Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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subgroup of Mo¨ bius transformations, and their com-

mutation relations are simply

½L

; L

¼ðm  nÞL

mþn

½4

In fact, [4] describes also the commutation relations

of all generators L

with n 2 Z: this is the Lie

algebra of (locally defined) 2D conformal transfor-

mations – it is called the Witt algebra.

The General Structure of Conformal

Field Theory

A 2D conformal field theory is determined (like any

other field theory) by its space of states and the

collection of its correlation functions (vacuum

expectation values). The space of states is a vector

space H (which, in many interesting examples, is a

Hilbert space), and the correlation functions are

defined for collections of vectors in some dense

subspace of H. These correlation functions are

defined on a 2D (Euclidean) space. We shall mainly

be interested in the case where the underlying 2D

space is a closed compact surface; the other

important case concerning surfaces with boundaries

(whose analysis was pioneered by Cardy) will be

reviewed elsewhere (see the article Boundary Con-

formal Field Theory). The closed surfaces are

classified (topol ogically) by thei r genus g, which

counts the number of handles; the simplest such

surface which we shall mainly consider is the sphere

with g = 0, the surface with g = 1 is the torus, etc.

One of the special features of conformal field

theory is the fact that the theory is naturally defined

on a Riemann surface (or complex curve), that is, on

a surface that possesses suitable complex coordi-

nates. In the case of the sphere, the complex

coordinates can be taken to be those of the complex

plane that cover the sphere except for the point at

infinity; complex coordinates around infinity are

defined by means of the coordinate function

(z) = 1=z that maps a neighborhood of infinity to

a neighborhood of zero. With this choice of complex

coordinates, the sphere is usually referred to as the

Riemann sphere, and this choice of complex

coordinates is, up to Mo¨ bius transformations,

unique. The correlation functions of a conformal

field theory that is defined on the sph ere are thus of

the form

h0jVð

; z

;



ÞVð

; z

;



Þj0i½5

where V( , z,



z) is the field that is associated to the

state , and z

and



are compl ex conjugates of one

another. Here j0i denotes the SL(2, C)=Z

-invariant

vacuum. The usual locality assumption of a 2D

(bosonic) Euclidean quantum field theory implies

that these correlation functions are independent of

the order in which the fields appear in [5].

It is conventional to think of z = 0 as describing

‘‘past infinity,’’ and z = 1 as ‘‘future infinity’’; this

defines a time direction in the Euclidean field theory

and thus a quantization scheme (radial quantiza-

tion). Furthermore, we identify the space of states

with the space of ‘‘incoming’’ states; thus, the state

is simply

¼ Vð ; 0; 0Þj0i½6

We can think of z

and



in [5] as independent

variables, that is, we may relax the constraint that



is the complex conjugate of z

. Then we have two

commuting actions of the conformal group on these

correlations functions: the infinitesimal action on

the z

variables is described (as before) by the L

generators, while the generators for the action on

the



variables are



. In a conformal field theory,

the space of states H thus carries two commuting

actions of the Witt algebra. The generator L



can be ident ified with the time-translation operator,

and thus describes the energy operator. The space of

states of the physical theory should ha ve a bounded

energy spectrum, and it is thus natural to assume

that the spectrum of both L

and



is bounded

from below; representations with this property are

usually called positive-energy representations. It is

relatively easy to see that the Witt algebra does not

have any unitary positive-energy representations

except for the trivial representation. However, as is

common in many instances in quantum theory, it

possesses many interesting projective representa-

tions. These projective representations are conven-

tional representations of the central extension of the

Witt algebra

½L

; L

¼ðm nÞL

mþn

mðm

1Þ

m;n

½7

which is the famous Virasoro algebra. Here c is a

central element that commutes with all L

;itis

called the central charge (or conformal anomaly).

Given the actions of the two Virasoro algebras

(that are generated by L

and



), one can

decompose the space of states H into irreducible

representations as

H¼





½8

where H

(



) deno tes the irreducible representations

of the algebra of L

(



), and M

2 N

describe the

multiplicities with which these combination s of

representations occur. (We are assuming here that

the space of states is completely reducible with

318 Two-Dimensional Conformal Field Theory and Vertex Operator Algebras

respect to the action of the two Virasoro algebras;

examples where this is not the case are the so-called

logarithmic conformal field theories.) The positive-

energy representations of the Virasoro algebra are

characterized by the value of the central charge, as

well as the lowest eigenvalue of L

; the state

whose L

eigenvalue is smallest is called the highest-

weight state, and its eigenvalue L

= h is the

conformal weight. The conformal weight determines

the conformal transformation properties of : under

the conformal transformation z 7!f (z),



z 7!



f (



z), we

have

Vð ; z;



zÞ

7! f

ðzÞðÞ



zÞ





Vð ; f ðzÞ;



f ð



zÞÞ ½9

where L

= h and



h . The corresponding

field V( ; z,



z) is then called a primary field; if [9]

only holds for the Mo¨ bius transformations [3], the

field is called quasiprimary.

Since L

with m > 0 lowers the conformal weight

of a state (see [7]), the highest-weight state is

necessarily annihilated by all L

(and



)withm > 0.

However, in general the L

(and



) with m < 0

do not an nihilate ; they generate the descendants

of that lie in the same representation. Their

conformal transformation property is more compli-

cated, but can be deduced from that of the primary

state [9], as well as the commutation relations of

the Virasoro algebra.

The Mo¨ bius symmetry (whose generators annihi-

late the vacuum) determines the 1-, 2- and 3-point

functions of quasiprimary fields up to numerical

constants: the 1-point function vanishes, unless

h =



h = 0, in which case h0jV( ; z,



z)j0i= C, inde-

pendent of z an d



z. The 2-point function of

and

vanishes unless h

= h

and



; if the

conformal weights agree, it takes the form

h0jVð

; z

;



ÞVð

; z

;



Þj0i

¼ C ðz

 z

2h







2



½10

Finally, the structure of the 3-point function of three

quasiprimary fields

, and

h0jVð

; z

;



ÞVð

; z

;



ÞVð

; z

;



Þj0i

¼ C

i<j

ðz

 z

ðh

h















½11

where for each pair i < j, k labels the third field, that

is, k 6¼ i and k 6¼ j. The Mo¨ bius symmetry also

restricts the higher correlation function of quasi-

primary fields: the 4-point function is determined up

to an (undetermined) function of the Mo¨ bius

invariant cross-ratio, and similar statements also

hold for n-point functions with n  5. The full

Virasoro symmetry must then be used to restrict

these funct ions further; however, since the genera-

tors L

with n 2 do not annihilate the vacuum

j0i, the Virasoro symmetry leads to Ward identities

that cannot be easily evaluated in genera l. (In typical

examples, the Ward identities give rise to differential

equations that must be obeyed by the correlation

functions.)

Chiral Fields and Vertex Operator

Algebras

The decomposition [8] usually contains a special

class of states that transform as the vacuum state

with respect to



; these states are the so-called

chiral states. (Similarly, the states that transform as

the vacuum state with respect to L

are the

antichiral states.) Given the transformation proper-

ties described above, it is not difficult to see that the

corresponding chiral fields V( ; z,



z) only depend on

z in any correlation function, that is V( ; z,



z) 

V( , z). (Similarly, the antichiral fields only depend



z.) The chiral fields always contain the field

corresponding to the state L

2

j0i, that describes a

specific component of the stress–energy tensor.

In conformal field theory, the product of two

fields can be expressed again in terms of the fields of

the theory. The conformal symmetry restricts the

structure of this operator product expansion:

Vð

; z

;



ÞVð

; z

;



ðz

 z











r;s0

Vð

r;s

; z

;



Þðz

 z







½12

where 

and 

are real numbers, and r, s 2 N

(Here i labels the conformal representations that

appear in the operator-product expansion, while r

and s label the different descendants.) The actual

form of this expansion (in pa rticular, representations

that appear) can be read off from the correlation

functions of the theory since the identity [12] has to

hold in all correlation functions.

Given that the chir al fields only depend on z in all

correlation functions, it is then clear that the

operator-product expansion of two chiral fields

again only contains chiral fields. Thus, the subspace

of chiral fields closes under the operator-product

expansion, and therefore defines a consistent (sub)-

theory by itself. This subtheory is sometimes referred

to as a meromorphic conformal field theory (Goddard

1989). (Obviously, the same also applies to the

subtheory of antichiral fields.) The operator-product

Two-Dimensional Conformal Field Theory and Vertex Operator Algebras 319

expansion defines a product on the space of mero-

morphic fields. This product involves the complex

parameters z

in a nontrivial way, and therefore does

not directly define an algebra structure; it is, however,

very similar to an algebra, and is therefore usually

called a vertex operator algebra in the mathematical

literature. The formal definition involves formal

power series calculus and is quite complicated; details

can be found in (Frenkel–Lepowski–Meurman 1988).

By virtue of its definition as an identity that holds

in arbitrary correlation functions, the operator-

product expansion is associative, that is,

Vð

; z

;



ÞVð

; z

;



ÞðÞVð

; z

;



¼ Vð

; z

;



Þ Vð

; z

;



ÞVð

; z

;



ÞðÞ½13

where the brackets indicate which operator-product

expansion is evaluated first. If we consider the case

where both

and

are meromorphic fields, then

the associativity of the operator-product expansion

implies that the states in H form a representation of

the vertex operator algebra. The same also holds for

the vertex operator algebra associated to the anti-

chiral fields. Thus the meromorphic fields encode in

a sense the symmetries of the underlying theory: this

symmetry always contains the conformal symmetry

(since L

2

j0i is always a chiral field, and



2

j0i

always an antichiral field). In general, however, the

symmetry may be larger. In order to take full

advantage of this symmetry, it is then useful to

decompose the full space of states H not just with

respect to the two Virasoro algebras, but rather with

respect to the two vertex operator algebras; the

structure is again the same as in [8], where,

however, each H

and



is now an irreducible

representation of the chiral and antichiral vertex

operator algebra, respectively.

Rational Theories and Zhu’s Algebra

Of particular interest are the rational conformal

field theories that are characterized by the property

that the corresponding vertex operator algebras only

possess finitely many irreducible representations.

(The name ‘‘rational’’ stems from the fact that the

conformal weights and the central charge of these

theories are rational numbers.) The simplest exam-

ple of such rational theories are the so-called

minimal models, for which the vertex operator

algebra descr ibes just the conformal symmetry:

these models exist for a certain discrete set of

central charges c < 1 and were fir st studied by

Belavin, Polyakov, and Zamolodchikov in 1984.

(Their paper is contained in the reprint volume of

Goddard and Olive (1988).) It was this seminal

paper that started many of the modern develop-

ments in conformal field theory. Another important

class of examples are the Wess–Zumino–Witten

(WZW) models that describe the world-sheet theory

of strings moving on a compact Lie group. The

relevant vertex operator algebra is then generated by

the loop group symmetries. There is some evidence

that all rational conformal field theories can be

obtained from the WZW models by means of two

standard constructions, namely by considering

cosets and taking orbifolds; thus rational conformal

field theory seems to have something of the flavor of

(reductive) Lie theory.

Rational theories may be characterized in terms of

Zhu’s algebra that can be defined as follows. The

chiral fields V( , z) that only depend on z must by

themselves define local operators; they can therefore

be expanded in a Laurent expansion as

Vð ; zÞ¼

n2Z

ð Þz

nh

½14

where h is the conformal weight of the state . For

example, for the case of the holomorphic compo-

nent of the stress–energy tensor one finds

TðzÞ¼

n2Z

n2

½15

where the L

are the Virasoro generators. By the

state/field correspondence [6], it then follows that

ð Þj0i¼0 for n > h ½16

and that

h

ð Þj0i¼ ½17

(For an example of the above component of the

stress–energy tensor, [16] implies that L

1

j0i=

j0i= L

j0i= 0 for n  0 – thus the vacuum is in

particular SL(2, C)=Z

invariant. Furthermore, [17]

shows that L

2

j0i is the state corresponding to this

component of the stress–energy tensor.) We denote

by H

the space of states that can be generated by

the action of the modes V

( ) from the vacuum j0i.

On H

we consid er the subspace O(H

) that is

spanned by the states of the form

ðNÞ

ð Þ; N > 0 ½18

where V

(N)

( ) is defined by

ðNÞ

ð Þ¼

n¼0



nN

ð Þ½19

and h is the conformal weight of . Zhu’s algebra is

then the quotient space

A ¼H

=OðH

Þ½20

320 Two-Dimensional Conformal Field Theory and Vertex Operator Algebras

It actually forms an associative algebra, where the

algebra structure is defined by

?¼ V

ð0Þ

ð Þ ½21

This algebra structure can be identified with the

action of the ‘‘zero-mode algebra’’ on an arbitrary

highest-weight state.

Zhu’s algebra captures much of the structure of

the (chiral) conformal field theory: in particular, it

was shown by Zhu in 1996 that the irreducible

representations of A are in one-to-one correspon-

dence with the representations of the full vertex

operator algebra. A conformal field theory is thus

rational (in the above, physicists’, sense) if Zhu’s

algebra is finite dimensional. (In the mathematics

literature, a vertex operator algebra is usually called

rational if in addition every positive-energy repre-

sentation is completely reducible. It has been

conjectured that this is equivalent to the condition

that Zhu’s algebra is semisimple.)

In practice, the determination of Zhu’s algebra is

quite complicated, and it is therefore useful to

obtain more easily testable conditions for rational-

ity. One of these is the so-called C

condition of

Zhu: a vertex operator algebra is C

-cofinite if the

quotient space H

) is finite dimensional,

where O

) is spanned by the vectors of the form

nh

ð Þ; n  1 ½22

It is easy to show that the C

-cofiniteness condition

implies that Zhu’s algebra is finite dimensio nal.

Gaberdiel and Neitzke have shown that every

-cofinite vertex operator algebra has a simple

spanning set; this observation can, for example, be

used to prove that all the fusion rules (see below) of

such a theory are finite.

Fusion Rules and Verlinde’s Formula

As explained above, the correlation function of three

primary fields is determined up to an overall

constant. One important question is whether or not

this constant actually vanishes since this determines

the possible ‘‘couplings’’ of the theory. This infor-

mation is encoded in the so-called fusion rules of the

theory. More precisely, the fusion rules N

2 N

determine the multiplicity with which the represen-

tation of the vertex operator algebra labeled by k

appears in the operator-product expansion of the

two representations labeled by i and j.

In 1988, Verlinde found a remarkable relation

between the fusion rules of a vertex operator

algebra and the modular transformation properties

of its characters. To each irreducible representation

of a vertex operator algebra, one can define the

character



ðÞ¼tr

ðc=24Þ



; q ¼e

2i

½23

For rational vertex operator algebras (in the math-

ematical sense) these charac ters transform under the

modular transformation  7!1= as

ð1=Þ¼



ðÞ½24

where S

are constant matrices. Verlinde’s formula

then states that, at least for unitary theories,



½25

where the ‘‘0’’ label denotes the vacuum representa-

tion. A general argument for this formula has been

given by Moore and Seiberg in 1989; very recently,

this has been made more precise by Huang.

Modular Invariance and the Conformal

Bootstrap

Up to now, we have only considered conformal field

theories on the sphere. In order for the theory to be

well defined also on higher-genus surfaces, it is

believed that the only additional requirement comes

from the consistency of the torus amplitudes. In

particular, the vacuum torus amplitude must only

depend on the equivalence class of tori that is

described by the modular parameter  2 H,upto

the discrete identifications that are generated by the

usual action of the modular group SL(2, Z ) on the

upper half-plane H. For the theory with decomposi-

tion [8] this requires that the function

Zð; Þ¼



ðÞ

ðÞ½26

is invariant under the action of SL(2, Z). This is a

very powerful constraint on the multiplicity matrices

that has been analyzed for variou s vertex

operator algebras. For example, Cappelli, Itzykson,

and Zuber have shown that the modular invariant

WZW models corresponding to the group SU(2)

have an A–D–E classification. The case of SU(3) was

solved by Gannon, using the Galois symmetries of

these rational conformal field theories.

The condition of modular invariance is relatively

easily testable, but it does not, by itself, guarantee that

a given space of states H comes from a consistent

conformal field theory. In order to construct a

consistent conformal field theory, one needs to solve

the conformal bootstrap, that is, one has to determine

Two-Dimensional Conformal Field Theory and Vertex Operator Algebras 321

all the normalization constants of the correlators so

that the resulting set of correlators is local and

factorizes appropriately into 3-point correlators

(crossing symmetry). This is typically a difficult

problem which has only been solved explicitly for

rather few theories, for example, the minimal models.

Recently, it has been noticed that the conformal

bootstrap can be more easily solved for the corre-

sponding boundary conformal field theory. Further-

more, Fuchs, Runkel, and Schweigert have shown that

any solution of the boundary problem induces an

associated solution for conformal field theory on

surfaces without boundary. This construction relies

heavily on the relation between 2D conformal field

theory and 3D topological field theory (Turaev 1994).

See also: Boundary Conformal Field Theory;

Compactification of Superstring Theory; Current Algebra;

Knot Theory and Physics; String Field Theory;

Superstring Theories; Symmetries in Quantum Field

Theory of Lower Spacetime Dimensions.

Further Reading

Di Francesco P, Mathieu P, and Se´ne´chal D (1997) Conformal

Field Theory. New York: Springer.

Frenkel I, Lepowski J, and Meurman A (1988) Vertex Operator

Algebras and the Monster. Boston, MA: Academic Press.

Gaberdiel MR (2000) An introduction to conformal field theory.

Reports on Progress in Physics 63: 607 (arXiv:hep-th/

9910156).

Gannon T (1999) Monstrous moonshine and the classification of

CFT, arXiv:math.QA/9906167.

Gannon T (2006) Moonshine beyond the Monster: The Bridge

Connecting Algebra, Modular Forms and Physics (to appear).

Cambridge: Cambridge University Press.

Gawedzki K (1999) Lectures on conformal field theory. In:

Quantum Fields and Strings: A Course for Mathematicians,

vol. 2. Providence, RI: American Mathematical Society.

Ginsparg P (1988) Applied Conformal Field Theory. Lectures

Given at the Les Houches Summer School in Theoretical

Physics. Elsevier.

Goddard P (1989) Meromorphic conformal field theory. Infinite

Dimensional Lie Algebras and Lie Groups: Proceedings of the

CIRM Luminy Conference, 1988, 556. Singapore: World

Scientific.

Goddard P and Olive DI (1988) Kac–Moody and Virasoro

Algebras, A Reprint Volume for Physicists. Singapore: World

Scientific.

Kac VG (1998) Vertex Algebras for Beginners. Providence, RI:

American Mathematical Society.

Pressley A and Segal GB (1986) Loop Groups. Oxford: Clarendon.

Schweigert C, Fuchs J, and Walcher J (2000) Conformal field

theory, boundary conditions and applications to string theory,

arXiv:hep-th/0011109.

Turaev VG (1994) Quantum Invariants of Knots and 3-Manifolds.

Berlin: de Gruyter.

Two-Dimensional Ising Model

B M McCoy, State University of New York at Stony

Brook, Stony Brook, NY, USA

Introduction

The Ising model is a model of a classical ferro-

magnet on a lattice first introduced in 1925 in the

one-dimensional case by E Ising. At each lattice site

there is a ‘‘spin’’ variable , which takes on the

values þ1 (spin up) and 1 (spin down). The mutual

interaction energy of the pair of spins 



and 



where  and 

are nearest neighbors, is E(, 

)if





= 



and is E(, 

)if



= 



. In addition, the

spins can interact with an external magnetic field as

H



. On a square lattice, where j specifies the row

and k specifies the column, the i nteraction energy

for the homogeneous case where E

(, 

)and

(, 

) are independent of the position , 

may

be explicitly written as

EðHÞ¼

j;k

½E



j;k



j;kþ1

þ E



j;k



jþ1;k

þ H

j;k

½1

This very simple model [1] has the remarkable

property that in two dimensions at H = 0 many

properties of physical interest can be computed

exactly. Furthermore, the model has a ferromagnetic

phase transition at a critical temperature T

,at

which the specific heat diverges and the magnetic

susceptibility diverges to infinity and below which

there is a nonzero spontaneous magnetization. In

addition, the microscopic correlations between spins

can also be exactly computed. These exact calcula-

tions are the basis of the modern theory of second-

order phase trans itions used to analyze real ferro-

magnets and real fluids near their critical points in

both two and three dimensions. The model may also

be interpreted as a lattice gauge theory .

Solvability

The solvability of the Ising model at H = 0 was

discovered by Onsager in 1944 in one of the most

profound and inventive papers ever written in

mathematical physics. Onsager discovered that the

model possesses an infinite-dimensional symmetry,

which allowed him to exactly compute the free

322 Two-Dimensional Ising Model

energy per site. This symmetry is generated by the

relations

½A

; A

¼4G

lm

½G

; A

¼2A

mþl

 2A

ml

½G

; G

¼0

½2

This algebra of Onsager is a subalgebra of what is

now called the loop algebra of the Lie algebra Sl

and it is the first infinite-dimensional algebra to be

used in physics.

In the 60 years since Onsager first computed the

free energy, several other methods of exact solution

have been found. In 1949, Kaufman reduced the

computation of the free energy to a problem of free

fermions. A closely related combinatorial method

was invented by Kac and Ward, Hurst and Green,

and by Kastelyn. Baxter (1982) has computed the

free energy by means of star triangle equations and

functional equations in his book.

The fermionic and the combinatorial methods are

powerful enough to compute the correlation func-

tions but are not generalizable to other models. The

functional equation methods of Baxter generalize to

many other important models but they do not give

correlation functions. The re are still aspects of

Onsager’s method that remain unexplored.

The free energy per site in the thermodynamic

limit is defined as

F ¼k

T lim

N!1

1

ln Z

ðHÞ½3

where N is the total number of sites of the lattice

and the partition function Z

(H) is defined as

ðHÞ¼

all ¼1

EðHÞ=k

½4

with the sum being over all values 

j, k

= 1 and k

is Boltzmann’s constant. The result of Onsager is

that, at H = 0,

F=k

T ¼ln 2 þ

8

2

d

2

d

cosh 2E

 cosh 2E

T sinh 2E

T cos 

sinh 2E

T cos 

½5

This free energy has a singularity at a temperature

defined from

sinhð2E

Þsinhð2E

Þ¼1 ½6

and near T

the specific heat diverges as

c 



sinh

þ 2E



þE

sinh



ln j1  T=T

j½7

The next property to be computed was the

spontaneous magnetization, which is usually

defined as



¼ lim

H!0

MðHÞ½8

However, because solution is only available at

H = 0, this definition cannot be used and instead



is computed from an alternative definition in

terms of the spin–spin correlation function

<

0;0



M;N

> ¼

ð0Þ

¼1



0;0



M;N

Eð0Þ=k

½9



¼ lim

þN

<

0;0



M:N

> ½10

The result for M



, first announced by Onsager in

1949, is



ð1  k

1=8

for T  T

0 for T > T



½11

where

k ¼ðsinh 2E

T sinh 2E

TÞ

1

½12

A key point in the computation of the magnetiza-

tion [11] from [9] is that the spin–spin correlation

function can be written as a determi nant. In fact,

there are many such differe nt, but equal, determi-

nental representations and the size of the smallest

one in general is 2(jMjþjNj). The simplest case is

the diagonal correlation

<

0;0



N ;N

> ¼

1

2

 a

1N

1

 a

2N

 a

3N



N1

N2

N3

 a



½13

where

2





de

in

1  ke

i

1  ke

i



1=2

½14

Determinants of the form [13], where the elements

on each diagonal are equal, are called Toeplitz.

The study of the spin–spin correlations of the

Ising model provides a microscopic picture of the

behavior of the ferromagnet near the phase transition

temperature T

, and an entire branch of mathematics

has developed from the study of the behavior of

Toeplitz determinants when the size is large. The first

such mathematical advance was the discovery by

Szego¨ of a general formula for the limit as N !1,

from which the magnetization [11] is computed.

Two-Dimensional Ising Model 323

The simplest result for the approach to this N !1

limit is the behavior of the diagonal correlation at

T = T

(k = 1), where [13] exactly reduces to

<

0;0



N ;N

> ¼





N1

l¼1

1 



lN

½15

which behaves as N !1as

<

0;0



N ;N

>  AN

1=4

½16

where A  0.6450  is a transcendental constant.

Further results for large N and T fixed are for

T < T

(k < 1),

<

0;0



N ;N

>  M



1 þ

2ðNþ1Þ

N

ðk

2

1Þ

þ

()

½17

and for T > T

(k > 1),

<

0;0



N;N

> 

N

ðNÞ

1=2

ð1  k

2

1=4

þ  ½18

By comparing [16] with [17] and [18], we see that

at T = T

the correlations decay algebraically but for

T 6¼ T

the decay is exponential. It is useful to write

the exponential in [17] for T < T

N

¼ e

N=



with 

1

¼ln k ½19

and in [18] for T > T

¼ e

N=

with 

1

¼ ln k ½20

The quantity  is called the correlation length and as

T ! T

the correlation length diverges as





j1  kj

1

¼ const: jT  T

1

½21

A more profound property of the correlations is

that they satisfy differential and difference equa-

tions. It was found by Jimbo and Miwa (1980) that

the diagonal correlation function satisfies the non-

linear differential equation related to the sixth

Painleve´ function

tðt  1Þ



¼ N

ðt  1Þ

d

 



 4

d

ðt  1Þ

d

  



d

 



½22

where for T < T

we set t = k

2

and



ðtÞ¼tðt  1Þ

ln <

0;0



N;N

> 

½23

and for T > T

we set t = k

and



ðtÞ¼tðt  1Þ

ln <

0;0



N;N

> 

½24

Furthermore it was found by McCoy et al. (1981)

that for a given temperature the general two spin

correlation function and all multipoint correlations

satisfy quadratic nonlinear partial difference equa-

tions in the locations of the spins.

Scaling Theory

It is evident that the results [17] and [18] do not

reduce to [16] when k !1. Therefore, in order to

uniformly characterize the behavior of the correla-

tion function in the critical region near T

,itis

necessary to introduce what is called the scaling

function. This uniform expansion is obtained by

introducing a scaled length defined as

r ¼ N= ½25

and considering the joint (scaling limit) where

N !1 and T !1 with r fixed ½26

We define the scaled correlation function as



ðrÞ¼ lim

scaling

2



<

0;0



N;N

> ½27

where the subscr ipt  means that the limit is taken

from T > T

or T < T

, respectively, M



is the

spontaneous magnetization [11] and

¼ðk

2

 1Þ

1=8

½28

This concept of the scaling limit and scaling

function is very general and can be defined for any

system with a criti cal point that has an order

parameter like M



that vanishes at T

and a

correlation length that diverges at T

. However,

the Ising model has the further remarkable property

discovered by Wu et al. (197 6) that the scaled

correlation function may be explicitly expressed in

terms of a function which satisfies an ordinary

nonlinear differential equation. Specifically,



ðrÞ¼

½1   ðr=2Þðr=2Þ

1=2

 exp

r=2



2

½ð1  

ð

½29

where the function (r) satisfies the Painleve´ III

equation



ð





þ 

 

1

½30

324 Two-Dimensional Ising Model

with the boundary condition that

ð rÞ1  2K

ð2rÞ as r !1 ½31

where K

(r) is the modified Bessel function of the

third kind and

 ¼ 1= ½32

The leading behavior of G



(r) for r !1is

ðrÞK

ðrÞ½33



ðrÞ1 þ 



½K

ðrÞK

ðrÞ

 rK

ðrÞK

ðrÞþ

ðrÞ



½34

where K

(z) is the modified Bessel function of the

third kind. When  is given by [32] these r !1

limits of G



(r) agree with the behavior of

<

0, 0



N, N

> for N  1 and jT  T

j small with

NjT  T

j1 which is obtained from [18] and

[17]. The behavior of G



(r) for r !0 with the value

of  given by [32] is



ðrÞ¼const: r

1=4

½35

where the constant agrees with that computed from

the result [16] for <

0, 0



N, N

> at T = T

for N  1.

For other values of the boundary condition constant ,

the scaling function G



(r) diverges with a power

which differs from 1/4. The computation of the

constant in [35] requires the evaluation of a nontrivial

integral involving the Painleve´ III function.

The agreement of the limits r !1 and r !0of

the function G



(r) with the lattice results near T

means that this scaling function uniformly inter-

polates between T 6¼ T

and T = T

and that the

lattice size (defined here as unity) and the self-

generated correlation length  are the only two

length scales in the theory. This feature that the

system generates only one new length scale near T

is referred to as one length scale scaling.

Susceptibility

The final quantity of macroscopic thermodynamic

interest is the magnetic susceptibility

ðTÞ¼

@MðH Þ



H¼0

½36

which is expressed in terms of the spin–spin

correlation function as

ðTÞ¼

M;N

f<

0;0



M;N

>  M



g½37

The susceptibility may be studied by using the

determinental expression for the correlation func-

tion. The simplest result is obtained (for the

isotropic case, E

= E

) by using the scaling form

[27] to find for T  T

that

T



ðTÞM





2

drrfG



 



g½38

where 

= 0 and 



= 1. and thus 



(T) diverges

at T ! T





ðTÞC



jT  T

7=4

½39

where C



are transcendental constants given as

integrals over the scaling function G



(r), which

were first evaluated by Barouch et al. in 1973 as



¼ 0:0255369719 ...;

¼ 0:9625817322 ...

½40

Critical-Exponent Phenomenology

From the behavior for the Ising model of the

specific heat, magnetization, susceptibility, corre-

lation length, and the correlation at T

given

above we abstract for general systems the phe-

nomenological critical-exponent parametrization

for T !T

 of

c  A



jT  T





½41

M  A

 Tj



½42

  A





jT  T





½43

  A





jT  T





½44

and at T = T

for R !1

<



>  A



d2þ

where d is the dimension ½45

The exponents 



,



, 



above and below T

are usually

found to be equal, and the exponent  is usually called the

anomalous dimension. If it is assumed that the scaling

function [27] exists and that one length scale scaling holds

then the exponents are related by what are called scaling

laws, such as

2 ¼ 



ðd  2 þ Þ½46





þ 2  



¼ 2 ½ 47

d



¼ 2  



½48

Thus, from the properties of the Ising model near

, we have obtained a phenomenology for use on

all systems near the critical point.

Two-Dimensional Ising Model 325

Fuchsian Equations and Natural

Boundaries for Susceptibility

This critical phenomenology, howev er, has not

taken into account the fact that the susceptibility

is a much more complicated function than either

the spontaneous magnetization [11] or the free

energy [5], which have only isolated singularities at

= 1, and that there is more structure to the

susceptibility than the singularity of [39].

For arbitrary T, the susceptibility was shown by

Wu, McCoy, Tracy, and Barouch to be expressible

in the form





ðTÞ¼M





ðjÞ

ðTÞ½49

where in the sum j is odd (even) for T above (below)

. The quantities



(j)

(T) are explicitly given as

j-fold integrals of algebraic functions and thus will

satisfy linear differential equations with polynomial

coefficients. Such functions can have only isolated

singularities. The function



(1)

(T) is elementary and

has a double pole at T

and



(2)

(T) is given in terms

of complete elliptic integrals. Quite recently,

remarkable Fuchsian linear differential equations

for



(3)

(T) and



(4)

(T) of sevent h and tenth orders,

respectively, have been obtained by Zenine, Bouk-

raa, Hassani, and Maillard for the isotropic lattice.

Furthermore, it was shown by Orrick et al. (2001)

that



(j)

has singularities in the complex T plane at

coshð2E

=kTÞcoshð2E

=kTÞ

 sinhð2E

=kTÞcosð2=jÞ

 sinhð2E

=kTÞcosð2 m

=jÞ¼0 ½50

with m, m

= 1, 2, ..., j. The form of the singularity



(j)

(T) for T > T

is as



ðj

3Þ=2

ln  ½51

and, for T < T

,itisas



ðj

3Þ=2

½52

where  measures the deviation from the singular

point [50]. These singularities become dense as

j !1 and, therefore, the singularity at T = T

not isolated and instead the critical point is

embedded in a natural boundary. Such a function

cannot satisfy a linear differential equation of finite

order with polynomial coefficients.

The existence of the natural boundary in the

susceptibility is a new phenomenon which is not

seen in either the free energy or magnetization and

leads to the speculation that in the presence of a

magnetic field the one length scale scaling property

of the model at H = 0 may fail. If this proves to be

correct, there will be physical effects which are not

incorporated in the phenomenological scaling theory

of critical phenomena.

Impure Ising Models

The Ising model may also be studied when the

interaction energies at sites j, k are not chosen to be

independent of position but are allowed to vary

from site to site. When these interactions are chosen

randomly out of some probability distribution, this

is a model of a ferromagnet with frozen (quenched)

impurities. All real systems will be impure to some

extent, so the study of such dirty systems is of great

practical importance.

The special case where the interactions are transla-

tionally invariant in the horizontal direction but are

allowed to vary in a layered fashion from row to row

was introduced by McCoy and Wu in 1968 and

found to be dramatically different from the pure Ising

model described above. In particular, what is a

critical temperature T

in the pure case is now spread

out into a region bounded by the temperatures the

pure model would be critical if all the bonds took on

the minimum or maximum value allowed by the

probability distribution. In this new region, the

correlations (in the direction of translational invar-

iance) are found to decay as a power law which

depends on the temperature; the specific heat is never

infinite but the susceptibility is infinite in an entire

temperature region that includes the temperature at

which the spontaneous magnetization first appears as

T is lowered. The existence of this new region for

Ising models with a general randomness in two and

three dimensions has been demonstrated by Griffiths.

More recently, this effect has been reinterpreted in

terms of impurities in quantum spin chains.

Quantum Field Theory

The Ising model of [1] may be reinterpreted as a two-

dimensional lattice gauge theory of the gauge field

jþ1=2;k

¼1

on the vertical link between ðj; kÞ and ðj þ 1; kÞ

j;kþ1=2

¼1

on the horizontal link between ðj; kÞ

and ðj; k þ 1Þ½53

and a ‘‘Higgs’’ field



j:k

¼1 on the site ðj; kÞ½54

326 Two-Dimensional Ising Model

with the action

¼E

j;k

jþ1=2;k

jþ1;kþ1=2

jþ1=2;kþ1

j;kþ1=2

E

j;k

ð

j;k

jþ1=2;k



jþ1;k

þ

j;k

j;kþ1=2



j;kþ1

Þ½55

If we define

g;h

¼ tanh E

g;h

T ½56

the partition function of the gauge theory is expressed

in terms of the Ising model partition function as

¼½8coshðE

TÞcosh

ðE

TÞz

1=2



ðHÞ½57

where we make the identification

H=k

T ¼

ln z

and E=k

T ¼

ln z

½58

This identification may be extended to correlation

functions. Of particular interest for the gauge theory is

the plaquette–plaquette correlation < P

0, 0

j, l

>,where

j;k

¼ s

jþ1=2;k

jþ1;kþ1=2

jþ1=2;kþ1

j;kþ1=2

½59

which is expressed in terms of the Ising correlations

at H 6¼ 0as

0;0

j;k

>  <P

0;0

¼ sinh

ð2H=k

TÞð<

0;0



j;k

>  <

0;0

Þ½60

To study this correlation further, we need to study

the correlations of the Ising model in nonzero

magnetic field. This has been done by McCoy and

Wu in the scaling limit H !0, T !T

with

h ¼

jT  T

15=8

fixed ½61

for T < T

, where it is found that the scaling

function G(r, h) for small h and large r if

Gðr; hÞ

ahK

ð2 þ h

2=3



Þr

 

1=2

1=2

2r

hae

rh

2=3



½62

where 

are the solutions of

1=3



3=2



þ J

1=3



3=2



¼ 0 ½63

with J

(z) the Bessel function of order n and K

(z)

the modified Bessel function of the third kind.

A field theory is said to possess a particle spectrum

if the Fourier transform of the two-point function

Gðk; hÞ¼

ikr

Gðr; hÞ½64

has poles of the form A

=(k

þ m

), where m

is the

mass of the lth particle. If we note that the Fourier

transform of K

(r)is

ikr

ðrÞ¼

2

þ 1

½65

we see that the Fourier transform of [62] is the sum of

an infinite number of poles. This is to be compared

with the Fourier transform of the scaled corre lation

function G



(r)atH = 0 and T < T

[34], which does

not contain any poles at all and may instead be

interpreted as having a two-particle cut. This phe-

nomenon of a cut at h = 0 breaking up into an infinite

number of poles for h > 0 is a signal that at h = 0 the

theory has free unconfined two-particle states which

become weakly confined by a linear confining

potential for h > 0. This confinement is thought to

be a characteristic of most gauge theories.

See also: Eight Vertex and Hard Hexagon Models;

Holonomic Quantum Fields; Painleve

Equations;

Percolation Theory; Phase Transitions in Continuous

Systems; Statistical Mechanics and Combinatorial

Problems; Toeplitz Determinants and Statistical

Mechanics; Yang–Baxter Equations.