Light-cone spreading of perturbations and the butterfly effect in a classical spin chain
We find that localised perturbations in a chaotic classical many-body system– the classical Heisenberg spin chain – spread ballistically with a finite speed and little change in form as a function of distance from the origin of the perturbation even when the local spin dynamics is diffusive. We study this phenomenon by shedding light on the two complementary aspects of this butterfly effect– the rapid growth of perturbations and its simultaneous ballistic (light-cone) spread, as characterised by the Lyapunov exponents and the butterfly speed respectively. We connect this to recent studies of the out-of-time-ordered correlator (OTOC), which have been proposed as an indicator of chaos in a quantum system. We provide a straightforward identification of the OTOC with a correlator in our system and demonstrate that most of the interesting qualitative features – with the exception of the physics of entanglement – are present in the classical system. We present mostly results for the infinite temperature case but also discuss finite temperatures, where our results remain valid.
The butterfly effect lorenz1996essence; hilborn2004sea; lorenz2000butterfly is a vivid picture for the sensitivity of a spatially extended chaotic many-body system to arbitrarily small changes to its initial conditions. In this picture, this exquisite sensitivity – the proverbial butterfly wingbeat is enough to make the difference between presence or absence of a tornado – perhaps takes precedence over the fact that these changes are global – tornado activity is toggled in a place far away from the butterfly.
While this sensitivity to initial conditions is well-studied and quantified via the (positive) Lyapunov exponents, the spatial spreading of the perturbation has received somewhat less attention. This spreading, if ballistic, is characterised by a butterfly speed. Butterfly speed and Lyapunov exponents thus encode two complementary aspects of the butterfly effect. It is then natural to ask under what conditions a non-zero butterfly speed exists, and what determines its magnitude.
These questions have acquired additional interest thanks to many recent studies of quantum many body systems, in particular with relation to ideas involving the scrambling of quantum information sekino2008fast; Shenker:2013pqa; brown2012scrambling; Lashkari2013; maldacena2016bound; rozenbaum2017lyapunov; tsuji2017bound; iyoda2017scrambling; kukuljan2017weak; kurchan2016quantum; roberts2016lieb; swingle2016measuring; swingle2017slow; bohrdt2016scrambling; stanford2016many; banerjee2017solvable; Shenker2014; Hosur2016. In this setting, the out-of-time-ordered correlator (OTOC) is shown to quantify the growth and spread of perturbations in a class of these systems kitaev; maldacena2016bound; banerjee2017solvable. For two Hermitian operators and localised around at time and at time respectively, the OTOC, defined as , estimates the effect of the operator, on the operator, , at a later time and a different location. In a class of large gauge theories it was found that, for a given and , the OTOC is generically characterised by an exponent, and a velocity , which are respectively the measure of the exponential growth and the speed of spreading of the initial perturbation. Analogous to classical dynamical systems, the former is often identified with the largest Lyapunov exponent, and the latter with the butterfly speed.
Furthermore, the idea that at temperature , the obtained from the OTOC has a universal quantum bound suggests a limiting rate of quantum scrambling maldacena2016bound. This bound is saturated for models like the Sachdev-Ye-Kitaev model PhysRevLett.70.3339; kitaev and large quantum field theories gu2017local. Several recent studies explored these ideas in a wide variety of problems which include bounds on chaos maldacena2016bound; rozenbaum2017lyapunov; tsuji2017bound, butterfly/scrambling speed swingle2016measuring; lucas2017energy and the Lieb-Robinson bound roberts2016lieb, equilibration times, information propagation iyoda2017scrambling, bounds on transport coefficients and the Loschmidt echo kurchan2016quantum.
Interestingly, the above features are present even when the usual probe for relaxation and equilibration in a many-body system, the two-point functions , are diffusive and hence does not capture the ballistic spread of perturbations. This was observed in a study of the OTOC in a system with diffusive energy transport– the one-dimensional Bose-Hubbard chain at finite temperature bohrdt2016scrambling; PhysRevB.96.054503, and recently in the context of random unitary circuits khemani2017operator; rakovszky2017diffusive, which lend themselves to a considerable degree of analytical and numerical insightvon2017operator.
Given this background, it is natural to ask which of these features are intrinsically quantum mechanical, i.e. to separate the purely quantum effects from the classical ones. This is all the more important as issues of the propagation and growth of an initially localised perturbation is central to problems such as equilibration and thermalization of a many-body system.
Here, we study such spatio-temporal propagation of classical perturbations in a well-known diffusive classical many-body system – the classical Heisenberg spin-chain at high temperatures. We first identify a classical correlator which embodies the semiclassical limit of an OTOC, and which turns out to be a very simple quantity: the deviation of the overlap between the system and its perturbed copy under time evolution.
With this in mind, through detailed numerical simulations and analytical calculations, we indeed find that–(a) the exponential growth, and (b) the ballistic spread of perturbations are captured by the classical limit of OTOC. As advertised above, we characterise these via the Lyapunov exponents and a finite butterfly speed so that information on the action of the perturbation is transmitted ballistically. Thus, even in this classical setting, both quantities (a) and (b) lie at the very heart of butterfly effect of classical many-body chaos.
This main finding is summarised in Fig. 1. We round this off by a detailed analysis, in particular to connect the many-body chaos setting with the standard lore of classical chaos. We would like to emphasise that the finite butterfly speed is in stark contrast with the entirely diffusive spin dynamics as recorded by the regular two point correlator (Fig. 2).
The Heisenberg spin chain:
We consider a one-dimensional lattice of spins described by the Heisenberg Hamiltonian
where and are unit three component classical vectors and we take periodic boundary conditions . We assume a classical precessional dynamics with the equations of motion given by where the spin-Poisson bracket is defined as for arbitrary functions of the spin variables.
Classical OTOC analogue and Lyapunov exponents:
We consider two spin configurations which differ from each other only at site by an infinitesimal rotation, , about an axis (where is the unit vector along global -axis) such that . We consider the spreading of such a localised perturbation. For small , the change at some point is given by , where, the last step follows from the definition of the spin Poisson-bracket. To measure the evolution of the perturbation we define
where, throughout this paper, denotes averaging over spin configurations chosen from the equilibrium distribuition and is the partition function. Denoting the two initial spin configurations discussed above by and , we can obtain a simpler expression as
Hence we expect that for any , this quantity starts from the value and asymptotes to at . Apart from , we also calculate the usual dynamic spin-correlation function
At this point, it is useful to understand the connection between and OTOC. On canonical quantisation of the theory obtained by replacing the Poisson bracket with the commutator, i.e. , we do get where are now quantum operators. This is nothing but the finite temperature generalisation of the OTOC introduced earlier with and .
Finally we note that the usual definition of the Lyapunov exponents is in terms of the eigenvalues of where (and is transpose of ) is the Jacobian matrix. We denote the largest eigenvalue as and for a fixed initial spin configuration. From the definitions, it is expected that this will be related to the growth of with time, and we examine this in a precise way by using numerical simulations.
Numerical Results :
We now present the representative results of our numerical simulation of the Heisenberg spin chain with periodic boundary conditions. The simulations were performed using a fourth-order Runge-Kutta (RK4) numerical integration scheme for the dynamics and Monte Carlo techniques for equilibration. For the numerical simulations, the energy-scale is measured in units of . The time-step in RK4 was taken to be such that the energy/site and magnetisation/site were conserved upto order . The configuration averaging was done over equilibrated initial conditions for and for . The largest Lyapunov exponent, , was obtained by using standard QR decomposition methods geist1990comparison.
The classical spin chain does not order at any finite temperature PhysRevLett.17.1133 and therefore the static spin-spin correlator is exponentially decaying fisher1964magnetism. The dynamic spin-spin correlator, (Eq. 4), shows diffusive behaviour at long times as the total spin is conserved chaikin2000principles. This is clearly shown in the insets of Fig. 2 which shows the behaviour with the expected collapse at long times.
In Fig. (3) we show the temporal growth of the signal at different points on the lattice at . The values of in plane (Fig. 1) clearly show that the signal propagates ballistically and simultaneously grows with time. To characterise this propagation, we ask when does the signal reach a threshold value at a given . It is clear from Fig. 1 that is sizeable in a region – the light cone – whose size grows ballistically as . This gives the natural definition of the light-cone velocity in the sense of a “classical Lieb-Robinson bound” lieb1972finite; PhysRevLett.112.210601; Marchioro1978. We would like to note that since ours is a non-relativistic system, strictly for any for , but its value is arbitrarily small PhysRevLett.112.210601; Marchioro1978. So the actual speed of the front depends on the threshold , as shown in the inset of Fig. 4, and is expected to go to infinity as at fixed . We use to calculate in Fig. 4.
Beyond the arrival time , grows exponentially until it saturates to . In addition to , it is therefore useful to characterise the speed of the spreading front within the regime of the exponential growth of . This velocity, signifying the onset of a butterfly effect at a given is identified with the butterfly speed, . We calculate by using a threshold of in Fig. 4 where we plot both and as a function of and find they appear to saturate at large .
Having characterised the velocities, we now turn to the regime of exponential growth (as a function of at a given ) of . In particular we expect , where is a constant and is the largest Lyapunov exponent of the system. In Fig. 5 we plot for different values of vs. to show this linear behaviour. From this we estimate an exponentially growing temporal regime where , with . Furthermore, for to reach a magnitude , the required time is given by, as shown in the inset of Fig. 5.
At this point it is useful to ask how does compare with the largest Lyapunov exponent calculated from linearised system. To answer this question we need to take care of effects arising from the order of averaging over initial conditions rozenbaum2017lyapunov. For a fixed initial configuration, we compare the rate of the exponential growth of , with the largest Lyapunov exponent, , evaluated using the standard QR Method geist1990comparison. The comparison is shown in Fig. (6), where we see that after initial transients, s approach the value in the regime of exponential growth for different but small before finally decaying as due to saturation of at longer times. Since , obtained from the Jacobian matrix, corresponds to the limit , it reaches the saturation value of , as expected de2012largest. With this, We now note that in the regime of exponential growth, , by applying Jensen’s inequality jensen1906. This means as seen in Fig. 7. This completes our description of the butterfly effect at .
The above features of the butterfly effect, viz. exponential growth and ballistic propagation, both survive at as shown in Fig. 7. In this regime, the equilibrium static spin-spin correlation length is of the order of lattice spacing and the velocities renormalises marginally for . However, interestingly, the Lyapunov exponent decreases with while the spin-spin correlation length increases, signalling the suppression of the rate of exponential growth with the increase of spin correlation. It has recently been suggested that the diffusion coefficient is related to and as bohrdt2016scrambling; davison2017thermoelectric; hartman2017upper; gu2017local and we find that this approximately holds for high temperatures. At lower temperatures, reaching the diffusive regime requires study of larger system sizes and longer times and these results require further study.
We have studied the butterfly effect in a classical Heisenberg spin chain at infinite and finite temperatures and have shown that a systematic understanding of this effect includes two simultaneous, but logically complementary aspects– the exponential growth and ballistic spread of an infinitesimal local perturbation determined by the Lyapunov exponents and the butterfly speed, respectively. Both these effects are quantified by an appropriately defined measure that is naturally related to the OTOC recently studied in context of scrambling in quantum many-body systems. The above central aspects of the butterfly effect in classical system are analogous to those of the recently studied quantum system except for the absence of the speed associated with the spread of quantum entanglement, which does not have a classical counterpart.
Notably, the above ballistic spread of perturbation is present while the usual two-point dynamic spin-spin correlator shows diffusion and hence does not reflect correlations spreading with the butterfly speed. A natural question then pertains to the nature of correlators that are directly sensitive to this ballistic effect. A closely related issue is in regards to an analytical derivation of the equation of motion for the propagating ballistic front. In our case, the ensemble-averaged overlaps between perturbed and unperturbed states toggle from the unstable identical case to the stable uncorrelated one. Wave-like solutions are known to exist in non-linear dynamic many-body systems that describe front propagation between stable and unstable field configurations fisher1937wave; tikhomirov1991study, and an analysis of a microscopic quantum model of electrons aleiner2016microscopic has indeed unearthed such combustion wave behaviour. The detailed connection between such equations and the butterfly front in our case requires further study. The ubiquity of the present results, particularly in context of many-body systems directly motivated from condensed matter physics, would shed light on issues related to thermalisation in these systems.
The authors acknowledge A. Baecker, S. Banerjee, C. Dasgupta, D. Dhar, R. Govindarajan, J. Kurchan, M. Kastner, A. Polkovnikov, S. Ramaswamy and S. Sabhapandit for useful discussions. SB and RM acknowledges MPG for funding through the partner group at ICTS. SB acknowledges MPIPKS for hospitality and support of the DST (India) project ECR/2017/000504. A.Dhar and AK acknowledge support from the Indo-French Centre for the promotion of advanced research (IFCPAR) under Project No. 5604-2. SSR acknowledges the support of the DST (India) project ECR/2015/00036. The numerical calculations were done on the cluster Mowgli at the ICTS-TIFR.