Aging in the long-range Ising model

TL;DR

This paper investigates aging phenomena in a two-dimensional long-range Ising model, revealing how interaction range influences aging dynamics and autocorrelation exponents through Monte Carlo simulations.

Contribution

It provides the first detailed study of aging in long-range interacting systems, extending understanding beyond short-range models.

Findings

Aging follows simple scaling across all studied interaction ranges.

Autocorrelation exponents vary with interaction range, consistent with theoretical predictions.

Finite-size effects are significant for very long-range interactions, affecting interpretation.

Abstract

The current understanding of aging phenomena is mainly confined to the study of systems with short-ranged interactions. Little is known about the aging of long-ranged systems. Here, the aging in the phase-ordering kinetics of the two-dimensional Ising model with power-law long-range interactions is studied via Monte Carlo simulations. The dynamical scaling of the two-time spin-spin autocorrelator is well described by simple aging for all interaction ranges studied. The autocorrelation exponents are consistent with $\lambda=1.25$ in the effectively short-range regime, while for stronger long-range interactions the data are consistent with $\lambda=d/2=1$. For very long-ranged interactions, strong finite-size effects are observed. We discuss whether such finite-size effects could be misinterpreted phenomenologically as sub-aging.

Equations

$$\mathcal{H}=-\frac{1}{2}\sum _i\sum _{j\ne i}J(r_{ij})s_i{s}_j,\text{with}J(r_{ij})=\frac{1}{r_{ij}^{d+\sigma }},$$

$$C(t,t_w)=\frac{a_0}{2}{\int }_{{\mathbb{R}}^d}\mathrm{d}\vec{R}{\mathcal{R}}^{(3)}(t,t_w,t_{\mathrm{micro}};\vec{R})$$

$$C(y{t}_w,t_w)$$

$$\mathbf{\Psi }(w)$$

$$\mathbf{\Psi }(w)=\frac{S_d}{2}{\left(\frac{2}{\mathcal{M}}\right)}^{d/2}{\int }_0^{\infty }\mathrm{d}\varrho e^{-w\varrho }{\varrho }^{(d-2)/2}\Psi \left(\varrho \right)$$

$$f_C(y)$$

$${\int }_0^{\infty }\mathrm{d}\varrho e^{-w\varrho }{\varrho }^{d/2}\Psi (\varrho )=-\frac{1}{w}{\int }_0^{\infty }\mathrm{d}\varrho \frac{\mathrm{d}}{\mathrm{d}\varrho }\left(e^{-w\varrho }\right){\varrho }^{d/2}\Psi (\varrho )$$

$$\frac{{\mathbf{\Psi }}^{\prime }(w)}{\mathbf{\Psi }(w)}=-\frac{{\int }_0^{\infty }\mathrm{d}\varrho e^{-w\varrho }{\varrho }^{d/2}\Psi (\varrho )}{{\int }_0^{\infty }\mathrm{d}\varrho e^{-w\varrho }{\varrho }^{d/2-1}\Psi (\varrho )}\le -\frac{1}{w}\frac{d}{2}.$$

$$f_C(y)\le \mathbf{\Psi }(1)y^{-{\lambda }_C/2}\left[1-\left(2\frac{d}{2}-\lambda \right)\frac{1}{y}+\dots \right].$$

$$A\ge d-\lambda .$$

$$f_C(y)$$

Full text

Supplemental Material: Aging in the long-range Ising model

Henrik Christiansen

henrik.christiansen@itp.uni-leipzig.de

Institut für Theoretische Physik, Universität Leipzig, IPF 231101, 04081 Leipzig, Germany

  

Suman Majumder

suman.majumder@itp.uni-leipzig.de

Institut für Theoretische Physik, Universität Leipzig, IPF 231101, 04081 Leipzig, Germany

  

Malte Henkel

malte.henkel@univ-lorraine.fr

Laboratoire de Physique et Chimie Théoriques (CNRS UMR 7019), Université de Lorraine Nancy, 54506 Vandœuvre-lès-Nancy Cedex, France

Centro de Física Teórica e Computacional, Universidade de Lisboa, 1749-016 Lisboa, Portugal

  

Wolfhard Janke

wolfhard.janke@itp.uni-leipzig.de

Institut für Theoretische Physik, Universität Leipzig, IPF 231101, 04081 Leipzig, Germany

I Methods

For the Monte Carlo (MC) simulation of the LRIM given by the Hamiltonian

[TABLE]

we introduce the kinetics via single-spin flips. A randomly chosen spin is flipped according to the standard Metropolis update with probability $\min\left[1,\exp(-\Delta E/T)\right]$, with the Boltzmann constant $k_{B}$ set to unity. Here, $T$ is the temperature and $\Delta E$ is the change in energy before and after the flip. $N=L^{d}$ (where $L$ is the linear size of a hyper-cubic lattice) such attempts constitute one MC sweep, setting the time scale. Obviously, for the LRIM the calculation of the energy change is the rate limiting step, as it involves all the spins in the considered lattice. However, following our recent approach of storing the effective field for each spin and updating it only when a spin flip is accepted makes such simulation significantly faster Christiansen et al. (2019). Furthermore, to allow for simulations of system size up to $L=4096$ in $d=2$ dimensions, this update was parallelized using the shared-memory API OpenMP framework. Since systems with long-range interaction suffer severely from finite-size effects we additionally use Ewald summation Ewald (1921); Horita et al. (2017); Flores-Sola et al. (2017); Janke et al. (2019) to implement periodic boundary conditions and thereby to increase the effective system size. An effective $J_{ij}\equiv J(r_{ij})$ is calculated once at the beginning of the simulation.

As an initial configuration at high temperature, we chose a square lattice with randomly distributed equal proportion of up and down spins. We chose $T=0.1T_{c}$ as the quench temperature, where we extract $T_{c}$ from the data presented in Ref. Horita et al. (2017). Using the scaling relation $C(r,t)\equiv\tilde{C}\left[r/\ell(t)\right]$ for the equal-time two-point correlation function $C(r,t)=\langle s_{i}s_{j}\rangle-\langle s_{i}\rangle\langle s_{j}\rangle$ one can estimate the characteristic length scale $\ell(t)$ from the decay of $C(r,t)$ as intersection with a constant value where here we choose $C\left[r=\ell(t),t\right]=0.5$. All considered quantities such as $\ell(t)$ and $C(t,t_{w})$ are averages over independent time evolutions, indicated, e.g., in Eq. (3) of the main article by $\langle\ldots\rangle$. The presented results are averaged over $50$ independent runs for $L\leq 2048$ and $30$ for $L=4096$ (using different random number seeds).

II Illustration of the Loss of Time-Translational Invariance

Figure S1 shows the two-time correlator $C(t,t_{w})$ versus $t-t_{w}$, explicitly demonstrating the loss of time-translational invariance during coarsening. The data for larger $t_{w}$ decay slower, i.e., the older the system is at the waiting time $t_{w}$, the longer in terms of $t$ it needs to decorrelate.

III Finite-size effects of the Autocorrelation Function

In Fig. S2 we show $C(yt_{w},t_{w})$ versus $y$ for (a) $\sigma=1.5$, (b) $\sigma=0.8$, and (c) $\sigma=0.6$ with fixed $t_{w}=20$ and varying $L$. For $\sigma=1.5$ the data show the bulk behavior over a large $y$ range, and only for $L=512$ the data deviate by bending down at $y\approx 200$. The available data for $L=1024$ and $L=2048$ do not deviate, i.e., there are no detectable finite-size effects. For $\sigma=0.8$ and $\sigma=0.6$, the data for both $L=512$ and $L=1024$ undershoot from the bulk curve. This happens at larger $y$, the larger $L$. For $L=2048$ this effect is hence less pronounced and for $L=4096$ it can only be anticipated from these plots. Finally, because eventually the system reaches a configuration with spontaneous magnetization $m_{\mathrm{eq}}(T)$, the overlap and thereby autocorrelation function approaches a constant. Note that the data for smaller systems even cross the data of the bigger systems. This effectively limits the extent to which the data can undershoot from the bulk behavior for smaller $y$.

IV Alternative Form of Sub-aging

Instead of using the analytically derived form of sub-aging with $h(t)$ as defined in the main article, one may use the more phenomenological form of $t/t_{w}^{\mu}$ (or $\ell(t)/\ell(t_{w})^{\mu}$) to modify the scaling variable. In Fig. S3 we present $C(t,t_{w})$ vs. $t/t_{w}^{\mu}$ for $\sigma=0.6$ and $L=4096$ with $\mu=0.970,0.976,0.982,$ and $0.99$. Compared to using $h(t)$ the data collapse is worse and one effectively only shifts the crossing point of data for different $t_{w}$. This approach does thus not lead to better collapse.

V Two-time autocorrelators from local scale-invariance with $z=2$

According to local scale-invariance Henkel (1994); Henkel and Pleimling (2010); Henkel (2017) the generic dynamical scaling which arises especially in aging systems far from equilibrium can be extended to a larger group of dynamical symmetries. For the phase-ordering kinetics of systems with short-ranged interactions, it is known that the dynamical exponent $z=2$ Bray and Rutenberg (1994); Bray (2002). Then the Schrödinger group, which arises as dynamical symmetry of the free diffusion equation, is an example of an extended dynamical symmetry Henkel (1994). Numerous systems which physically realize Schrödinger invariance have been found, most notably phase-ordering kinetics in short-ranged Ising models in $d=1,2,3$ dimensions, see Henkel and Pleimling (2010); Henkel (2017) and references therein. Here we discuss how the requirement of Schrödinger invariance restricts the two-time autocorrelator in phase-ordering kinetics.

Physically, it is the two-time or multi-time response functions which transform co-variantly under local scale-transformations. Turning to the two-time autocorrelator $C(t,t_{w})$, after a quench to $T<T_{c}$ from a fully disordered initial state, it can be expressed as Picone and Henkel (2004)

[TABLE]

where ${\cal R}^{(3)}$ is a three-point response function which in the context of Janssen-de Dominicis theory could be expressed as an average $\left\langle\psi(t,\vec{0})\psi(t_{w},\vec{0})\widetilde{\psi}(t_{\rm micro};\vec{R})^{2}\right\rangle$ involving the order parameter $\psi$ and the conjugate response operator $\widetilde{\psi}$. The form of that three-point response in turn is fixed up to a scaling function Henkel (1994). Furthermore, $t_{\rm micro}$ is a microscopic time scale and the amplitude $a_{0}$ measures the width of the initial correlator. Since for phase-ordering kinetics the temperature $T$ is an irrelevant variable Bray (2002), the thermal heat bath merely furnishes corrections to scaling. In the dynamical scaling regime, the autocorrelator of phase-ordering kinetics can then be written as follows Picone and Henkel (2004),

[TABLE]

where the undetermined scaling function $\Psi(\varrho)$ comes from the three-point response function mentioned above. Because of the known asymptotics $f_{C}(y)\sim y^{-\lambda/2}$ for $y\to\infty$, it follows that $\mathbf{\Psi}(1)$ exists and is finite. Denoting by $S_{d}$ the surface of the hypersphere in $d$ dimensions, Eq. (S.4) is re-written in spherical coordinates as

[TABLE]

and we also made explicit the non-universal metric factor $\cal M$. We recognize from this that $\mathbf{\Psi}$ is the Laplace transform of the function $\varrho^{(d-2)/2}\Psi(\varrho)$. Since it is well-known that a Laplace transform is infinitely often differentiable wherever it is defined, we can asymptotically expand in $y$ [or equivalently around $w=1$, see (S.3)] and find (the prime denotes the derivative)

[TABLE]

where we identified the constant $A$. This is the form (5) used in the text.

In order to estimate the amplitude $A$, we require some more input on the scaling function $\Psi(\varrho)$ in (S.4). First, we assume that for $\varrho\to\infty$, $\Psi(\varrho)$ grows more slowly than exponentially which is consistent with $\mathbf{\Psi}(1)$ being finite. Second, we recall that for $\varrho\to 0$ consistency with the asymptotic scaling of $f_{C}(y)$ requires that $\Psi(\varrho)\sim\varrho^{\lambda-d/2}$ Picone and Henkel (2004). Because of the known bound $\lambda\geq d/2$ Fisher and Huse (1988); Yeung et al. (1996), $\Psi(\varrho)$ increases when $\varrho\ll 1$. We strengthen this to the requirement $\Psi^{\prime}(\varrho)\geq 0$ also when $\varrho$ is finite. Next, the integral representation (S.5) will become useful, via the following estimate

[TABLE]

where the two assumptions made on $\Psi(\varrho)$ were used explicitly and also for the estimation of the boundary terms after partial integration. With (S.5) we have

[TABLE]

Setting $w=1$, we then have the bound $\mathbf{\Psi}^{\prime}(1)/\mathbf{\Psi}(1)\leq-d/2$. For the scaling function $f_{C}(y)$ of (S.6), this gives

[TABLE]

This upper bound on $f_{C}(y)$ gives a lower bound on the amplitude in (S.6)

[TABLE]

Indeed, it was argued long ago by Fisher and Huse Fisher and Huse (1988) that $\lambda\leq d$. In models which respect this bound, (S.10) implies that $A\geq 0$. The validity of this Fisher-Huse bound was discussed in detail for phase-ordering systems Majumdar and Huse (1995). However, for phase-separating model-B dynamics, this Fisher-Huse bound does not hold Yeung et al. (1996); Brown et al. (1999).

Equation (S.6), along with (S.10), is reproduced in several exactly solvable models of phase-ordering with nearest-neighbour interactions and $z=2$, see Henkel and Pleimling (2010) for details.

For the 1D Glauber-Ising model at $T=0$, we have

[TABLE]

Since $\lambda=1$, the bound (S.10) $A\geq 0$ is consistent with the exact result $A=1/6$.

For the spherical model in $d>2$ dimensions and quenched to $T<T_{c}$, we have

[TABLE]

with the equilibrium magnetization $m_{\rm eq}^{2}=1-T/T_{c}$. Since $\lambda=d/2$, the bound (S.10) $A\geq d/2$ coincides with the exact result $A=d/2$.

Equation (S.6) can also be used as an ansatz for the spherical model with long-ranged interactions. There is a phase transition in the long-range universality class at a non-vanishing $T_{c}$ provided $0<\sigma<\min(d,2)$ and $z=\sigma$. The scaling function of the two-time autocorrelator is

[TABLE]

Hence, $\lambda=d/2$ is $\sigma$-independent and we note that once more $A>0$.