dynamics and domain walls: is the “landscape paradigm” instructive?

ELSEVIER Physica D 107 (1997) 204--217

PHY$1CA

Dynamics and domain walls: Is the "landscape paradigm" instructive? Daniel S. Fisher 1

Physics Department, Harvard University, Cambridge, MA 02138, USA

Abstract

General questions concerning the dynamics of systems with many degrees of freedom are discussed, focussing on approach to equilibrium in simple statistical mechnical models. The contrasts between the behavior of systems with short-range interactions and mean-field pictures in terms of energy "landscapes" are emphasized. It is shown how much of the behavior of systems with short-range interactions can be described in terms of domain walls and their dynamics. Simple Ising magnetic models, both ferromagnets and spin glasses, are used as a paradigm.

Keywords: Domain walls; Landscape; Ordering; Ageing; Coarsening

O. Introduction

In this conference, many speakers have shown

sketches like that in Fig. 1 meant to represent some-

how "rough energy landscapes". In some cases, such

as in the mean field theory of spin glasses, these

landscapes are said to have some kind of "hierarchi-

cal" structure, with the valleys being "states" [ 1 ]. But

what are such pictures really supposed to represent?

Are they a good caricature of some systems with frus-

trated interactions that is instructive for understanding

their physics? If so, do they represent energy or free energy and what does this imply about the dynamics?

The purpose of this lecture is to explore briefly some

of these questions.

The main message will be that for a broad class of physical systems - and probably for most others

- such pictures are at best only very poor caricatures and often, in fact, very misleading.

1 Fax: + 617 495-416; e-mail: [email protected].

Fig. 1. A rough energy landscape.

1. Systems

The type of systems we will focus on have many de-

grees of freedom, with an underlying equilibrium distribution of the configurations given by a Boltzmann

factor Z -1 e x p ( - ~ / T ) controlled by a hamiltonian

with N degrees of freedom and, for simplicity, relaxational dynamics with thermal noise. Although at

long times the system will converge to equilibrium, we will be interested in history dependent phenomena such as evolution from random initial conditions, or

0167-2789/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved PII S0167-2789(97)00088-2

D.S. Fisher/Physica D 107 (1997) 204-217 205

responses to changes in parameters such as the tem-

perature T or applied fields. We will, in particular,

be interested in large systems which are those large

enough that the longest time scale of experiments on

the system is much less than the time for the full sys-

tem to reach equilibrium, i.e.

expt equilibrium tmax << r N . (1)

(This is in contrast to "small" systems, which can equi-

librate fully in experimental time scales, the primary

example featured in this conference being small pro-

tein molecules [2].)

The interactions in our systems will be assumed to

be local in space: an important feature of many sys-

tems which makes them behave, as we shall see, very

differently from infinite range mean-field-like carica-

tures. In physical dimensions d = 1, 2 or 3, the vol-

ume of the system, proportional to N, scales with its

characteristic spatial extent as

N ~ £d. (2)

What we would like to be able to develop is a ro-

bust phenomenological description of some of the long

time phenomena in such systems which is sufficiently

robust and flexible to be extendable to other - includ-

ing non-equilibrium - systems.

Since one of the main lessons from the study of

equilibrium phase transitions is that to simplify com-

plex systems, one should first understand the complex-

ities of simple systems, we will focus on the simplest

interacting systems: Ising magnets. Thus we consider

N dynamical spin variables a = 4-1 located on the

sites of a linear, square, or cubic lattice with fixed in-

teractions J/j between nearest neighbor pairs (i j ) gov-

erued by the hamiltonian

n = - E J/J ' J - H (3) ij i

We will sometimes be interested in the effects of a

magnetic field H that tries to align the spins, but will

generally set H = 0. For many choices of the {J/j} such Ising models have a phase transition in zero field

to an ordered phase that exists for T less than a crit-

ical temperature, Tc. The simplest dynamics are just

relaxational: spins can flip to lower the energy but flips

in the opposite direction are caused by thermal noise

with a rate controlled by T.

The simplest interesting non-equilibrium situation

to consider is the approach of the system towards equi-

librium starting from disordered (e.g. totally random

ai 's) initial conditions. Starting at time t = 0, one

can wait for a time t~ and then probe the system over

some time interval of length tin. Such experiments are

often called "aging" and one of the issues we would

like to address is in what ways they might - or might

not - tell us about the "structure of the phase space"

of these systems.

2. Pure two-d imens iona l Ising model

The simplest of all systems with ordering transitions

is the two-dimensional ferromagnetic Ising model with

all the Jij equal to J and H = 0. For temperatures less

than Tc <x J , the mean field picture of the system is

very simple. It is often characterized by the behavior

of the free energy of the system F as a function of the

magnetization

1 m = N ai, (4)

i

which in equilibrium is simply (ai); F(m) is sketched

in Fig. 2(a). The minimum free energy can be achieved

with two possible values of m, m = ±rn0, correspond-

ing, respectively, to the two equilibrium states of the

system, one with mostly up spins denoted " + " and the

other with mostly down spins denoted " - " . The free

energy barrier between them occurs at m = 0 and has

free energy B --~ N above the equilibrium free energy.

In this mean field picture, the system started in a dis-

ordered state will rapidly "roll downhill" to one of the

two minima. But this mean field picture is quite wrong!

In reality, the free energy of the system with a con-

strained m is like that shown in Fig. 2(b): again, there

are two minima at +m0, but the region between them

is very flat and much less far above the minima; indeed

the barrier free energy is proportional (in two dimensions) not to the volume, but to the linear dimension:

B ~ £ ~ N I/2. (5)

206 D.S. Fisher/Physica D 107 (1997) 204-217

F

+ - 7 ~ B

x¢,

L

0

(a)

I'n

- - B

0

(b)

tTI

-I-

(c)

Fig. 2. Free energy F as a function of the magnetization, m, of an Ising ferromagnet, with B indicating the barrier between the "+" and " - " states; (a) is for a mean field system while (b) is for a two- or three-dimensional system. In (c) a domain wall configuration is shown that corresponds to a magnetization in the plateau region of (b).

This arises from a very simple physical phenomena

that is absent in a mean field picture: the minimum

free energy state of the system for - m 0 < m < m0

consists of a region of " + " state separated from a

region of " - " state by a domain wall in which the

excess free energy is concentrated; this is shown in

Fig. 2(c). The barrier is then just the free energy of a

domain wall spanning the system.

How, then, does order evolve from random initial

conditions? The flatness of Fig. 2(b) perhaps sug-

gests that the evolution will be rather slow and in-

deed this is the case [3]. The simplest situation is zero

temperature for which spins can only flip to lower the

energy or to keep it unchanged. The initial configu-

ration can be simply represented as a dense network

of (microscopic) domain walls separating regions of

up and down spins. Since the excess energy is just

proportional to the "area" (i.e. in two dimensions, the

length) of the domain walls, the system will evolve

to decrease the domain wall area and hence increase

the typical distance, R(t), between domain walls - a

crude measure of the linear size of the domains. A

computer simulation of the growth of domains with

time is shown in Fig. 3. As can be seen, at long times

the lattice structure does not appear to be important

as the walls will evolve in a way determined by their

curvature: the local velocity of a section of domain

wall will be proportional to its radius of curvature. The

time for a domain (or "droplet") of radius r to shrink

and vanish is determined by the resulting equation of

motion,

dr 1 - - ( x - - ( 6 ) dt r '

2 which yields r ---* 0 at time tr "~ rinitia I • From this,

one can guess the famous result that the pattem of

domains will "coarsen" with time as

R(t) ~ t 1/2. (7)

This linkage of characteristic time scales to charac-

teristic length scales is a general and essential feature

of systems with finite range interactions that is com-

pletely missing in mean field theory.

Because of the energy of the walls, the energy of

the system E(t) will decay very slowly, with

N E (t) - Egroun d state t 1 /2 ' (8 )

note that this is always much larger than the en-

ergy barrier between the ground states which is

only of order N (a-1)/a. Indeed there is, in princi-

ple, always enough energy to overcome this barrier

- it is just not in the right place: the excess free


Fig. 3. Simulation of the growth of domains m a two-dimensional Ising ferromagnet at zero temperature (from Ref. [3]). The dark regions are up spins and the light regions are down spins. From top left to bottom right, the time after the random initial condition is t = 5, 15, 60 and 200.

energy is concentrated in well separated, slowly

evolving domain walls. This behavior is in strik-

ing contrast to mean field theory, and persists at

all temperatures T < Tc. Within each domain, the

system is in local equilibrium, an important con-

cept that does not exist in infinite range mean field

models.

While its basic physics is simple, this non-

equilibrium behavior of Ising ferromagnets is highly

non-trivial. The pattern of domain walls at long times,

when rescaled by R(t) , reaches a statistical steady

state whose properties are not known analytically.

The self-similarity is reflected, for example, in the

decay of the memory of the initial conditions via the

208

correlation function

1 (ffi (0)tTi (t)) "~ - - (9)

[R(t)] ~-

with ~. an unknown non-equilibrium exponent in the range ½d < ~. < d which is found numerically, to

be ~. ~ 5 in two dimensions [4]. This simple system

also exhibits aging. If after waiting for a time t, the ac susceptibility is measured, i.e. the response

Om(w) xt(w) - (10)

O H (og)

to a magnetic field at frequency o9, this is found to be

dominated for o9 >> 1/t by the motion of the domain

walls as these now have a force on them proportional

to H. This yields, for o9 >> l / t ,

1 Xt(og) (11)

-iogR(t)

the I /R factor arising from the density of domain

walls per unit volume. In contrast, the equilibrium susceptibility goes to a (temperature dependent) constant

at low frequencies. Generally, the domains will con-

tribute the equilibrium susceptibility and the walls a contribution like Eq. (11). Thus, the total susceptibil-

ity will roughly take the form at long times

Xt(og) ~'~ Xeq(og) "{- R(t)X(ogt) (12)

with X a scaling function with argument ogt, that in-

cludes the effects of crossover from the o9 >> l i t regime of an almost static wall pattern to the o9 "~ 1/t regime of evolving walls. Note that at any fixed fre-

quency, at long times Xt(og) --* Xeq(og) because of convergence to local equilibrium within the domains.

3. Geometry of phase space

We have seen that even pure Ising ferromagnets evolve in phase space in quite a complicated way - very different from mean field theory - with both rel- atively slow dynamics and aging phenomena.

In a beautiful recent paper, Kurchan and Laloux [5]

have shown that behavior of this kind is ubiquitous to systems with many degrees of freedom and deterministic dynamics that continuously lowers the energy

D.S. Fisher/Physica D 107 (1997) 204-217

"~ basin boundary

Fig. 4. Schematic of flows in phase space.

("gradient descent" dynamics). (One such example of this is a continuous spin Ising ferromagnet ("~b 4

model") in the continuum which behaves similarly at

long times to the lattice model discussed above.) They

consider the general geometrical structure of high-

dimensional phase spaces; shown schematically in

Fig. 4. These can be characterized by (local) minima;

basins of attraction of these minima; basin boundaries of these on which saddle points are attractors; second-

order saddles with two unstable directions which separate the domains of attraction of two saddles on

a basin boundary; etc. In general, extremal points of

the energy, at which the system will be stationary, can be characterized by their "index", the number of

unstable eigen-directions of perturbations away from

the extremal points: minima have index I = 0; simple

saddles, ! : l; second-order saddles I = 2, etc. As is general in high (phase space) dimensions (at

least if the number of minima is not too large), most of the phase space volume of each basin will be near

basin boundaries; most of the basin boundaries will be near boundaries of domains of attraction of the simple saddles; etc. Thus, in some sense, most of phase space will be near very high index points ! Kurchan and Laloux argue that the dynamics is then controiled by the index of nearby saddles with this index decreasing


o

Fig. 5. A situation in an Ising ferromagnet in which small fluctuations can determine whether the region will be in the "+" or " - " state.

with time, leading, quite generally, to slow dynamics

at long times.

Several important features emerge: the energy of the system E(t) is always O(N) above the minima,

thus the system is never clearly "in" the basin of a

single minimum as the barrier between minima are <<

O(N). Concomitantly, the system is always sensitive

to thermal fluctuations with small noise at any time (as N ~ oo) being sufficient to change the basin into

which the system will eventually fall. This can be seen

schematically for the 2D Ising model by considering a configuration with two straight walls that cross: as

shown in Fig. 5, which way they cross will determine

the future evolution of a whole region.

We refer the interested reader to this paper [5] for a

detailed exposition and its application to the dynamics of several systems - including Ising ferromagnets and

some mean field models.

209

free energies - hard to define precisely out of equi-

librium - and the dynamics should act - on average

- to lower the free energy. Changing larger regions of a system from one meta-stable configuration to

another will generally involve intermediate configurations with more spins in unfavorable orientations.

We thus expect that the energy (really free energy)

barriers to change a region of length scale L will grow as

B ~ L ¢. (13)

(Note that we have already seen a simple example

of this: the barriers to flip all of an isolated region

of a pure Ising ferromagnet scales as Eq. (13) with

~ = d - 1 . )

The time for thermal fluctuations to overcome a high barrier is "-~ exp(B/T) thus large length scales will in-

volve very long time scales. But because of variations from region-to-region due to the randomness, BL will

vary yielding big variations in time scales. The length- time scaling in a random system implied by Eq. (13), is therefore

L ~ In rL ~ - - . (14)

T

4. Random magnets and barriers

How are these general features of evolution of order changed in the presence of random interactions? The

simplest cases to consider are random ferromagnets for which the { Jij } of Eq. (3) are independent random

variables that are all strictly positive; later we will

consider spin glasses for which the {J i j} can be either sign.

The behavior at zero temperature immediately

changes drastically: There are now exponentially many, O(eCXN), local minima of 7-( in which the

system will quickly become stuck. At any positive temperature, fluctuations will enable motion from one minimum to another but barriers will have to

be overcome. Yet as soon as thermal fluctuations are introduced, we really should consider something like

We thus see that random systems at low tempera-

tures (and generally in ordered phases) will have ex-

tremely broad distributions of characteristic relaxation times.

How, then, can one understand the long time dynamics of such a system? The natural hope is that

one can somehow coarse-grain in space and time, to

give dynamics determined by some kind of effective hamiltonian 7~ (really a free energy) which includes

large barriers, but in which small barriers have been

overcome and local equilibrium reached. In general, there is no known systematic way of doing this

except rather schematically; this will be discussed shortly for random magnets. But it is instructive to first consider a very simple system for which a coarse graining procedure can really be carried out systematically.

210

5. Random walks in random environments

The simplest random system that exhibits some of

the effects we are interested in, is a single particle moving in a one-dimensional random potential (for a

recent review, see [6]). (This is, in fact, the dynamics of

a single domain wall in a one-dimensional "spin glass"

with the Jij's randomly + J with equal probability, and a uniform magnetic field H, but this connection

is not needed for what follows.)

We consider a random walker x( t ) in a random

potential:

dx - - f [ x ( t ) ] + rl(t ) (15)

dt

with r/(t) Gaussian white noise with temperature T,

and a locally random force f ( x ) with mean zero

and variations cutoff at the scale of the lattice con-

stant. The force is just the gradient of a random potential,

dV(x) f ( x ) -- (16)

dx

with

V ( x ) -- V ( y ) "~ ( x - y)l/2 (17)

arising from integrating the random force. This potential is shown schematically in Fig. 6(a). If the particle

starts at some point x( t = 0), it will rapidly fall to

the nearest local minimum and then fluctuate around this, occasionally going over a barrier to a nearby local minimum. The height of the barriers that must be

overcome to move a distance L, will typically be of order B .,. L 1/2 but with variations in their magnitudes

also of order L 1/2. On short time and length scales the

behavior will be quite stochastic but at large times it will be very different!

We are interested in the position of the particle after a time, t. Roughly speaking, in this time it can overcome all barriers with

B < Bt = T i n t (18)

and will be in (local) equilibrium within regions in which there are no barriers larger than Bt. This sug- gests coarse-graining the random potential as shown


K L )i x(t=0) x

(a)

x(tl ) x(O)

(b)

x(t2) x(0) x

(c)

Fig. 6. (a) Random potential in which random walker starting at x(0) moves. Typical scales of barriers, B, as a function of length scales, L, are indicated. (b) and (c) arc coarse grained versions of the potential f rom (a) at late times t2 >> t l >> 1. With high probabil i ty the particle at t ime t w i l l sit near the bottom of the valley in the effective potential l~t that contains x(0).

schematically in Fig. 6. First, the smallest up or down

section is chosen - corresponding to the smallest bar-

tier - and it is replaced by a straight section as shown in Fig. 6(a). The next smallest barrier is then similarly

decimated and so on until all remaining sections have heights larger than Bt yielding an effective potential, ~'t(x). The system is now divided into "valleys" ap- propriate for time t and the corresponding dynamics

is very simple: at time h, the particle position, X(tl), is very likely to be near the bottom of the valley in

I~" h (x) that includes its initial position, x(0), as shown in Fig. 6(b). The particle will typically stay in this valley for a long time.


But we can now continue to coarse grain the po-

tential until a much longer time t2 by which time the particle can overcome one or both of the barriers bor-

dering its valley at time tl, and the particle then falls

to the bottom of a deeper valley in ~'t2(x) as shown in Fig. 6(c). One crucial feature is that as the length scale grows, the time scales grow exponentially - with

the typical

x ( t ) ~ (lnt) 2 (19)

from inverting Eq. (17). A second feature is that the

behavior becomes more and more deterministic at long

times! This is because the distribution of the remain-

ing barriers in lT"t (x) becomes broader and broader as

t increases. At any fixed long time scale, t, almost

all barriers are thus either much bigger than Bt and

hence extremely unlikely to be overcome, or substan-

tially less than Bt and hence contribute to the local equilibrium distribution within the valley in which the particle is trapped at time t.

This pseudo-deterministic motion of the particle,

with x ( t ) fluctuating for a long time only slightly

around a valley bottom but appearing to jump around

to different valleys as a function of In t, was first

found by rigorous mathematical analysis by Kesten

[7]. Remarkably, a renormaiization group procedure

recently developed for a very different problem [8] can be used - essentially as described above - to ob-

tain many asymptotically exact results, including the

effects of a small drift force ( f g: 0), and aging phe-

nomena as the particle explores deeper and deeper valleys [9].

The main lesson from this, for our purposes, is that broad distributions of barriers that grow with length

scales naturally lead to very good separation o f time

scales at long times with most processes either in local

equilibrium, or not accessible on that time scale. The

pseudo-determinism found here is probably much less ubiquitous, at least in the strong form seen with the

one-dimensional phase space. Indeed, even with phase spaces that are slightly higher dimensional (perhaps three, such as for three domain walls in one dimension) some of this behavior may already be lost; but this is a subject of current research.

6. Random ferromagnets

We now return to random ferromagnets in two (or

three) spatial dimensions and try and understand the evolution of order from random initial conditions as we did previously for the pure Ising model. The equi-

librium properties of this system are rather simple: just

as in the pure case, for H = 0 below Tc, it has two equilibrium states, mostly up ("+") and mostly down

C - " ) - (although rare regions of the sample make the mean equilibrium dynamic auto-correlations decay as

a non-universal power of time, but that is beyond the

scope of this lecture). Finding the ground states of the

system is thus trivial for a physicist or computer, but the poor system left to its own devices has much more

difficulty doing so. 2 As in the pure ferromagnet, after

a short time the configuration will look like a pattern of

domain wails as in Fig. 3, separated by domain walls

(with an effective width, due to thermal fluctuations, of the order of the equilibrium correlation length). The

excess free energy of a flattish section of domain wall of linear size L will have two components

Fwall ~ L d- 1 4_ random part, (20)

where the first term is the interfacial free energy and

the second smaller random part is due to the tendency

of the wall to pass through weaker bonds and the vari- ation of the strengths of such bonds from region to

region. This randomness will also make the wall have

some scaie-dependent roughness characterized by a roughness experiment, ( , although it looks flatter on

longer scales since ( < 1. To move such a section of domain wall, it will have to pass through less favor-

able regions of the random bonds and hence have to overcome a barrier B ~ L ¢' which will be at least as

big as the random part of Eq. (20). On large scales, the

curvature of the walls provides a way for the system to lower its free energy by getting rid of this curvature

via moving sections of a wall over barriers. As shown

first by Villain [11], the size of the section that can be moved grows with the characteristic domain size

2 See e.g. [10] for a discussion of the absence of a connection between algorithmic difficulty and difficulty in reaching equilibrium.


R(t) (although more slowly than linearly) so that the

process is slower and slower at long times. As for the random walk in a random environment, this results in

a logarithmic growth of the scale of the domain pattern with time 3

R(t) ~, (In t) 1/~ (21)

with

~, - (22) 2 - ( "

It is interesting to note that the statistics of the domain

patterns, rescaled by R(t) , are probably the same at

long times as in the pure case even though the dynam-

ics is very different. Indeed, for macroscopic times,

the domains can reach at best mesoscopic sizes in the random system, while in the pure case, they can reach

macroscopic sizes.

As in the pure case, the responses of the random ferromagnet will depend on its "age", t, although the

effect is now much more striking. Again, the response

Xt (w) to a small ac magnetic field will be dominated by the domain walls for a range of frequencies. But

now because of the logarithmic connection between

time and length scales the contribution from the walls

will have a very different scaling form, with

Xt(w) ~x X \ lnt ] (23)

so that the aging effects will persist and vary slowly over very wide frequency and equilibration-time ranges.

From the discussion of the random walk in random

environment and the dynamics and noise sensitivity of

domain wall evolution in the pure ferromagnet, two

questions arise immediately: Is the dynamics at long

times pseudo-deterministic in the sense of not depend- ing crucially on noise at late times? And, on the theoretical side, can the long time dynamics be described in terms of some effective hamiltonian and/or effective equations of motion? Both of these seem fruitful avenues of research on this deceptively simple system.

3 For d = 2, ~ _< N _< ½ with the lower bound possibly an exact equality, and in d = 3, ½ < ~ _< 1

7. Spin glasses

We now turn to the most interesting - and contro-

versial - kind of Ising magnets: spin glasses. The ideal

spin glass has the Jij 's drawn independently from a distribution that is symmetric in Jij --> --Ji j ; this im-

plies that the ensemble of systems has a statistical

symmetry. As is well known, this immediately gives

rise to "frustration", most simply seen by considering

a square of four nearest neighbor spins with three neg-

ative and one positive coupling around the square. If

three spins are chosen to minimize the corresponding

bond energies, the fourth spin does not really know in

which direction to point. More generally, the frustration makes finding the ground states of all-but-very-

small systems extremely difficult since they involve

delicate balances between energy gains on some bonds

and losses on others. Indeed, some thought is needed even to define sensibly ground states in a robust way

for infinite systems (of course, infinite systems are

needed for equilibrium phase transitions to be possi-

ble). A simple definition - which is clearly a necessary condition - is that a ground state configuration must

be stable to any finite changes, i.e. that the energy can-

not be decreased by changing any finite set of {t:r i }. A critical question is: How many ground states ex-

ist? Clearly they must come in pairs, since reversing all spins in a ground state yields another ground state.

The simplest scenario, proposed by David Huse and this author and supported by various heuristic argu-

ments, is that there is exactly one pair o f ground states;

i.e. a configuration {tr/c } which we denote "G" and its spin reversed partner G - {-or/6}.

As we will briefly review, this, together with gen-

eral properties implied by the statistical symmetries

of spin glasses, gives rise to a very rich and robust

phenomenological picture of the spin glass ordered phase, often called the "droplet" picture [4,12]. Un- fortunately, to this author's knowledge, there is at this point no consistent scenario that has been proposed for short-range spin glasses other than the droplet picture (or slight modifications of it), thus there are not really competing sets of experimental predictions for most quantities of physical interest in large systems.

213

Prob

The droplet picture builds up, aided by the renor-

malization group framework, an understanding of the ordered phase of spin glasses in terms of the

ground states plus excitations - as is usually done for conventional pure systems. The basic excitations

are "droplets": connected sets of spins bounded by a

domain wall in which the configuration is reversed

from the ground state outside; i.e. inside the wall is G instead of G. On a scale L, the lowest energy excita-

tion in a given region will consist of a fractal domain

wall with fractal dimension ds in the range

d - 1 < ds < d , (24)

bounding a reasonably compact droplet, as shown

schematically in Fig. 7. The energies of such exci-

tations, AL, will be broadly distributed as shown in

Fig. 8, although of course all must be positive. By

analogy with ferromagnets, we expect the typical AL

to scale as a power of L with the scaling determined

by the "stiffness" exponent, 0:

AL ~ L °. (25)

If 0 is negative, large excitations will be easily ex- cited thermally and occur all over the system thereby

destroying the broken spin-reversal symmetry present

at zero temperature. This is the case in d = 2 for which there is no spin glass transition [13].

L

,k

r L °


A L

Fig. 8. Schematic of the probability distribution of the free energies A L of droplets at scale L. A typical droplet will have A L ~ t 0 but rare droplets in the regime A L < T, shown shaded, will be thermally active.

If, on the other hand, 0 is positive, most large exci-

tations will be costly and hence not be thermally ex-

cited; the broken symmetry will then be preserved at

low positive temperature. This is believed to be the case in d = 3 for which 0 appears, from numerical

simulations [ 13], to be slightly positive, but in general all we know about 0 analytically is that

0 < ½ ( d - l ) (26)

(the ,,1,, reflecting, roughly, central-limit-like sums of

many random energy differences). For positive 0, then,

we expect a broken-spin-reversal-symmetry ordered

phase to persist up to a critical temperature Tc. The distribution of AL'S shown in Fig. 8 implies

that a small fraction of even the largest excitations will

have AL _< T and hence be thermally active. These

rare, large active droplets will dominate much of the

near equilibrium physics of the ordered phase, but we will not be concerned with this here.

As for random ferromagnets, there will be length scale dependent barriers to create excitations, since

unfavorable configurations of the wall will need to be

passed through to create a droplet. The barriers will scale as B ~ L ¢ with

~p >_ 0 (27)

Fig. 7. Schematic of droplet excitation in a spin glass on length scale L. Inside the domain wall the spins are reversed from the ground state configuration.

(an upper bound of ~p < d - 1 also obtains). How does the order develop in a spin glass? At zero

temperature, like the random ferromagnet, the system


GT 6--- R---~

Fig. 9. Schematic of evolving configuration in a spin glass started from random initial conditions; domain walls between the two equilibrium states GT and -aT are shown.

will quickly get stuck in a local minimum of 7-/. But

at non-zero temperature, barriers can be overcome -

albeit slowly - and the system will evolve. Of course it will evolve not towards the ground states G or G, but

towards a pair of finite temperature equilibrium states

that include (mostly small) thermal excitations; these

we denote GT and its spin reversed partner GT -- - G r .

At long times, t, after starting from random initial

conditions (here achievable, effectively, by turning off an initially large magnetic field), the system will look

like an interpenetrating pattern of domains of GT and

GT separated by domain walls - with approximate local equilibrium within the domains. This behavior is

shown schematically in Fig. 9 (although the ordered

phase cannot actually occur in the two-dimensional

case shown). From the growth of barriers with length scale, we expect that the characteristic scale of the domains will grow as

R(t) ~ (In t) l/q, (28)

roughly as for random ferromagnets [4]. But the behavior is different in several very interesting ways.

First, the domain wails in the spin glass will always be fractal so that they do not become smoother at long times, although the fraction of the system they occupy will decrease as 1/R(t) d-as. Second, and more cru-

cially, it is not possible to determine locally whether or not a domain wall is present unless the equilibrium states are somehow known in advance!

In random ferromagnets, the domain walls have a

finite energy per unit area (or length) of the wails. But

in a spin glass, the excess free energy of a section

of scale L of a large wall (i.e. L << R) is of order R ° (L/R) ds which vanishes for large R. Thus the walls

are very ephemeral objects in spin glasses!

Nevertheless, the lack of equilibrium due to the presence of the walls does cause a variety of aging

phenomena. In particular, the response of the system

to an ac magnetic field will (as for random ferromagnets) have a scaling form

Xt(og) o c - ~ ( l n ° J ~ \ In t :/ (29)

(but with a different form and prefactors than Eq. (22)).

Indeed this kind of "activated dynamic scaling" is

ubiquitous in random systems: it is due to the control of the scaling by the ratio of the length scales of the

dominant processes and the logarithmic dependence of

these length scales on the associated time scales [14]. The third and most dramatic difference between the

domain wall evolution in spin glasses and that in fer-

romagnets, is the effects of changes in temperature (or other parameters) on the system. The root cause of

this is the dependence of the equilibrium states of spin

glasses on temperature changes. The entropy of small

droplet fluctuations (which are locally random) can be

thought of as modifying the effective couplings of the

system (after some coarse graining) to modified couplings of temperature T, J/)ff (T). Most of the spins in

the equilibrium states, +Gr, will point in the same di-

rection as the corresponding spin of the ground states of the effective hamiltonian with couplings Ji~ff(T), i.e. for low T

+G{Jiejff (T)} (30) GT{Jij}

Since the effective couplings will change with temperature, and ground states are very sensitive to changes in couplings (e.g. by a droplet energy becoming neg-

ative), the states GT and GT+ST will differ dramati- cally even for a small temperature change ST!

On small scales, the relative orientation of nearby spins will, because of the small changes in Ji~ff(T), tend to have the same relative orientation in Gr+~r as in GT. But on scales larger than an "overlap


length", lST, the relative orientations at even two

nearby temperatures will become uncorrelated. This

can be parametrized by the behavior of the averaged

(over randomness) overlap correlation function.

(cri~rj)T(tYicrj)T+3T ~ 0 for li -- Jl >> I3T, (31)

with, for small 3T, the overlap length

1 l ~T 3 T ( d , / 2 _ o ) . (32)

How does this temperature dependence arise? Phys-

ically, it is quite simple: the states GT+3T and GT+3T

can be considered to be made up of a patchwork of do-

mains of Gr and GT separated by what-would-have-

small ( ~r )

(a)

been domain walls at temperature T, but at T + 3T

just represent the locally optimal configuration.

The hypersensitivity to temperature changes has

striking effects on evolution of order. If the system is

first aged at temperature T for a time t, there will be

domains of GT and GT of size R(t), and concomi-

tant excess free energy. A small change in 6T after

time t, yields l~T >> R(t); this will result in only

large-scale changes in GT+aT and hence only a few

altered, and a few extra, domain walls in the system

measured relative to the new equilibrium GT+6T and

GT+~T, as shown in Fig. 10(a). The evolution can

then continue to proceed without much modification.

But if a larger temperature change is made, so that

lrT << R(t), the system will revert to a situation, rela-

tive to the new equilibrium at T + 6T, with a network

of domain walls of scale l~T as shown in Fig. 10(b).

There is now once again substantial excess free en-

ergy, and the system will have to proceed essentially

from scratch: its evolution towards equilibrium at the

old temperature, T, having been futile!

Predictions for the effects of temperature changes -

with both positive and negative 3T - on spin glasses

have been made [4] and generally confirmed by exper-

iments, indicating the existence - at least indirectly -

of a 3T-dependent overlap length [15].

(b)

Fig. 10. Schematic effects of a temperature change on the spin glass configuration of Fig. 9. Domain walls between the new equilibrium states GT+,~T and -GT+&T are indicated with the thin solid lines, while those between the old states are shown dashed. Where these coincide, they are indicated by thick lines; (a) corresponds to a small 3T and large overlap length l~T while (b) corresponds to a large 3T and hence small overlap length l?~ T .

8. Conclusions

We have seen that much of the long-time behav-

ior of simple large systems with local interactions and

ordered phases can be understood in terms of do-

main walls. Many features are quite general: conver-

gence towards local equilibrium; behavior dominated

by fewer degrees of freedom on long time scales asso-

ciated with coarse graining of the spatial structure; a

rich phase space geometry that is accessible even for

simple ferromagnets; and various history dependent

"aging" phenomena as a system equilibrates.

Quenched randomness generally causes very many energy minima at zero temperature separated by energy barriers. At positive temperature, the barriers can

be overcome. But in an ordered phase the barriers grow

as powers of the length scale of the process yielding


a relationship between characteristic time and length

scales

In t ~ L ~. (33)

The concomitant broad distribution of barriers on long length scales gives strong separation, at long time

scales, between processes that are in equilibrium, and

those that are essentially frozen, at that time scale.

These features - and others - are in fact ubiquitous for

random statistical mechanical systems controlled by

zero temperature renormalization group fixed points, just as power law scaling is ubiquitous for conven-

tional critical points.

All these features can occur in simple systems

like random ferromagnets that do not have frus-

trated ground states. But frustration, as occurs in spin glasses, leads to some novel additional physics in-

cluding hypersensitivity of the system to changes in

parameters, and "unobservable" domain walls. The picture of spin glasses that was reviewed here

is linked to the assumption of only one pair of equi-

librium states for spin glasses. It is thus the simplest

possible scenario, but is already very rich. Whether a more complicated phenomenology with many equilib-

rium states can be constructed for spin glass models

with local interactions is an important open question; whether such a picture could bear some resemblance

to the Parisi picture of the infinite range Sherrington- Kirkpatrick model, and what its experimentally ob-

servable consequences would be, are completely

unclear. What is clear, especially from some recent mathematical work [16], is that the "standard" in-

terpretation of the Parisi picture [13] for short-range spin glasses cannot be correct - indeed as we have

given examples of here, the whole set of questions

for systems with local interactions are different. At this point, one should ask what classes of systems

can usefully be studied at low temperatures in terms of domain walls. Systems with discrete broken symmetries, or those in which there are naturally a small set of possible local equilibria (such as near most first- order transitions) can naturally be described this way. More surprisingly, such systems as X Y magnets with random anisotropy or random magnetic fields, and vor- tex lattices in superconductors pinned by impurities,

can also be fruitfully analyzed in terms of domain

walls. Even if there were infinitely many states in spin glasses, general considerations with local interactions

imply that the number of possible states in a region is much less than the number of configurations. There-

fore an effective description on long scales should in-

volve some thinning out of degrees of freedom and

quite likely domain-wall-like concepts will be needed

to understand (at the least) the non-equilibrium phenomena such as those that dominate experiments. Cur-

rently, the theoretical tools for studying these types

of systems are still very limited. But I hope that the

power of simple scaling arguments and a qualitative

renormalization group framework has been illustrated by some of the results in this talk.

In the future there are, in addition to the approach-

to-equilibrium issues discussed here, many other

problems that involve interplay between thermal fluc-

tuations, randomness, and non-equilibrium dynamics - the closest to those studied here involving driven

domain walls [17] and hysteresis loops in magnets

[18]. It is hoped that the ideas developed for studying deceptively simple issues in simple model random

systems will be useful for these and other richer and

more interesting phenomena. For the phenomenologi-

cal description used here is indeed very robust: many modifications are possible to include the effects of

additional physics, and - most crucially - the phe-

nomenology contains the seeds of its own failure in

situations in which it breaks down.

Acknowledgements

Much of the author's understanding of these sub-

jects is thanks to interactions with David Huse which are gratefully acknowledged. This work was supported

in part by the National Science Foundation via grants DMR-91006237, DMR-9630064 and Harvard Univer- sity's Materials Science and Engineering Center.

References

[1] D. Sherrington, Physica D (1997), to appear. [2l P. Wolynes, Physica D (1997), to appear.


[3] A.J. Bray, Adv. Phys. 43 (1994) 357. [4] D.S. Fisher and D.A. Huse, Phys. Rev. B 38 (1988) 373

and references thererin. [5] J. Kurchan and L. Laloux, preprint. [6] J.-P. Bouchaud and A. Georges, Phys. Rep. 195 (1990)

127. [7] H. Kesten, Physica A 138 (1986) 299. [8] D.S. Fisher, Phys. Rev. B 51 (1995) 6411. [9] D.S. Fisher and P. LeDoussal, unpublished.

[10] D.A. Huse and D.S. Fisher, Phys. Rev. Lett. 57 (1986) 2203.

[11] J. Villain, Phys. Rev. Lett. 52 (1984) 1543.

[12] D.S. Fisher and D.A. Huse, J. Phys. Lett. A 20 (1987) LI005; Phys. Rev. B 38 (1988) 386.

[13] K. Binder and A.P. Young, Rev. Mod. Phys. 58 (1986) 801.

[14] D.S. Fisher, J. Appl. Phys. 61 (1987) 3672. [15] J.O. Andersson, J. Mattsson and P. Nordblad, Phys. Rev.

B 48 (1993) 13977. [16] C.M. Newman and D.L. Stein, preprint Condmat 96 03 001

and references therein. [17] O. Narayan and D.S. Fisher, Phys. Rev. B 48 (1993) 7030. [18] K. Dahmen and J.P. Sethna, Phys. Rev. Lett. 71 (1993)

3222.

dynamics and domain walls: is the “landscape paradigm” instructive?

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