#  Nobel Lecture: Multiple equilibria

Giorgio Parisi<EMAIL_ADDRESS>Dipartimento di Fisica, Università di
Roma La Sapienza,
INFN, Sezione di Roma I, CNR-NANOTEC UOS Roma
Piazzale A. Moro 2, I-00185 Roma, Italy

###### Abstract

This is an extended version of my Nobel Lecture, delivered on December $8$
2021\. I will recall the genesis of the concept of multiple equilibria in
natural sciences. I will then describe my contribution to the development of
this concept in the framework of statistical mechanics. Finally, I will
briefly mention the cornucopia of applications of these ideas both in physics
and in other disciplines.

##  I The interplay of disorder and fluctuations in physical systems from
atomic to planetary scales

The Nobel Prize in Physics 2021 was awarded for groundbreaking contributions
to our understanding of complex physical systems with one half jointly to
Syukuro Manabe and Klaus Hasselmann for the physical modeling of Earth’s
climate, quantifying variability and reliably predicting global warming and
the other half to Giorgio Parisi for the discovery of the interplay of
disorder and fluctuations in physical systems from atomic to planetary scales.

When I gave my 25 minutes Nobel lecture ”Mulitple Equilibria” I was unable to
cover all the work that I have done on systems from atomic to planetary scale.
This impossibility persists in this written version, also because I would like
to avoid adding too much material to the oral version. For this reason, I will
not mention many very interesting subjects:

  * •

Random field Ising model and dimensional reduction.

  * •

Intermittency in turbulence and multifractals.

  * •

Stochastic interfacial motion (i.e. the Kardar-Parisi-Zhang equation).

  * •

Stochastic quantization.

  * •

The flight of starlings.

Other very important related topics I will briefly mention are:

  * •

Stochastic resonance.

  * •

Non-equilibrium fluctuations.

  * •

Granular matter (hard spheres).

  * •

Random Laser.

  * •

Theoretical aspects of finite-dimensional spin glasses.

  * •

Large scale simulations of spin glasses.

  * •

Optimization theory, constraint satisfaction problems: 3-SAT, coloring, etc.

  * •

Neural networks.

A review of various developments of these ideas is contained in the one
hundred authors’ book Spin Glass Theory and Far Beyond – Replica Symmetry
Breaking after 40 Years THEBOOK .

##  II The antefact

In 1972 Niles Eldredge and Stephen Jay Gould proposed the evolutionary theory
of punctuated equilibria ELDGOU1972 . In a nutshell the evolution of the
species is not a continuous gradual process: there are long periods of stasis
with practically no changes in the morphology and these periods are punctuated
by rare bursts of evolutionary change. The stability of a system composed of
many random components started to be an interesting mathematical issue MAY  in
the same years.

Many other different systems have long periods of equilibrium separated by
fast transitions to a new equilibrium point. This may happen for example in
ecosystems, in climate (glaciations), geological eras, and so on. In the
original stochastic resonance model for glaciations, the climate has two
equilibrium states.

Generally speaking, we could say that a complex system can stay in many
different equilibrium states, while a simple system may stay only in one or a
few equilibrium states. For example, an animal like a dog can do many
different actions (e.g. play, sleep, eat, hunt …); an animal can switch from
one state to another state in a very short time as the effect of a small
perturbation, e.g. suddenly waking up after hearing a suspicious noise.

Similar considerations can be done in Hebb’s theory HEBB  of memory (1949)
which was modelized with great success first by Little LITTLE  (1974) and
finally by Hopfield HOPFIELD  (1982) with his very successful theory of
associative neural networks. The part of the brain that is responsible for
memory may stay in an extremely large number of different equilibrium states
(attractors AMIT ), each corresponding to the recalling of a different item:
it may remain in that situation for a long period, switching from one memory
to the other as an effect of an external perturbation. The extremely large
number of items that we can memorize and recall is related to the extremely
large number of possible equilibrium states.

Similar ideas have been put forward by Goldstein GOLDSTEIN  in 1969 to
understand the physics of standard structural glasses (like window glass): as
explained in CavagnaPedestrian  Goldstein’s idea is that at low enough
temperatures, a supercooled liquid explores the phase space mainly through
activated jumps between different amorphous minima, separated by potential
energy barriers.

I notice en passant that ”multiple equilibria” is a well-known concept to
economists: for example in 1970 the Nobel Laureate Gerard Debreu wrote the
paper Economies with a Finite Set of Equilibria DEBREU . The title of the
Nobel lecture in the economics of 2022 given by Philip Dybvig was ”Multiple
equilibria”: the same title in two consecutive years but in two different
disciplines.

##  III Spin glasses: the starting point

The concept of systems with many equilibrium points was floating in different
parts of science in the seventies, including physics. However, it was not
clear how to attach the problem using the standard tools of physics. The
systems were rather complex and so different from the others; people working
in glasses did not know the punctuated equilibra theory; moreover, such a
piece of knowledge would be useless: it would have been too difficult to
construct an appropriate model.

To make progress physicists need to find the simplest problem, model it, and
understand the properties of the model in mathematical terms: only at a later
stage, they can use the acquired knowledge as a trampoline for other problems.

Things started to change unexpectedly in 1971. After a long series of
experiments, it was proved Cannella  that an alloy of gold with a few percent
of iron undergoes a phase transition: at a very small magnetic field, the
magnetic susceptibility had a sharp pick, indicating a transition to some kind
of antiferromagnetic order (see fig. (1)).

Figure 1:  The magnetic susceptibility of spin glasses: the results for Au-Fe
alloys in the region of low Fe concentration, from Cannella .

How do these materials behave? In standard ferromagnetic systems, a pair of
spins decrease its energy when the two spins point in the same direction: this
is what happens when two iron atoms are in contact.

In the spin glass case, the iron atoms are in random positions inside the gold
matrix: at a low percentage of iron, they are not in contact and the
interaction is mediated by the presence of the gold atoms. However, the sign
of the interactions depends on the distance of the atoms of iron: sometimes
the interaction wants to put a pair of spins in the same direction, and
sometimes in an opposite direction.

A triplet of spins is said to be frustrated if we cannot satisfy all the
requests: for example, if spin $A$ want to be in the same direction as spin
$B$, spin $B$ want to be in the same direction as spin $C$, and spin $C$ want
to be in the opposite direction of spin $A$, we cannot satisfy all the
requirements simultaneously because they are not compatible.

In the same way, if John and Fred want to sit at the same table, Fred and Bob
want to sit at the same table, but John and Bob hate to sit at the same table
MPV ; PaBook2  we have frustration as stressed by Toulouse Tou77  (and in real
life a lot of discussions). Of course, frustration, i.e. non-compatible
requirements, is ubiquitous in the real world, from Shakespeare’s tragedies to
the problem of putting pieces of baggage in the trunk. Disorder and
frustration are essential features of spin glasses.

Theorists started to work and models were constructed. Edwards and Anderson
(1975) EA  arrived at the first simple model: the spins ($\sigma_{i}$) are
located on a $D$-dimensional lattice (the index $i$ denotes the lattice
point): their number $N$ is equal to $L^{D}$, $L$ being the side of a
$D$-dimensional cube. In the simplest versions, the spins $\sigma$’s are Ising
variables (they can take the values $\pm 1$).

The Hamiltonian of the system is

$H_{J}[\\{\sigma\\}]=-\frac{1}{2}\sum_{i,k=1,N}J_{i,k}\sigma_{i}\sigma_{k}+h\sum_{i}\sigma_{i}\,,$
(1)

where the couplings $J_{i,k}$ are different from zero only if the point $i$
and $k$ are nearest neighbors: they are independent random variables [Gaussian
distributed or binary ($\pm 1$)] and $h$ is a constant magnetic field.

The paper is remarkable for very important progress:

  * •

We expect that at low-temperature spin glasses develop a spontaneous
magnetization: $m_{i}\equiv\langle\sigma_{i}\rangle$, where
$\langle\cdot\rangle$ denotes the thermal expectation value. However, each
sample is characterized by different $J$’s and it will have different values
of the $m_{i}$. We face the problem of studying the statistical behavior of an
ensemble of different systems and new ideas were needed to reach this goal.

Their great idea was to consider as an order parameter the quantity
$q_{EA}\equiv\mbox{Av}(m_{i}^{2})$, where Av denotes the average over the
system [i.e. $\mbox{Av}(m_{i}^{2})\equiv N^{-1}\sum_{i=1,N}m_{i}^{2}]$.

In principle, $q_{EA}$ could depend on $J$ but this possibility was correctly
disregarded. At zero magnetic field ($h=0$), the order parameter $q_{EA}$ is
zero at temperatures above the critical temperature and it becomes positive at
temperatures below the critical temperature. Naively the magnetic
susceptibility $\chi$ is given by $\chi=\beta(1-q_{EA})$, so that the change
of behavior in the magnetic susceptibility at the critical point can be
interpreted as the appearance of a non-zero $q_{EA}$.

  * •

We face the problem of computing the average free energy $F$. Let me write
down a few definitions:

$\displaystyle
Z_{J}(\beta,N)=2^{-N}\sum_{\\{\sigma\\}}\exp\left(-H_{J}[\\{\sigma\\}]\right)\,,$
$\displaystyle
F(\beta)=-\lim_{N\to\infty}\frac{\overline{\log(Z_{J}(\beta,N))}}{\beta N}\,,$
(2)

where the overline denotes the average over the $J$. To lighten the notation
in the text we will not indicate the obvious dependence on $N$ and $\beta$ of
these two quantities.

The average over the $J$ is rather complex: however, it can be done in a
simple way using an outrageous trick.

We first notice that $Z_{J}^{n}$ is the product of $n$ factors $Z_{J}$. In
this way $Z_{J}^{n}$ is the partition function of a system with $nN$ spins:
each system of $N$ spins is replicated $n$ times. Using this representation we
can do the average over $J$ and we get formulae that involve the integral over
$n(n-1)/2$ variables.

The second step is to use the formula

$\overline{\log(Z_{J}(\beta,N))}=\lim_{n\to
0}\overline{\frac{Z_{J}(\beta,N)^{n}-1}{n}}\,.$ (3)

It is clear that we are cheating: at the beginning, the number of factors
($n$) must be an integer (half a factor or $\pi$ factors do not make sense)
and in the second step $n$ is a real number. However, physicists often do
illegal manipulations, hoping that in the end, the results remain correct.

This approach is called the replica method (or the replica trick) because we
consider the partition functions of $n$ identical replicas of the same
systems.

Equivalently we can also define

$F_{n}(\beta)=-\lim_{N\to\infty}\frac{\overline{Z_{J}(\beta,N)^{n}}}{\beta
nN}\,,\quad F(\beta)=\lim_{n\to 0}F_{n}(\beta)\,.$ (4)

Once the free energy $F_{n}$ is computed for integer $n$ (in the infinite $N$
limit), we analytically continue $F_{n}$ to $n=0$ in order to get the average
free energy $F$.

The function $F_{n}$ is well defined for non-integer $n$, however, this naive
approach fails if the function $F_{n}$ is not analytic in $n$ and it has a
singularity (i.e. a point of non-analyticity) for non-integer $n$ (e.g. for
$0<n<1$). Such a singularity can be seen only in the infinite $N$ limit.

A subsequent simplification was taken by Sherrington and Kirkpatrick SK : the
Hamiltonian has the same form as before with the difference that all pairs of
the spins have a direct interaction; the average of the couplings is given by
$\overline{J^{2}}=1/N$ where $N$ is the total number of spins.

According to the folklore in statistical mechanics, when all the components
have a direct infinitesimal interaction in the limit $N\to\infty$, each
component feels only the collective behavior of the system: physical space is
no more present and the correlations have an infinite range. Such a model
should be soluble, in the sense that one can arrive at a closed set of
equations in the infinite volume limit. However, the solution is correct only
if we have correctly identified the collective behavior of the system.

The solution of the infinite range model should provide the useful mean field
theory of the model: the large dimension limit of the lattice-defined theory
should be described by this mean-field theory, as has been proved in many
cases.

The paper SK  was a turning point: the goal became to obtain the solution of
the SK model. The title Solvable model of a spin-glass was correct, the model
is actually solvable, but the paper did not contain the correct solution
because, as stressed by the authors, the reported solution of the model
produced a negative value of the entropy at low temperature: it was an
internal contradiction because the entropy by definition is non-negative. Some
subtle mistake was made.

The derivation of SK was simple. After some computations based on Gaussian
integrations, it was possible to write exactly for a system of $N$ spins:

$\overline{Z_{J}(\beta,N)^{n}}=\int dQ\exp(-N\beta
F_{n}(Q))\approx\exp\left(-N\beta F_{n}(Q_{n}^{*})\right),$ (5)

where $Q$ denotes a symmetric $n\times n$ matrix that has diagonal elements
equal to zero; the integral runs over all these matrices. The function
$F_{n}(Q)$ has an explicit form that it is not worthwhile to write here and
$Q_{n}^{*}$ is the value of $Q$ that minimizes $F_{n}(Q)$. The simplest option
for finding candidates for a minimum is to look for the solutions of the
stationary point equation $\partial F_{n}(Q)/\partial Q=0$;

Now Sherrington and Kirkpatrick worked directly at $n=0$; they made also an
extra innocent-looking assumption

$Q_{a,b}^{*}=q,\ \mbox{for}\ a\neq b\,.$ (6)

This assumption was quite natural: both the integrand and the integral in eq.
(5) are invariant under the reshuffling of the $n$ indices (i.e. the action of
the permutation group of $n$ elements, $S_{n}$, called also the replica
group).

In a nutshell this symmetry stems from the commutativity of the
multiplication: when we write $Z_{J}^{n}$ as the product of $n$ factor
$Z_{J}$, we attach a label $a$ ($a=1,n$) to each of these factors, but the
order in which we perform the multiplication is irrelevant. The matrix $Q^{*}$
in eq. (6) is the only one that is symmetric under the action of the replica
group, so it is the obvious choice.

After two years De Almeida and Thouless proved DEATHO  that this innocent-
looking assumption was wrong: the value of $Q^{*}$ chosen by SK was not a
minimum; more precisely the replica symmetric matrix of eq. (6) is a minimum
for $n>n_{c}$, but it no longer a minimum for $n<n_{c}$, with $0<n_{c}<1$. So
the worst-case scenario of the non-analyticity of the function $F_{n}$ was
realized.

The reader should notice that the precise definition of minimum is unclear in
the context. However, everybody agreed that a stationary point of the function
$F$ (i.e. a point such that $\partial F/\partial Q_{a,b}=0$) is a candidate to
be a minimum only if the Hessian matrix of second derivatives

${\cal H}_{a,b;c,d}=\frac{\partial^{2}F}{\partial Q_{a,b}\partial Q_{c,d}}$
(7)

is non-negative, (i.e. its eigenvalues are non-negative). This statement
generalizes the well-known one-dimensional result that if
$f^{\prime}(x^{*})=0$ and $f^{\prime\prime}(x^{*})<0$ the function $f(x)$ has
a maximum at $x=x^{*}$, certainly not a minimum.

The subtle mistake was found by de Almeida and Thouless, but, unfortunately,
these authors were not able to suggest which form could have the minimum; it
was clear the saddle point matrix $Q^{*}$ could not be left invariant by the
action of the replica group: the replica symmetry should be spontaneously
broken, but the space of non-symmetric matrices is very very large.

A natural possibility was that the correct saddle-point matrix $Q^{*}$ is
symmetric under a subgroup of the replica group $S_{n}$, but also the number
of different subgroups is very large, especially in the slippery limit $n\to
0$. Another possibility was to abandon the replica method but it was unclear
what method to use in place of the replica method.

A few attempts to find the correct minimum were done, but they were
unsuccessfully 13 ; 14 . Important progress on the physics of the problem was
made in the TAP paper (Touless, Anderson, and Palmer TAP ), where it was shown
that the low-temperature behavior of the model was very different from that of
SK. More results in this direction were contained in the very influential Les
Houches lecture notes by P.W Anderson ILL .

##  IV The exact solution of the model.

The history of the solution of the model and the consequent discovery of
replica symmetry breaking is quite complex. The interested reader can find a
detailed exposition in the monumental and choral work DZ ; HIST . Here I will
expose mainly the history from my personal viewpoint, which is exposed at
length in my interview in DZ .

My interest in the problem started at the end of 1978. I was working with
Sourlas and Drouffe on lattice QCD in high dimensions DPS . In our formalism,
the corrections to mean field theory for lattice gauge theories were related
to the properties of interacting branched polymers.

Looking at the literature in the library, I encountered the replica method in
the study of branched polymers LUB  and I learned that the replica method was
giving incorrect results in the case of the SK model. I immediately wanted to
understand why. I started to study, to look for a different form of the matrix
$Q^{*}$, but none were satisfactory.

After many trials I had an intuition: in other papers, 13 ; 14 , the $n$
indices were divided into $n/m$ groups of $m$ elements each and the value of
the matrix elements of $Q$ depends on the group. Everybody was assuming that
$m$ was an integer: Blandin and collaborators 13  were using $m=2$; De
Dominicis and Garel DG  generalized this approach to a generic integer $m$ and
considered also the $m\to\infty$ limit: in this limit, the entropy problem was
solved, but other low-temperature properties were not in agreement with the
simulations either with the theoretical analysis of TAP.

I made the bold assumption that $m$ could be a non-integer number, more
precisely a number in the interval [0-1]. For example, I was dividing the $n$
replicas into $2n$ groups of $1/2$ replicas each. Of course, that is crazy,
but my viewpoint was I should first check if this crazy idea was leading to
correct results and postpone other questions to a later stage. It was clear to
me that also non-integer angular momentum is a crazy idea, but this idea led
to Regge poles.

This approximation is what is now called one-step replica symmetry breaking.
Everything was nearly perfect. The theory predicted values of the specific
heat and the energy in agreement with the numerical simulations. The entropy
remained a problem, but instead of being $-0.16$ at zero temperature, it was
$-0.01$: still negative but quite near zero. I wrote the results in a short
note GP79  that I submitted to Physics Letters A.

At the end of the paper, I added the observation that one could improve the
theory by dividing the $n/m$ groups into $(n/m)/m_{1}$ groups of replicas,
where $m_{1}$ was a new variational parameter. I was also conjecturing the
correct solution was obtained when the procedure was repeated an infinite
number of times.

Some fancy group theory arguments also were added, arguing the permutation
group of zero objects is an infinite group because it contains itself as a
proper subgroup. Indeed the subgroup of $S_{n}$ that does not change the
matrix $Q^{*}$ is the semidirect product of the permutation group of $n/m$
elements $S_{n/m}$) with the direct product of $n/m$ copies of the permutation
group of $m$ elements ($S_{m}$). For $n=0$ also $n/m=0$, so that $S_{0}$
contains as a subgroup the semi-direct product of $S_{0}$ with the direct
product of zero factors $S_{m}$. This sentence looks like a nonsensical text
written by modern AI.

The response of the referee was remarkable. In a nutshell: The approach does
not make sense, but the numbers coming from the formulae are reasonable, so it
can be published. The last observation is not worth the paper on which it is
written and it should be removed.

I laughed because in the meanwhile I extended the computation to an infinite
number of subdivisions: I found that I could associate a continuous function
$q(x)$ (where $x$ belongs to the interval $[0-1]$ to the matrix $Q$ when one
repeats the process of symmetry breaking an infinite number of times. In other
words, I arrived at the formula

$F=\max_{q(x)}F[q(x)]\,,$ (8)

where $F[q(x)]$ is a functional of the function $q(x)$. I first computed it
near the critical temperature GP79b  and finally at all temperature GP79c .

At the end of 1979, I got the expression for the free energy of the model. At
zero magnetic field, we have

$F[q(x)]=-\frac{\beta}{4}\left[1+\int_{0}^{1}\,dx\,q^{2}(x)-2q(1)\right]-\frac{1}{\beta}f(0,0)\,,$
(9)

where $f(x,h)$ is an auxiliary function defined in the strip $0\leq x\leq 1$
and it is the solution to the non-linear antiparabolic equation

$\frac{\partial f(x,h)}{\partial x}=-\frac{1}{2}\frac{{\rm d}q}{{\rm
d}x}\left[\frac{\partial^{2}f}{\partial h^{2}}+x\left(\frac{\partial
f}{\partial h}\right)^{2}\right]\,,$ (10)

with the initial condition $f(1,h)=\ln(2\cosh(\beta h))$. The zero temperature
entropy was zero, as it should be.

It is remarkable that in the final formula, the craziness disappeared: we come
back to respectable mathematics with one big surprise: in all known mean field
theory the order parameter was a number, here it is a function MPV ;
parisibook2 .

But what is the physical meaning of the function $q(x)$? How could we derive
the relatively simple final formulas without using the nonsensical mathematics
described before? Most importantly are the final formula for the free energy
correct?

At that time there was no suggestion of how this result could be rigorously
proved: the very existence of the free energy density in the large $N$ limit
was lacking rigorous proof. The difficulty was to prove that all the random
systems have the same free energy density when $N$ goes to infinity: we needed
a clever generalization of the law of large numbers.

There were some hints on the meaning of having a non-constant $q(x)$. If
$q(x)$ were constant, i.e. $q(x)=q$, we would recover the original SK
solution. By an ingenious trick (adding an infinitesimal term that breaks the
replica symmetry) in 13  it was argued that when replica symmetry was broken,
$q_{EA}$ is given by the largest matrix element of the matrix $Q$. In this
context, this result leads to:

$q_{EA}=\max_{x}q(x)\,,\quad\chi=\beta\int_{0}^{1}dx(1-q(x))\,.$ (11)

Unfortunately, a back-of-the-envelope calculation gave $\chi=\beta(1-q_{EA})$
which was not the right result. These formulas and this discrepancy were
suggesting something, but what?

Despite this hint, I had no sound idea of the meaning of $q(x)$ and I was
unable to make further advances. At the end of 1979, I started to study other
problems, waiting for inspiration for further progress.

The belief in the correctness of the approach was strongly reinforced by two
papers. In the first paper TAK  it was shown that near the critical
temperature, the values of negative eigenvalues of the Hessian in the one-step
formulation were reduced by a factor of 9 indicating that one was moving in
the correct direction. Finally in a later and conclusive paper DK0  it was
shown that in the formulation with an infinite number of replica symmetry
breaking, the negative eigenvalues of the Hessian are no more present: one
remains with a rather intricate structure of zero modes and nearly zero modes.

##  V The discovery of complexity, with the help of my friends

In the summer of 1982, I lectured at the Les Houches summer school on spin
glasses LH  and in the fall I went to Paris, at the IHES, for a two-month
visit. I had many discussions about spin glasses which was a hot subject: and
a lot of progress was made. People had already realized that spin glasses
could be in many stationary states Bray ; DGGO , most of them having higher
free energy (i.e. metastable equilibrium states). However, the connection of
this fact with the replica symmetry-breaking solution described above was not
clear DY .

In the end, I found that the replica symmetry-breaking solution implies that a
spin glass may have many equilibrium states GP83 , labeled by an index
$\alpha$ ($\alpha=0,1\cdots\infty$): each of these states appears at
equilibrium with a probability $w_{\alpha}$.

Each state is characterized by its own magnetizations $m_{i}^{\alpha}$. One
can define the matrix of overlaps

${\cal
Q}_{\alpha,\gamma}=\frac{\sum_{i}m_{i}^{\alpha}m_{i}^{\gamma}}{N}\equiv\mbox{Av}(m_{i}^{\alpha}m_{i}^{\gamma})\,.$
(12)

The Edwards Anderson order parameter ($q_{EA}$) is given by ${\cal
Q}_{\alpha,\alpha}$.

Each sample has its own matrix ${\cal Q}$ and its own weights $w$. In other
words for each sample, we could define a descriptor ${\cal D}_{J}$ that is
formed by the matrix ${\cal Q}$ and the weighs $w$’s.

To describe the connection with the function $q(x)$, it is convenient to
consider the probability of finding two states with overlap $q$ in a given
sample: for very large, but finite, $N$, it is given by

$P_{J}(q)=\sum_{\alpha,\gamma}w_{\alpha}w_{\gamma}\delta\left({\cal
Q}_{\alpha,\gamma}-q\right)\,.$ (13)

The function $P_{J}(q)$ depends on the system. The simplest quantity to
consider is its average over the samples:

$P(q)=\overline{P_{J}(q)}.$ (14)

The connection with my theory was surprising: if we define

$x(q)=\int_{0}^{q}dq^{\prime}P(q^{\prime})\,,$ (15)

one could prove that $x(q)$ is the inverse of the function $q(x)$ that I
introduced in the solution using replicas.

The interpretation of $q(x)$ was now clear. The quantity $x$ is a probability:
indeed it is the probability that two replicas of the system have an overlap
less than or equal to $q$. This probabilistic interpretation of $x$ explains
why $x$ is in the range from zero to one.

The study of the statistical properties of the function $P_{J}(q)$ and of the
descriptor ${\cal D}_{J}$ were done together with very good friends of mine 5a
; 5b . The results were so unexpected that when I saw the first results of our
computations, my first thought was my theory must be wrong if it produces such
nonsense.

After a lot of work, we realized that the predictions made a lot of sense:

  * •

The probability distribution of the overlap changes from system to system also
in the infinite volume limit. This is quite amazing because usually intensive
quantities do not fluctuate. However, the self overlaps ${\cal
Q}_{\alpha,\alpha}$ do not depend on $\alpha$ and on the system. It is
possible to show that ${\cal Q}_{\alpha,\alpha}=q_{EA}$.

  * •

The probability distribution of the $w$’s follows simple mathematical rules
that I will not describe here.

  * •

The number of equilibrium states is infinite. This was the first case of an
infinitely numerable number of different equilibrium states. This is a quite
strange situation because it violates the Gibbs rule.

  * •

The most surprising result was that in presence of an arbitrarily small non-
zero magnetic field, the states satisfied the ultrametricity property

$d_{\alpha,\gamma}\leq\min(d_{\alpha,\beta},d_{\beta,\gamma})\quad\forall\beta\,,$
(16)

where the distance between two states is defined in a natural way:

$d_{\alpha,\gamma}^{2}={\cal Q}_{\alpha,\alpha}+{\cal
Q}_{\gamma,\gamma}-2{\cal
Q}_{\alpha,\gamma}=\mbox{Av}\left(m_{i}^{\alpha}-m_{i}^{\gamma}\right)^{2}\,.$
(17)

Without entering into the mathematical details, ultrametricity implies that
the states can be assigned to the leaves of a tree and that the distance
between states is proportional to the height you have to climb the tree for
going from one leaf to another leaf.

An explicit probabilistic construction of this tree is given in PRY , while a
rigorous construction of the tree was done in Ruelle . In fig. (2) I show an
example, where only a finite number of the branches are depicted, i.e. those
with higher probability.

A model that by inspection has the same distribution of weights of the one-
step replica broken case is the Random Energy Model of Derrida REM  that
generalizes to the GREM model for more than one-step replica symmetry breaking
GREM .

Ultrametricity and taxonomy are essentially related: a standard taxonomy, i.e.
a hierarchical classification, is possible only if the relevant properties
have an ultrametric structure. In the standard taxonomic classification of
living beings, the distance is related to the history of evolution. In spin
glasses, the taxonomy is intrinsic to the static equilibrium properties of the
system and it is not related to evolution in time.

Figure 2: An example of the tree of states in the case where the overlap takes
only three possible values: taken from PRY . The reader should notice that the
tree is upside down: the root is at the top and the leaves are at the bottom.

In spin glasses and other systems, where the function $q(x)$ contains a
continuous part with a non-zero derivative, the function $P(q)$ is non-zero in
an interval. This situation is called continuous replica symmetry breaking.
Here the tree of the states is branching at any level: in any height interval,
there is an infinite number of branches. It is an extremely complex
construction that appeared unexpectedly. The presence of such construction for
the probability of the descriptors ${\cal P}({\cal D})$ reflects the extreme
complexity of the rugged landscape of spin glasses.

In other cases the function $P(q)$ is the sum of two or more delta functions:
for example, if the function $P(q)$ contains only two delta functions, we have
the so-called one-step replica symmetry breaking, which was the first case we
considered. When there are three deltas, we have two steps replica symmetry
breaking (a very rare case). Finally, when only one delta function is present
we are in the usual case where there is no replica symmetry breaking.

The crazy form of the $Q$ matrix is equivalent to this extremely complex
probability distribution of the descriptor. Indeed it was shown MPVART  that
the formula for the free energy can be derived using probabilistic arguments
starting from the assumption that the probability of the descriptors is just
the one given by the replica formula.

##  VI Mathematicians come to the rescue

At this stage, at the beginning of the nineties, we faced an impasse.
Numerical simulations for the SK model were in agreement with the predictions
of the replica symmetry broken approach BY . However it was possible that the
correct form of the matrix $Q$ was different and more complex, and the correct
form of the probability of the descriptors [${\cal P}({\cal D})$] was much
more intricate. It was difficult to conclude in a definite way.

A possible approach could be to consider a free energy that depends on ${\cal
P}({\cal D})$, i.e. ${\cal F}[{\cal P}]$. We could conjecture that the correct
free energy is given by

$F=\max_{\cal P}F[{\cal P}]\,.$ (18)

Unfortunately, there was no idea of how to prove the previous formula. Why
”$\max$” and not the usual ”$\min$”? How to justify this strange choice?

Moreover, the space of the descriptors is infinite-dimensional and the space
of possible functions on an infinite-dimensional space is extremely large. How
can we explore such a big space? The very complicated ultrametric ${\cal
P}({\cal D})$ of the replica approach is just the simplest example of a very
large family of distributions.

I was very skeptical about the possibility of finding proof that all the
assumptions done were correct. Fortunately, the situation changed rapidly.

In 2002 Guerra GT  introduced a new mathematical technique (Guerra
interpolation) that allowed him to prove that $F\geq F_{R}$, where $F_{R}$ is
the free energy computed via the replica formula GuerraI , vindicating the use
of ”$\max$”. A few months later Talagrand Tala  beautifully twisted Guerra’s
argument and he proved that $F\leq F_{R}$, arriving at the conjectured result
$F=F_{R}$.

Unfortunately, this sophisticated proof did not have direct implications on
the form of ${\cal P}({\cal D})$. The problem was solved differently. It was
shown that equation (18) has a precise mathematical meaning ASS . Moreover
Ghirlanda and Guerra GuerraSS  proved some identities (the GG identities) that
restricted very strongly the form of ${\cal P}({\cal D})$ (see also ACON ).

In a longly-awaited paper after many partial results, Panchenko PANU  was able
to show that the GG identities implied ultrametricity. There is only one
possible form of ${\cal P}({\cal D})$ that satisfies the GG identities and
this form is parametrized as a functional of the function $q(x)$. Using the
Panchenko results one obtains directly to replica formula after some nowadays
standard computations. This second proof PanBook  is more connected to
physical intuitions.

Unfortunately at the present moment, nobody is able to transform the original
approach with replicas into rigorous proof. There are some ideas on a possible
approach CPV ; PBOOK , however, it is still not easy to transform these half-
baked ideas into a mathematical proof. It seems to me PBOOK  that there must
be some relevant deep mathematical theorems that have not yet been proven.

In the nutshell, the work of the mathematicians has been fundamental to
clarifying the physical hypothesis at the basis of the replica approach and to
a any possible doubts about the correctness of the results of this approach.

##  VII Marginal Stability

We have seen that the existence of multiple equilibrium states is the
distinctive hallmark of glassiness MPV ; parisibook2 . While at finite
temperatures these equilibrium states may be metastable (or maybe not,
depending on the system), the issue of metastability becomes irrelevant in the
zero temperature limit.

A crucial question is how these states differ one from the other and how they
are distributed in the phase space (the phase space becomes an infinite
dimensional space in the thermodynamic limit). Moreover, in this situation a
small perturbation produces a large rearrangement of the relative free energy
of the equilibrium states: after a perturbation, the lowest equilibrium state
may be quite different from the previous one. The linear response theorem has
to be modified to take care of the existence of many equilibrium states.

Two main possible scenarios have very different properties as discussed for
example in REVISITED .

  * •

Different equilibrium states are scattered in phase space in a nearly random
fashion: they stay at a fixed non-zero distance from the others CGP1 . In this
situation, if you live inside one equilibrium state, you do not feel the
existence of other equilibrium states: the barriers are very high MPV . We can
call this scenario the stable glass: it corresponds to one-step replica
symmetry breaking. In mean field theory the barriers for going from one state
to another are exponentially large in the system size CGP2 ; CGP3 . Beyond
mean field theory they are expected to be exponentially large in the
appropriate parameter.

  * •

The distribution of equilibrium states in phase space is very far from a
random one. Each state is surrounded by a large number of other equilibrium
states that are arbitrarily near to that given state:

$\min_{\gamma\neq\alpha}d_{\alpha,\gamma}=0\,.$ (19)

One could say the equilibrium states form a kind of fractal set. The barriers
for going from one state to another are much smaller than in the previous
case.

If you live in one equilibrium state, you feel that you can move in many
directions (typically in the direction of nearby states) without increasing
too much the free energy: in other words, there are nearly flat directions in
the potential exactly like at a second order transition critical point MPV .

The spectrum of small oscillations within an equilibrium state has an excess
of low-frequency modes that in some cases may dominate the one coming from
other more conventional sources (e.g. phonons or magnons). The precise form of
this anomalous low-frequency spectrum and the localization properties of the
eigenvalues crucially depend on the details of the theory.

These glassy systems are self-organized critical systems. We call this
scenario the marginally stable glass that corresponds to continuous replica
symmetry breaking. (I notice en passant that expression marginal stability was
also used to characterize some off-equilibrium metastable states KWT  with a
slightly different meaning).

For reasons of space, I will consider here only the case of the marginally
stable glass.

A great step forward in the study of marginal stable glasses was taken by De
Dominicis and Kondor DK1 . They computed the matrices elements of the
resolvent of the Hessian ($1/({\cal H}+\lambda)$): in the framework of the
mean field approximation, if we put $\lambda=p^{2}$, we obtain the correlation
functions of the spins in momentum space. These correlations (in position
space) are defined as

$\langle\sigma_{a}(x)\sigma_{b}(x)\sigma_{c}(0)\sigma_{d}(0)\rangle\equiv
C_{a,b;c,d}(x)\,.$ (20)

Unfortunately, they depend on four indices and this makes the whole analysis
quite complex.

In mean field theory, when replica symmetry is broken continuously these
correlations are long-range with a power law decay that has a very complex
structure as a function of the four replica indices. Depending on the replica
indices, in Fourier space we have the following possible singular behavior at
small momentum $p$:

$A_{2}/p^{2}\,,\qquad A_{3}/p^{3}\,,\qquad A_{4}/p^{4}\,,$ (21)

the $A$’s being constants that have been evaluated.

The existence of long-range correlations (although we do not know the
exponents outside mean field theory) is one of the fundamental predictions of
continuous replica symmetry breaking and marginal stability.

Fortunately, there are very clear experimental observations of a large
correlation length in spin glasses ORBACH  that increases with the age of the
system. These experiments have also been reproduced in silico in great detail
JANUSCORR ; JANUSCORR1 ; JANUSCORR2 .

The De Dominicis-Kondor computation (a real tour de force) was done in the
framework of mean field theory, but it underlines a general relation:
continuous replica symmetry breaking implies marginal stability and power-
decaying correlations at large distances. The detailed results should be valid
in high dimensions. They clearly show the existence of long-range correlations
that decay at a large distance with a dimensional-dependent exponent: the
correlation length is infinite.

For not-too-high dimensional spin glasses, the situation is less clear, also
because we have a limited command of the perturbation theory around the mean-
field theory. The computations are technically very demanding: they have never
been done without approximations, but I hope that they will be done in the
future.

I believe that one should better understand Ward’s identities related to
replica symmetry breaking W1 ; W2 . Ward identities are the right tools to
address cancellations due to symmetries. The theoretical situation in a
magnetic field is still worse because we do not know the upper critical
dimension BrayR , i.e. the dimension at which the exponents start to be non-
trivial, although there is a recent suggestion that the upper critical
dimension is 8 D8 .

The sad outcome is that we do not know at which dimensions the spin glass
phase disappears. We have experimental and numerical evidence that there is a
transition at zero magnetic field in three dimensions and that the transition
is absent in two dimensions. It was suggested theoretically that the
transition temperature goes to zero in dimension $D_{c}=2.5$ FPV . Obviously,
we cannot (for the moment) do numerical simulation in non-integer dimensions,
however, we can resort to extrapolation or interpolation of critical
quantities as a function of the dimensions of the space.

The value $D_{c}=2.5$ is in very good agreement with numerical interpolation
of the zero temperature exponents for the Edward Anderson model (the value
$D_{c}=2.4986$ is estimated in Bo ) (a much less precise extrapolation of the
critical temperature PR3  gives $D_{c}\approx 2.65$). Numerical estimations
for the free energy barriers MP3  in dimensions 3 are in good agreement with
the approximation that predicts $D_{c}=2.5$.

When the magnetic field is zero, numerical experiments in dimension 3 or
higher are in very good and detailed agreement with the qualitative
predictions of mean-field theory. For example in finite dimensions marginal
stability predicts the existence of long-range correlations DK1  that are
clearly observed in many simulations (e.g. Janus ) together with many others
predictions CINQUE . One can also measure numerically the correlation
functions that are related to the presence of ultrametricity in dimension 3
MPY .

More than ten years ago it was suggested that also in some two and three-
dimensional structural glasses, in particular, hard-sphere systems at infinite
pressure (i.e., in the jamming limit), are marginally stable, in the
restricted sense that they are unstable under an infinitesimal perturbation
WYART0 ; WYART1 ; WYART2 ; LN .

Following an original idea of Kirkpatrick and Wolynes KW , quite recently PZR
; LARGED1 ; FINAL ; BOOKPUZ , as discussed in detail in the next section, a
mean field model of hard spheres has been constructed and solved in the
infinite-dimensional limit ($D\to\infty$, where $D$ is the dimension of the
space where the spheres move). The model has many features similar to the SK
model: all the stigmata of marginal stability are present here, suggesting
that a similar situation also holds for some finite-dimensional glasses.

##  VIII Some more experimental and numerical confirmations

Although experiments are the ultimate source of confirmation of a theory,
numerical simulations have proven to be a remarkable tool. They are crucial
for studying quantities that are not accessible experimentally and for quickly
disproving wrong theories.

Indeed the core prediction of the replica theory is the existence of multiple
equilibrium states that have the same macroscopic properties but they differ
microscopically. Unfortunately at the present moment, it is impossible to
measure simultaneously the value of a large number of spins in experiments, so
this key property may be directly observed only in simulations.

Another reason for the importance of simulations is the possibility to explore
the behavior of the system in a short time window, that is not accessible to
experiments: in this way, simulations are complementary to experiments.

###  VIII.1 Spin glass susceptibilies

The magnetic susceptibility measures how the magnetization changes by adding a
magnetic field. However, in the low-temperature region, the magnetization
depends on the protocol we use to thermalize the system and to add the
magnetic field (a form of hysteresis): consequently, we can define protocol-
dependent (or history-dependent) magnetization.

A clear prediction of the theory is the existence of two susceptibilities in
two extreme cases:

  * •

When we add a small magnetic field and we force the system to remain in the
same state, we measure the linear response susceptibility that is given by
$\chi_{LR}=\beta(1-q_{EA})$.

  * •

When we add a magnetic field and we allow the system to jump to the
thermodynamically favored state, we measure the thermodynamic susceptibility
that is given by $\chi_{eq}=\beta\int dx(1-q(x))$.

Figure 3: The analytic results of the mean-field approximation for the linear
response susceptibility ($\chi_{LR}$, lower curve) and the field cooled
susceptibility ($\chi_{eq}$, upper curve). They coincide in the high-
temperature region.

Figure 4: The linear response susceptibility ($\chi_{LR}$, lower curve) and
the field cooled susceptibility ($\chi_{eq}$, upper curve). The experimental
results are taken from DJN

.

The two susceptibilities have been measured experimentally in spin glasses and
depicted in figures (3) and (4) (mean-field theory and experiments
respectively).

The experimental protocols we consider are the following:

  * •

The linear response susceptiibly ($\chi_{LR}$) is measured by adding a very
small magnetic field when the system is already at the final low temperature.
This extra field should be small enough to neglect non-linear effects. In this
situation, when we change the magnetic field, the system remains inside a
given state and it is not forced to jump from one state to another state: this
is the ZFC (zero-field cooled) susceptibility, which corresponds to
$\chi_{LR}$.

  * •

The second susceptibility ($\chi_{eq}$) can be approximately measured by
cooling the system in presence of a small magnetic field and comparing the
observed magnetization with the one measured without this small magnetic
field. In this case, the system has the ability to choose the state that is
most appropriate in presence of the applied field. This susceptibility, the
so-called FC (field-cooled) susceptibility, is experimentally nearly
independent from the cooling rate. The quasi-independence of the field-cooled
magnetization on the cooling rate confirms that the field-cooled magnetization
is near to the equilibrium one: it is considered to be a good proxy of
$\chi_{eq}$.

Summarizing, one can identify $\chi_{LR}$ and $\chi_{eq}$ with the ZFC
susceptibility and with the FC susceptibility respectively.

The theoretical plot of the two susceptibilities is shown in fig. (3). As we
discussed above

$\chi_{LR}=\beta(1-q_{EA})\,,\quad\chi_{eq}=\beta\int dx(1-q(x))\,.$ (22)

The experimental plot of the two susceptibilities is shown in fig. (4). They
are equal in the high-temperature phase while they differ in the low-
temperature phase.

The experimental data are in clear qualitative agreement with the theoretical
predictions, especially if we consider that some of the data are also in the
critical region, where the critical exponents are far from those of mean-field
theory. I think that these data provide a quite conclusive argument for the
relevance of mean-field theory for the experiments.

###  VIII.2 Spin glass simulations

How to measure directly $q(x)$ in simulations?

This can be done by measuring the probability distribution of the overlap:

$P(q)=\overline{P_{J}(q)}\,,$ (23)

where the overline denotes the average over the different instances of the
system. To this end, we can introduce two clones of the same system that we
call $\sigma$ and $\tau$.

Let us consider a system of size equal to $L^{D}$, that we can thermalize by
numerical simulations. We can measure the probability distribution of the
clones of the same system. We define the instantaneous overlap of the two
clones as

$q={\sum_{i}\sigma_{i}\tau_{i}\over V}\,.$ (24)

For each sample $J$ we can measure the probability distribution $P_{J}(q)$. We
finally have:

$x_{J}(q)=\int_{0}^{q}dq^{\prime}P_{J}(q^{\prime})\,,\quad\overline{x_{J}(q)}=x(q)\,.$
(25)

In other words, the probability distribution of $q$ for a given system gives
the function $P_{J}(q)$. We must now average the function $P_{J}(q)$ over a
large number of instances of the systems (at least one thousand if we want not
too large errors) to get average probability $P(q)$.

The computations are numerically hard because the computer time need for
thermalization increases very fast with the side of the system $L$. For
example in three dimensions ($D=3$) in the low-temperature phase, the
thermalization time increases like $L^{z}$ with $z\approx 12$ JTIME  if we use
the parallel tempering algorithm. Using standard Monte Carlo simulations the
thermalization time would be exponentially large in $L$ to the appropriate
power .

The signature of replica symmetry breaking is that in the infinite volume
limit the function $P(q)$ is different from two delta functions at $q=\pm
q_{EA}$. In fig. (5) we show the results for the average function $P(q)$ from
the Janus collaboration Janus  in three dimensions for $L=8,16,24,32$: the
plateaux at $q=0$ has no tendency to disappear by increasing $L$. These smooth
curves are the average of $P_{J}(q)$ functions that are very different from
one another (see fig. (6)): each sample has its own $P_{J}(q)$ function. The
sample-to-sample fluctuations of $P_{J}(q)$ follow very well the theoretical
prediction as shown in SamSam . A comparison of finite dimensional numerical
simulations with infinite range model Aspel  was done in Com0 ; Com1 .

Figure 5: The results from the Janus collaboration Janus  for the average
function $P(Q)$ in three dimensions for $L=8,12,16,24,32$ in the low
temperature regions.

Figure 6: Four different examples of the $P_{J}(q)$ function in three
dimensions for $L=32$ corresponding to different samples in the low
temperature region Janus . The average over one thousand different samples
gives the smooth function $P(q)$ of fig. (5).

Unfortunately, I cannot present more results on large-scale simulations of
spin glasses. I have worked on this problem for more than 20 years: I consider
numerical simulations to have the same cognitive value as experiments.
Unfortunately, a detailed presentation of the many results would bring us too
far.

###  VIII.3 Fluctuation-dissipation relations

We can also define and measure experimentally susceptibilities that
interpolate between $\chi_{LR}$ and $\chi_{eq}$: they can be directly related
to the function $q(x)$.

The definition is simple: we add a small magnetic field and we impose that the
perturbed system remains at an overlap greater than or equal $q$ from the
original one: the corresponding susceptibility BohBoh  is given by:

$\chi(q)=\beta\int_{q}^{1}dq^{\prime}\,x(q^{\prime})$ (26)

This formula reproduces the values of the two previously defined
susceptibilities ($\chi_{LR}$ and $\chi_{eq}$) if we set $q=q_{EA}$ and $q=0$
respectively:

$\displaystyle\chi_{LR}=\chi(q_{EA})$ $\displaystyle=$
$\displaystyle\beta(1-q_{EA})\,,$ $\displaystyle\chi_{eq}=\chi(0)$
$\displaystyle=$ $\displaystyle\beta\int dx(1-q(x))\,.$ (27)

The experimental measure of $\chi(q)$ can be done in an off-equilibrium
setting and using generalized fluctuation-dissipation relations CuKu .

I present here a very schematic version of the Cugliando-Kurchan theory CuKu ;
NOI .

Let us consider a system that has been carried to the final temperature at
time 0, we wait a time $t_{W}$ before measurements start. If the waiting time
$t_{W}$ is large but finite, the system is slightly off equilibrium. We can
look at the magnetic response at a large time $t$ after $t_{W}$.

Both times are macroscopic, much larger than the characteristic microscopic
time: in the experiments, they could range from a few seconds to a few hours.

We define the correlation:

$C(t,t_{W})=\mbox{Av}\left(\sigma_{i}(t_{W})\sigma_{i}(t_{W}+t)\right)\,.$
(28)

For large times we have the modified fluctuation-dissipation relations CuKu ;
NOI :

$\frac{d\chi(t_{W},t)}{dt}=-\beta X(C(t_{W},t))\frac{dC(t_{W},t)}{dt}\,,$ (29)

where $\chi(t_{W},t)$ is the response function at the time $t_{W}+t$ after
that an infinitesimal magnetic field has been introduced at time $t_{W}$.

We can eliminate the time parametrically and consider $\chi(t_{W},C)$. For
very large waiting time $t_{W}$ we should have:

$\frac{d\chi(t_{W},C)}{dC}=-\beta X(C)\,.$ (30)

In other words, we find that when $t_{W}\to\infty$ the quantity
$d\chi(t_{W},C)/dC$ has a well-defined limit that is equal to $X(C)$.

At the end of the day, one finds that this dynamically introduced quantity
$X(C)$ must be equal to the equilibrium $x(q)$, which has the meaning of a
probability:

$X(C)=x(q)\biggr{|}_{q=C}\,,\quad\chi(C)=\chi(q)\biggr{|}_{q=C}\,.$ (31)

The interpretation of these results in terms of a modified Onsager postulate
has been done in FrVi .

These results are very important as they open an experimental window on the
determination of the function $q(x)$ and give a theoretical framework to study
the off-equilibrium behavior.

The theory has been confirmed in a beautiful experiment OCIO , that was done
20 years ago: the main results for the response function versus correlation is
depicted in fig. 7. It would be extremely interesting to repeat the experiment
with modern ad accurate technologies, taking advantage of the progress that
has been done in the theory and in numerical simulations.

Similar results have been obtained in very careful numerical simulations,
where a direct comparison with the theory is possible because the function
$P(q)$ is known from equilibrium simulations (see fig. (5)). The very large
time span (12 orders of magnitude) helps to put under control JFDT  systematic
errors related to the infinite time extrapolation.

Figure 7: Experimental raw results (full symbols) and extrapolations to the
infinite time limit (open symbols) for the response function (defined in
eq.(30)) versus the correlation (defined in eq.(29)). The different curves
span the waiting times studied: $t_{w}=100\ s,t_{w}=200\ s,t_{w}=500\
s,t_{w}=1000\ s,t_{w}=2000\ s$ (from (OCIO ))

Other impressive off-equilibrium phenomena in spin glass are memory and
rejuvenation FS , unfortunately, I cannot discuss them due to a lack of space.
Fortunately, also these effects have been partially reproduced in accurate
simulations RecentJanus ; RecentJanus1  where one can obtain much more
accurate information using a very wide range of waiting times.

###  VIII.4 Granular Material and hard spheres

Classical granular matter WYART0 ; WYART1 ; WYART2 ; LN ; 9  is a problem of
high interest in experimental and theoretical physics. If we neglect friction
and the objects are spherical it reduces to the hard-sphere model which has
been extensively studied. It was well known that by increasing the density (or
the pressure) one enters a glassy phase, where nothing was supposed to happen
by subsequent compression. This phenomenon corresponds to the appearance of a
one-step replica symmetry breaking.

It was a real surprise when it was discovered that the analytic solution of
the hard spheres thermodynamics in the mean field approximation LARGED1 ;
FINAL ; BOOKPUZ  predicted the existence at high pressure of a transition (the
Gardner transition) to a marginal stable phase starting from the region where
one replica symmetry was broken at one step according to KWT .

Approaching this transition by increasing the pressure leads to a divergent
correlation time and to a divergent correlation length. This possibility of a
transition from one-step replica symmetry breaking to a continuous replica
symmetry breaking, with the consequent appearance of a marginal stable phase
was discussed firstly in GKS  in the framework of a spin glass type model, but
the transition was identified in GA , where its properties were computed. When
we cool a glass, it enters a non-equilibrium region and the possibility of a
transition to continuous replica symmetry breaking in this off-equilibrium
region was first discussed in P1 ; P2 .

This unexpected prediction was confirmed in detailed numerical analyses 16 ;
37 . This marginal phase predicted by the replica theory of glasses, has been
directly observed experimentally in a slowly densifying colloidal glass Lexp ;
Jin  and in two dimensional hard disks SD ; 2D .

A spectacular result was the computation of the mean-field exponents for the
jamming transition LARGED1 ; FINAL ; BOOKPUZ  that happens in the phase where
replica symmetry is spontaneously broken. For example in the mean field
approximation at jamming the correction function $g(r)$ of hard spheres of
diameter 1, at distance $r$ slightly greater than 1 behaves as

$g(r)\propto\frac{1}{(r-1)^{\gamma}}\,,\quad\gamma=0.41269\cdots\,,$ (32)

where the value of $\gamma$ is obtained by solving non-linear equations.

This prediction is correct not only in high dimensions (as it should be), but
it has been verified also in three and quite likely in two dimensions where
some logarithmic corrections may be present TANTI .

###  VIII.5 Random Laser

The theoretical interest in random lasers in connection with replica symmetry
breaking started in 2006 ACRZ . Fortunately, experimental evidence of replica
symmetry breaking has been provided 30 ; 33 ; 99 . In random lasers is
possible to observe directly the occupancy of different harmonic modes and
therefore one can measure directly the $P_{J}(q)$ function.

Many different kinds of lasers have since been studied: not only the standard
solid disordered lasers but also random fiber lasers A ; B , random laser
suspensions in very viscous liquids C  heterogeneous random lasers in highly
porous fibers D .

Remarkably, similar phenomena are present also in nonlinear optical
propagation through photorefractive disordered waveguides 90 .

##  IX The spin glass cornucopia

In 1988 P.W. Anderson published seven columns in ”Physics Today” discussing
various issues on spin glasses. In one of the columns he described spin
glasses as an amazing cornucopia PWA : To me, the key result here is the
beautiful revelation of the structure of the randomly ”rugged landscape” that
underlies many complex optimization problems (..) Physical spin glasses and
the SK model are only a jumping-off point for an amazing cornucopia of wide-
ranging applications of the same kind of thinking.

Anderson was right. Here I will try to sketch some of them.

We have seen many developments in physics. I will mention here only a few
examples.

  * •

Structural glasses: replica symmetry breaking is relevant for the study of the
glass transition.

A very important step forward was done in the eighties using the mode coupling
theories ModeCou . However, it was realized that the same kind of equations
can be obtained in the framework of generalized spin glass models KWT ; BB .
This new approach was complemented by the discovery of the new replica-based
thermodynamical potentials FP . In this way, it was possible to identify the
mode-coupling transition as a dynamic transition, where the correlation time
goes to infinity without any thermodynamics counterpart, and to understand the
behavior at low temperatures.

It was immediately clear that in reality, the dynamical transition is an
artifact of mean-field theory and it is a cross-over region. In this
framework, one can see the emergence of a large dynamical correlation length
FP3  that has an experimental counterpart in the large non-linear
susceptibility NONLIN .

The discovery of nearly analytically soluble non-disordered models, where
replica computations could be successfully done RIT , was very important (at
least to me) for showing the relevance of the replica approach to structural
glasses. Finally, first principle computations of the glass transition and the
glassy properties in the low-temperature region have been successfully done
MPMOL ; BV .

Saddle points in the free energy landscape control the tunneling among
different minima of the potentials, as can be seen also in mean field models
SELLE5 .

Beyond mean-field theory, the dynamic transition corresponds to the crossover
from dynamics dominated by narrow flat directions to dynamics dominated by
barriers It was also possible to simplify the theory in such a way as to
compute the critical exponents in high dimensions PRR .

  * •

Very important progress have been made in off-equilibrium theory, the approach
is quite complex because we lack Boltzmann-Gibbs statistical mechanics. As
stressed in many papers (e.g. in BOH ), the situation is different for systems
that are only slightly out of equilibrium. For example, there are systems that
within the experimental timescale cannot reach equilibrium because of high
free-energy barriers (that may be of energetic or entropic nature): this
situation typically applies to disordered systems, such as spin glasses and
structural glasses.

These systems approach equilibrium slowly, by jumping from one metastable
state to another, and they remain out of equilibrium forever if continually
perturbed by a slowly changing external field.

In these systems, we can expect a separation, by many orders of magnitude,
between the microscopic time scale of the system (for example, that
represented by the vibrations of individual atoms) and the macroscopic time
needed to cross the barrier (for example, changes in the structure of the
system itself).

The systems can then be considered to be essentially thermalized inside a
metastable state, and so fluctuation-dissipation ideas can still be applied:
the slowly changing overall state of the system is considered to be a small
perturbation. As we have seen the standard fluctuation theorem cannot be
applied to jumps from one equilibrium state to the other and we have to use
the modified fluctuation-dissipation relations.

The extremely long time needed to thermalize a glassy system induced great
interest in off-equilibrium phenomena like aging and many new ideas have been
introduced. Maybe the most significant advance on conceptual ground was the
introduction of the modified fluctuation-dissipation relation on long-time
scales for quasi-adiabatic dynamics, whose application to spin glasses I have
discussed in the previous section CuKu . A general theory of off-equilibrium
quasi-adiabatic dynamics has also been proposed QUASI .

We have also seen many developments outside physics. I will mention only a few
examples.

  * •

Optimization theory was strongly affected. The replica approach enables
scientists to compute analytically the properties of the optimal solution for
large random instances. This was done for many problems: e.g. the random
traveling salesman problem or the assignment problem. In this last problem, we
have a set of $N$ cities and $N$ wells. The cost of connecting a city to a
well is a random number in the interval [0-1]. One could show that the cost of
the globally optimal solution in the large $N$ limit is $\pi^{2}/6$ plus
computable $1/N$ corrections MPBIP ; PARAT ; SIC .

  * •

Similar arguments MP ; MP2  allow scientists to study constraint satisfaction
problems where the task is to find a configuration that satisfies all the
randomly chosen constraints: the most simple task to visualize is to color a
graph with $M$ colors in such way that two nodes with the same color are not
in contact. One question is how many colors are needed for a large random
graph depending on the statistical properties of the graph COL .

The most spectacular results are the results of the 3SAT problem, i.e. a
satisfaction problem with random clauses with three elements MZ ; MZK ; MPZ .
The 3SAT problem is the archetypical NP-complete problem.

  * •

There have been many applications of these ideas in a biological context. Let
me mention only the folding of heteropolymers of biological interest (proteins
OW ; Sha ; Manifold , RNA RNA ) and the study of the many equilibria of
ecological environments where many competing species coexist in a sometimes
fragile equilibrium E0 ; E1 ; E2 .

  * •

These ideas helped scientists to develop new algorithms. The most famous one
is the simulated annealing model of Kirkpatrick (the ”K” of SK), Gelatt, and
Vecchi SIMANN . This algorithm evolved into the simulated tempering SIMTEMP
and the parallel tempering PARTEMP , which is the state-of-the-art algorithm
in many simulations problems.

Among algorithms in many different contexts (e.g. 3SAT) let me mention the
survey propagating algorithm MPZ  and the backtracking survey propagation BP .
The same ideas have been applied to many others problems (e.g. compressed
sensing ConSen , matrix factorization MatFac …).

  * •

We have seen that the ideas of Hebb on memory did materialize into a concrete
model in the seminal model of Hopfield HOPFIELD . In two fundamental papers
Amit, Gutfreund, and Sompolisky AGS0 ; AGS  applied spin glass theoretical
tools to derive analytically the properties of the Hopfield model.

These papers were extremely important because they showed that the Hopfield
model could be completely understood analytically: they were the trampoline
for the analytic study of more complex and more realistic variations of the
Hopfield model.

Bringing neural network theory closer to biological realism was a major drive
in the subsequent work of the late Daniel Amit who wrote a very influential
book: Modeling Brain Function: The World of Attractor Neural AMIT . This was
the starting point of many other analytic studies also intending to produce
more realistic models relevant to the actual brain behavior and neurobiology
PDG .

This intellectual milieu was the origin of deep learning which now dominates
artificial intelligence.

  * •

At the same time, there were many developments in the statistical mechanics of
learning on different types of architecture. Remarkable results were the upper
bounds of learning abilities: among them, I would like to recall the capacity
of the standard perceptron G , of the binary perceptron KM , of recurrent
neural networks DeGa  and of feed-forward neural networks FUZ .

It is interesting to note that in the more complex cases, there is a region
where replica symmetry is broken near the transition point from perfect to
partial learning. This transition has many points in common with the jamming
of hard spheres and in some cases, it is characterized by the same exponents
FUZ .

##  X On complexity

As we have seen multiple equilibria are at the root of my studies of complex
systems. Complex systems theory has thus been applied to various systems, the
most interesting feature of complex systems is the existence of a large number
of different equilibrium states.

In a nutshell what does not change over time (or changes irreversibly) is not
complex, while a system that can take many different forms or behaviors
certainly is. If we look around, look at ourselves, animals, ecosystems, the
Earth, and the climate, we have complexity all around us. One of the most
interesting outputs of my work was to find some of the physical tools needed
to deal with complexity in the framework of systems with an energy function
where the Boltzmann-Gibbs statistical does apply.

In many cases frustration is crucial: when frustration is present a given
actor receives contradictory requests from the other actors, that cannot be
simultaneously satisfied so that many compromises are possible and
consequently many different equilibrium states PSTAR . When we look at real
complex systems, such as a living cell, a brain, a society, or a complete
living being such as an animal, we always see that in these systems there is
constant competition (frustration), but also cooperation between a very large
number of elements that (depending on the case) can be proteins, neurons, or
people. These systems are never in equilibrium, but they oscillate and
fluctuate around some defined state of equilibrium IK .

In this situation the system is flexible and malleable, it can adapt to
changes in the environment by transitioning between various possible states
without thereby losing identity: in other words, we sleep, wake up, etc., but
we do not change identity. Switching among many different equilibrium states
gives a living being the possibility of tuning his behavior in such a way as
to adapt itself to changing needs of a fluctuating environment.

Systems, to remain complex, must have an internal balance, frustration should
be high enough to avoid the formation of a ferromagnetic order. However, as
stressed in IK , it is possible that this delicate mechanism of cooperation,
competition, stimulation, and inhibition,is violated: in that case frustration
decreases, the behavior of the system changes, the correlations between the
subsystems change, the overall system begins to be no longer complex, and
something abnormal happens. Examples are a tumor in the cell, a disease in the
nervous system, and a dictatorship in society. At this point, the complexity
of the system degrades and the functioning of the system, as a whole, is
severely damaged or even eliminated.

So I would like to conclude with a quote from my friend Imre Kondor, who
asserts that the loss of complexity is dangerous, and recall the warning
attributed to a great nineteenth-century historian, Jacob Burckhardt, who
studied political and social processes in depth: The denial of complexity is
the essence of tyranny IK .

## Appendix
On the early history of the replica method

Nicola d’Oresme was the greater scientist of the fourteenth century: he
discovered, among many other things, what we call now Galilean invariance,
i.e. the impossibility of detecting motion with local measurements. In other
words, he stated that there are no absolute velocities, but only relative
velocities, hence nothing contradicts Earth’s motion around the Sun.

Around 1355 he discovered that

$\sqrt{x}=x^{1/2}\,.$ (33)

If we define the $x^{n}$ as the multiplication of $n$ factors $x$, $x^{1/2}$
does not make sense. However, if we want to define $x^{n}$ for non-integer
$n$, we can assume that the equation

$\left(x^{n}\right)^{m}=x^{(n\ m)}$ (34)

is valid also for non-integer $n$. In this way, we can compute $x^{n}$ for any
rational $n$. Indeed if we put $n=1/2$ and $m=2$, we get

$\left(x^{1/2}\right)^{2}=x^{(1/2\ 2)}=x\,.$ (35)

This is the definition of the square root.

In modern times the first use of replicas was done by Robert Brout in 1959
Brout . He aimed to treat the quenched disorder and to compute the average
(over the disorder) of the free energy, exactly the task we face in spin
glasses. He wrote the equivalent of eq. (2).

He used the replica method for simplifying the diagrammatics without
discussing too much about the meaning of taking $n=0$. Indeed he worked in a
perturbative setting. At each order of perturbation theory, the results are
polynomials in $n$: he showed that the term proportional to $n^{0}$, i.e. the
limit at $n=0$ of the polynomial, gives the needed result, i.e. the average
over the disorder. The results could also be obtained by explicit
computations. For him, replicas were a tool for organizing a complex
combinatorial problem.

In 1972 de Gennes used replicas for self-avoiding polymers DeGen  extending
the $O(n)$-symmetric $g\phi^{4}$ to $n=0$. In this way, he was able to compute
the critical exponents for polymers using the standard renormalization group.
Also de Gennes used replicas in perturbation theory: the whole computation
could have been done using the standard perturbative expansions for polymers.
However, in the replica approach, it was possible to use all the results of
field theory. In this case, the replica group is $O(n)$; we have seen in the
general case the replica group is $S(n)$, which is a subgroup of $O(n)$.

The idea of developing a $O(n)$ symmetric $g\phi^{4}$ field theory in terms of
closed trajectories (polymers) is also contained in the paper by Symanzik of
1969 in a completely different context Sy ; PaStat .

The same theory ($g\phi^{4}$ at $n=0$) can be also formulated as a Laplacian
plus a random (imaginary) potential model; for negative $g$ this theory
corresponds to Anderson’s theory for random potentials. The same approach can
be applied to study localization, where the replica group is the non-compact
group $O(n,n)$ NonCon . As well known from PaSou , we can reformulate the
theory introducing Fermions and this leads to the use of the $O(2/2)$
Fermionic group Efetov  instead of the undefined $O(0)$ group.

De Gennes’s method was generalized in the seventies to the study of many other
problems (e.g. polymers LUB ). This activity was done in a perturbative
framework: all the computations could be done without replicas, but in a more
cumbersome, error-prone way.

It is remarkable that in those years the idea of analytic continuation from
integers appeared in many different contexts:

  * •

In 1959 Tullio Regge Regge  introduced complex angular momentum. Here,
Carlson’s theorem was crucial to prove the uniqueness of the analytic
continuation.

  * •

In 1972 Fortuin-Kasteleyn FK  introduced the random cluster model for the $q$
state Potts model: in the limit $q=1$ one obtains lattice percolation.

  * •

Always in the same year (1972) Giambiagi and Bollini GB , and ’t Hooft and
Veltman HV  introduced dimensional regularization (non-integer dimensional
spaces).

The idea of spaces with non-integer dimension was independently at the basis
of the very successful $\epsilon$ expansion for the critical exponents in
$4-\epsilon$ expansion WF  in the framework of Wilson’s renormalization group.
Ken Wilson obtained the Nobel Prize a few years later.

A rigorous definition of rotationally invariant Euclidean space is lacking;
some suggestions have been done in ParTwo .

Remarkably, the replica method for statistical mechanics was introduced by
Roger Brout, who died before receiving the Nobel prize together with Englert
and Higgs; others four Nobel Laureates (de Gennes, Anderson, Thouless, and
Kosterlitz) contributed to the development of the method.

## Acknowledgements

I am very grateful to the hundreds of people that have worked with me on these
subjects, but they are too many to list here. However, I would like to thank
Luca Leuzzi, Andrea Maiorano, Enzo Marinari, Marc Mézard, Federico Ricci-
Tersenghi, Juan-Jesus Ruiz-Lorenzo, and Francesco Zamponi for reading the
manuscript and for many useful suggestions.

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