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where < , >

diff

denotes the inner product in the

Hilbert space of solutions of the vector constraint,

and the exponential has been expanded in power s in

the last expression on the right-hand side.

From early on, it was realized that smooth loop

states are naturally annihilated by

S (indepen-

dently of any quantization ambiguity). Conse-

quently,

S acts only on spin network nodes.

Generically, it does so by creating new links and

nodes modifying the underlying graph of the spin

network states (Figure 11).

Therefore, each term in the sum [30] represents a

series of transitions – given by the local action of

at spin network nodes – through different spin

network states interpolating the boundary states s

and s

, respectively. The action of

S can be

visualized as an ‘‘interaction vertex’’ in the ‘‘time’’

evolution of the node (Figure 11). A s in the explicit

3D case, eqn [30] can be expressed as sum over

‘‘histories’’ of spin networks pictured as a system of

branching surfaces described by a 2-complex whose

elements inherit the representation labels on the

intermediate states. The value of the ‘‘transition’’

amplitudes is controlled by the matrix elements of

S. Therefore, although the qualitative picture is

independent of quantization ambiguities, transition

amplitudes are sensitive to them.

Before even considering the issue of convergence

of [30], the problem with this definition is evident:

every single term in the sum is a divergent integral!

Therefore, this way of presenting spin foams has to

be considered as formal until a well-defined regular-

ization of [2] is provided. That is the goal of the spin

foam approach.

Instead of dealing with an infinite number of

constraints Thiemann recently proposed to impose

one single master constraint de fined as

M ¼



ðxÞq

ðxÞV

ðxÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

det qðxÞ

½31

Using techniques developed by Thiemann, this

constraint can indeed be promoted to a quantum

operator acting on H

kin

. The physical inner product

is given by

<s; s

:¼ lim

T!1

<s;

T

dt e

> ½32

A spin foam representation of the previous expres-

sion could now be achieved by the standard

skeletonization that leads to the path-integral repre-

sentation in quantum mechanics. In this context,

one splits the t -parameter in discrete steps and

writes

¼ lim

N!1

½e

M=N



¼ lim

N!1

½1 þ it

M=N

½33

The spin foam representation follows from the fact

that the action of the basic operator 1 þ it

M=N on a

spin network can be written as a linear combination

of new spin networks whose graphs and labels have

been modified by the creation of new nodes (in a

way qualitatively analogous to the local action

shown in Figure 11). An explicit derivation of the

physical inner product of 4D LQG along these lines

is under current investigation.

Spin Foams from the Covariant Formulation

In 4D, the spin foam representation of the dynamics

of LQG has been investigated more intensively in

the covariant formulation. This has led to a series of

constructions which are referred to as spin foam

models. These treatments are related more closely to

the construction based on the covariant path-

integral approach of the last section. Here we

illustrate the formulation which has captured much

interest in the literature: the Barrett–Crane (BC)

model.

Spin foam models for gravity as constrained quan-

tum BF theory The BC model is one of the most

extensively studied spin foam models for quantum

gravity. To introduce the main ideas involved, we

concentrate on the definition of the model in the

Riemannian sector. The BC model can be formally

N(x

nop

N(x)S(x)

Figure 11 The action of the scalar constraint and its spin foam representation. N(x

) is the value of N at the node and S

nop

are the

matrix elements of

652 Spin Foams

viewed as a spin foam quantization of SO(4)

Plebanski’s form ulation of general relativity. Ple-

banski’s Riemannian action depends on an SO(4)

connection A, a Lie-alg ebra-valued 2-form B , and

Lagrange multiplier fields  and . Writing explicitly

the Lie algebra indices, the action is given by

S½B; A;;



^ F

ðAÞþ

IJKL

^ B

þ 

IJKL



IJKL



½34

where  is a 4-form and 

IJKL

= 

JIKL



IJLK

= 

KLIJ

is a tensor in the internal space.

Variation with respect to  imposes the constraint



IJKL



IJKL

= 0on

IJKL

. The Lagrange multiplier

tensor 

IJKL

has then 20 independent components.

Variation with respect to  imposes 20 algebraic

equations on the 36 components of B. The (non-

degenerate) solutions to the equations obtained by

varying the multipliers  and  are

¼

IJKL

^ e

and

¼e

^ e

½35

in terms of the 16 remaining degrees of freedom of

the tetrad field e

. If one substitutes the first solution

into the original action, one obtains Palatini’s

formulation of general relativity; therefore, on shell

(and on the right sector), the action is that of

classical gravity.

The key idea in the definition of the model is that

the path integral for the theory corresponding to the

action S[B, A, 0, 0], namely

topo

D½BD½Aexp i

^ F

ðAÞ





½36

can be given a meani ng as a spin foam sum, [4],in

terms of a simple generalization of the construction

of the previous sect ion. In fact, S[B, A, 0, 0] corre-

sponds to a simple theory known as BF theory that

is formally very similar to 3D gravity (see BF

Theories). The result is independent of the chosen

discretization because BF theory does not have local

degrees of freedom (just as 3D gravity).

The BC model aims at providing a definition of

the path integral of gravity pursuing a well-posed

definition of the formal expression

D½BD½A B ! 

IJKL

^ e



 exp i

^ F

ðAÞ





½37

where D[B]D[A](B !

IJKL

^ e

) means that one

must restrict the sum in [36] to those configurations

of the topological theory satisfying the constraints

B =  (e ^ e) for some tetrad e. The remarkable fact

is that this restriction can be implemented in a

systematic way directly on the spin foam configura-

tions that define P

topo

In P

topo

spin foams are labeled with spins corre-

sponding to the unitary irreducible representations of

SO(4) (given by two spin quantum numbers (j

, j

)).

Essentially, the factor ‘‘(B ! 

IJKL

^ e

)’’ restricts

the set of spin foam quantum numbers to the so-

called simple representations (for which j

= j

= j).

This is the ‘‘quantum’’ version of the solution to the

constraints [35]. There are various versions of this

model. The simplest definition of the transition

amplitudes in the BC model is given by

Pðs



sÞ¼

fjg

f 2 F

s!s

ð2j

þ 1Þ



v2F

s!s





½38

where we use the notation of [22], the graphs denote

15j-symbols, and 

are half-integers labeling SU(2)

normalized 4-intertwiners. No rigorous connection

with the Hilbert space picture of LQG has yet been

established. The self-dual version of Plebanski’s

action leads, through a similar construction, to

Reisenberger’s model.

The simplest amplitude in the BC model corre-

sponds to a single 4-simplex, which can be viewed

as the simplest triangulation of the 4D spacetime

given by the interior of a 3-sphere (the correspond-

ing 2-complex is shown in Figure 12). States of the

4-simplex are labeled by ten spins j (labeling the ten

edges of the boundary spin network, see Figure 12)

which can be shown to be related to the area in

Figure 12 The dual of a 4-simplex.

Spin Foams 653

Planck units of the ten triangular faces that form the

4-simplex. A first indication of the connection of the

model with gravity was that the large-j asymptotics

appeared to be dominated by the exponential of the

Regge action (the action derived by Regge as a

discretization of general relativity). This estimate

was done using the stationary-phase approximation

to the integral that gives the amplitude of a

4-simplex in the BC model. However, more detailed

calculations showed that the amplitude is dominated

by configurations corresponding to degenerate

4-simplexes. This seems to invalidate a simple

connection to general relativity and is one of the

main puzzles in the model.

Spin Foams as Feynman Diagrams

The main problem with the models of the previous

section is that they are defined on a discretization 

of M and that – contrary to what happens with a

topological theory, for example, 3D gravity

(eqn [29]) – the amplitudes depend on the discretiza-

tion . Various possibilities to eliminate this reg-

ulator have been discussed in the literature but no

explicit results are yet known in 4D. An interesting

proposal is a discretization-independent definition of

spin foam models achieved by the introduction of an

auxiliary field theory living on an abstract group

manifold – Spin(4)

and SL(2, C)

for Riemannian

and Lorentzian gravity, respectively. The action of

the auxiliary group field theory (GFT) takes the form

S½¼





ð5Þ

½½39

where M

(5)

[] is a fifth-order m onomial, and

G is the corresponding group. In the simp-

lest m odel, M

(5)

[] = (g

, g

)(g

, g

)

(g

, g

)(g

, g

)(g

, g

). The

field  is required to be invariant under the

(simultaneous) right action of the group on its

four arguments in addition to other sym metries

(not described here f or simplicity). The perturba-

tive expansion in  of the GFT Euclidean path

integral is given by

P ¼

D½e

S½



sym½F



A½F

½40

where A[F

] corresponds to a sum of Feynman-

diagram amplitudes for diagrams with N interaction

vertices, and sym[F

] denotes the standard symme-

try facto r. A remarkable property of this expansion

is that A[F

] can be expressed as a sum over spin

foam amplitudes, that is, 2-complexes labeled by

unitary irreduci ble representations of G. Moreover,

for very simple interaction M

(5)

[], the spin foam

amplitudes are in one-to-one correspondence to

those found in the models of the previous section

(e.g., the BC model). This duality is regarded as a

way of providing a fully combinatorial definition of

quantum gravity where no reference to any dis-

cretization or even a manifold structure is made.

Transition amplitudes between spin network states

correspond to n-point functions of the field theory.

These models have been inspired by generalizations

of matrix models applied to BF theory.

Divergent transition amplitudes can arise by the

contribution of ‘‘loop’’ diagrams as in standard

quantum field theory. In spin foams, diagrams

corresponding to 2D bubbles are potentially divergent

because spin labels can be arbitrarily high leading to

unbounded sums in [4]. Such divergences do not occur

in certain field theories dual (in the sense above) to the

BC model. However, little is known about the

convergence of the series in  and the physical meaning

of this constant. Nevertheless, Freidel and Louapre

have shown that the series can be re-summed in certain

models dual to lower-dimensional theories.

Causal Spin Foams

Let us con clude by presenting a fundamentally

different construction leading to spin foams. Using

the kinematical setting of LQG with the assumption

of the existence of a microlocal (in the sense of

Planck scale) causal structure, Markopoulou and

Smolin define a general class of (causal) spin foam

models for gravity. The elementary transition ampli-

tude A

Iþ1

from an initial spin network s

another spin network s

Iþ1

is defined by a set of

simple combinatorial rules based on a definition of

causal propagation of the information at nodes. The

rules and amplitudes have to satisfy certain causal

restrictions (motivated by the standard concepts

in classical Lorentzian physics). These rules gene-

rate surface-like excitations of the same kind one

encounters in the previous formulations. Spin foams

are labeled by the number of times, N, these

elementary transitions take place. Transition

amplitudes are defined as

; s

i¼

AðF

Þ½41

which is of the generic form [4]. The models are not

related to any continuum action. The only guiding

principles in the construction are the restrictions

imposed by causality, and the requirement of the

existence of a nontrivial critical behavior that

reproduces genera l relativity at large scales. Some

indirect evidence of a possible nontrivial continuum

limit has been obtained in certain versions of these

models in 1 þ 1 dimensions.

654 Spin Foams

See also: Algebraic Approach to Quantum Field Theory;

BF Theories; Canonical General Relativity; Chern–

Simons Models: Rigorous Results; Lattice Gauge

Theory; Loop Quantum Gravity; Quantum Dynamics in

Loop Quantum Gravity; Quantum Geometry and its

Applications; Topological Quantum Field Theory:

Overview.

Further Reading

Ashtekar A (1991) Lectures on Nonperturbative Canonical

Gravity. Singapore: World Scientific.

Ashtekar A and Lewandowski J (2004) Background independent

quantum gravity: a status report.

Baez J (1998) Spin foams. Classical and Quantum Gravity 15:

1827–1858.

Baez J (2000) An introduction to spin foam models of quantum

gravity and BF theory. Lecture Notes in Physics 543: 25–94.

Baez J and Muniain JP (1995) Gauge fields, Knots and Gravity.

Singapore: World Scientific.

Oriti D (2001) Spacetime geometry from algebra: spin foam

models for nonperturbative quantum gravity. Reports on

Progress in Physics 64: 1489–1544.

Perez A (2003) Spin foam models for quantum gravity. Classical

and Quantum Gravity 20: R43.

Rovelli C Quantum Gravity. Cambridge: Cambridge University

Press (to appear).

Thiemann T Modern Canonical Quantum General Relativity.

Cambridge: Cambridge University Press (to appear).

Spin Glasses

F Guerra,Universita

di Roma ‘‘La Sapienza’’,

Rome, Italy

Introduction

From a physical point of view, spin glasses, as dilute

magnetic alloys, are very interesting systems. They

are characterized by such features as exhibiting a new

magnetic phase, where magnetic moments are frozen

into disordered equilibrium orientations, without any

long-range order. See, for example, Young (1987) for

general reviews, and also Stein (1989) for a very

readable account about the physical properties of

spin glasses. The experimental laboratory study of

spin glasses is a very difficult subject, because of their

peculiar properties. In particular, the existence of

very slowly relaxing modes, with consequent memory

effects, makes it difficult to realize the very basic

physical concept of a system at thermodynamical

equilibrium, at a given temperature.

From a theoretical point of view some models

have been proposed, which try to capture the

essential physical features of spin glasses, in the

frame of very simple assumptions.

The basic model has been proposed by Edwards

and Anderson (1975) many years ago. It is a simple

extension of the well-known nearest-neighbor Ising

model. On a large region  of the unit lattice in d

dimensions, we associate an Ising spin (n) to each

lattice site n, and then we introduce a lattice

Hamiltonian



ð; JÞ¼

ðn;n

Jðn; n

ÞðnÞðn

Þ½1

Here, the sum runs over all couples of nearest-

neighbor sites in , and J are quenched random

couplings, assumed for simplicity to be independent

identically distributed random variables, with cen-

tered unit Gaussian distribution. The quenched

character of the J means that they do not contribute

to thermodynamic equilibrium, but act as a kind of

random extern al noise on the coupling of the 

variables. In the expression of the Hamiltonian, we

have indicated with  the set of all (n), and with J

the set of all J(n, n

). The region  must be taken

very large, by letting it invade all lattice in the limit.

The physical motivation for this choice is that for

real spin glasses the interaction between the spins

dissolved in the matrix of the alloy oscillates in sign

according to distance. This effect is taken into

account in the model through the random character

of the couplings between spins.

Even though very drastic simplifications have

been introduced in the formulation of this model,

as compared to the extremely complicated nature

of physical spin glasses, nevertheless a rigorous

study of all properties emerging from the static

and dynamic behavior of a thermodynamic system

of this kind is far from being complete. In particular,

with reference to static equ ilibrium properties, it

is not yet possible to reach a completely substan-

tiated description of the phases emerging in the

low-temperature region. Even physical intuition

gives completely different guesses for different

people.

In the same way as a mean-field version can be

associated to the ordinary Ising model, so it is possible

for the disordered model described by [1].Nowwe

consider a number of sites i = 1, 2, ..., N, and let each

spin (i)atsitei interact with all other spins, with the

intervention of a quenched noise J

. The precise form

of the Hamiltonian will be given in the following.

This is the mean-field model for spin glas ses,

introduced by Sherrington and Kirkpatrick (1975).

Spin Glasses 655

It is a celebrated model. Numerous articles have

been devoted to its study during the years, appearing

in the theoretical physics literature.

The relevance of the model stems surely from the

fact that it is intended to represent some important

features of the physic al spin glass systems, of great

interest for their peculiar properties, at least at the

level of the mean-field approximation.

But another important source of interest is

connected with the fact that disordered systems, of

the Sherrington–Kirkpatrick type, and their general-

izations, seem to play a very important role for

theoretical and practical assessments about hard

optimization problems, as it is shown, for example,

by Me´zard et al. (2002).

It is inte resting to remark that the original paper

was entitled ‘‘Solvable model of a spin-glass,’’ while

a previous draft, as told by David Sherrington,

contained the even stronger designation ‘‘Exactly

solvable.’’ However, it turned out that the very

natural solution devised by the authors is valid only

at high temperatures, or for large external magnetic

fields. At low temperatures, the proposed solution

exhibits a nonphysical drawback given by a negative

entropy, as properly recognized by the authors in

their very first paper.

It took some years to find an acceptable solution.

This was done by Giorgio Parisi in a seri es of

papers, marking a radical departure from the

previous meth ods. In fact, a very intense method of

‘‘spontaneous replica symmetry breaking’’ was

developed. As a consequence, the physical content

of the theory was encoded in a functional order

parameter of new type, and a remarkable structure

emerged for the pure states of the theory, a kind of

hierarchical, ultrametric organization. These very

interesting developments, due to Parisi, and his

coworkers, are explained in a brilliant way in the

classical book by Me´zard et al. (1987). Part of this

structure will be recalled in the following.

It is important to remark that the Parisi solution is

presented in the form of an ingenious and clever

‘‘ansatz.’’ Until few years ago, it was not known

whether this ansatz would give the true solution for

the model, in the so-called thermodynamic limit,

when the size of the system becomes infinite, or it

would be only a very good approximation for the

true solution.

The general structures offered by the Parisi solu-

tion, and their possible generalizations for similar

models, exhibit an extremely rich and interesting

mathematical content. Very appropriately, Talagrand

(2003) has used a strongly suggestive sentence in the

title to his recent book: ‘‘Spin glasses: a challenge for

mathematicians.’’

As a matter of fact, how to face this challenge is a

very difficult problem. Here we would like to recall

the main features of a very powerful method, yet

extremely simple in its very essence, based on a

comparison and interpolation argument on sets of

Gaussian random variables.

The method found its first simple application in

Guerra (2001), where it was shown that the

Sherrington–Kirkpatrick replica symmetr ic approxi-

mate solution was a rigorous lower bound for the

quenched free energy of the system, uniformly in

the size. Then, it was possible to reach a long-

awaited result (Guerra and Toninelli 2002): the

convergence of the free energy density in the

thermodynamic limit, by an intermediate step

where the quenched free energy was shown to be

subadditive in the size of the system.

Moreover, still by interpolation on families of

Gaussian random variables, the first mentioned result

was extended to give a rigorous proof that the

expression given by the Parisi ansatz is also a lower

bound for the quenched free energy of the system,

uniformly in the size (Guerra 2003). The method gives

not only the bound, but also the explicit form of the

correction in a complex form. As a recent and very

important result, along the task of facing the challenge,

Michel Talagrand has been able to dominate these

correction terms, showing that they vanish in the

thermodynamic limit. This milestone achievement was

first announced in a short note, containing only a

synthetic sketch of the proof, and then presented with

all details in a long paper (Talagrand 2006).

The interpolation method is also at the basis of

the far-reaching generalized variational principle

proved by Aizenman et al. (2003).

In our presentation, we will try to be as self-

contained as possible. We will give all definitions,

explain the basic structure of the interpolation

method, and show how some of the results are

obtained. We will concentrate mostly on questions

connected with the free energy, its properties of

subadditivity, the existence of the infinite-volume

limit, and the replica bounds.

For the sake of comparison, and in order to

provide a kind of warm-up, we will recall also some

features of the standard elementary mean-field

model of ferromagnetism, the so-called Curie–

Weiss model. We will concentrate also here on the

free energy, and systema tically exploit elementary

comparison and interpolation arguments. This will

show the strict analogy between the treatment of the

ferromagnetic model and the developments in the

mean-field spin glass case. Basic roles will be played

in the two cases, but with different expressions, by

positivity and convexity propert ies.

656 Spin Glasses

Then, we will consider the problem of connecting

results for the mean-field case to the short-range case.

An intermediate position is occupied by the so-called

diluted models. They can be studied through a

generalization of the methods exploited in the mean-

field case, as shown, for example, in De Sanctis (2005).

The organization of the paper is as follows. We

first introduce the ferromagnetic model and discuss

behavior and properties of the free energy in the

thermodynamic limit, by emphasizing, in this very

elementary case, the comparison and interpolation

methods that will be also exploited, in a differe nt

context, in the spin glass case.

The basic features of the mean-field spin glass

models are discussed next, by introducing all

necessary definitions. This is followed by the

introduction, for generic Gaussian interactions, of

some important formulas, concerning the derivation

with respect to the strength of the interaction, and

the Gaussian comparison and interpolation method.

We then give simple applications to the mean-field

spin glass model, in particular to the existence of the

infinite-volume limit of the quenched free energy

(Guerra and Toninelli 2002), and to the proof of

general variational bounds, by following the useful

strategy developed in Aizenman et al. (2003).

The main features of the Parisi representation are

recalled briefly, and the main theorem concerning

the free energy is stated. This is followed by a brief

mention of results for diluted models.

We also attack the problem of connecting the

results for the mean-field case to the more realistic

short-range models.

Finally we provide conclusions and outlook for

future foreseen developments.

Our treatment will be as simple as possible, by

relying on the basic structural properties, and by

describing methods of presumably very long lasting

power. The emphasis given to the mean-field case

reflects the status of research. After some years from

now this review would perhaps be written according

to completely different patterns.

A Warm-up. The Mean-field

Ferromagnetic Model: Structure

and Results

The mean-field ferromagnetic model is among the

simplest models of statistical mechanics. How ever, it

contains very interesting features, in particular a

phase transition, characterized by spontaneous

magnetization, at low temperatures. We refer to

standard textbooks for a full treatment and a

complete appreciation of the model in the frame of

the theory of ferromagnetism. Here we first consider

some properties of the free energy, easily obtained

through comparison methods.

The generic configuration of the mean-field

ferromagnetic model is defined through Ising spin

variables 

= 1, attached to each site i = 1,

2, ..., N.

The Hamiltonian of the model, in some external

field of strength h, is given by the mean-field expression

ð; hÞ¼

ði;jÞ



 h



½2

Here, the first sum extends to all N(N  1)=2 site

couples, and the second to all sites.

For a given inverse temperature , let us now

introduce the partition function Z

(, h) and the

free energy per site f

(, h), according to the well-

known definitions

ð; hÞ¼



...

expðH

ð; hÞÞ ½3

f

ð; hÞ¼N

1

E log Z

ð; hÞ½4

It is also convenient to define the average spin

magnetization

m ¼



½5

Then, it is immediately seen that the Hamiltonian

in [2] can be equivalently written as

ð; hÞ¼

 h



½6

where an unessential constant term has been

neglected. In fact, we have

ði;jÞ



i;j;i6¼j



N ½7

where the sum over all couples has been equivalently

written as one half the sum over all i, j with i 6¼ j,

and the diagonal terms with i = j have been added

and subtracted out. Notice that they give a constant

because 

= 1.

Therefore, the partition function in [3] can be

equivalently substituted by the expression

ð;hÞ¼



...

exp

Nm



exp h



½8

which will be our starting point.

Our interest will be in the lim

N!1

1

log Z

(, h).

To this purpose, let us establish the important

subadditivity property, holding for the splitting of the

Spin Glasses 657

large-N system in two smaller systems with N

and N

sites, respectively, with N = N

þ N

log Z

ð; hÞlog Z

ð; hÞþlog Z

ð; hÞ½9

The proof is very simple. Let us denote, in the most

natural way, by 

, ..., 

the spin variables for the

first subsystem, and by 

þ1

, ..., 

the N

spin

variables of the second subsystem. Introduce also the

subsystem magnetizations m

and m

, by adapting

the definition [5] to the smaller systems, in such a

way that

Nm ¼ N

þ N

½10

Therefore, we see that the large system magnetiza-

tion m is the linear convex combination of the

smaller system ones, according to the obvious

m ¼

½11

Since the mapping m ! m

is convex, we also have

the general bound, holding for all values of the 

variables



½12

Then, it is enough to substitute the inequality in the

definition [8] of Z

(, h), and recognize that we

achieve factorization with respect to the two sub-

systems, and therefore the inequality Z

 Z

So we have established [9]. From subadditivity, the

existence of the limit follows by standard arguments.

In fact, we have

lim

N!1

1

log Z

ð; hÞ¼inf

1

log Z

ð; hÞ½13

Now we will calculate explicitly this limit, by

introducing an order parameter M, a trial function,

and an appropriate variational scheme. In order to

get a lower bound, we start from the elementary

inequality m

 2mM  M

, holding for any value

of m and M. By inserting the inequality in the

definition [8] we arrive at a factorization of the sum

over ’s. The sum can be explicitly calculated, and

we arrive immediately to the lower bound, uniform

in the size of the system,

1

log Z

ð; hÞ

 log 2 þ log cosh ðh þ MÞ

M

½14

holding for any value of the trial order parameter M.

Clearly, it is convenient to take the supremum over M.

Then, we establish the optimal uniform lower bound

1

log Z

ð; hÞ

 sup

log 2 þ log cosh ðh þ MÞ

M



½15

It is simple to realize that the supremum coincides

with the limit as N !1. To this purpose we follow

the following simple procedure. Let us consider all

possible values of the variable m. There are N þ 1of

them, corresponding to any number K of possible

spin flips, starting from a given  configuration,

K = 0, 1, ..., N. Let us consider the trivial decom-

position of the identity, holding for any m,

1 ¼



½16

where M in the sum runs over the N þ 1 possible

values of m,and is Kroneker delta, being equal to 1

if M = N, and zero otherwise. Let us now insert [16]

in the definition [8] of the partition function inside

the sum over ’s, and invert the two sums. Because of

the forcing m = M given by the , we can write

= 2mM  M

inside the sum. Then if we neglect

the , by using the trivial   1, we have an upper

bound, where the sum over ’s can be explicitly

performed as before. Then it is enough to take the

upper bound with respect to M, and consider that

there are N þ 1 terms in the now trivial sum over M,

in order to arrive at the upper bound

1

log Z

ð; hÞ

 sup



log 2 þ log cosh ðh þ MÞ



M



þ N

1

logðN þ 1Þ½17

Therefore, by going to the limit as N !1, we can

collect all our results in the form of the following

theorem giving the full characterization of the

thermodynamic limit of the free energy.

Theorem 1 For the mean-field ferromagnetic

model we have

lim

N !1

1

log Z

ð; hÞ¼inf

1

log Z

ð; hÞ½18

¼ sup

log 2 þ log cosh ðh þ MÞ

M



½19

This ends our discussion about the free energy in

the ferromagnetic model.

Other properties of the model can be easily

established. Introduce the Boltzmann–Gibbs state

ðAÞ

¼ Z

1



...

A exp

Nm



exp



h





½20

where A is any function of 

...

The observable m() becomes self-averaging under

, in the infinite-volume limit, in the sense that

lim

N!1

ððm  Mð; hÞÞ

Þ¼0 ½21

658 Spin Glasses

This property of m is the deep reason for the success

of the strategy exploited earlier for the convergence

of the free energy. Easy consequences are the

following. In the infinite-volume limit, for h 6¼ 0,

the Boltzmann–Gibbs state becomes a factor state

lim

N!1

ð

...

Þ¼Mð; hÞ

½22

A phase transition appears in the form of sponta-

neous magnetization. In fact, while for h = 0 and

  1 we have M(, h) = 0, on the other hand, for

>1, we have the discontinuity

lim

h!0

Mð; hÞ¼lim

h!0



Mð; hÞMðÞ > 0 ½23

Fluctuations can also be easily controlled. In fact,

one proves that the rescaled random variable

ﬃﬃﬃﬃﬃ

(m  M(, h)) tends in distribution, under !

to a centered Gaussian with variance given by the

susceptibility

ð; hÞ

Mð; hÞ

ð1  M

1  ð1  M

½24

Notice that the variance becomes infinite only at the

critical point h = 0,  = 1, wher e M = 0.

Now we are ready to attack the much more

difficult spin glass model. But it will be surprising to

see that, by following a simple extension of the

methods described here, we will arrive at similar

results.

Basic Definitions for the Mean-Field Spin

Glass Model

As in the ferromagnetic case, the generic configura-

tion of the mean-field spin glass model is defined

through Ising spin variables 

= 1, attached to

each site i = 1, 2, ..., N.

But now there is an external quenched disorder

given by the N(N  1)=2 independent and identical

distributed random variables J

, defined for each

pair of sites. For the sake of simplicity, we assume

each J

to be a centered unit Gaussian with averag es

E(J

) = 0, E(J

) = 1. By quenched disorder we mean

that the J have a kind of stochastic external

influence on the system, without contributing to

the thermal equilibrium.

Now the Hamiltonian of the model, in some

external field of strength h, is given by the mean-

field expression

ð; h; JÞ¼

ﬃﬃﬃﬃﬃ

ði;jÞ



 h



½25

Here, the first sum extends to all site pairs, and the

second to all sites. Notice the

ﬃﬃﬃﬃﬃ

, necessary to

ensure a good thermodynamic behavior to the free

energy.

For a given inverse temperature , let us now

introduce the disorder-dependent partition func-

tion Z

(, h, J) and the quenched average of the

free energy per site f

(, h), according to the

definitions

ð; h; JÞ¼



...

expðH

ð; h; JÞÞ ½26

 f

ð; hÞ¼N

1

E log Z

ð; h; JÞ½27

Notice that in [27] the average E with respect to the

external noise is made ‘‘after’’ the log is taken. This

procedure is called quenched averaging. It represents

the physical idea that the external noise does not

contribute to the thermal equilibrium. Only the ’s

are thermalized.

For the sake of simplicity, it is also convenient to

write the partition function in the following equiva-

lent form. First of all let us introduce a family of

centered Gaussian random variables K(), indexed

by the configurations , and characterized by the

covariances

EðKðÞKð

ÞÞ ¼ q

ð; 

Þ½28

where q(, 

) are the overlaps between two generic

configurations, defined by

qð; 

Þ¼N

1



½29

with the obvious bounds 1  q(, 

)  1, and

the normalization q(, ) = 1. Then, starting from

the definition [25], it is immediately seen that the

partition function in [26] can also be written, by

neglecting unessential constant terms, in the form

ð; h; JÞ



...

exp 

ﬃﬃﬃﬃﬃ

KðÞ

exp h



½30

which will be the starting point of our treatment.

Basic Formulas of Derivation

and Interpolation

We work in the following general setting. Let U

be a family of centered Gaussian random variables,

i = 1, ..., K, with covariance matrix given by

E(U

)  S

.Wetreattheindexi now as configura-

tion space for some statistical mechanics system, with

partition function Z and quenched free energy given by

E log

expð

ﬃﬃ

ÞE log Z ½31

Spin Glasses 659

where w

 0 are generic weights, and t is a

parameter ruling the strength of the interaction.

It would be hard to underestimate the relevance of

the following derivation formu la

E log

expð

ﬃﬃ



1

expð

ﬃﬃ





2

expð

ﬃﬃ

Þ:

 expð

ﬃﬃ

ÞS



½32

The proof is straightforward. First we perform

directly the t-derivative. Then, we notice that the

random variable s appear in expressions of the form

E(U

F), where F are functions of the U’s. These can

be easily handled through the following integration

by parts formula for generic Gaussia n random

variables, strongly reminiscent of the Wick theorem

in quantum field theory,

EðU

FÞ¼



½33

Therefore, we see that always two derivatives are

involved. The two terms in [32] come from the

action of the U

derivatives, the first acting on the

Boltzmann factor, and giving rise to a Kronecker 

the second acting on Z

1

, and giving rise to the

minus sign and the duplication of variables.

The derivation form ula can be expressed in a

more compact form by introducing replicas and

suitable averages. In fact, let us introd uce the state !

acting on functions F of i as follows

!ðFði ÞÞ ¼ Z

1

expð

ﬃﬃ

ÞFðiÞ½34

together with the associated product state  acting

on replicated configuration spaces i

, i

, ..., i

.By

performing also a global E average, finally we define

the averages

hFi

 EðFÞ½35

where the subscript is introduced in order to recall

the t dependence of these averag es.

Then, eqn [32] can be written in a more compact

form

E log

expð

ﬃﬃ

Þ¼

i

i½36

Our basic comparison argument will be based on

the following very simple theorem.

Theorem 2 Let U

and

, for i = 1, ..., K, be

independent families of centered Gaussian random

variables, whose covariances satisfy the inequalities

for generic configurations

EðU

ÞS

 Eð

Þ

½37

and the equalities along the diagonal

EðU

ÞS

¼ Eð

Þ

½38

then for the quenched averages we have the inequal-

ity in the opposite sense

E log

expðU

ÞE log

expð

Þ½39

where the w

 0 are the same in the two

expressions.

Considerations of this kind are present in the

mathematical literature, as mentioned, for example,

in Talagrand (2003).

The proof is extremely simple and amounts to a

straightforward calculation. In fact, let us consider

the interpolating expression

E log

expð

ﬃﬃ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1  t

Þ½40

where 0  t  1. Clearly, the two expressions under

comparison correspond to the values t = 0andt = 1,

respectively. By taking the derivative with respect to

t, with the help of the previous derivation formula,

we arrive at the evaluation of the t derivati ve in

the form

E log

expð

ﬃﬃ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1  t

1

expð

ﬃﬃ

ÞðS



2

expð

ﬃﬃ

 expð

ﬃﬃ

ÞðS



½41

From the conditions assumed for the covariances,

we immediately see that the interpolating function is

nonincreasing in t, and the theorem follows.

The derivation formula and the comparison

theorem are not restricted to the Gaussian case.

Generalizations in many directions are possi ble. For

the diluted spin glass mod els and optimization

problems we refer, for example, to Franz and

Leone (2003), and to De Sanctis (2005), and

references therein.

660 Spin Glasses

Thermodynamic Limit and the

Variational Bounds

We give here some striking applications of the basic

comparison theorem. Guerra and Toninelli (2002)

have given a very simple proof of a long-awaited

result, about the convergence of the free energy per

site in the thermodynamic limit. Let us show the

argument. Let us consider a system of size N and

two smaller systems of sizes N

and N

respectively,

with N = N

þ N

, as before in the ferromagnetic

case. Let us now compare

E log Z

ð; h; JÞ

¼ E log



...

exp 

ﬃﬃﬃﬃﬃ

KðÞ

 exp h



½42

with

E log



...

exp 

ﬃﬃﬃﬃﬃﬃﬃ

ð1Þ

ð

ð1Þ

 exp 

ﬃﬃﬃﬃﬃﬃﬃ

ð2Þ

ð

ð2Þ

exp h



 E log Z

ð; h; JÞþE log Z

ð; h; JÞ½43

where 

(1)

stands for 

, i = 1, ..., N

, and 

(2)

for



, i = N

þ 1, ..., N. Covariances for K

(1)

and K

(2)

are expressed as in [28], but now the overlaps are

substituted with the partial overlaps of the first and

second block, q

and q

, respectively. It is very

simple to apply the comparison theorem. All one has

to do is to observe that the obvious

Nq ¼ N

þ N

½44

analogous to [10], implies, as in [12],



½45

Therefore, the comparison gives the superaddivity

property, to be compared with [9],

E log Z

ð; h; JÞ

 E log Z

ð; h; JÞþE log Z

ð; h; JÞ½46

From the superaddivity property the existence of the

limit follows in the form

lim

N!1

1

E log Z

ð; h; JÞ

¼ sup

1

E log Z

ð; h; JÞ½47

to be compared with [13].

The second application is in the form of the

Aizenman–Sims–Starr generalized variatio nal princi-

ple. Here, we will need to introduce some auxiliary

system. The denumerable configuration space is

given by the values of  = 1, 2, .... We introduce

also weights w



 0 for the  system, and suitably

defined overlaps between two generic configurations

p(, 

), with p(, ) = 1.

A family of centered Gaussian random variables

K(), now indexed by the configurations , will be

defined by the covariances

Eð

KðÞ

Kð

ÞÞ ¼ p

ð; 

Þ½48

We will also need a family of centered Gaussian

random variables 

(), indexed by the sites i of our

original system and the configurations  of the

auxiliary system, so that

Eð

ðÞ

ð

ÞÞ ¼ 

pð; 

Þ½49

Both the probability measure w



, and the overlaps

p(, 

) could depend on some additional external

quenched noise, which does not appear explicitly in

our notation.

In the following, we will denote by E averages

with respect to all random variables involved.

In order to start the comparison argument, we

will con sider first the case where the two  and 

systems are not coupled, so as to appear factorized

in the form

E log



...



exp 

ﬃﬃﬃﬃﬃ

KðÞ

 exp 

ﬃﬃﬃﬃﬃ

KðÞ

exp h



 E log Z

ð; h; JÞþE log



 exp 

ﬃﬃﬃﬃﬃ

KðÞ

½50

In the second case, the K fields are suppressed and

the coupling between the two systems will be taken

in a very simple form, by allowing the  field to act

as an external field on the  system. In this way

the ’s appear as factorized, and the sums can

be explicitly performed. The chosen form for the

second term in the comparison is

E log



...



exp 



ðÞ

exp h



 N log 2 þ E log



ðc

...c

Þ½51

Spin Glasses 661