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arXiv:2512.17333 · Physical review. B./Physical review. B

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

Andrea Pelissetto, Davide Rossini, Ettore Vicari

cond-mat.stat-mechquant-phSubmitted Dec 19, 2025• Announced Dec 22, 2025• 1 citation
v1 · Dec 19, 2025v2 · Jan 29, 2026
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Abstract

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 ${\hat H}(g,h_f)$ 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).

Journal reference
Phys. Rev. B 113, 144316 (2026)
DOI
10.1103/plrd-q8vq
Comments
24 pages

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We study the out-of-equilibrium dynamics of one-dimensional quantum Ising-like systems, arising from sudden quenches of the Hamiltonian parameter ggg driving quantum transitions between disordered and ordered phases. In particular, we consider quenches to values of ggg around the critical value gc{g}_{c}gc​, and mainly address the question whether, and how, the quantum transition leaves traces in the evolution of the transverse and longitudinal magnetizations during such a deep out-of-equilibrium dynamics. We shed light on the emergence of singularities in the thermodynamic infinite-size limit, likely related to the integrability of the model. Finite systems in periodic and open boundary conditions develop peculiar power-law finite-size scaling laws related to revival phenomena, but apparently unrelated to the quantum transition, because their main features are generally observed in quenches to generic values of ggg. We also investigate the effects of dissipative interactions with an environment, modeled by a Lindblad equation with local decay and pumping dissipation operators within the quadratic fermionic model obtainable by a Jordan-Wigner mapping. Dissipation tends to suppress the main features of the unitary dynamics of closed systems. We finally address the effects of integrability breaking, due to further lattice interactions, such as in anisotropic next-to-nearest-neighbor Ising (ANNNI) models. We show that some qualitative features of the post-quench dynamics persist, in particular, the different behaviors when quenching to quantum ferromagnetic and paramagnetic phases, and the revival phenomena due to the finite size of the system.
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We study the out-of-equilibrium Kibble-Zurek (KZ) dynamics in quantum Ising chains in a transverse field, driven by a time-dependent longitudinal field h(t)=t/tsh(t)=t/t_sh(t)=t/ts​ (tst_sts​ is the time scale of the protocol), across their first-order quantum transitions (FOQTs) at h=0h=0h=0. The KZ protocol starts at time ti<0t_i<0ti​<0 from the negatively magnetized ground state for hi=ti/ts<0h_i = t_i/t_s<0hi​=ti​/ts​<0. Then, the system evolves unitarily up to a time tf>0t_f > 0tf​>0, such that the magnetization of the state at time tft_ftf​ is positive. In finite-size systems, the KZ dynamics develops out-of-equilibrium finite-size scaling (OFSS) behaviors. Their scaling variables depend either exponentially or with a power law on the size, depending on the boundary conditions (BC). The OFSS functions can be computed in effective models restricted to appropriate low-energy (magnetized and/or kink) states. The KZ scaling behavior drastically changes in the thermodynamic limit (TL), defined as the infinite-size limit keeping ttt and tst_sts​ fixed, which appears substantially unrelated with the OFSS regime, because it involves higher-energy multi-kink states, which are irrelevant in the OFSS limit. The numerical analyses of the KZ dynamics in the TL show the emergence of a quantum spinodal-like scaling behavior at the FOQTs for all considered BC, which is independent of the BC. The longitudinal magnetization changes sign at h(t)=h∗>0h(t)=h*>0h(t)=h∗>0, where h∗h*h∗ decreases with increasing tst_sts​, as h∗∼1/ln⁡tsh*\sim 1/\ln t_sh∗∼1/lnts​. Moreover, in the large-tst_sts​ limit, the time-dependence of the magnetization is described by a universal function of Ω=t/τsΩ= t/τ_sΩ=t/τs​, with τs=ts/ln⁡tsτ_s = t_s/\ln t_sτs​=ts​/lnts​.
There has been much recent interest in the study of out of equilibrium quantum systems due to experimental advances in trapped cold atomic gases that allow systematic studies of these effects.In this work we study a quantum Ising model in the presence of transverse and longitudinal magnetic fields.The model lacks an analytical solution in the most general case and has to be treated numerically.We first investigate the static properties of the model, including the ground state energy and the zero temperature phase diagram, using the Density Matrix Renormalization Group (DMRG) method.We identify ordered and disordered phases separated by a quantum phase transition.We then consider the time evolution of the system following an abrupt quench of the fields.We study the response of the order parameter to the quenches within the ordered and disordered phases as well as quenches between the phases using the time-dependent DMRG method.The results are in qualitative agreement with experiments and the implemented numerical method in this work can be extended to allow for more general interactions.iii I would like to thank my supervisor, Dr. Malcolm Kennett, whose constant support and hope in the most disappointing situations, his helpful guidance, and endless energy were all new and very invaluable to me.I like to thank my