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).