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arXiv:1803.00280 · Physical review. B./Physical review. B

Aging dynamics in quenched noisy long-range quantum Ising models

Jad C. Halimeh, Matthias Punk, Francesco Piazza

Published Jul 6, 2018• 11 citations
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Abstract

We consider the $d$-dimensional transverse-field Ising model with power-law interactions $J/{r}^{d+\ensuremath{\sigma}}$ in the presence of a noisy longitudinal field with zero average. We study the longitudinal-magnetization dynamics of an initial paramagnetic state after a sudden switch-on of both the interactions and the noisy field. While the system eventually relaxes to an infinite-temperature state with vanishing magnetization correlations, we find that two-time correlation functions show aging at intermediate times. Moreover, for times shorter than the inverse noise strength $\ensuremath{\kappa}$ and distances longer than $a{(J/\ensuremath{\kappa})}^{2/\ensuremath{\sigma}}$ with $a$ being the lattice spacing, we find a critical scaling regime of correlation and response functions consistent with the model A dynamical universality class with an initial-slip exponent $\ensuremath{\theta}=1$ and dynamical critical exponent $z=\ensuremath{\sigma}/2$. We obtain our results analytically by deriving an effective action for the magnetization field including the noise in a nonperturbative way. The above scaling regime is governed by a nonequilibrium fixed point dominated by the noise fluctuations.

DOI
10.1103/physrevb.98.045111

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We consider the ddd-dimensional transverse-field Ising model with power-law interactions J/rd+\ensuremathσJ/{r}^{d+\ensuremath{\sigma}}J/rd+\ensuremathσ in the presence of a noisy longitudinal field with zero average. We study the longitudinal-magnetization dynamics of an initial paramagnetic state after a sudden switch-on of both the interactions and the noisy field. While the system eventually relaxes to an infinite-temperature state with vanishing magnetization correlations, we find that two-time correlation functions show aging at intermediate times. Moreover, for times shorter than the inverse noise strength \ensuremathκ\ensuremath{\kappa}\ensuremathκ and distances longer than a(J/\ensuremathκ)2/\ensuremathσa{(J/\ensuremath{\kappa})}^{2/\ensuremath{\sigma}}a(J/\ensuremathκ)2/\ensuremathσ with aaa being the lattice spacing, we find a critical scaling regime of correlation and response functions consistent with the model A dynamical universality class with an initial-slip exponent \ensuremathθ=1\ensuremath{\theta}=1\ensuremathθ=1 and dynamical critical exponent z=\ensuremathσ/2z=\ensuremath{\sigma}/2z=\ensuremathσ/2. We obtain our results analytically by deriving an effective action for the magnetization field including the noise in a nonperturbative way. The above scaling regime is governed by a nonequilibrium fixed point dominated by the noise fluctuations.
We study the dynamics of thermalization resulting from a time-dependent noise in a quantum Ising chain subject to a sudden quench of the transverse magnetic field. For weak noise the dynamics shows a prethermalized state at intermediate time scales, eventually drifting towards an asymptotic infinite temperature steady state characterized by diffusive behavior. By computing analytically the density of kinks, as well as the transverse and longitudinal magnetic field correlators, we characterize these two regimes, their observability, and their signatures in the various physical quantities.
The theoretic renormalization-group approach is applied to the study of short-time critical behavior of the ddd-dimensional spin systems (model A) in the presence of quenched impurities with a long-range correlations decaying as ${r}^{\ensuremath{-}(d\ensuremath{-}\ensuremath{\rho})}$. The asymptotic scaling laws are studied in the frame of a double expansion in $ϵ=4\ensuremath{-}d$ and $\ensuremath{\rho}$ with $\ensuremath{\rho}$ of order ϵϵϵ. In d<4d<4d<4, the initial slip exponents ${\ensuremath{\theta}}^{\ensuremath{'}}$ of the magnetization and $\ensuremath{\theta}$ of the response function, are calculated up to two-loop order. The crossover between fixed points is obtained. The long-time limit of the fluctuation-dissipation ratio is found in the aging regime, and its connection to equilibrium quantities is discussed. The comparison of our results with those of other systems without long-range correlated quenched impurities is also investigated.
Based on the obtained exact results we systematically study the quench dynamics of a one-dimensional spin-1/2 transverse field Ising model with zero- and finite-temperature initial states. We focus on the magnetization of the system after a sudden change of the external field and a coherent time-evolution process. With a zero-temperature initial state, the quench magnetic susceptibility as a function of the initial field strength exhibits strongly similar scaling behaviors to those of the static magnetic susceptibility, and the quench magnetic susceptibility as a function of the final field strength shows a discontinuity at the quantum critical point. This discontinuity remains robust and always occurs at the quantum critical point even for the case of finite-temperature initial systems, which indicates a great advantage of employing quench dynamics to study quantum phase transitions.
Abstract We study an exactly solvable long-range (LR) transverse-field Ising model with a power-law decaying interaction characterized by the decay exponent α . For 1 < α < 2 , the model exhibits two quantum critical points with distinct critical exponents. In the thermodynamic limit, the system is adiabatically driven across these critical points in the presence of noise, from the paramagnetic phase with all spins down to one with all spins up. In the noiseless case, within the LR regime, the steady-state properties are primarily influenced by the modes near the k c = π region, while the low-energy modes around k c = 0 remain nearly adiabatic and contribute negligibly. The defect density follows Kibble–Zurek (KZ) scaling law and scales as n ∝ τ Q − 1 / 2 , implying KZ scaling exponent independent of decay exponent. However, the magnitude of the defect density increases with increasing the range (i.e. decreasing the value of α ). On the other hand in the presence of noise, the low-energy modes around k c = 0 become highly susceptible to excitation, making them the dominant source of excitations, with their contribution gradually diminishing as the interaction range increases. This differs from the short-range model, where previous studies have shown that under noisy condition modes around k = π / 2 play a significant role. The contrasting influence of LR interactions under noisy and noiseless driving, is also observed in spin–spin correlation function and defect distribution, highlighting how LR interactions fundamentally reshape the interplay between adiabatic driving and noise.
We study the dynamics caused by transport of transverse magnetization in one dimensional transverse Ising chain at zero temperature. We observe that a class of initial states in fermionic momentum-space produce spatial variation in transverse magnetization. Starting from such a state, we obtain the transverse magnetization analytically and then observe its dynamics in presence of a homogeneous constant field Γ\GammaΓ. In contradiction with general expectation, whatever be the strength of the field, the magnetization of the system does not become homogeneous even after infinite time. In each site, the dynamics is associated with oscillations having two different timescales. The envelope of the larger timescale oscillation decays algebraically having an exponent which is universal for the entire class of such initial states. Signature of quantum criticality is also embedded in the dynamics in two aspects : Behavioral change shown by the frequency of large timescale oscillation at any site and the corresponding magnetization at infinity when they cross the quantum critical point.
We study an infinite-range ferromagnetic Ising model in the presence of a transverse magnetic field, which exhibits a quantum paramagnetic-ferromagnetic phase transition at a critical value of the transverse field. In the thermodynamic limit, the low-temperature properties of this model are dominated by the behavior of a single large classical spin governed by an anisotropic Hamiltonian. Using this property, we study the quench and ac dynamics of the model both numerically and analytically, and develop a correspondence between the classical phase-space dynamics of a single spin and the quantum dynamics of the infinite-range ferromagnetic Ising model. In particular, we compare the behavior of the equal-time order-parameter correlation function both near to and away from the quantum critical point in the presence of a quench or ac transverse field. We explicitly demonstrate that a clear signature of the quantum critical point can be obtained by studying the ac dynamics of the system even in the classical limit. We discuss possible realizations of our model in experimental systems.