Glassy dynamics in disordered oscillator chains

TL;DR

This paper investigates how energy dissipates or localizes in a disordered chain of nonlinear oscillators, revealing fractal disorder effects and classical many-body localization phenomena through numerical simulations.

Contribution

It demonstrates that fractal disorder patterns lead to stretched exponential energy decay and provides evidence for classical many-body localization at low temperatures.

Findings

Energy decay fits stretched exponential with variable exponent

Fractal disorder influences energy trapping

Evidence of classical many-body localization at low temperature

Abstract

The escape of energy injected into one site in a disordered chain of nonlinear oscillators is examined numerically. When the disorder has a `fractal' pattern, the decay of the residual energy at the injection site can be fit to a stretched exponential with an exponent that varies continuously with the control parameter. At low temperature, we see evidence that energy can be trapped for an infinte time at the original site, i.e. classical many body localization.

Equations

$$H=\frac{m}{2}\sum _{l=1}^N[{\dot{x}}_l^2+{\omega }_l^2{x}_l^2]-J\sum _{l=1}^N\cos (x_l-x_{l-1})$$

$$a_l(t)={\dot{x}}_l(t)-i{\omega }_l{x}_l(t)$$

$${\dot{a}}_l=-i{\omega }_l{a}_l+\frac{J}{m}[\sin (x_{l+1}-x_l)+\sin (x_{l-1}-x_l)]$$

$$x_l=\frac{1}{2i{\omega }_l}[a_l^{\ast }(t)-a_l(t)].$$

$$\sim \frac{J}{m{\omega }_{l\pm 1}+(n-1){\omega }_l}\exp [i(m{\omega }_{l\pm 1}+n{\omega }_l)t].$$

$$\frac{J^r}{({\omega }_l-{\omega }_{l+r})({\omega }_{l+1}-{\omega }_{l+r})\dots ({\omega }_{l+r-1}-{\omega }_{l+r})}.$$

$$\delta {\omega }_{l,l+2^k}\le 2{\lambda }^k[1-\lambda -{\lambda }^2\dots ]=2{\lambda }^k\frac{1-2\lambda }{1-\lambda }=2{\lambda }^{k+1}$$

$$\Delta E_l(t)=\frac{1}{2}[E_l(t;l)-E_l(t;l+N/2)]$$

Full text

††preprint: APS/123-QED

The Fermi Pasta Ulam (FPU) model of coupled nonlinear oscillators fermi has served as a testing ground for basis ideas in statistical mechanics for more than half a century. After the initial result that energy did not distribute itself efficiently between the oscillator modes, there was a huge body of work ford , including the development of related integrable models soliton ; toda . The fact that the FPU model is approximately integrable is generally believed to result in very long equilibration times, often beyond what is observable numerically bennetin . At high energies, equilibration proceeds efficiently and such metastable behavior is not seen hightemp . The FPU model has also been used to study heat conduction in low dimensional systems, where a heat conductivity that diverges in the thermodynamic limit is seen for various models dhar .

Even when the FPU model does not equilibrate efficiently, the normal modes in which the the energy is concentrated are delocalized. It is possible to construct disordered and linearized versions of the FPU model for which the normal modes are localized, but the normal modes are all decoupled from each other. Following the great progress in the field of many body localization for quantum statistical systems huse , it is natural to ask if localization of energy can be achieved in interacting classical systems too. The study of heat conduction in disordered nonlinear oscillator chains suggests otherwise DL : even a small amount of nonlinearity is found numerically to result in normal heat conductivity for large chains. Although this implies that a localized energy packet will spread out, it is still possible that a fraction of the energy packet will remain at its original location.

In this paper, we consider a one-dimensional ring of linear oscillators, each with a different frequency, in which nearest neighbors are coupled together with a nonlinear potential. The system is initialized by equilibrating at a temperature $T.$ Thereafter, a packet of energy is deposited at one site, and the system is evolved using molecular dynamics. The excess energy at the site is measured as a function of time $t.$ If the system thermalizes, the difference of excess energy $\Delta E(t)$ should vanish as $t\rightarrow\infty.$

In order to avoid accidental resonances between oscillators that are far away from each other, or rare regions where clusters of nearby oscillators are resonant resulting in chaotic spots in the dynamics basko , the frequencies $\{\omega_{i}\}$ have to be chosen judiciously. We choose the frequencies in a ‘fractal’ manner: a large gap between the frequencies of sites near each other, and a small gap between the frequencies of sites far away from each other, with the gap size decaying as a power of the distance between the sites. This is made quantitative later in this paper.

As the nearest neighbor coupling constant $J$ is lowered at low temperature, we find that the decay of the excess energy $\Delta E_{J}(t)$ slows down substantially. The curves for $\Delta E(t)$ for various values of $J$ can be superimposed on top of each other if $\ln t$ and $\Delta E$ are scaled for each $J.$ Thus $\Delta E(t)=A(J)\hat{\Delta E}(t^{\beta(J)}).$ The numerical results are consistent with a stretched exponential form for the function $\hat{\Delta E};$ stretched exponential dynamics are often seen in experiments on glassy systems angell , and there are various theoretical models with traps and a range of time scales that obtain similar behavior phillips . (A power law form with a cutoff can also be fit to the data.)

The smallest $J$ value shows an essentially flat $\Delta E(t),$ and so we vary the temperature at this $J$ and measure the decay of the excess energy. The behavior of $\Delta E(t)$ is found to be a non-monotonic function of $T$: it decays slowly at both high temperature and low temperature, with a more rapid decay at intermediate temperatures. In the low temperature regime, the numerical results indicate that the system freezes at a non-zero temperature $T_{f},$ with $\Delta E_{T<T_{f}}(t\rightarrow\infty)\neq 0.$ To our knowledge, this is the first evidence of classical many body localization.

The Hamiltonian of the chain with ‘fractal’ disorder is

[TABLE]

with periodic boundary conditions. The particles in the chain all have equal masses and are tethered to their equilibrium positions by harmonic springs. The tethering ensures that momentum conservation is destroyed and there is no anomalous transport campbell ; ramaswamy . The frequency of each of the tethering harmonic oscillators is different. When $J=0,$ the energy is obviously localized at each lattice site. For $J\neq 0,$ the oscillators are coupled, but if $J$ is small, one might try to use perturbation theory. If we define

[TABLE]

then the dynamical equations can be expressed as

[TABLE]

with the supplementary equation

[TABLE]

This is now a set of coupled first order differential equations. The solution to zeroeth order in $J$ is trivial.

To first order in $J,$ $a_{l}$ is forced by terms that are of the form $\sim\exp[i(m\omega_{l\pm 1}+n\omega_{l})t],$ where $m$ and $n$ are integers. The terms with $m=0,n=-1$ shift the natural frequency of the oscillator $\omega_{l}$ by an amount that is $O(J).$ Other terms yield $O(J)$ corrections to $a_{l},$ of the form

[TABLE]

To next order in $J,$ each site is influenced by its next nearest neighbors, and so on. If there is a near degeneracy between the frequencies of two sites $l$ and $l+r$ that are $r$ steps apart, for small amplitudes (i.e. low temperatures), the leading correction to $a_{l}$ is of the order of

[TABLE]

If the $\{\omega_{i}\}$’s are chosen randomly, accidental near degeneracies can result in a small denominator in (and a breakdown of) the perturbation expansion. To avoid this, we choose the frequencies in the following manner. Let $N=2^{n}.$ First, the frequencies at all the odd and even sites are set to 1 and 3 respectively. Next, the frequencies of successive pairs of sites are increased or decreased by $\lambda,$ so that the frequencies are $\{1+\lambda,3+\lambda,1-\lambda,3-\lambda,\ldots\}.$ At the next step, the frequencies of successive groups of 4 sites are increased or decreased by $\lambda^{2}.$ This is carried out $n$ times, when all the $\{\omega_{i}\}$’s are non-degenerate. The frequency gap between two sites that are an odd multiple of $2^{k}$ apart satisfies

[TABLE]

for $N\rightarrow\infty$ if $\lambda=(5-\sqrt{17})/2.$ Then it is easy to see that if $J<<2\lambda^{2},$ the expression in Eq.(6) is small for all $r.$

Although this is necessary for the perturbation expansion to be well behaved, it is not sufficient. Regardless of the relationship between two frequencies, it is always possible to find values of $m$ and $n$ in Eq.(5) for which $m\omega^{\prime}+(n-1)\omega$ is as small as one wishes. This is the problem of small denominators that makes the KAM theorem KAM difficult. However, these terms are higher order in the amplitudes of the oscillators. One might expect them to be important when the amplitude of an oscillator happens to be large, but in that case, because the sine function is bounded, the coupling term on the right hand side of Eq.(3) is weak compared to the $-i\omega_{l}a_{l}$ term. This is not a proof that these terms are unimportant, and therefore we turn to numerical simulations.

Because energy transport in this system is at best very slow, one has to be careful to bring it to thermal equilibrium before a packet of energy is injected; simply running molecular dynamics for a long time is not sufficient. We initialized the velocities from a Gaussian distribution, and equilibrated the coordinates using Monte Carlo dynamics (with acceptance rate $\approx 0.5$). Equilibrium was considered to be achieved when the virial theorem was satisfied to within 10%.

Once in equilibrium, a heat packet of magnitude $E_{+}$ was injected in the system in the form of equal and opposite momentum between two neighboring sites. The system was then evolved dynamically with the Forrest-Ruth algorithm, using a time step of $h=0.01.$ If we denote $E_{l}(t;k)$ as the energy of sites $l$ and $l+1$ at time $t$ when the heat packet was injected at sites $k$ and $k+1$ at $t=0$, the residual energy at $l$ is defined as

[TABLE]

so that, if the system were to equilibrate, $\Delta E_{l}(\infty)$ would be zero. For the measurements reported here, the mass of each site was $m=1,$ the size of the ring was $N=64$ or 128, and the extra energy injected was $\Delta E(t=0)=5.0$.

Figure 1 shows the residual energy $\langle\Delta E_{l}\rangle_{l}$ averaged over $l,$ as a function of time, for the fractal oscillator model with various values of the coupling constant $J$. The system consisted of $N=64$ sites held at temperature $T=0.25$. The brackets $\langle...\rangle$ denote averages over initial conditions as well as averages over sites where the energy was injected and measured. The simulations were averaged over $30\times 64$ runs, with 30 runs for each site. As $J$ was reduced, the dynamics become steadily slower and slower. The curves collapse onto one another if the horizontal axis is scaled and the vertical axis is shifted for each curve by a different amount, and the result fits nicely to a stretched exponential. This implies that all the curves are of the form $\langle\Delta E_{l}(t)\rangle=A(J)\exp[-(t/t_{0})^{\beta(J)}],$ with $t_{0}=3\times 10^{5}.$ However, the curve for $J=0.25$ is essentially flat, making it very difficult to determine whether it fits the same stretched exponential form or if the decay takes infinitely long: $\Delta E(t\rightarrow\infty)\neq 0.$ In order to elucidate this further, we hold the coupling constant fixed at $J=0.25$ and vary the temperature.

Figure 2 shows the residual energy as a function of time for the same system (but with $N=128$) at various temperatures with $J=0.25.$ Unlike in Figure 1, the energy was only injected at the sites $l=(0,1).$ For $T>0.50$, the measurements were averaged over 1200 runs, while only 900 runs were realized for the lower temperature curves. In addition, the timestep for $T>5.0$ were reduced by a factor of 10 to account for the faster dynamics. Unexpectedly, the decay of the residual energy is slow at low and high temperatures, but not at intermediate temperatures. For the high temperature behavior, we argued earlier that the coupling between oscillators is weak, and the slow decay of the residual energy is not surprising. and at low temperatures, the linear disordered model (which is localized) has small corrections, with the same result.

Because of the possibility of metastability in oscillator chains bennetin , one has to be careful whether the slow decay really indicates energy being trapped for an infinite time. Therefore, the times $t_{I}(T)$ at which the residual energy drops to $E_{0}=2.1,$ approximately 80% of the energy originally injected, are calculated, and found to fit the Vogel Fulcher vogel form $t_{I}(T)=A\exp[B/(T-T_{f})]$ with $A=0.47\times 10^{5},$ $B=0.57$ and $T_{f}=0.34.$ This indicates that a finite residual energy remains at the original pair of sites when $T<T_{f},$ i.e. $\langle\Delta E_{l}(t\rightarrow\infty)\rangle\geq E_{0}.$

To confirm our argument that the bounded form of the interaction potential is (at least in part) responsible for energy localization, we carried out simulations on the fractal oscillator chain but with nearest neighbor potential $V(u)=J(u^{2}/2+u^{4}/4).$ With $N=64$ and $J=0.25,$ the measurements were averaged over 1200 runs for $T\leq 1.0$ and 4800 runs for $T>1.0.$ The results for this system are in Figure 3. All the curves collapse on top of each other if the $t$-axis is scaled differently for each $T,$ and the result fits to a stretched exponential (except for large $t$ and $T$). This implies $\langle\Delta E\rangle\sim A\exp[-(t/t_{0})^{\beta(T)}],$ with the best fit values of $A=2.8$ and $t_{0}=7\times 10^{3}.$ The data is consistent with $\beta(T)\sim T^{0.2}.$ Thus $\beta(T\rightarrow 0)$ appears to be zero, i.e. one cannot show that the dynamics freeze at $T\neq 0.$

Finally, in Figure 4 we compare $\Delta E(t)$ when energy is injected at various points in the ring, showing slow decay for some sites and fast decay for others. When energy is injected into the sites $(0,1)$, the case discussed so far, the decay of $\Delta E(t)$ is one of the slowest.

In conclusion, we have studied energy trapping in disordered classical oscillator chains, with disorder chosen to avoid resonances. If the coupling between oscillators has a cosine form, we see evidence that, at low temperatures, energy can be trapped at a site for infinite time, indicating classical many body localization. This is not seen when the coupling is polynomial.

We thank Richard Montgomery, David Huse and Sid Nagel for helpful discussions, and David Huse for suggesting this problem. O.N. thanks the International Center for Theoretical Sciences (ICTS) for their hospitality during the program Non-equilibrium Statistical Physics (ICTS/Prog-NESP/2015/10), where some of this work was carried out.