_f\) lies in the chaotic regime, where thermalization may emerge at asymptotically long times. We study the out-of-equilibrium dynamics for different values of g, finding qualitatively distinct behaviors for g > g_c (where the chain is in the disordered phase), for g = g_c (QQ across the CQT), and for g<g_c (QQ across the FOQT line).