Initial Slip Exponents Research Articles

Universal scaling at a prethermal dark state

Recent experimental and theoretical progress as well as the prospect of commercially viable quantum technologies have inspired great interest in the study of open quantum systems and their dynamics. Many open quantum systems are well described by an effective non-Hermitian Hamiltonian generating a time evolution that allows eigenstates to decay and dissipate to the environment. In this framework, quantum coherent scaling is traditionally tied to the appearance of dark states, where the effect of dissipation becomes negligible. Here, we discuss the universal dynamical scaling after a sudden quench of the non-Hermitian $O(N)$ model Hamiltonian. While universality is generally spoiled by non-Hermiticity, we find that for a given set of internal parameters short-time scaling behavior is restored with an initial slip exponent profoundly different from that of closed quantum systems. This result is tied to the compensation of dissipation by interaction effects at short times leading to a prethermal dark state, where coherent many-body dynamics can be still observed.

Critical properties of the prethermal Floquet time crystal

The critical properties characterizing the formation of the Floquet time crystal in the prethermal phase are investigated analytically in the periodically driven $O(N)$ model. In particular, we focus on the critical line separating the trivial phase with period synchronized dynamics and absence of long-range spatial order from the non-trivial phase where long-range spatial order is accompanied by period-doubling dynamics. In the vicinity of the critical line, with a combination of dimensional expansion and exact solution for $N\to\infty$, we determine the exponent $\nu$ that characterizes the divergence of the spatial correlation length of the equal-time correlation functions, the exponent $\beta$ characterizing the growth of the amplitude of the order-parameter, as well as the initial-slip exponent $\theta$ of the aging dynamics when a quench is performed from deep in the trivial phase to the critical line. The exponents $\nu, \beta, \theta$ are found to be identical to those in the absence of the drive. In addition, the functional form of the aging is found to depend on whether the system is probed at times that are small or large compared to the drive period. The spatial structure of the two-point correlation functions, obtained as a linear response to a perturbing potential in the vicinity of the critical line, is found to show algebraic decays that are longer ranged than in the absence of a drive, and besides being period-doubled, are also found to oscillate in space at the wave-vector $\omega/(2 v)$, $v$ being the velocity of the quasiparticles, and $\omega$ being the drive frequency.

Short-time Monte Carlo simulation of the majority-vote model on cubic lattices

We perform short-time Monte Carlo simulations to study the criticality of the isotropic two-state majority-vote model on cubic lattices of volume N=L3, with L up to 2048. We obtain the precise location of the critical point by examining the scaling properties of a new auxiliary function Ψ. We perform finite-time scaling analysis to accurately calculate the whole set of critical exponents, including the dynamical critical exponent z=2.027(9), and the initial slip exponent θ=0.1081(1). Our results indicate that the majority-vote model in three dimensions belongs to the same universality class of the three-dimensional Ising model.

Short-imaginary-time quantum critical dynamics in the J−Q3 spin chain

We study the short-imaginary-time quantum critical dynamics (SITQCD) in the J-Q$_3$ spin chain, which hosts a quasi-long-range-order phase to a valence bond solid transition. By using the scaling form of the SITQCD with a saturated ordered phase, we are able to locate the critical point at $q_{\rm c}=0.170(14)$. We also obtain the critical initial slip exponent $\theta=-0.507(3)$ and the static exponent $\beta/\nu=0.498(2)$. More strikingly, we find that the scaling dimension of the initial order parameter $x_{0}$ is close to zero, which suggests that the initial order parameter is a marginal operator. As a result, there is no initial increase behavior of the order parameter in the short-imaginary-time relaxation process for this model, which is very different from the relaxation dynamics in the Ising-type phase transitions. Our numerical results are realized by the projector quantum Monte Carlo algorithm.

Universal Prethermal Dynamics in Gross-Neveu-Yukawa Criticality.

We study the prethermal dynamics of the Gross-Neveu-Yukawa quantum field theory, suddenly quenched in the vicinity of a critical point. We find that the universal prethermal dynamics is controlled by two fixed points depending on the size of the quench. Besides the usual equilibrium chiral Ising fixed point for a shallow quench, a dynamical chiral Ising fixed point is identified for a deep quench. Intriguingly, the latter is a nonthermal fixed point without any equilibrium counterpart due to the participation of gapless fermionic fields. We also find that in the scaling regime controlled by the equilibrium fixed point, the initial slip exponent is rendered negative if there are enough flavors of fermions, thus providing a unique signature of fermionic prethermal dynamics. We then explore the temporal crossover between the universal scaling regimes governed by the two universality classes. Possible experimental realizations are also discussed.

Universal short-time quantum critical dynamics of finite-size systems

We investigate the short time quantum critical dynamics in the imaginary time relaxation processes of finite size systems. Universal scaling behaviors exist in the imaginary time evolution and in particular, the system undergoes a critical initial slip stage characterized by an exponent $\theta$, in which an initial power-law increase emerges in the imaginary time correlation function when the initial state has zero order parameter and vanishing correlation length. Under different initial conditions, the quantum critical point and critical exponents can be determined from the universal scaling behaviors. We apply the method to the one- and two-dimensional transverse field Ising models using quantum Monte Carlo simulations. In the one-dimensional case, we locate the quantum critical point at $(h/J)_{c}=1.00003(8)$ \thirdrevise{in the thermodynamic limit}, and estimate the critical initial slip exponent $\theta=0.3734(2)$, static exponent $\beta/\nu=0.1251(2)$ \thirdrevise{by analyzing data on chains of length $L=32\sim 256$ and $L=48\sim 256$, respectively}. For the two-dimensional square-lattice system, the critical coupling ratio is given by $3.04451(7)$ \thirdrevise{in the thermodynamic limit} while the critical exponents are \thirdrevise{$\theta=0.209(4)$ and $\beta/\nu=0.518(1)$ estimated by data on systems of size $L=24\sim 64$ and $L=32\sim 64$, correspondingly.} Remarkably, the critical initial slip exponents obtained in both models are notably distinct from their classical counterparts, owing to the essential differences between classical and quantum dynamics. The short time critical dynamics and the imaginary time relaxation QMC approach can be readily adapted to various models.

Universal short-time dynamics: Boundary functional renormalization group for a temperature quench

We present a method to calculate short-time non-equilibrium universal exponents within the functional renormalization-group scheme. As an example, we consider the classical critical dynamics of the relaxational model A after a quench of the temperature of the system and calculate the initial slip exponent which characterizes the non-equilibrium universal short-time behaviour of both the order parameter and correlation functions. The value of this exponent is found to be consistent with the result of a perturbative dimensional expansion and of Monte Carlo simulations in three spatial dimensions.

Critical initial-slip scaling for the noisy complex Ginzburg–Landau equation

We employ the perturbative fieldtheoretic renormalization group method to investigate the universal critical behavior near the continuous non-equilibrium phase transition in the complex Ginzburg–Landau equation with additive white noise. This stochastic partial differential describes a remarkably wide range of physical systems: coupled nonlinear oscillators subject to external noise near a Hopf bifurcation instability; spontaneous structure formation in non-equilibrium systems, e.g., in cyclically competing populations; and driven-dissipative Bose–Einstein condensation, realized in open systems on the interface of quantum optics and many-body physics, such as cold atomic gases and exciton-polaritons in pumped semiconductor quantum wells in optical cavities. Our starting point is a noisy, dissipative Gross–Pitaevski or nonlinear Schrödinger equation, or equivalently purely relaxational kinetics originating from a complex-valued Landau–Ginzburg functional, which generalizes the standard equilibrium model A critical dynamics of a non-conserved complex order parameter field. We study the universal critical behavior of this system in the early stages of its relaxation from a Gaussian-weighted fully randomized initial state. In this critical aging regime, time translation invariance is broken, and the dynamics is characterized by the stationary static and dynamic critical exponents, as well as an independent ‘initial-slip’ exponent. We show that to first order in the dimensional expansion about the upper critical dimension, this initial-slip exponent in the complex Ginzburg–Landau equation is identical to its equilibrium model A counterpart. We furthermore employ the renormalization group flow equations as well as construct a suitable complex spherical model extension to argue that this conclusion likely remains true to all orders in the perturbation expansion.

Universal short-time quantum critical dynamics in imaginary time

We propose a scaling theory for the universal imaginary-time quantum critical dynamics for both short times and long times. We discover that there exists a universal critical initial slip related to a small initial order parameter $M_0$. In this stage, the order parameter $M$ increases with the imaginary time $\tau$ as $M\propto M_0\tau^\theta$ with a universal initial slip exponent $\theta$. For the one-dimensional transverse-field Ising model, we estimate $\theta$ to be $0.373$, which is markedly distinct from its classical counterpart. Apart from the local order parameter, we also show that the entanglement entropy exhibits universal behavior in the short-time region. As the critical exponents in the early stage and in equilibrium are identical, we apply the short-time dynamics method to determine quantum critical properties. The method is generally applicable in both the Landau-Ginzburg-Wilson paradigm and topological phase transitions.

Nonequilibrium Critical Relaxation in the 2D Random Field Ising Model

A particular type of nonequilibrium critical dynamics is discussed in this paper, in which the initial state is generated in the square lattice Ising model with random fields of zero mean and variance h. Thus we interpolate between the ordered (h = 0) and disordered (h → ∞) initial states. In particular we consider the square lattice Ising model in which for weak enough random fields the clusters of parallel spins are percolating in the initial state. We aim to study the effect of correlations in the initial state on the nonequilibrium dynamics, so we measure the nonequilibrium relaxation of the magnetization and the autocorrelation function. The relaxation of the magnetization, characterized by the initial slip exponent, θ, is not altered by the percolating nature of the initial state. However, the starting part of the magnetization curve is affected by a cluster dissolution effect leading to a reentrance in time.

Nonequilibrium critical relaxation in the presence of extended defects

We study nonequilibrium critical relaxation properties of systems with quenched extended defects, correlated in ${\ensuremath{\varepsilon}}_{d}$ dimensions and randomly distributed in the remaining $d\ensuremath{-}{\ensuremath{\varepsilon}}_{d}$ dimensions. Using a field-theoretic renormalization-group approach, we find the scaling behavior of the nonequilibrium response and correlation functions and calculate the initial slip exponents $\ensuremath{\theta}$ and ${\ensuremath{\theta}}^{\ensuremath{'}},$ which describe the growth of correlations during the initial stage of the critical relaxation, in the two-loop approximation.

SHORT-TIME DYNAMICS OF THE RANDOM n-VECTOR MODEL WITH LONG-RANGE INTERACTION

The short-time critical behavior of the random n-vector model with long-range interaction is studied by the theoretic renormalization-group approach. After a sudden quench to the critical temperature from the high temperature phase, the system is released to an evolution within model A dynamics. The initial slip exponents and the dynamic exponent are calculated to two-loop order.

Dynamics of Non-interacting System with Long-Range Correlated Quenched Impurities**The project supported by National Natural Science Foundation of China under Grant No. 19772074 and Guangzhou University

The theoretic renormalization group approach is applied to the study of the critical behavior of non-interacting system with long-range correlated quenched impurities, which has a power-like correlations . To two-loop order, the asymptotic scaling laws and the critical exponents are studied in the frame of a double expansion with ρ of order . In , it is argued that the initial slip exponent θ = 0 together with the dynamic exponent is exact in this kind of random system.

Short-time dynamics of a random Ising model with long-range interaction.

Short-time critical dynamics of a random Ising model (model A) with long-range interaction decaying as r(-(d+sigma)) (where sigma is the parameter controlling the range of the interaction), is studied by the theoretic renormalization-group approach. In dimensions d<2sigma, the initial slip exponents theta(') describing the initial increase of the order parameter, and theta for the growth of the response function, which govern the short-time scaling behaviors, are calculated to the second order in sqrt[epsilon] with epsilon=2sigma-d. The crossover between the long-range interaction and the short-range interaction, which occurs at some sigma>2, is also discussed.

Short-Time Dynamics of the Random n-Vector Model**The project supported by National Natural Science Foundation of China under Grant No. 19772074 and Advanced Research Center of Zhongshan University

Short-time critical behavior of the random n-vector model is studied by the theoretic renormalization-group approach. Asymptotic scaling laws are studied in a frame of the expansion in ϵ=4-d for n≠1 and for n=1 respectively. In d<4, the initial slip exponents θ′ for the order parameter and θ for the response function are calculated up to the second order in ϵ=4-d for n≠1 and for n=1 at the random fixed point respectively. Our results show that the random impurities exert a strong influence on the short-time dynamics for d<4 and n<nc.

Short-Time Critical Behavior Affected by Weakly Long-Range Interactions**The project supported by National Natural Science Foundation of China under Grant No. 19772074 and Advance Research Center of Zhongshan University

The theoretic renormalization group approach is applied to the study of short-time critical behavior of the Ginzburg–Landau model with weakly long-range interactions . The system initially at a high temperature is firstly quenched to the critical temperature and then released to an evolution with a model A dynamics. A double expansion in and with of order is employed, where is the spatial dimension. The asymptotic scaling laws and the initial slip exponents and for the order parameter and the response function respectively are calculated to the second order in for close to 2.

Short-time critical behaviour of anisotropic cubic systems with long-range interaction

The renormalization group approach is applied to the study of the short-time critical behaviour of the d-dimensional n-component anisotropic cubic spin systems with long-range interaction of the form pσsps-p in momentum space. Firstly, the system is quenched from a high temperature to the critical temperature and then relaxes to equilibrium within the model A dynamics. The asymptotic scaling laws and the initial slip exponents θ' and θ of the order parameter and the response function, respectively, are calculated to the second order in = 2σ-d. For 1≤d nc, the cubic anisotropy affects the short-time critical behaviour.

SHORT-TIME CRITICAL BEHAVIOR OF THE GINZBURG–LANDAU MODEL WITH QUENCHED DISORDER

The theoretic renormalization-group approach is applied to the study of short-time dynamics of the d-dimensional n-component spin systems with long-range interactions r-(d+σ) and quenched disorder which has long-range correlations r-(d-ρ). Asymptotic scaling laws are obtained in a frame of double expansions in ∊=2σ-d and ρ with ρ of the order ∊. The static exponents are obtained exactly to all the order. The initial slip exponents θ′ for the order parameter and θ for the response function, as well as the dynamic exponent z, are calculated upto the first order in ∊. In d=2σ, in contrast to the unique logarithmic decay in the long-time regime which does not depend on σ, ρ, n and the disorder, we find rich scaling structures including logarithmic and exponential-logarithmic scalings in the short-time regime. Non-universal critical scalings of Ising systems are also discussed for d=2σ.

Short-time critical behavior of anisotropic cubic systems

The renormalization group approach is applied to the study of the short-time critical behavior of the d-dimensional n-component spin systems with cubic anisotropy. First, the system is quenched from high temperature to the critical temperature and then relaxes to equilibrium within model A dynamics. The asymptotic scaling laws and the initial slip exponents ${\ensuremath{\theta}}^{\ensuremath{'}}$ and $\ensuremath{\theta}$ of the order parameter and the response function, respectively, are calculated to the second order in $\ensuremath{\epsilon}=4\ensuremath{-}d.$ Logarithmic corrections to short-time behaviors of both the autocorrelation and the order parameter are found in $d=4$ dimension.

The short-time critical behaviour of the Ginzburg-Landau model with long-range interaction

The renormalisation group approach is applied to the study of the short-time critical behaviour of the d-dimensional Ginzburg-Landau model with long-range interaction of the form in momentum space. Firstly the system is quenched from a high temperature to the critical temperature and then relaxes to equilibrium within the model A dynamics. The asymptotic scaling laws and the initial slip exponents and of the order parameter and the response function respectively, are calculated to the second order in .