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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.
2The 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...
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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)
2.1 The 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.
2.2.1The 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)
2.2.2 Continuum 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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2.2.3The 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.
2.2.4The 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
Page 16
B2
B3
σσ,l
= 0.641409...
σσ,l
= −0.300749...
(42)
In the next section we shall see which contributions must be added to
this expression as a consequence of the irrelevant fields.
3Contribution from irrelevant operators to
the spin-spin correlator in presence of a
magnetic field.
There are two main sources of contributions due to irrelevant operator to
eq.(41) above. They have different origin and contribute in a different way
to the spin-spin correlator. The first correction can be understood as a per-
turbative contribution to the spin-spin correlator due to irrelevant operators.
We shall discuss it in subsec. 3.1. The second correction arises from the new
terms in the relation between σland σ, due to the presence of a magnetic
field. We shall discuss it in subsec. 3.2.
3.1Irrelevant operators in the perturbed Hamiltonian.
The contributions to the scaling function due to the (possible) presence of
additional irrelevant operators in the lattice Hamiltonian can be studied by
means of standard perturbative expansions around the CFT fixed point.
In order to address the problem in presence of an external magnetic field we
must consider the CFT pertaining to the critical Ising model as perturbed by
a mixed relevant/irrelevant perturbation. Hence there will appear correction
terms to the scaling behaviour which both depend on the external magnetic
field and show a non-trivial (i.e. non scaling) dependence on the spin-spin
distance rij. Moreover if the irrelevant operator in which we are interested
breaks the rotational invariance (and this is the case for instance of the
(T2+¯T2) operator discussed above) we must expect a dependence on the
angle θ between the principal axes of the lattice and the direction of the
correlator. General symmetry arguments and dimensional analysis strongly
constrain these terms. If we keep in our analysis only the first two irrelevant
15
Page 17
operators T¯T and (T2+¯T2) the most general expression for the scaling
function turns out to be:
?
?
?
?σl(0)σl(r)?|rij|1/4
= B1
σσ,l
1 +−1 + a1h2
|rij|2
1 +b(θ)
|rij|2
1 +c(θ)
|rij|2
+(3 + a2h2)cos(4θ)
|rij|2
t8/15
l
?
+ B2
σσ,l
?
?
+ B3
σσ,l
t16/15
l
+ O(t2
l) (43)
where a1, a2are unknown constants and b(θ), c(θ) are unknown functions of
the form
b(θ) = b1+ b2cos(4θ), (44)
c(θ) = c1+ c2cos(4θ),(45)
in the angular variable θ.
Let us analyze the above corrections in detail.
• Correction to B1
At the lowest order in the irrelevant coupling, we can identify a scalar
correction proportional to |rij|−2and a spin 4 correction proportional
to cos(4θ) |rij|−2. Both these terms acquire a h dependence due to the
magnetic relevant perturbation. The most general expression, compat-
ible with the symmetries of the model is:
?
σσ,l:
B1
σσ,l
1 +P1(h)
|rij|2+P2(h)cos(4θ)
|rij|2
?
(46)
where P1(h) and P2(h) are even polynomials in the magnetic field h
which in the h → 0 limit must agree with the asymptotic expansion
reported in eq.(25). Expanding the polynomials up to the first term in
h we find
P1(h) = −1 + a1h2
P2(h) = 3 + a2h2.(47)
from which the first term in eq.(43) follows. It is easy to see that the
terms proportional to a1and a2are highly suppressed due to the h2
16
Page 18
power. They behave as the O(t2) that we have systematically neglected
in the previous section. As a matter of fact we have not been able to
see these terms in our numerical data and we shall neglect them in the
following.
• Corrections to B2
Following the same line discussed above we find in this case:
σσ,land B3
σσ,l:
B2
σσ,l
?
?
1 +b(θ)
|rij|2
1 +c(θ)
|rij|2
?
?
t8/15
l
B3
σσ,l
t16/15
l
(48)
where b(θ) and c(θ) are the most general mixture of scalar and spin
4 terms. They can be expanded in powers of cos(4θ). keeping only
the first two orders in the expansion we end up with the expressions
reported in eq.s (44) and (45) where b1,b2,c1and c2are unknown con-
stants.
While the term proportional to c(θ) cannot be detected within the
precision of our data (we shall further discuss this point at the end
of the next section), the magnitude of the one proportional to b(θ) is
definitely larger than our numerical uncertainties. Thus we expect to
be able to observe such a correction.
3.2 Irrelevant operators in σl.
We have seen above (see eq.(36)) that in presence of a magnetic field the
relation between lattice and continuum spin operator becomes more com-
plicated and a term proportional to hǫ appears. Strictly speaking ǫ is not
an irrelevant operator, however this is only an accident, the following terms
(those that we neglect) in the correspondence between σland σ would indeed
be irrelevant operators. Moreover, exactly as it happens for the irrelevant
operators, also the hǫ term gives a subleading contribution to the spin-spin
correlator. For these reasons we have included also this correction in this
paper.
Inserting eq. (36) in the spin-spin correlator we find
?σl(0)σl(r)? = R2
σ?σ(0)σ(r)? + 2hRσp1?σ(0)ǫ(r)? + h2p2
1?ǫ(0)ǫ(r)? (49)
17
Page 19
Since we are interested in keeping only terms below O(t2) the last term in
eq.(49) can be neglected. Using the known result (see [20] for details)
?σ(0)ǫ(r)?|r|9/8= B1
σǫt1/15+ B2
σǫt + B3
σǫt23/15+ O(t31/15) (50)
with
B1
B2
B3
σǫ
= Aσ?
=∂hC1
= Aǫ
Cσ
σǫ
=0.63879...
σǫ
?
∂hCǫ
σǫ
?
=3.29627...
=− 1.82085...
σǫσǫ
we can rewrite eq.(49) as
?σl(0)σl(r)?|r|1/4
= R2
+ 2hRσp1|r|−7/8(B1
σ(B1
σσ+ B2
σσt8/15+ B3
σǫt1/15+ B2
σσt16/15)
σǫt + B3
σǫt23/15) (51)
Only the first term in the second line of eq.(51) gives a contribution below
O(t2) thus we end up with
?σl(0)σl(r)?|r|1/4
= R2
+ 2Rσp1t|r|−22/8B1
σ(B1
σσ+ B2
σσt8/15+ B3
σǫt1/15
σσt16/15)
(52)
We see that the new term can be considered as a correction, proportional
1
|r|22/8to the t16/15term of eq.(41):
to
?σl(0)σl(r)?|rij|1/4
= B1
σσ,l+ B2
σσ,lt8/15
l
+ (R2
σB3
σσ+ 2Rσp1|r|−22/8B1
σǫ)/R16/15
h
t16/15
l
+ O(t2
l) (53)
Inserting the values of the various constants and of p1we end up with
?σl(0)σl(r)?|rij|1/4= B1
with a3= 0.0307...
Unfortunately this new contribution is too small to give a reliable signa-
ture within the precision of our data. In fact the amplitude of the correction
a3|r|−22/8t16/15
(or at most of the order of) 10−4and cannot be disentangled from the other
σσ,l+B2
σσ,lt8/15
l
+(B2
σσ,l+a3|r|−22/8)t16/15
l
+O(t2
l) (54)
l
for distances greater than one lattice spacing is always below
18
Page 20
sources of corrections within the precision of our data which ranges from
10−5to 10−4. The situation is very similar to that of the correction propor-
tional to c(θ) discussed in the previous section which in fact has the same t
dependence.
However it is interesting to observe that both these corrections are al-
most in threshold to be observed and it is well possible that with the next
generation of simulation they could be detected. For this reason we included
also their analysis in the present paper.
4 The simulation
We studied the model with a set of Montecarlo simulations using a Swendsen-
Wang type algorithm, modified so as to take into account the presence of an
external magnetic field. We used the same program and simulation setting
of ref. [20] to which we refer for a detailed discussion of the performances of
the algorithm and of finite size effects. We simulated the model for twelve
different values of the magnetic field, ranging from hl = 4.4069 × 10−4to
hl= 8.8138 × 10−3.
For all the values of hl that we simulated, we studied the correlator:
?σ(0)σ(r)?, for r = 1,···,10 both in the diagonal direction and along the
axes. This is an important feature of the present analysis, since it is exactly
by comparing the correlator along the axes versus the diagonal one that we
can extract the spin of the perturbing operators.
For all the simulations the lattice size was chosen to be L = 200 (the
analysis of [20] shows that for all the values of hl that we studied this is
enough to avoid finite size effects). Each two measures of the correlators were
separated by five SW sweeps. For each single correlator we performed 4×105
measures. The values of hlthat we studied are reported in tab.4 where we
also listed for completeness the corresponding values of the correlation length
ξ. This quantity is very important , since it defines in a quantitative way
what we mean as “short distance behaviour”. Short distance means “smaller
than the correlation length”. This is in fact the regime in which we expect
that the IRS analysis should hold and in which we may hope to observe
effects due to the irrelevant operators. ξ is given (in lattice units) by:
ξ(hl) = 0.24935...h
−8
l
15
(55)
19
Page 21
(see [27] for details on this equation and on the continuum to lattice conver-
sion of ξ).
Table 1: The values of hl(measured in units of βc) that we studied and the
corresponding correlations lengths.
hl
ξ
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.01
0.015
0.02
15.4
10.6
8.5
7.3
6.5
5.9
5.4
5.1
4.8
4.5
3.6
3.1
5Comparison between predictions and nu-
merical results.
Let us start this section by writing the expression that we expect for the
spin-spin correlator in a magnetic field, according to the analysis performed
in the sections 2 and 3 above. We have:
?σl(0)σl(r)?|rij|1/4
= ?σl(0)σl(r)?h=0|rij|1/4+
?
?
a1+a2(θ)
|rij|2+ ...
?
t8/15
l
+b1+b2(θ)
|rij|2+ ...
?
t16/15
l
+ O(t2
l)(56)
where ?σl(0)σl(r)?h=0|rij|1/4is given by eq.(25); a1≡ B2
while a2(θ) and b2(θ) are in principle unknown functions of the angular vari-
able θ.
σσ,land b1≡ B3
σσ,l,
20
Page 22
In this section we shall compare this expression with our numerical estimates
for the correlators along the diagonal and the axis. This will allow us to ob-
tain a rather precise estimate of the function a2(θ) in the two cases θ = 0
and θ = π/4. With these two informations we shall also be able to study the
spin content of the operators which generate this contribution.
In the following we shall confine ourselves to the study of a2(θ) since precision
of our data is not enough to extend our analysis also to the function b2(θ).
We performed our analysis in three steps.
1] As a first step we constructed the combinations
G∆(r,hl) ≡ ?σl(0)σl(r)? − ?σl(0)σl(r)?h=0
using the fact that the critical point correlator is exactly known. Thus
we have:
(57)
G∆(r,hl)|rij|1/4
= (a1+a2(θ)
|rij|2+ ...)t8/15
|rij|2+ ...)t16/15
l
+ (b1+b2(θ)
l
+ O(t2
l).(58)
2] Then we study, at fixed value of r, the hldependence of G∆(r,hl). Ac-
cording to eq.(58) above we should choose as fitting function
G∆(¯ r,hl) = A(¯ r)h8/15
l
+ B(¯ r)h16/15
l
(59)
(the notation ¯ r indicates that we are fitting at a fixed value of |rij|).
However keeping only the first two terms in the scaling function of the
correlator is a too rough approximation. Within the range of values of
hlthat we studied and with the precision that we obtained, this choice
in general led to very high χ2values. Fortunately, thanks to the IRS
analysis we know the functional form of the next to leading orders in
the hlexpansion of the correlator. It turns out that adding one more
term is enough for the correlators ranging from 1 to 7 lattice spacings
both along the axis and the diagonal directions, while for the remaining
correlators we have to go up to the fourth term in the expansion. The
general form of the scaling function at this order turns out to be:
G∆(¯ r,hl) = A(¯ r)h8/15
l
+ B(¯ r)h16/15
l
+ C(¯ r)h30/15
l
+ D(¯ r)h32/15
l
(60)
21
Page 23
(see [20] for a discussion of this scaling function and the origin of the
two new exponents h30/15
l
and h32/15
l
we found good confidence levels. The values of A(r) that we obtained
in this way can be found in table 5. Notice that the data in each fit
are completely uncorrelated since they belong to different simulations.
). In this way for all the values of r
3] As a last step we address the r dependence of the function A(r). Accord-
ing to eq.(58) we expect the following behaviour:
A(r)
r3/4= a1+a2(θ)
r2
(61)
Fixing a1to its known IRS value we end up with a one parameter fit
which in both cases of θ = 0 and θ = π/4 has a very good confidence
level. Our final results are:
• axis correlator
a2(0) = (−0.062 ± 0.004) (62)
• diagonal correlator
a2(π/4) = (−0.014 ± 0.007)(63)
These results, and in particular the fact that we have different corrections
in the two directions, unambiguously show that the contribution is at least
partially due to a spin 4 operator. The natural candidate for this role is
again the T2+¯T2operator. Our result also suggests that a scalar operator is
present in the game. We see two possible reasons for this contribution. The
first is that, since we are working in the IRS framework we are compelled
to use the “continuum limit reference frame”. As we have seen in sect.2.1.5
this choice induces a T¯T term and consequently a scalar type correction.
The second is that one cannot exclude the presence in the magnetic scaling
field of a (rotationally invariant) momentum dependent term, which would
manifest itself exactly as a scalar correction of the type that we observe2.
This non trivial possibility has been recently discussed in [15] (see the note
at pag.8161) to explain the corrections of order t2(where t is the reduced
temperature) in the second moment of the spin spin correlator. Most proba-
bly both mechanisms are at work in the present case. We cannot distinguish
between them within the precision of our data.
2We thank A.Pelissetto for pointing to us this interesting possibility.
22
Page 24
Table 2: The lattice spacing N is intended along the axis for Aaxisand along
the diagonal for Adiag.
N
1
2
3
4
5
6
7
8
9
10
Aaxis(N)
(0.581 ± 0.003)
(1.048 ± 0.004)
(1.446 ± 0.005)
(1.802 ± 0.005)
(2.134 ± 0.006)
(2.448 ± 0.006)
(2.746 ± 0.007)
(3.039 ± 0.009)
(3.316 ± 0.009)
(3.58 ± 0.01)
Adiag(N)
(0.823 ± 0.003)
(1.396 ± 0.004)
(1.894 ± 0.005)
(2.354 ± 0.007)
(2.777 ± 0.008)
(3.19 ± 0.01)
(3.58 ± 0.01)
(3.96 ± 0.01)
(4.32 ± 0.03)
(4.67 ± 0.04)
6Concluding remarks
The role of the irrelevant operators in the two dimensional Ising model has
attracted much interest in these last months due to the results on the mag-
netic susceptibility at h = 0 recently reported in [9, 10]. While it is by now
clear that contributions due to irrelevant operators are present in the free
energy of the Ising model at h = 0 nothing is known on the case in which the
magnetic perturbation of the critical model is chosen. Moreover the charac-
terization of these irrelevant operators and, possibly, their identification with
quasiprimary fields of the Ising CFT is still an open problem. In this paper
we tried to make some progress in this direction. We studied the corrections
induced by the presence of irrelevant operators in the spin-spin correlator of
the 2d Ising model in presence of an external magnetic field.
Our main results are:
• The 1/r2corrections which are present in the correlator at the critical
point survive unchanged also if the magnetic field is switched on. The h
independent part of this contribution is much larger than our statistical
errors and can be observed very precisely. It is completely due to the
spin 4 irrelevant operator.
• Several new terms appear when the magnetic field is switched on. By
23
Page 25
using standard perturbative methods one can evaluate the amplitude of
the corrections that they induce in the spin-spin correlator. In general
these terms are very small.However in one case the amplitude of
the expected correction turned out to be definitely larger than our
statistical errors. It is the case of the term proportional to
1
|r|1/4
?t8/15
r2
?
.(64)
for which we could indeed observe and measure such correction with
good confidence. The comparison between the two values of this cor-
rection for the correlator along the diagonal and the one along the
axis shows that, besides the expected contribution due to the spin-
4(T2+¯T2) operator, there is also a scalar term which may have a
twofold origin. It could be due the T¯T operator, but it could also be
the signature of a momentum dependent term in the magnetic scaling
field of the model.
Other contributions due to these or other irrelevant operators are for the
moment beyond our resolution, but it could be possible to observe them in
the next generation of simulations.
24
Page 26
ADetermination of p1
The simplest way to fix the value of p1is to look at the expectation value
?σl?. We expect:
?σl? = Rσ?σ? + p1?ǫ?
from which we have (using the notations of [27])
(A.1)
?σl? = RσAMh
1
15+ p1AEh
23
15
(A.2)
from [27] we see that
?σl? =16
15Al
fh
1
15+38
15Al
fAl
f,2h
23
15
(A.3)
Using the numerical values reported in [27], i.e.
Al
f= 0.9927995....(A.4)
Al
f,2∼ 0.021...(A.5)
we obtain
p1∼ 0.0345 (A.6)
Acknowledgements We thank M. Hasenbusch, A. Pelissetto and E. Vi-
cari for useful discussions and correspondence on the subject.
25
Page 27
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