A quadratic well plays a central role in a wide variety of modern physical theories and applications. In this work, we investigate many-body dynamics in a quadratic well, using an Ising chain as a paradigmatic example. In contrast to a uniform Ising chain, where the quantum phase transition is driven by the field strength, the present system exhibits spatially varying quantum phases along the chain. Through analysis of the Majorana representation, we obtain exact solutions for localized modes, revealing a topologically degenerate spectrum in the thermodynamic limit. In the case of a finite-size quantum phase region, the Kramers-like degeneracy is lifted by a constant shift, leading to periodic oscillations for a finite-temperature thermal initial state. Numerical simulations of the magnetization, local density of state, and quench fidelity support our conclusions. Our findings enrich the understanding of many-body dynamics in a trapping field.
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=gc and
h=0, and first-order quantum transitions (FOQTs) driven by
h along the line
h=0 (
g<gc). 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
hi<0 to a positive value
hf>0. We focus on values of
hf such that the spectrum of the post-QQ Hamiltonian
H(g,hf) 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>gc (where the chain is in the disordered phase), for
g=gc (QQ across the CQT), and for
g<gc (QQ across the FOQT line).
We study the real-time dynamics of a quantum Ising chain driven periodically by instantaneous quenches of the transverse field between $+{\ensuremath{\Gamma}}_{0}$ and ${\ensuremath{\Gamma}}_{0}$ back and forth in equal intervals of time. Two interesting phenomena are reported and analyzed. (i) We observe dynamical many-body freezing (DMF), i.e., strongly nonmonotonic freezing of the response with respect to the driving parameters (pulse width and height) resulting from coherent suppression of dynamics of all quasiparticle modes. For certain combinations of the pulse height and the period, maximal freezing (DMF peaks) is observed, where a massive collapse of the entire Floquet spectrum occurs and the many-body system remains frozen extremely close to the initial state for all time. (ii) Second, away from the freezing peak, we observe the emergence of a distinct oscillation with a single nontrivial frequency, which can be much lower than the driving frequency. This remarkable slow oscillation involving many high-energy modes dominates the response in the limit of long observation time. We identify this slow oscillation as the unique survivor of destructive quantum interference between the many-body modes. The oscillation tends to decay algebraically with time to a constant value. All the key features are demonstrated analytically with numerical evaluations for specific results.
To create a more comprehensive understanding of nonequilibrium dynamics of open quantum many-body systems, we visit an exactly solvable example, that is a quenched transverse-field Ising chain coupled to Markovian baths, which act locally on the Jordan-Wigner fermionic space. Performing explicit calculations on the heat transfer, transverse magnetization, and kink density, we find that the imbalance of two opposite damping mechanisms play a crucial role in constructing a nontrivial nonequilibrium steady state accompanied with a dissipative quantum phase transition, also that the competition between unitary drive and decoherence does not necessarily lead to a quasi-stationary state or prethermalization under certain ordinary relaxation.
For a system at absolute zero temperature, all thermal fluctuations are frozen out, while quantum fluctuations prevail.These microscopic quantum fluctuations can induce a quantum phase transition (QPT) in the ground state of a many-body system when the relative strength of two competing energy terms is varied across a critical value.This paper presents a detailed and quantitative study of the magnetic properties of Ising two-spin system at zero temperature in the presence of longitudinal, transverse and mixed fields, and also at finite temperatures.At the critical longitudinal magnetic field, the system is found to undergo a QPT to a ferromagnetic state.For this system, an infinitesimal departure from zero transverse field is found to trigger off QPT and also remove the degeneracies that persisted even in the presence of longitudinal field.These quantum fluctuations, leading to QPT are more evident in the mixed field than the separate effect of longitudinal and transverse fields.The observed quantum effects at zero temperature for this system are found to be completely wiped out at finite temperatures due to thermal fluctuations.A temperature increase favours antiferromagnetic alignment of spins, while a decrease in temperature favours ferromagnetic alignment.
We study the generation and propagation of local perturbations in a quantum many-body spin system. In particular, we study the Ising model in transverse field in the presence of a local field defect at one edge. This system possesses a rich phase diagram with different regions characterized by the presence of one or two Majorana zero modes. We show that their localized character (i) enables a characterization of the Ising phase transition through a local-only measurement performed on the edge spin, and (ii) strongly affects the propagation of quasiparticles emitted after the sudden removal of the defect, so that the dynamics of the local magnetization show clear deviations from a ballistic behavior in the presence of the Majorana fermions.
We study the response of a thermal state of an Ising chain to a nonlocal non-Hermitian perturbation, which coalesces the topological Kramer-like degeneracy in the ferromagnetic phase. The dynamic responses for initial thermal states in different quantum phases are distinct. The final state always approaches its half component with a fixed parity in the ferromagnetic phase but remains almost unchanged in the paramagnetic phase. This indicates that the phase diagram at zero temperature is completely preserved at finite temperatures. Numerical simulations for Loschmidt echoes demonstrate such dynamical behaviors in finite-size systems. In addition, it provides a clear manifestation of the bulk-boundary correspondence at nonzero temperatures. This work presents an alternative approach to understanding the quantum phase transitions of quantum spin systems at nonzero temperatures.