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arXiv:2008.08583 · Physical review. E

Quantum aging and dynamical universality in the long-range O(N→∞) model

Jad C. Halimeh, Mohammad F. Maghrebi

Published May 28, 2021• 13 citations

Abstract

Quantum quenches to or near criticality give rise to the phenomenon of aging, manifested by glassylike dynamics at short times and far from equilibrium. The recent surge of interest in the dynamics of quantum many-body systems has rejuvenated interest in this phenomenon. Motivated by the ubiquitous long-range interactions in emerging experimental platforms, it is vital to study quantum aging in such settings. In this paper, we investigate the dynamical universality and aging in the d-dimensional O(N) model with the long-range coupling 1/x^{d+σ} and in the mean-field limit N→∞ that allows an exact treatment. An immediate consequence of long-range coupling is the emergence of nonlinear light cones. We focus on the correlation and response functions, and identify a rich scaling behavior depending on how the corresponding space-time positions are located relative to each other, via a local light cone, and to the time of the quench via a global quench light cone. We determine the initial-slip exponent that governs the short-time dependence of two-point functions. We highlight the qualitative features of aging due to the long-range coupling, in particular in the region outside the light cones. As an important consequence of long-range coupling, the correlation function decays as 1/x^{d+σ} outside the quench light cone while increasing polynomially with the total time after quench. This is while, for short-time differences, the two-time response function "equilibrates" at all distances even outside this light cone. Our analytic findings are in excellent agreement with exact numerics, and provide a useful benchmark for modern experimental platforms with long-range interactions.

DOI: 10.1103/physreve.103.052142

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Quantum quenches to or near criticality give rise to the phenomenon of aging, manifested by glassylike dynamics at short times and far from equilibrium. The recent surge of interest in the dynamics of quantum many-body systems has rejuvenated interest in this phenomenon. Motivated by the ubiquitous long-range interactions in emerging experimental platforms, it is vital to study quantum aging in such settings. In this paper, we investigate the dynamical universality and aging in the d-dimensional O(N) model with the long-range coupling 1/x^{d+σ} and in the mean-field limit N→∞ that allows an exact treatment. An immediate consequence of long-range coupling is the emergence of nonlinear light cones. We focus on the correlation and response functions, and identify a rich scaling behavior depending on how the corresponding space-time positions are located relative to each other, via a local light cone, and to the time of the quench via a global quench light cone. We determine the initial-slip exponent that governs the short-time dependence of two-point functions. We highlight the qualitative features of aging due to the long-range coupling, in particular in the region outside the light cones. As an important consequence of long-range coupling, the correlation function decays as 1/x^{d+σ} outside the quench light cone while increasing polynomially with the total time after quench. This is while, for short-time differences, the two-time response function "equilibrates" at all distances even outside this light cone. Our analytic findings are in excellent agreement with exact numerics, and provide a useful benchmark for modern experimental platforms with long-range interactions.

The nonequilibrium dynamics of a system that is located in the vicinity of a quantum critical point is affected by the critical slowing down of order-parameter correlations with the potential for novel out-of-equilibrium universality. After a quantum quench, i.e., a sudden change of a parameter in the Hamiltonian, such a system is expected to almost instantly fall out of equilibrium and undergo aging dynamics, i.e., dynamics that depends on the time passed since the quench. Investigating the quantum dynamics of an

N

-component ${\ensuremath{\varphi}}^{4}$ model coupled to an external bath, we determine this universal aging and demonstrate that the system undergoes a coarsening, governed by a critical exponent that is unrelated to the equilibrium exponents of the system. We analyze this behavior in the large-

N

limit, which is complementary to our earlier renormalization-group analysis, allowing in particular the direct investigation of the order-parameter dynamics in the symmetry-broken phase and at the upper critical dimension. By connecting the long-time limit of fluctuations and response, we introduce a distribution function that shows that the system remains nonthermal and exhibits quantum coherence even on long time scales.

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.

We investigate the effects of fluctuations on the dynamics of an isolated quantum system represented by a ${\left({\mathbf{\ensuremath{\phi}}}^{2}\right)}^{2}$ field theory with

O(N)

symmetry after a quench in

d>2

spatial dimensions. A perturbative renormalization-group approach involving a dimensional expansion in $\ensuremath{\epsilon}=4\ensuremath{-}d$ is employed in order to study the evolution within a prethermal regime controlled by elastic dephasing. In particular, we focus on a quench from a disordered initial state to the critical point, which introduces an effective temporal boundary in the evolution. At this boundary, the relevant fields acquire an anomalous scaling dimension, while the evolution of both the order parameter and its correlation and response functions display universal aging. Since the relevant excitations propagate ballistically, a light cone in real space emerges. At longer times, the onset of inelastic scattering appears as secularly growing self-generated dissipation in the effective Keldysh field theory, with the strength of the dissipative vertices providing an estimate for the time needed to leave the prethermal regime.

The nonequilibrium response of a quantum many-body system defines its fundamental transport properties and how initially localized quantum information spreads. However, for long-range-interacting quantum systems little is known. We address this issue by analyzing a local quantum quench in the long-range Ising model in a transverse field, where interactions decay as a variable power law with distance ∝r(-α), α>0. Using complementary numerical and analytical techniques, we identify three dynamical regimes: short-range-like with an emerging light cone for α>2, weakly long range for 1<α<2 without a clear light cone but with a finite propagation speed of almost all excitations, and fully nonlocal for α<1 with instantaneous transmission of correlations. This last regime breaks generalized Lieb-Robinson bounds and thus locality. Numerical calculation of the entanglement spectrum demonstrates that the usual picture of propagating quasiparticles remains valid, allowing an intuitive interpretation of our findings via divergences of quasiparticle velocities. Our results may be tested in state-of-the-art trapped-ion experiments.

We study the dynamical scaling of long-range

\mathrm{O}(N)

models after a sudden quench to the critical temperature, using the functional renormalization group approach. We characterize both short-time aging and long-time relaxation as a function of the symmetry index

N

, the interaction range decay exponent

σ

and the dimension

d

. Our results substantially improve on perturbative predictions, as demonstrated by benchmarks against Monte Carlo simulations and the large-

N

limit. Finally, we demonstrate that long-range systems increase the performance of critical heat engines with respect to a local active medium.

The theoretic renormalization-group approach is applied to the study of short-time critical behavior of the

d

-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

ϵ

. In

d<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.