Abstract We study the post-quench unitary dynamics of a quantum sunburst spin model, composed of a transverse-field quantum Ising ring which is suddenly coupled to a set of independent external qubits along the longitudinal direction, in a way to respect a residual translation invariance and the Ising Z 2 symmetry. Starting from the different equilibrium quantum phases of the system, we characterize the decoherence and the energy storage in the external qubits, which may be interpreted as a probing apparatus for the inner Ising ring. Our results show that, in proximity of the quantum transitions of the Ising ring, either first-order or continuous, it is possible to put forward dynamic finite-size scaling frameworks which unveil peculiar scaling regimes, depending on the way in which the large-size limit is taken: either by fixing the number n of probing qubits, or their interspace distance b . In any case, the dependence of the various observables on n can be reabsorbed into a redefinition of the quench parameter by a n prefactor. We also address the role of a nearest-neighbor coupling between the external qubits.
We study the ground-state properties of a quantum sunburst model, composed of a quantum Ising spin ring in a transverse field, symmetrically coupled to a set of ancillary isolated qubits, to maintain a residual translation invariance and also a Z_{2} symmetry. The large-size limit is taken in two different ways: either by keeping the distance between any two neighboring ancillary qubits fixed or by fixing their number while increasing the ring size. Substantially different regimes emerge, depending on the various Hamiltonian parameters: For small energy scale δ of the ancillary subsystem and small ring-qubit interaction κ, we observe rapid and nonanalytic changes in proximity to the quantum transitions of the Ising ring, both first order and continuous, which can be carefully controlled by exploiting renormalization-group and finite-size scaling frameworks. Smoother behaviors are instead observed when keeping δ>0 fixed and in the Ising disordered phase. The effect of an increasing number n of ancillary spins turns out to scale proportionally to sqrt[n] for sufficiently large values of n.
We address the quantum dynamics of a system composed of a qubit globally coupled to a many-body system characterized by short-range interactions. We employ a dynamic finite-size scaling framework to investigate the out-of-equilibrium dynamics arising from the sudden variation (turning on) of the interaction between the qubit and the many-body system, in particular when the latter is in proximity to a quantum first-order or continuous-phase transition. Although the approach is quite general, we consider
d-dimensional quantum Ising spin models in the presence of transverse and longitudinal fields as paradigmatic quantum many-body systems. To characterize the out-of-equilibrium dynamics, we focus on a number of quantum-information-oriented properties of the model. Namely, we concentrate on the decoherence features of the qubit, the energy interchange between the qubit and the many-body system during the out-of-equilibrium dynamics, and the work distribution associated with the quench. The scaling behaviors predicted by the dynamic finite-size scaling theory are verified through extensive numerical computations for the one-dimensional Ising model, which reveal a fast convergence to the expected asymptotic behavior with increasing system size.
We study the nonequilibrium dynamics of an isolated bipartite quantum system, the sunburst quantum Ising model, under interaction quench. The prequench limit of this model is two noninteracting integrable systems, namely a transverse Ising chain and finite number of isolated qubits. As a function of interaction strength, the spectral fluctuation property goes from Poisson to Wigner-Dyson statistics. We chose entanglement entropy as a probe to study the approach to thermalization or lack of it in postquench dynamics. In the near-integrable limit, as expected, the linear entropy displays oscillatory behavior, while in the chaotic limit it saturates. Along with the chaotic nature of the time evolution generator, we show the importance of the role played by the coherence of the initial state in deciding the nature of thermalization. We further show that these findings are general by replacing the Ising ring with a disordered XXZ model with disorder strength putting it in the many-body localized phase.
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 out-of-equilibrium dynamics of one-dimensional quantum Ising-like systems, arising from sudden quenches of the Hamiltonian parameter
g driving quantum transitions between disordered and ordered phases. In particular, we consider quenches to values of
g around the critical value
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
g. 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.
We study the nonequilibrium dynamics of a homogeneous quantum Ising ring after a quench, in which the transverse field
g suddenly changes from zero to a nonzero value. The long-timescale reduced dynamics of a single spin and of two nearest-neighbor spins, which involves the evaluation of expectation values of odd operators that break the fermion parity, is exactly obtained for finite-size but large rings through the use of a recently developed Pfaffian method [N. Wu, Phys. Rev. E 101, 042108 (2020)]. Time dependence of the transverse and longitudinal magnetizations (${\ensuremath{\langle}{\ensuremath{\sigma}}_{j}^{z}\ensuremath{\rangle}}_{t}$ and ${\ensuremath{\langle}{\ensuremath{\sigma}}_{j}^{x,y}\ensuremath{\rangle}}_{t}$), single-spin purity, expectation value of the string operator ${X}_{j}={\ensuremath{\prod}}_{l=1}^{j\ensuremath{-}1}{\ensuremath{\sigma}}_{l}^{z}{\ensuremath{\sigma}}_{j}^{x}$ (${\ensuremath{\langle}{X}_{j}\ensuremath{\rangle}}_{t}$), several equal-time two-site correlators (${\ensuremath{\langle}{\ensuremath{\sigma}}_{j}^{x,z}{\ensuremath{\sigma}}_{j+1}^{x,z}\ensuremath{\rangle}}_{t}$, ${\ensuremath{\langle}{\ensuremath{\sigma}}_{j}^{x}{\ensuremath{\sigma}}_{j+1}^{y}\ensuremath{\rangle}}_{t}$, and ${\ensuremath{\langle}{\ensuremath{\sigma}}_{j}^{x}{\ensuremath{\sigma}}_{j+1}^{z}\ensuremath{\rangle}}_{t}$), and pairwise concurrence after quenches to different phases are numerically studied. Our main findings are that (i) The expectation value of a generic odd operator approaches zero in the long-time limit; (ii) ${\ensuremath{\langle}{X}_{j}\ensuremath{\rangle}}_{t}$ exhibits
j-independent exponential decay for a quench to
g=1 and the time at which ${\ensuremath{\langle}{X}_{j}\ensuremath{\rangle}}_{t}$ reaches its first maximum scales linearly with
j; (iii) The single-spin purity dynamics is mainly controlled by ${\ensuremath{\langle}{\ensuremath{\sigma}}_{j}^{x}\ensuremath{\rangle}}_{t}$ (${\ensuremath{\langle}{\ensuremath{\sigma}}_{j}^{z}\ensuremath{\rangle}}_{t}$) for a quench to
g<1 ($g\ensuremath{\ge}1$). For quenches to the disordered phase with $g\ensuremath{\gg}1$, the single-spin tends to be in the maximally mixed state and the transverse and longitudinal correlators ${\ensuremath{\langle}{\ensuremath{\sigma}}_{j}^{z}{\ensuremath{\sigma}}_{j+1}^{z}\ensuremath{\rangle}}_{t}$ and ${\ensuremath{\langle}{\ensuremath{\sigma}}_{j}^{x}{\ensuremath{\sigma}}_{j+1}^{x}\ensuremath{\rangle}}_{t}$, respectively, approaches $\ensuremath{-}0.25$ and
0.5 in the thermodynamic limit; (iv) The nearest-neighbor entanglement acquires a finite plateau value that increases with increasing
g, and approaches a saturated value $\ensuremath{\sim}0.125$ for $g\ensuremath{\gg}1$.