Page 1

arXiv:hep-th/0103263v1 30 Mar 2001

DFTT 4/2001

GEF-TH-04/2001

Correction induced by irrelevant operators

in the correlators of the 2d Ising model

in a magnetic field.

M. Casellea∗, P. Grinzaa†and N. Magnolib‡

aDipartimento di Fisica Teorica dell’Universit` a di Torino and

Istituto Nazionale di Fisica Nucleare, Sezione di Torino

via P.Giuria 1, I-10125 Torino, Italy

bDipartimento di Fisica, Universit` a di Genova and

Istituto Nazionale di Fisica Nucleare, Sezione di Genova

via Dodecaneso 33, I-16146 Genova, Italy

Abstract

We investigate the presence of irrelevant operators in the 2d Ising

model perturbed by a magnetic field, by studying the corrections in-

duced by these operators in the spin-spin correlator of the model. To

this end we perform a set of high precision simulations for the corre-

lator both along the axes and along the diagonal of the lattice. By

comparing the numerical results with the predictions of a perturbative

expansion around the critical point we find unambiguous evidences of

the presence of such irrelevant operators. It turns out that among

the irrelevant operators the one which gives the largest correction is

the spin 4 operator T2+¯T2which accounts for the breaking of the

rotational invariance due to the lattice. This result agrees with what

was already known for the correlator evaluated exactly at the critical

point and also with recent results obtained in the case of the thermal

perturbation of the model.

∗e–mail: caselle@to.infn.it

†e–mail: grinza@to.infn.it

‡e–mail: magnoli@ge.infn.it

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1Introduction

Since the seminal work of Belavin, Polyakov and Zamolodchikov [1] much

progress has been done in the description of 2d statistical models at the

critical point. In particular in all the cases in which the critical point theory

can be identified with one of the so called “minimal models” a complete list of

all the operators of the theories can be constructed. Moreover, it is possible

to write differential equations for the correlators and in some cases find their

explicit expression in terms of special functions. Following these remarkable

results, in these last years a lot of efforts have been devoted to extend them

also outside the critical point. This can be achieved by perturbing the CFT

with one (or more) of its relevant operators. Much less is known in this case.

The most important result in this context is that for some particular choices

of CFT’s and relevant operators these perturbations give rise to integrable

models [2, 3]. In these cases again we have a rather precise description of the

theory. In particular it is possible to obtain the exact asymptotic expression

for the large distance behavior of the correlators [4]. From this information

several important results (and in particular all the universal amplitude ratios)

can be obtained.

Similar efforts have also been devoted in trying to study vacuum expecta-

tion values, amplitude ratios and correlators involving irrelevant operators.

However progress in this direction is limited by the lack of results (both nu-

merical and analytical) from actual statistical mechanics realizations of the

models.

The only notable exception to this state of art is the 2d Ising model.

In this case, thanks to a set of remarkable results on the 2 and 4 point

correlators [5, 6, 7] several interesting results on the contribution of irrelevant

operators have been obtained both in the case of the model at its critical

point [8] and for T ?= Tc[9, 10] (i.e. for the theory perturbed by the energy

operator). Very recently these results have been used to study up to very high

orders the contribution of irrelevant operators to the magnetic susceptibility

for T ?= Tc[10]. In this paper we shall try to extend this analysis to the

case of the magnetic perturbation of the Ising model.

shall look at the contributions due to irrelevant operators to the spin-spin

correlator in presence of an external magnetic field. In particular we shall

mainly concentrate on the energy momentum tensor T¯T, and the operator

T2+¯T2(which appear as a consequence of the breaking due to the lattice of

In particular, we

1

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the full rotational symmetry) which are the most important (i.e. those with

smallest (in modulus) renormalization group eigenvalue) among the irrelevant

operators.

In this case there is no exact expression for the correlators or for the free

energy of the model and the only way that we have to judge of the validity

of the CFT predictions is to compare them with the results of numerical

simulations.Notwithstanding this there is a number of good reasons to

choose exactly this model and this observable. Let us look at them in detail.

• The Ising model in a magnetic field is known to be exactly integrable [3],

thus, even if it is not exactly solvable, a lot of important informations

can all the same be obtained, in particular very precise large distance

expansions exist for the correlators and the vacuum expectation values

of the relevant operators are known in the continuum limit [11].

• The model is an optimal choice from the point of view of numerical

simulations, since very fast and efficient algorithms exist to study it.

• The model can also be realized as a particular case of the so called

IRF (Interaction Round a Face) models [12]. In this framework several

interesting results can be obtained on the spectrum of the model [13].

• By looking at the spin-spin correlator instead of the susceptibility we

may study the effects of the non zero spin operators which appear as

a consequence of the breaking of the rotational symmetry due to the

lattice (see below for a precise definition). These operators also appear

in rotational invariant quantities like the susceptibility, but only at the

second order [14, 15], i.e. at such an high power of the perturbing

constant that they can be observed only if the exact solution of the

model is known (as in the papers [9, 10]) and cannot be seen if one can

only resort to numerical simulations.

• A very interesting result of [10] (which had already been anticipated

twenty years ago by Aharony and Fisher [16] and also confirmed in [17])

is that in the thermal perturbation of the Ising model the energy mo-

mentum tensor T¯T seems to be absent. The first correction which

involves irrelevant operators is thus due to the spin 4 operator T2+¯T2.

This result should also hold at the critical point where it can be inde-

pendently observed by using transfer matrix methods [14]. However,

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apparently, it disagrees with the explicit form of the correlator at the

critical point (which is exactly known on the lattice) which contains a

scalar correction which can only be interpreted as due to the T¯T term.

One of the aims of the present paper is to understand this puzzle.

• If one is interested in studying irrelevant operators it is mandatory to

look at the short distance behaviour of the correlator. In this respect

the natural framework in which one must operate is the so called IRS

expansion [18, 19]. This approach has been recently discussed in great

detail [20] exactly in the case of the Ising model in a magnetic field in

which we are interested here. It is only by using the results of [20] as

input of our analysis that we shall be able to reach the high level of

precision which is needed in order to observe the very small corrections

which are the signatures of the irrelevant operators.

This paper is organized as follows. In sect.2 we shall briefly summarize

some known results on the 2d Ising model in a magnetic field. In sect.3 we

shall discuss the most important contributions due to the irrelevant operators

to the spin-spin correlators and shall evaluate their magnitude and behaviour.

In sect.4 we shall present the numerical simulations that we have performed

and finally in sect.5 we shall discuss the comparison between numerical results

and theoretical predictions. Sect. 6 is devoted to some concluding remarks.

2 The 2d Ising model in a magnetic field.

We shall be interested in the following in the Ising model defined on a 2d

square lattice of size L with periodic boundary conditions, in presence of an

external magnetic field H. The model is defined by the following partition

function:

eβ(?

where the notation ?i,j? denotes nearest neighbour sites in the lattice. In

order to select only the magnetic perturbation, β must be fixed to its critical

value:

β = βc=1

Z =

?

σi=±1

?i,j?σiσj+H?

iσi)

(1)

2log(√2 + 1) = 0.4406868...

3

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by defining hl= βcH we have

Z =

?

σi=±1

eβc?

?i,j?σiσj+hl?

iσi

(2)

The magnetization M(h) is defined as usual:

M(h) ≡1

N

∂

∂hl(log Z)|β=βc= ?1

N

?

i

σi?. (3)

where N ≡ L2denotes the number of sites of the lattice.

Eq.(2) is the typical partition function of a perturbed critical model. With

the choice β = βcthe only perturbing operator is

σl≡1

N

?

i

σi

, (4)

We shall call in the following σl as the spin operator (more precisely the

lattice discretization of the spin operator). Notice that the mean value of σl

coincides with M(h):

?σl? ≡ M(h)

Our goal in this paper is to study the contribution of the irrelevant opera-

tors to the spin-spin correlator. To this end we shall first study the model

at the critical point (sect.2.1), we shall then switch on the magnetic field

(see sect.2.2) and discuss the modifications that it induces in the spin-spin

correlator

(5)

2.1The Ising model at the critical point

2.1.1Operator content.

The Ising model at the critical point is described by the unitary minimal

CFT with central charge c = 1/2 [1]. Its spectrum can be divided into

three conformal families characterized by different transformation properties

under the dual and Z2symmetries of the model. They are the identity, spin

and energy families and are commonly denoted as [I], [σ], [ǫ]. Each family

contains a “primary” field (which gives the name to the entire family) and an

infinite tower of “secondary” field (see below). The conformal weights of the

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primary operators are hI= 0, hσ= 1/16 and hǫ= 1/2 respectively. Thus we

see that in the Ising model the set of primary fields coincides with that of the

relevant operators of the spectrum (remember that the relationship between

conformal weights and renormalization group eigenvalues is: y = 2 − 2h).

This is a peculiar feature of the Ising model only, and is not shared by any

other minimal unitary model. Thus in this case the irrelevant operators

are bound to be secondary fields. Since in this paper we are particularly

interested in the irrelevant operators, let us study in more detail the structure

of the three conformal families.

• Secondary fields

All the secondary fields are generated from the primary ones by apply-

ing the generators L−iand¯L−iof the Virasoro algebra. In the following

we shall denote the most general irrelevant operator in the [σ] family

(which are odd with respect to the Z2symmetry) with the notation σi

and the most general operators belonging to the energy [ǫ] or to the

identity [I] families (which are Z2even) with ǫiand ηirespectively. It

can be shown that, by applying a generator of index k: L−kor¯L−kto

a field φ (where φ = I,ǫ,σ depending on the case), of conformal weight

hφ, a new operator of weight h = hφ+ k is obtained. In general any

combination of L−iand¯L−igenerators is allowed, and the conformal

weight of the resulting operator will be shifted by the sum of the indices

of the generators used to create it. If we denote with n the sum of the

indices of the generators of type L−iand with ¯ n the sum of those of type

¯L−ithe conformal weight of the resulting operator will be hφ+ n + ¯ n.

The corresponding RG eigenvalue will be y = 2 − 2hφ− n − ¯ n.

• Nonzero spin states

The secondary fields may have a non zero spin, which is given by the

difference n − ¯ n. In general one is interested in scalar quantities and

hence in the subset of those irrelevant operators which have n = ¯ n.

However on a square lattice the rotation group is broken to the fi-

nite subgroup C4(cyclic group of order four). Accordingly, only spin

0,1,2,3 are allowed on the lattice. If an operator φ of the continuum

theory has spin j ∈ N, then its lattice discretization φlbehaves as a

spin j (mod 4) operator with respect to the C4subgroup. As a conse-

quence all the operators which in the continuum limit have spin j = 4N

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with N non-negative integer can appear in the lattice discretization of

a scalar operator.

• Null vectors

Some of the secondary fields disappear from the spectrum due to the

null vector conditions. This happens in particular for one of the two

states at level 2 in the σ and ǫ families and for the unique state at

level 1 in the identity family. From each null state one can generate,

by applying the Virasoro operators a whole family of null states hence

at level 2 in the identity family there is only one surviving secondary

field, which can be identified with the stress energy tensor T (or¯T).

• Secondary fields generated by L−1

Among all the secondary fields a particular role is played by those gen-

erated by the L−1Virasoro generator. L−1is the generator of trans-

lations on the lattice and as a consequence it has zero eigenvalue on

translational invariant observables.

• Quasiprimary fields

A quasiprimary field |Q? is a secondary field which is not a null vector

(or a descendent of a null vector) and satisfies the equation

L1|Q? = 0 (6)

These fields play a central role in our analysis since they are the only

possible candidates to be irrelevant operators of the model.

By imposing eq.(6) it is easy to construct the first few quasiprimary

operators for each conformal family. For our analysis however the two

lowest ones are enough. They both belong to the conformal family of

the identity. Their expression in terms of Virasoro generators is:

Q1

2= L−2|1?

−2−3

(7)

Q1

4= (L2

5L−4)|1? (8)

(we use the notation Qη

the η family).

nto denote the quasiprimary state at level n in

From these fields we can construct two irrelevant operators, which both

have conformal weight 4 and RG eigenvalue −2.

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1] The first combination is Q1

identified with the energy momentum tensor T¯T.

2] The second combination is Q1

identified with the combination T2+¯T2.

2¯

Q1

2which has spin zero and can be

4+¯Q1

4which has spin 4 and can be

This last contribution appears as a consequence of the breaking of the

full rotational invariance due to the lattice1.

2.1.2 Structure constants.

Once the operator content is known, the only remaining information which is

needed to completely identify the theory are the OPE constants. The OPE

algebra is defined as

?

where with the notation {k} we mean that the sum runs over all the fields

of the conformal family [k]. The structure functions Ck

functions of r which must be single valued in order to take into account

locality. In the large r limit they decay with a power like behaviour

Φi(r)Φj(0) =

{k}

C{k}

ij(r)Φ{k}(0) (9)

ij(r) are c-number

Ck

ij(r) ∼ |r|−dim(Ck

ij)

(10)

whose amplitude is given by

ˆCk

ij≡lim

r→∞Ck

ij(r) |r|dim(Ck

ij).(11)

Several of these structure constants are zero for symmetry reasons. These

constraints are encoded in the so called “fusion rule algebra” which, in the

case of the Ising model is.

[ǫ][ǫ]

[σ][ǫ]

[σ][σ]

=

=

=

[1]

[σ]

[1] + [ǫ].

(12)

1Notice that if we would be interested in scalar quantities, this term would disappear

even on the lattice at the first order and could contribute only at the second order. This is

the case for instance of the susceptibility recently discussed in [10], in which this operator

gives a contribution only at order t4and not t2. However in the present case since we are

interested in a correlator, which defines a preferential direction on the lattice this term

can contribute already at the first order. This fact will be discussed in detail sect. 2.1.5

below.

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By looking at the fusion rule algebra we can see by inspection which are the

non-vanishing structure constants.

The actual value of these constants depends on the normalization of the

fields, which can be chosen by fixing the long distance behaviour of, for

instance, the σσ and ǫǫ correlators. In this paper we follow the commonly

adopted convention which is:

?σ(x)σ(0)? =

1

|x|

1

|x|2,

1

4

,|x| → ∞ (13)

?ǫ(x)ǫ(0)? =

|x| → ∞. (14)

With these conventions we have, for the structure constants among primary

fields

ˆCσ

σ,σ=ˆCσ

ǫ,ǫ=ˆCǫ

ǫ,σ= 0(15)

ˆC1

σ,σ=ˆCσ

σ,1=ˆC1

ǫ,ǫ=ˆCǫ

ǫ,1= 1 (16)

and

ˆCσ

σ,ǫ=ˆCǫ

σ,σ=1

2.(17)

2.1.3Continuum versus lattice operators.

Our main interest in this paper is the spin-spin correlator on the lattice. This

rises the question of the relationship between the lattice and the continuum

definitions of the operator σ. In the following we shall denote the lattice dis-

cretization of the operators with the index l. Thus σ denotes the continuum

operator and σlthe lattice one.

In general the lattice operator is the most general combination of contin-

uum operators compatible with the symmetry of the lattice one. If we are

exactly at the critical point this greatly simplifies the analysis, since only

operators belonging to the [σ] family are allowed. Moreover (due to the pe-

culiar null state structure of the spin family) the first quasiprimary operator

in the spin family appears at a rather high level and can be neglected in

our analysis. Thus as far as we are interested only at the critical point the

spin operators on the continuum and on the lattice are simply related by a

normalization constant which we shall call in the following Rσ. The simplest

way to obtain Rσis to look at the analogous of eq.(13) on the lattice. This

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is a well known result [21], which we report here for completeness. The large

distance behaviour of the correlator at the critical point is

?σiσj?h=0 =

R2

σ

|rij|1/4

(18)

where rijdenotes the distance on the lattice between the sites i and j and

R2

σ= e3ξ′(−1)25/24= 0.70338... (19)

By comparing this result with eq.(13) we find

σl = Rσσ = 0.83868...σ(20)

2.1.4The lattice Hamiltonian at the critical point

The last step in order to relate the continuum and lattice theories at the

critical point is the construction of the lattice Hamiltonian (let us call it

Hlat) at the critical point. As above, the lattice Hamiltonian will contain all

the operators compatible with the symmetries of the continuum one. In this

case all the operators belonging to the [σ] family are excluded due to the

Z2symmetry. Also the operators belonging to the [ǫ] family are excluded

for a more subtle reason. The Ising model (both on the lattice and in the

continuum) is invariant under duality transformations while the operators

belonging to the [ǫ] family change sign under duality, thus they also cannot

appear in Hlat(t = 0). Thus we expect

?

where HCFT is the continuum Hamiltonian and the u0

mentioned above we can keep in this expansion only the first two terms

which are respectively T¯T and T2+¯T2.

Hlat = HCFT + u0

i

d2xηi,ηi∈ [I],(21)

iare constants. As

2.1.5The spin-spin correlator on the lattice at the critical point.

We have already presented above, in eq.(18) the large distance expansion of

the spin-spin correlator on the lattice at the critical point.

Thanks to the exact results obtained in [5, 6, 7] (see [22] for a review) we

have much more informations on this correlator at the critical point. In order

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to discuss these results let us first introduce a more explicit notation for the

spins. We shall denote in the following with σM,Nthe spin located in the site

with lattice coordinates (M,N). By using the translational invariance of the

correlator we can always fix one of the two spin in the origin. Thus the most

general correlator can be written as: ?σ0,0σM,N?. A particular role will be

played in the following by the correlator along the diagonal: ?σ0,0σN,N? and

the one along the axis ?σ0,0σ0,N?.

Among the various results on the critical correlators two are of particular

relevance for us:

• Remarkably enough, exact expressions exist for the correlator both

along the diagonal and along the axis for any value of N. These ex-

pressions are rather cumbersome and we shall not report them here.

They can be found in [22].

• An asymptotic expansion exists for large values of the separation rij.

For a square lattice, the first three terms have been explicitly evaluated

in [8].

This expansion takes the following form (for further details see [8])

log?σ0,0σM,N? = logA −1

4logr + A1(θ)r−2+

+ A2(θ)r−4+ A3(θ)r−6+ O(r−8)(22)

where

A1(θ) = 2−8(−1 + 3cos4θ)

A2(θ) = 2−13(5 + 36cos4θ + 36cos8θ)

A3(θ) = 3−12−19(−524 − 324cos4θ + 24732cos8θ + 28884cos12θ)(23)

and

r2

=

1

2(M2+ N2)

= r

= r

M

N

√2sinθ

√2cosθ .(24)

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In the following we shall only be interested to the first correction since higher

order corrections are beyond the resolution of our data. Hence we end up

with

?σ0,0σM,N? =

r1/4

Remarkably enough this expansion perfectly agrees with what one finds by

assuming the presence in the Hamiltonian of the model of the two irrele-

vant operators discussed in sect.2.1.1. In fact both T¯T and T2+¯T2have

RG eigenvalue -2, thus, if they are present, they should contribute to the

correlator exactly with a term proportional to 1/R2. Moreover we expect

that the T2+¯T2operator, which has spin 4, should give a term proportional

to cos(4θ) while the scalar operator T¯T should give a contribution without

θ dependence. This is exactly the pattern that we find in eq.(25) (see for

instance pag. 218 of [23] for a detailed discussion of this point). However

this remarkable agreement also rises a non trivial problem. In fact the high

precision analysis of [10],[14] clearly exclude the presence in the lattice

Hamiltonian of the T¯T operator (while they both confirm that the T2+¯T2

is indeed present) It is thus not clear which could be the origin of the scalar

term in eq.(25). A possible solution to this puzzle is to notice that the results

of eq.(25) and those of of [10], [14] are obtained with two different choices

of the coordinates of the 2d plane.

In fact eq.(25) is written in terms of the “continuous” variable r (which, is

the one that we must choose if we want to match the lattice results with the

continuum limit ones). Both the results of [10] and [14] are instead obtained

in the “lattice reference frame” (which is the most natural variable on the

lattice) in which there is no√2 when comparing the distances along the axes

and along the diagonals. Hence, to compare the correction to the spin-spin

correlation function with the findings of of [10], [14] we must rewrite (25)

it in terms of lattice coordinates. This can be easily performed by using the

relations (24).

After some algebra we obtain

A

?

1 + 2−8(−1 + 3cos4θ) r−2+ ...

?

(25)

• axis correlator

?σ0,0σ0,N? =A 21/8

N1/4

?

1 +1

64N−2+ O(N−3)

?

(26)

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• diagonal correlator

?σ0,0σN,N? =

A

N1/4

?

1 −1

64N−2+ O(N−3)

?

. (27)

These same expansion can also be obtained by directly looking at the exact

lattice results for the correlators (see for instance [22] where these 1/N2terms

are obtained in full detail).

Looking at eqs.(26, 27) we see that with the lattice choice of coordinates

the 1/N2term exactly changes its sign as we move from the axis to the

diagonal. This is exactly what one would expect for the contribution of a

spin 4 operator, and thus it is apparent from eqs.(26, 27) that no scalar

correction appears at order 1/N2in perfect agreement with the results of

of [10], [14].

It is only the change of coordinates of eq.(24) which induces in the con-

tinuum limit a scalar term (which we may well identify in this limit with a

T¯T type contribution).

2.2 Adding the magnetic field.

2.2.1 The continuum theory.

The continuum theory in presence of an external magnetic field is represented

by the action:

A = A0+ h

where A0is the action of the conformal field theory.

As a consequence of the applied magnetic field the structure functions

acquire a h dependence so that we have in general

?

Also the mean values of the σ and ǫ operators acquire a dependence on

h. Standard renormalization group arguments (for un updated and thor-

ough review on renormalization group theory applied to critical phenomena

see [24]) allow one to relate this h dependence to the scaling dimensions of

the operators of the theory and lead to the following expressions:

?

d2xσ(x) (28)

Φi(r)Φj(0) =

{k}

C{k}

ij(h,r)Φ{k}(0). (29)

?σ?h= Aσh

1

15+ ...(30)

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?ǫ?h= Aǫh

8

15+ ... (31)

The exact value of the two constants Aσand Aǫcan be found in [25] and [26]

respectively

Aσ=

2C2

15(sin2π

3+ sin2π

5+ sinπ

15)= 1.27758227.., (32)

with

C =

4sinπ

?2

5Γ

?

?1

15

5

8

?

Γ

3

Γ

??

4π2Γ

?3

4

4

?

Γ2?

Γ2?13

3

16

16

?

?

Γ

?1

?

4

5

,(33)

and

Aǫ= 2.00314.... (34)

2.2.2Continuum versus lattice operators.

In presence of a magnetic field also the fields belonging to the energy and

identity families can appear in the relation between the lattice and the con-

tinuum version of the spin operator. The most general expression is

σl= fσ

0(hl)σ + hlfǫ

0(hl)ǫ + fσ

i(hl)σi+ hlfǫ

i(hl)ǫi+ hlfI

i(hl)ηi,i ∈ N (35)

where fσ

σi, ǫiand ηiwe denote the secondary fields in the spin, energy and identity

families respectively.This is the most general expression, however, as a

matter of fact, only the first term is relevant for our purposes (all the higher

terms give negligible contributions)

i(hl) fǫ

i(hl) and fI

i(hl) are even functions of hl. With the notation

σl= Rσσ + hp1ǫ(36)

The determination of p1 is rather non-trivial. We shall discuss it in the

appendix. It turns out that

p1∼ 0.0345

It is also important to stress that the lattice and continuum values of the

magnetic field do not coincide but are related by

(37)

hl= Rhh(38)

with Rh= 1.1923.. (see [20] for details).

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2.2.3 The lattice Hamiltonian.

As it happened for the conversion from σ to σlalso in the construction of

the perturbed Hamiltonian on the lattice, several new operators belonging

to the energy and spin family must now be taken into account. However

it turns out that all the quasiprimary fields of these two families appear at

rather high level and can be neglected in the present analysis. Thus, as far

as we are concerned, the only effect of switching on the magnetic field is that

the constants in front of the T¯T and T2+¯T2terms in eq.(21) acquire an h

dependence. For symmetry reasons these must be even analytic functions of

h.

2.2.4 The spin-spin correlator.

The short distance behaviour of the spin-spin correlator in presence of a

magnetic field can be obtained by using the so called IRS approach.

detailed discussion of this approach can be found in [18, 19] the particular

application to the Ising correlators are discussed in [20] to which we refer for

details. We only list here the results. Setting t ≡ |h| |r|15/8we have up to

O(t2)

A

?σ(0)σ(r)?|r|1/4= B1

σσ+ B2

σσt8/15+ B3

σσt16/15+ O(t2) (39)

with

B1

B2

B3

σσ

=

= Aǫ?

= Aσ

?

C1

σσ

Cǫ

∂hCσ

=1

σσ

σσ

?

=1.00157...

=− 0.51581...

σσσσ

(40)

This result holds in the continuum theory. By using the known conversion

between continuum and lattice units we can obtain the spin-spin correlator

on the lattice:

?σl(0)σl(r)?|rij|1/4= B1

Where rijis the distance between the two spins on the lattice, measured

in units of the lattice spacing, tlis defined as tl≡ |hl| |rij|15/8and

B1

σσ,l

= 0.703384...

σσ,l+ B2

σσ,lt8/15

l

+ B3

σσ,lt16/15

l

+ O(t2

l)(41)

14