Scale invariance implies conformal invariance for the three-dimensional Ising model

0.22 MB·English

by  Bertrand Delamotte

Scale invariance implies conformal invariance for the three-dimensional Ising model
Bertrand Delamotte,1 Matthieu Tissier,1 and Nicola´s Wschebor1,2
1LPTMC, UPMC, CNRS UMR 7600, Sorbonne Universit´es, 4 Place Jussieu, 75252 Paris Cedex 05, France
2Instituto de F´ısica, Faculdad de Ingenier´ıa, Universidad de la Repu´blica, J.H. y Reissig 565, 11000 Montevideo, Uruguay
(Dated: January 28, 2016)
Using Wilson renormalization group, we show that if no integrated vector operator of scaling
dimension −1 exists, then scale invariance implies conformal invariance. By using the Lebowitz
6
1 inequalities, we prove that this necessary condition is fulfilled in all dimensions for the Ising uni-
0 versality class. This shows, in particular, that scale invariance implies conformal invariance for the
2 three-dimensional Ising model.

n
a
I. INTRODUCTION taryandPoincar´einvarianttheories. Moreover,thereare
J
strongindicationsthatanon-perturbativeproofcouldbe
7
at reach in this dimension [17–19].
2 Conformalsymmetry playsa considerablerole bothin
high energy and condensed matter physics. There has Despite decadesofeffort,itisstillanopenquestionto

] been a renewed interest in recent years, particularly be- know whether a typical statistical model is conformally
h
invariant at criticality in three dimensions. The aim of
c cause of the AdS/CFT conjecture [1] and the successful
this article is twofold. First, we derive a sufficient condi-
e use of conformal methods in three-dimensional critical
m physics [2–7]. The groundbreaking papers of the 1970s tion which, when fulfilled, ensures that scale invariance
implies conformal invariance. In the second part of the
- and 1980s [8–13] solved two fundamental issues in two
at dimensions: First, scale invariance implies conformal in- paper, we prove that this condition is fulfilled in any di-
mension for the euclidean Z model.
t variance under mild assumptions [12, 13] and, second, 2
s
. conformalsymmetryenablesustosolvemostofthescale The rest of the paper is organized as follows. In
at invariant problems that is, to determine critical expo- Sect. II, we make a brief review of the nonperturbative
m nents and correlation functions [14]. renormalization-group. We then recall in Sect. III the
relationbetween scale invariance and the NPRG. By us-
- An important ingredientfor the exactsolution oftwo-
d ingthesamemethods,wegeneralizetheseconsiderations
dimensional conformalmodels is the existence of aninfi-
n to the case of conformal invariance in Sect. IV. In Sect.
nite number of generators of the conformal group. In
o
V, we finally derive a sufficient condition for the valid-
c higher dimensions, the number of generators is finite,
ityofconformalinvarianceinscaleinvariantmodels. We
[ andwecouldnaivelyconcludethatsymmetryarguments
show, on general arguments that this condition is ex-
alone are not sufficient to solve a model in the critical
4 pectedto be fulfilled in O(N)models (andingeneraliza-
regime. However, it is well-known that scale-invariant
v
tionsthereof). InSectVI,theseconsiderationsaremade
6 theoriesareinaone-to-onecorrespondencewiththefixed
rigourous for the Ising universality class. We give our
7 points of the Wilson Renormalization Group (RG) [15],
conclusions in Sect VII.
7 and that the fixed point of a theory completely deter-
1 mines all the correlationfunctions of a criticalmodel for
0
sufficiently small wavenumbers. Therefore, at the level
.
1 of principles, scale (and a fortiori conformal) invariance II. NONPERTURBATIVE
0 is sufficient to determine all the universal critical prop- RENORMALIZATION-GROUP FORMALISM
5 erties of a model. Of course, in practice, the computa-
1
tion of these critical properties requires us to solve the
: The proof of conformal invariance in all dimensions
v functional Wilson RG equations. This is a formidable
presented below is intimately related to the deep struc-
Xi task that we do not know how to carry out without ap- ture of the Wilson RG 1 and scale invariance. We there-
proximations. Any supplementary information, even if fore start by recalling, in the case of the Z model,
r 2
a redundant, is therefore welcome and this is what con- the formalism of the modern formulations (sometimes
formal invariance could provide. A breakthrough in this
calledNonperturbativeRG,orfunctionalRG)oftheWil-
directionhasbeenachievedtheselastyearswiththecon-
son RG [22–26]. The coarse-graining procedure at some
formal bootstrap program [2–4], which led to the exact
RG scale k is implemented by smoothly decoupling the
(although numerical) computation of the critical expo-
long-wavelength modes ϕ(|q| < k) of the system, also
nents of the Ising model in three dimensions assuming,
called the slow modes, by giving them a large mass,
among other things, conformal invariance.
while keeping unchanged the dynamics of the short-
Inparallel,alargeactivityhasbeendevotedto under-
standingtherelationbetweenscaleandconformalinvari-
ancein–orcloseto–fourdimensions. Ithasbeenproven
to all orders of perturbation theory that scale invariance 1 The history of the relation between conformal invariance and
impliesconformalinvariance[16]infour-dimensionaluni- WilsonRGislong,seeforexample[20,21].
2
wavelength/rapid ones ϕ(|q| > k). This decoupling is model described by an action S . As we discuss in
scal
convenientlyimplementedbymodifyingtheactionorthe the following,the existence of such a model is not neces-
Hamiltonian of the model: S[ϕ]→S[ϕ]+∆S [ϕ] where sary for our proof but it is convenient to imagine that it
k
∆S [ϕ] is quadratic in the field and reads, in Fourier existstomotivatetheformoftheexpectedWardidentity
k
space, ∆S [ϕ] = 1/2 R (q2)ϕ(q)ϕ(−q). The precise (WI)ofscaleinvariance. Ifsuchamodelexisted,thisWI
k q k
shape ofR (q2)does nRotmatter forwhatfollowsaslong wouldbe obtainedbyperformingthefollowinginfinitesi-
k
as it can be written as malchangeofvariablesϕ(x)→ϕ(x)+ǫ(Dφ+xµ∂µ)ϕ(x)
inthe functionalintegral,withD the scalingdimension
φ
Rk(q2)=Zkk2r(q2/k2) (1) ofthefield,usuallywrittenintermsoftheanomalousdi-
mension η as D =(d−2+η)/2. Actually, the analysis
where Z is the field renormalization factor and r is a φ
k of this model and of its WI faces both UV and infrared
function that, (i) falls off rapidly to 0 for q2 > k2 – the
(IR) problems. In the IR regime, non-analyticities are
rapidmodesϕ(|q|>k)arenotaffectedby∆S –and(ii)
k present and, in the UV, it is difficult to control math-
goestoaconstantforq2 =0–theslowmodesϕ(|q|<k)
ematically the continuum limit: Λ → ∞. Let us first
acquire a mass of order k and thus smoothly decouple.
discuss the IR aspect. Since ∆S [φ] acts as an IR regu-
The partition function, which now depends on the RG k
lator,theWilsonRGoffersasolutiontotheIRproblem:
parameter k, reads:
theregularizedmodel,Eq. (2),isnotscaleinvarianteven
iftheoriginalmodelassociatedwithS wereand,thus,
scal
Zk[J]=Z Dϕ exp(cid:18)−S[ϕ]−∆Sk[ϕ]+ZxJϕ(cid:19) (2) all Γk(n>)0({pi}) are regular contrarily to Γk(n=)0({pi}). The
price to pay for regularity is the breaking of scale invari-
where the field J is a source term which corresponds to
ance that manifests itself through a modification of the
themagneticfieldintheIsingmodelandwheretheultra-
WI [see [28, 29] for situations where R breaks symme-
k
violet (UV) regime of the functional integral is assumed
tries]. By enlarging the space of cutoff functions R to
k
toberegularizedatamomentumscaleΛ,seeforinstance
arbitraryfunctions R (x,y) that are neither constrained
k
[27] for a lattice regularization in this formalism. It is
to satisfy (1) nor to be invariant under rotations and
convenient to define the free energy W [J] = lnZ [J]
k k translations [as also done in [30]], this modified WI for
and its (slightly modified) Legendre transform by
scale invariance, obtained from Eqs. (2,3), reads
1
Γk[φ]+Wk[J]=ZxJφ−2ZxyRk(|x−y|)φ(x)φ(y), (3) Zxy(Dx+Dy +DR)Rk(x,y)δRδkΓ(xk,y)
wtriatnhsfφo(rxm)=ofδRWkk(/qδ2)J(axn)d, Rwkh(e|xre−thye|)ltahset itnevrmersheaFsoubreieenr +Z (Dx+Dφ)φ(x)δδφΓ(xk) =0 (6)
x
added for the following reason. When k is close to Λ, all
modes are completely frozen by the ∆S term because, where Dx = x ∂x and D = 2d−2D is the scaling
k µ µ R φ
for all q, R (q2) is very large. Thus, Z can be com- dimension of R which implies that the field renormal-
Λ k→Λ k
putedbythesaddle-pointmethodanditisthenstraight- ization in Eq. (1) behaves as Z ∝k−η. By constraining
k
forward to show that the presence of the last term in R (q) to be of the form (1), Eq. (6) can be conveniently
k
Eq.(3)leadstoΓ [φ]≃S[ϕ=φ]. Onthecontrary,when rewritten (following [29]):
Λ
k = 0, the definition of R implies that R (q2) ≡ 0
k k=0
andthe originalmodelis recovered: Zk=0[J]=Z[J] and ∂ Γ [φ]=− (Dx+D )φ(x) δΓk[φ]. (7)
t k φ
Γ [φ]=Γ[φ], with Γ[φ] the usual Gibbs free energy or Z δφ(x)
k=0 x
generating functional of one-particle-irreducible correla-
Introducing dimensionless and renormalized quantities
tion functions.
(denoted with a tilde)
The exact RG equation for Γ reads [23–25]:
k
1 x=k−1x˜ (8)
∂ Γ [φ]= ∂ R (|x−y|)G [φ], (4)
t k t k k,xy
2Zxy φ(x)=kDφφ˜(x˜) (9)
where t = ln(k/Λ), and G [φ] is the field-dependent Γ˜ [φ˜]=Γ [φ] (10)
k,xy k k
propagator:
Eq. (7) rewrites:
δ2Γ [φ]
G =(Γ(2)+R )−1 , Γ(2) [φ]= k . (5)
k k k k,xy δφ(x)δφ(y) ∂ Γ˜ [φ˜]=0. (11)
t k
Eq. (11) means that an hypothetical scale-invariant ac-
III. SCALE INVARIANCE tion S would lead, in its regularized and not scale-
scal
invariant version S +∆S , to a RG flow where Γ˜ [φ˜]
scal k k
We now discuss how scale invariance emerges in the would be at a fixed point Γ˜∗[φ˜] for all values of t:
NPRG formalism. We first consider a scale-invariant ∂ Γ˜∗[φ˜]=0.
t
3
The very structure of the Wilson RG (or NPRG) also discrete character of the eigenperturbations around a
solves the UV problem. Actual models have a natural fixed point has been studied intensively by perturba-
UV cutoff Λ (e.g. a lattice spacing) at which is defined tive means. As for the Wilson RG, it has been stud-
theirmicroscopicactionS. Themomentumintegralsare ied in detail in the particular case of the O(N) models
thereforecut-offatΛandarethusUVfinite. Atscalesof in [34–36]. Although obtained within the derivative ex-
order Λ (i.e., if we consider correlation functions, in the pansion of the exact RG flow (4), its discrete character
regimewhereatleastoneexternalmomentumorkorthe certainly remains true beyond this approximation. The
magnetic field in appropriate units, is of the order of Λ), eigenvalues λ are obtained from the flow by substitut-
themodelisnotscaleinvariant. Infactscaleinvarianceis ing Γ˜ [φ˜]→Γ˜∗[φ˜]+ǫ exp(λt)γ˜[φ˜] and retaining only the
k
an emergentproperty that appears in the IR when some O(ǫ)terms. (Withourdefinitionoft,arelevantoperator
parameter(suchasthetemperature)hasbeenfine-tuned. has a negative eigenvalue.) This leads to the eigenvalue
We call S the corresponding action. In the RG for- problem
crit
malism, scale invariance emerges in the IR regime when
δγ˜
integrating the RG flow starting at Γ [φ] = S [ϕ = φ] λγ˜[φ˜]= (Dx˜+D )φ˜(x˜)
becauseΓ˜ getsclosetoafixedpointfΛorlargencreitgativet, Zx˜ φ δφ˜(x˜)
k
that is, k ≪Λ. As discussed above, the fixed-point con- − 1 [(Dx˜+D )r(x˜−y˜)]G˜∗ γ˜(2)G˜∗ . (12)
dition coincides with the WI for scale invariance in pres- 2Z R x˜z˜ z˜w˜ w˜y˜
x˜i
ence of a regulator,see Eqs. (7,11). As a consequence, if
where x˜ ={x˜,y˜,z˜,w˜}, G˜∗[φ˜]is the dimensionless renor-
the RG flow is attracted towardsan IR fixed-point, then i
scale invariance emerges in the universal, long-distance malizedpropagatoratthefixedpoint: G˜∗ =(Γ˜∗(2)+r)−1
regime 2. We stress that this discussion does not rely and r(x˜) is the dimensionless inverse Fourier transform
onthe actualexistence ofa well-definedcontinuum limit of r(q2/k2) defined in Eq. (1).
associated with a scale-invariant action S which is, We conclude fromthe abovediscussionthatregularity
per se, an interesting issue, but that we doscanlot need to selects among all the fixed-point functionals Γ˜∗[φ˜] those
address in the present work. that are physical, that is, that can be reached by an RG
flowfromaphysicalactionS andthathavea discrete
When the microscopic action is slightly different from crit
spectrum of eigenperturbations.
the critical one (e.g. choosing a temperature slightly
awayfromthecriticalone),theRGtrajectoryapproaches
the fixed point and then stays close to it for a long RG
IV. SPECIAL CONFORMAL
“time”, before departing. In this situation, the corre-
TRANSFORMATIONS
lation length ξ is finite but large and the WI is almost
fulfilled for momenta Λ ≫ p ≫ ξ−1. This defines the
Let us now study conformal invariance by following
critical regime of the theory. The closer the microscopic
the same method as above. As in the case of scale in-
actionis tuned to the criticalone, the largerthe correla-
variance, to motivate the form of the WI we imagine
tion length ξ, and the better the WI is fulfilled.
that a conformally-invariant model, associated with an
Let us now make two comments. First, when
action S [ϕ] written in terms of a primary field ϕ,
k → 0, ∂ Γ [φ] becomes negligible compared to any k- conf
t k exists. We put aside for now the problem of the ex-
independent finite scale and Γ → Γ. In this limit,
k istence of this model since, as discussed below, we do
Eqs. (7,11) become the usual WI of scale invariance, as
not need that it actually exists for our proof. If such
expected. Second, the above analysis shows that among
a model existed, the modified WI would follow from
the continuous infinity of solutions of the fixed point
equation ∂ Γ˜ [φ˜] = 0, only those that are regular for all performing the infinitesimal change of variables ϕ(x) →
t k ϕ(x)+ǫ (x2∂ −2x x ∂ +2αx )ϕ(x) in Eq. (2). By
fields are acceptable since they must be the limit when µ µ µ ν ν µ
k → 0 of the smooth evolution of Γ˜ [φ˜] from S [φ˜]. consideringgeneralcutofffunctionsasinEq.(6),wefind
k crit that it reads:
There is generically only a discrete, often finite, number
of such physical fixed-points.3 (Kx−D x +Ky−D y )R (x,y) δΓk
A characteristic feature of the physical fixed-points Zxy µ R µ µ R µ k δRk(x,y)
is that the linearized flow around them has a discrete δΓ
+ (Kx−2D x )φ(x) k =0, (13)
spectrum of eigenvalues from which some critical ex- Z µ φ µ δφ(x)
x
ponents can be straightforwardly obtained [15]. The
with Kx = x2∂x − 2x x ∂x. By specializing R to
µ µ µ ν ν k
functions of the form Eq. (1) and requiring again that
Z ∝ k−η, Eq. (13) can be rewritten
k
2 A−a∂fitxruleodng-npZinokgi.ntIatvniaoslmuoenallwoyuhasicrohduiinsmdneotnhtsheioinfingxecedalsnpeotbihneatntdhηea.fitnηekdabpyproηakche=s 0= Σµk[φ]≡Zx(Kµx−2Dφxµ)φ(x)δδφΓ(xk)
3 Twowell-knownexceptionstothisrulearethelineoffixedpoints 1
of the O(2) model in d = 2 [31] and the discrete infinity of − ∂tRk(|x−y|)(xµ+yµ)Gk,xy. (14)
2Z
(multicritical)Z2-invariantfixedpointsind=2[32,33]. xy
4
Again, this identity boils down to the usual WI of con- V. A SUFFICIENT CONDITION FOR
formal invariance in the limit k →0 where R →0. CONFORMAL INVARIANCE
k
At any fixed point, the scaling dimension of Σµ[φ] is
k
fixed by Eq. (14) to be −1. We thus define Σ˜µ[φ˜] = Letus now considera physicalmodelatcriticality. At
k
kΣµ[φ]. Its flow equation reads: thescaleΛ,Γ =S =S andthemodelisneithercon-
k Λ crit
formally nor scale invariant. However, when k ≪Λ, the
δΣ˜µ regularized model gets close to a fixed point and thus
∂tΣ˜µk[φ˜]−Σ˜µk[φ˜]=Zx˜(Dx˜+Dφ)φ˜(x˜)δφ˜(x˜k) G˜k ≃ G˜∗ and Σ˜µk[φ˜] ≃ Σ˜µ∗[φ˜]. The key point of our
proof is that at the fixed point, Eq. (15) is identical to
1
− 2Z (Dx˜+DR)r(x˜−y˜) G˜k,x˜z˜Σ˜µk,z(˜2w˜)G˜k,w˜y˜, (15) (12) with γ˜[φ˜]→Σ˜µ∗[φ˜] and λ=−1 although these two
x˜(cid:2)i (cid:3) equations have different physical meanings. A sufficient
conditionforprovingconformalinvarianceisthereforeto
where Σ˜µ(2)[φ˜] is the second functional derivative of Σ˜µ.
k k showthatthereisnointegratedvectoreigenperturbation
itiAsnthimeipnotretgarnatlpofroapdeertnysiotfyΣwµkitihstnhoate,xaptlitchitedfiexpeednpdoeinncte, Vµ = xVµ ofΓ˜∗[φ˜]ofscalingdimensionDV =−1. Ifno
sucheRigenperturbationexists,the conformalWI is satis-
onx . Indeed,observethattheleft-hand-sidesofEqs.(6)
µ fied, which means that the system is conformally invari-
and (13) can be interpreted as the action of the genera-
ant at criticality in the long distance regime. Moreover,
tors of dilatations D and conformal transformations K
µ the form of the conformal WI (14) fixes the transforma-
on Γ .
k tion law fulfilled by ϕ, which is the one of a primary
field.
δΓ
DΓk = (Dx+Dφ)φ(x) k To understand how conformal invariance is related
Z δφ(x)
x (16) with the scaling dimension of the vector eigenperturba-
δΓ
+ (Dx+Dy +D )R (x,y) k tions, it provesuseful to consider three simple examples.
R k
Zxy δRk(x,y) Thefirstoneisthe Isingmodelind=4. Thefixedpoint
being gaussian,the eigenvalues are trivially given by the
canonical dimensions. By using the fact that Σ˜µ is Z −
δΓ k 2
K Γ = (Kx−2D x )φ(x) k andtranslation-invariant,seeEqs.(14)and(20), wefind
µ k Z µ φ µ δφ(x)
x by inspection that the vector operator with lowest di-
+Zxy(Kµx− DRxµ+Kµy−DRyµ)Rk(x,y)δRδkΓ(xk,y), m+3en.sNioonteretahdastRtxhφer∂eµφex(∂isφts)2l.ocItalhvaesctthoersreofopreeradtiomresnwsiiothn
(17) lowerscalingdimensions[φ∂µφ,φ3∂µφandφ∂µφ(∂νφ)2)].
However these are total derivatives and are not associ-
thatis,Σµ =KµΓk. Similarexpressionscanbe obtained ated with integrated vector operators. In the absence of
for the generators of translations Pµ and rotations Jµν: vector operator of dimension −1, we retrieve the well-
known property that this model is conformally invariant
δΓ
k at criticality in the long distance regime [13, 16]. Using
P Γ = ∂ φ(x)
µ k µ
Zx δφ(x) standardmethods,wecancomputethecorrectionstothe
(18)
δΓ scalingdimensionofthevectoroperator φ∂ φ(∂φ)2 in
+ (∂x+∂y)R (x,y) k x µ
Zxy µ µ k δRk(x,y) a systematic expansion in ǫ = 4−d. WRe performed the
calculation at one loop and found that the correction
vanishes. The scaling dimension is therefore 3+O(ǫ2).
δΓk This analysis can be extended to the O(N) mod-
J Γ = x ∂ φ(x)
µν k hZx µ ν δφ(x) els. In d = 4, there exists now two independent
integrated vector operators, φa(∂ φa)(∂ φb)2 and
+Zxy(xµ∂νx+yµ∂νy)Rk(x,y)δRδkΓ(xk,y)i (19) xφa(∂νφa)(∂µφb)(∂νφb), with lRoxwest sµcalingνdimension
R3. As in the Ising case, there exist local operators with
−[µ↔ν]
lower scaling dimensions, that are however total deriva-
tives. Our sufficient condition is again fulfilled and we
Itcaneasilybe checkedthatthe generatorssatisfythe
recoverthewell-knownfactthatthesemodelsareconfor-
algebra of the conformal group. In particular, applying
mally invariant in the critical regime for d = 4. At one
[P ,K ] = 2δ D+2J to a translation, rotation and
µ ν µν µν loop the degeneracy of the scaling dimensions is lifted
dilatation invariant Γ yields
k and we obtain 3+O(ǫ2) and 3− 6ǫ +O(ǫ2).
N+8
PµΣνk =0. (20) The third example involvesa vector fieldAµ(x) andis
described by the (euclidean) action
Thus, at the fixed point, Σµ is the integral of a density
k 1 α
that does not have an explicit dependence on x. This S = (∂ A −∂ A )2+ (∂ A )2 . (21)
µ ν ν µ µ µ
Z (cid:20)4 2 (cid:21)
densityonlydependsonthefieldanditsderivatives. This x
proof generalizes trivially to other scalar models. This model is interesting because it is scale invariant
5
λ λ
Thepreviousargumentsarecompellingbutnotmathe-
(a)
matically rigorous. In particular,assuming that the the-
(b)
ory can be properly defined in noninteger dimensions,
3 (c) 3
which is standard but not obvious, it is hard to control
d d the analytic structure of the critical exponents in d. In
−1 −1 thenextsection,wegiveaproofinthe physicallyimpor-
tantcaseof the Ising model ind=3 whichdoes not rely
on such arguments.
FIG. 1: Possible behavior of the lowest eigenvalue DV asso-
ciated with avectorperturbationasafunction ofdimension.
VI. PROOF IN THE ISING UNIVERSALITY
Left panel: (a) and (b) correspond to typical behavior, (c)
CLASS
to the exceptional case where DV = −1 right in d = 3. In
thethreecases, conformal invarianceholds. Rightpanel: the
shaded area represents a continuum of eigenvalues and the WeconcentratehereontheIsinguniversalityclassand
curveaneigenvalueDV havingaplateauat−1aroundd=3. show that, in this case, the smallest eigenvalue DV as-
In these cases, conformal invariance can bebroken. sociated with an integrated vector perturbation is larger
than −1 for d < 4. Using our necessary condition, this
proves that, for the Ising universality class, the critical
regime is conformally invariant.
but generically not conformally invariant, except for
For simplicity, we consider a lattice version of the
α=α =(d−4)/d[37,38]. Thissituationcanbeunder-
c
Ginzburg-Landaumodelwhosedynamicsisdescribedby
stoodinourcontextbyconsideringthecontrapositionof
the hamiltonian (or action)
oursufficientcondition,whichstatesthat,assumingthat
A is a primary field [38], a necessary condition for hav-
µ
S =−J ϕ(i)ϕ(j)+ U(ϕ(i)). (22)
ing scale invariance without conformalinvariance is that
thereexistsanintegratedvectoroperatorwithscalingdi- Xhiji Xi
mension −1. It is actually easy to find such an operator
(whichisunique,uptoanormalization): C Aµ(∂ A ). where the index i labels the lattice sites, the ϕ(i) take
x ν ν values in the real domain and U(ϕ) is an even function
We can understand the particular case α=Rαc by an ex-
plicit calculation of Σµ[A ] from Eq. (14). This shows that diverges for |ϕ|→∞. We choose here a hypercubic
k ν lattice with lattice spacing a. The original Ising model
that C = αd+4−d which, as expected, vanishes when
canberecoveredbyconsideringapotentialU(ϕ)strongly
α=α .
c
peaked around ϕ=±1 but the Ginzburg-Landau model
To conclude this section, we discuss the plausibility of
which is in the Ising universality class for a generic po-
not having conformal invariance in d = 3 for the O(N)
tential is more convenient for what follows.
models at criticality. The only possibility would be to
The use of Ginzburg-Landau model has another ad-
have a vector eigenperturbation with eigenvalue −1 in
vantage. In the case of a quartic potential,
this dimension, see Fig. 1. This would mean that the
d = 3 model has an integer critical exponent, a prop- U(ϕ(i))= r0ϕ2(i)+ u0ϕ4(i) (23)
erty that is highly improbable. Let us anyhow suppose 2 4!
that one of these eigenvalues crosses −1 right in d = 3,
and for d<4, the model is super-renormalizable. Its ul-
as in curve (c) of Fig. 1. Then, for any dimension in-
travioletbehaviour is therefore controlledby a Gaussian
finitesimally smaller or larger than three, there would
fixed-point. In this case, the existence of a controlled
exist no eigenperturbation of dimension −1. The criti-
scaling limit seems to be under control even if, to the
calsystemwouldexhibitconformalsymmetryaboveand
best of our knowledge, there is no mathematical proof
below d = 3. Since correlation functions of the criti-
of its existence [39]. To compare different models in the
cal theory are expected to be continuous functions of d,
Ising universality class we assume below that this scal-
we conclude that, even in this highly improbable situ-
ing limit does exist in the following precise sense. Let us
ation, the model would exhibit conformal invariance at
considertwolocaloperatorsO (i,a)andO (i,a). Letus
criticality in d = 3. We are thus led to the more strin- 1 2
also introduce (i) a smooth interpolating field φinterp(x)
gentnecessary(butnotsufficient)condition: foracritical
withx∈Rd thatcoincidesonthelatticepointswithφ(i)
model not to be conformally invariant, there must exist
and (ii) two interpolating operators Ointerp(x,a) defined
a vector perturbation of scaling dimension −1 in a finite 1,2
interval of dimensions containing d=3. This could hap- by: Ointerp = O (φinterp). Of course the construction
i i
peneitherbecauseadiscreteeigenvalueisindependentof of these interpolating operators is not unique. We now
the dimension in some range of dimension around three consider the particular case where O and O are such
1 2
or because there exists a continuum of eigenvalues, see that Ointerp → Ointerp when a → 0. (We notice that
1 2
Fig.1. Suchabehavioris,tosaythe least,notstandard. this limit is independent of the choice of interpolation
To our knowledge, this has never been observed in any used to define φinterp). Our assumption, that we call for
interacting model. short“scalinglimit”, is thatthere exists a multiplicative
6
factor Z (a) depending on the lattice spacing such that where ∂ is a lattice discretization of the partial deriva-
O µ
the correlation functions of the operators O (x,a) and tive. (Notethatthefirstthreeoperatorsaretotalderiva-
1
Z (a)O (x,a) are the same for distances much larger tives.) Indimensiond<4,althoughwedo notknowthe
O 2
than a: explicit form of the eigenoperators, we can, in principle,
(n)
decompose any vector operator on the basis {V } and
µ
hO (x,a)O (y )...O (y )i
1 3 3 n n generically, there is a nonvanishing overlap with each of
∼ZO(a)hO2(x,a)O3(y3)...On(yn)i (24) the eigenoperators:
where {O3,...,On} are arbitrary local operators and Vµ = αnVµ(n). (31)
where the equivalence occurs for a ≪ min{|y3 − Xn
x|,...,|y − x|}. This hypothesis is a pre-requisite of
n
As a consequence, the critical regime of the two-point
all Monte Carlo simulations, and is, of course, satisfied
correlationfunctionis dominatedbythe smallestcritical
toallordersofperturbationtheoryinanyrenormalizable
dimension
theory. We assume here that it is also valid nonpertur-
batively. α2
hV (x)V (y)i ∼ 0 . (32)
Our strategy is to study correlation functions of local µ µ c |x−y|2DV(0)
vector operators V (x) and use their critical behavior to
µ
find a bound on the scaling dimension of the integrated Westressthatthelistofquantumnumbersmustinclude
operator Vµ = ad iVµ(i). We rightaway mention two those associated with lattice isometries. For example,
difficulties. P we require scalar (respectively vector) operators defined
First, there are local vector operators that are total on the lattice to be even (respectively odd) under parity
derivatives and which are therefore not associated with transformations.
an integrated one. As discussed before, the vector oper- The proof is organized as follows. Using Lebowitz in-
ator ∂µ(ϕ2) is such an operator. Note that its scaling equalities[40,41], wederivealowerboundforD(0) from
dimension near d = 4 is lower than that of the vec- V
which follows a lower bound for the scaling dimension of
tor operators which are not total derivatives (such as
the integratedvectoroperators. The proofthatthe scal-
∂ (ϕ)2(∂ ϕ)2).
µ ν ing dimension D of any integrated vector operator is
V
Second, operators with the same quantum numbers
different from −1 for d≤ 4 is then a direct consequence
typically mix together in the calculation of correlation
of this bound.
functions. As a consequence, the critical behavior of a
As a first step, we derive a bound for the correlation
two-point function of some vector operator is governed function hϕ2(x)ϕ2(y)i . We use here the Lebowitz in-
c
by the lowest scaling dimension of the class of opera-
equalities [40] which state that, considering two decou-
tors with which it mixes. To be more precise, let us call
pled copies of the ferromagnetic system (that we note ϕ
Vµ(n) the local vector eigenoperator of scaling transfor- and ϕ′), both described by the action (22), and consid-
mations with scaling dimension D(n) (ordered such that ering two sets A and B of lattice points,
V
D(0) ≤ D(1) ≤ ···). The associated two-point correla-
V V h [ϕ(i)+ϕ′(i)][ϕ(j)−ϕ′(j)]i≤
tion function behaves, in the critical regime, as:
i∈AY,j∈B
(33)
hV(n)(x)V(n)(y)i ∼ 1 , (25) h [ϕ(i)+ϕ′(i)]ih [ϕ(j)−ϕ′(j)]i,
µ µ c |x−y|2DV(n) iY∈A jY∈B
h [ϕ(i)+ϕ′(i)][ϕ(j)+ϕ′(j)]i≥
where the subscript c indicates connected correlation
functions, defined as: i∈AY,j∈B
(34)
h [ϕ(i)+ϕ′(i)]ih [ϕ(j)+ϕ′(j)]i.
hX(x)Y(y)i =hX(x)Y(y)i−hX(x)ihY(y)i (26)
c iY∈A jY∈B
andwhereanappropriatenormalizationhasbeenchosen. In particular, this implies that:
Ind=4,the eigenproblemcanbe solvedandthe scaling
dimensions are the canonical dimensions. In particular, h(ϕ(x)+ϕ′(x))2(ϕ(y)−ϕ′(y))2i≤
the eigenoperators with lowest scaling dimensions are: h(ϕ(x)+ϕ′(x))2ih(ϕ(y)−ϕ′(y))2i.
(35)
V(0,d=4) ∝∂ φ2 D(0,d=4) =3 (27)
µ µ V
Expandingthebinomials,wereadilyobtainthefollowing
V(1,d=4) ∝∂ φ4 D(1,d=4) =5 (28)
µ µ V identity:
V(2,d=4) ∝∂ (∂ φ)2 D(2,d=4) =5 (29)
µ µ ν V hϕ2(x)ϕ2(y)i ≤2G2(x−y) (36)
c
V(3,d=4) ∝(∂ φ2)(∂ φ)2 D(3,d=4) =7 (30)
µ µ ν V where we have used the fact that the average of an odd
···
number of fields vanishes for temperatures higher than
7
(or equal to) the critical temperature. This implies that Now,anylocalvectoroperatorV onthelatticeisalin-
µ
the connected correlationfunction hϕ2(x)ϕ2(y)i cannot earcombinationofvectoroperatorsoftheform(40). For
c
decrease more slowly than the square of the propagator instance,adiscretizationofthe operator∂ (φ2)(∂ φ)2 is
µ ν
at long distances. This inequality can be generalized to given by:
arbitrary even powers of the fields:
φ(x)
0≤hϕm(x)ϕn(y)ic ≤CG2(x−y) (37) 16a3 (φ(x+µˆ)−φ(x−µˆ))Xν (φ(x+νˆ)−φ(x−νˆ))2
(42)
where C is a positive constant (that depends on m and
where the sum runs over all the nearest neighbors of x.
n) as shown in the Appendix A.
Using the triangular inequality, we conclude that:
Inthecriticalregime,scaleinvarianceimpliesthatcon-
nected two-point correlation functions behave as power-
Z
µν
laws. In particular: |hVµ(x)Vν(y)ic|≤ |x−y|2(d−1+η). (43)
A
hϕm(x)ϕn(y)ic ∼ |x−y|ℵm+ℵn (38) where Zµν is a positive constant. Using Eq. (32) this
implies that, for all n:
with A a positive constant (see, for example [14]). The
D(n) ≥d−1+η (44)
inequality (37)implies thatℵn ≥d−2+η. We canthen V
deduce the asymptotic behavior of the matrix of second
We conclude that the scalingdimensionD =D −dof
derivatives of this correlationfunction: V V
anyintegratedvectoroperatorisnotsmallerthan−1+η.5
h∂x(ϕm(x))∂y(ϕn(y))i ∼ 1 Bδ Using the unitarity of the Minkowskian φ4 theory, one
µ ν c |x−y|ℵm+ℵn+2(cid:16) µν can prove that η ≥0 [42] for the Ising universality class.
(39)
(x−y) (x−y) Moreover,aninteractingmasslesstheorysuchasthecrit-
µ ν
+C . ical Ising model for d < 4, has a non-zero η [43]. As a
(x−y)2
(cid:17)
consequence, our necessary condition is fulfilled and we
where B and C are some constants. conclude that scale invariance implies conformal invari-
We now consider two local vector operators that are ance for the Ising universality class for all d≤4.
the product of one power of ∂ ϕ(x) and an odd (finite)
µ
number of fields evaluated at points in a finite neighbor-
hood of x: VII. CONCLUSIONS
m−1
W(1)(x)= 1[∂ ϕ(x)] ϕ(x+se(1)). (40) Let us now point out some directions of research for
µ 2 µ i the future. It is clear that the condition of conformalin-
sX=±1 iY=1
variance(14)canbe straightforwardlyextendedto other
n−1
W(2)(x)= 1[∂ ϕ(x)] ϕ(x+se(2)). (41) theories (involving scalar,fermionic or vector fields) and
µ 2 µ i it would be interesting to conclude on the fate of confor-
sX=±1iY=1
malinvarianceinthiswiderclassofmodels. Inthesesys-
where e(1) and e(2) are some constant lattice vectors4. tems,itismuchmoredifficulttofindrigorousboundson
i i correlationfunctions(thatwouldgeneralizetheLebowitz
(1) (2)
The operators W (x) and W (x) are, up to a multi-
µ µ inequalities). It wouldthenproveuseful to approachthe
plicative constant, other discretizations of, respectively,
problemby computing the scalingbehaviorofvectorop-
theoperator∂x(ϕm(x))and∂x(ϕn(x)). Accordingtothe
µ µ erators by Monte-Carlo simulations.
assumptionoftheexistenceofthescalinglimit,Eq.(24),
Another promising line of investigation consists in
the connected correlation function hWµ(1)(x)Wν(2)(y)ic making use of the conformal invariance in the Wilson
has the same asymptotic behavior as in Eq. (39) up to framework to perform actual calculations of universal
a multiplicative factor depending on the lattice spacing. quantities. On the one hand, and in the best case, this
Indeed,when|x−y|ismuchlargerthanthe latticespac- wouldleadtoclosedandnumericallytractableequations
ing a, the vectors e(1) and e(2) can be neglected in (40) for the critical exponents. On the other hand, the ap-
i i
and(41)andthelocaloperatorsW(1) andW(2) arethen proximation schemes currently used for solving the Wil-
µ µ
proportional to ∂ (ϕm) and ∂ (ϕn) respectively as ex- sonRGflowequationbeingincompatiblewithexactcon-
µ ν
formal invariance,we can expect that constraining them
plained before.
to be conformally invariant at the fixed point would im-
prove their accuracy.
4 SinceΣµ,definedinEq.(14),isoddunderparity,itisimportant
in what follows to consider only vector operators that are also
odd. This is the reason why the sum over s is necessary in the 5 Thisboundalsoappliesforcorrelationfunctionsofmoregeneral
definitions(40)and(41). operators.
8
Notice finally that, at first sight, our approach could We first consider the case where m and n are even.
seem similar to the one based on the energy-momentum Using the Lebowitz inequality [40] [see Eq. (33)]
tensor and on the analysis of the virial current. This is
not the case although there is perhaps a relationshipbe- h[ϕ(x)+ϕ′(x)]m[ϕ(y)−ϕ′(y)]ni≤
tween the two. Σµk is a functional of φ and not of ϕ; it h[ϕ(x)+ϕ′(x)]mih[ϕ(y)−ϕ′(y)]ni
is built from Γ and not from S; what matters is that
k (A4)
its density vanishes up to a surface term and not that
it is conserved. Moreover, as we already explained, we
aswellastranslationinvarianceandthebinomialexpan-
only deal with a regularized theory which enables us to
sion, we obtain:
consider only the analytic candidates for Σ˜µ∗ contrary
to what should be done in a nonregularized theory. In m n
m n
any case, a clarification of the relation between the two (−1)b hϕa(x)ϕb(y)i hϕm−a(x)ϕn−b(y)i
c c
approacheswouldbe welcome. In this respect, our proof Xa=0Xb=0 (cid:18)a(cid:19)(cid:18)b(cid:19)h
ofthenon-existenceoflocalvectoroperatorofscalingdi- +hϕa(0)ihϕb(0)ihϕm−a(x)ϕn−b(y)i
c
mensiond−1(conservedornot)mightbeofinterestalso
when applied to a hypothetical conserved virial current. +hϕa(x)ϕb(y)i hϕm−a(0)ihϕn−b(0)i ≤0
c
i
(A5)
Acknowledgments Writing explicitly the terms with a ∈ {0,m} and b ∈
{0,n}, we get the following bound:
The authors acknowledge financial support from the
ECOS-Sud France-Uruguay program U11E01, and from 2hϕm(x)ϕn(y)i ≤ m n ×
c
the PEDECIBA. BD and MT also thank the Universi- (cid:18)a(cid:19)(cid:18)b(cid:19)
a∈{1,3X,···,m−1}b∈{1,3X,···,n−1}
daddelaRepu´blica(Uruguay)forhospitalityduringthe
completionofthis work,andNW the LPTMCfor hospi- hϕa(x)ϕb(y)ichϕm−a(x)ϕn−b(y)ic
talityduringhissabbaticalyear2012-2013. WethankA. (A6)
Abdesselam, L. Messio, T. Morris, V. Rivasseau and V.
Rychkov for useful discussions. We thank O. Rosten for where we have used the fact that connected correlation
pointingoutthereference[43]andV.Rychkovforpoint- functions, as wellas hϕm(0)i arenon-negative[44, 45]to
ing out an error in a previous version of the manuscript. eliminate the terms with even b. We observe that there
appearsintheright-handsideonlyconnectedcorrelation
functionswithatmostm−1powersofϕ(x)andatmost
n−1powersofϕ(y). Byhypothesis,properties(A1)and
Appendix A: Bound for correlation functions
hϕm(x)ϕn(y)i (A2) are thus valid for these correlation functions. Fur-
thermore, we observe that the right-hand side involves a
sum of terms that are a product of two correlationfunc-
In this appendix, we derive bounds on the correlation
tions with odd powers of the fields. In both cases, by
functions hϕm(x)ϕn(y)i , in the symmetric phase (T ≥
c usingproperties(A1)and(A2),thisquantityisbounded
T ),withmandnarbitraryintegerswiththesameparity
c by some positive constant times G2(x−y).6 This con-
(otherwise the correlationfunction vanishes).
cludes the proof of the induction hypothesis in the case
We want to show that:
of even m and n.
We now turn to the case where m and n are odd. We
hϕa(x)ϕb(y)i ≤C G(x−y) for odd a, b (A1)
c 1 now make use of the Lebowitz inequality
hϕa(x)ϕb(y)i ≤C G2(x−y) for even a, b (A2)
c 2
h[ϕ(x)+ϕ′(x)]m−1[ϕ(x)−ϕ′(x)][ϕ(y)−ϕ′(y)]ni≤
whereC andC aresomestrictlypositiveconstantsand
1 2 h[ϕ(x)+ϕ′(x)]m−1ih[ϕ(x)−ϕ′(x)][ϕ(y)−ϕ′(y)]ni
(A7)
G(x−y)=hϕ(x)ϕ(y)i. (A3)
which is of interest if m > 1 (we can obviously derive
Note that for odd a and b, the connected and discon-
a similar inequality with {m,x} ↔ {n,y} which can be
nected correlationfunctions are equal, see Eq. 26.
applied in the case m = 1). We again use the binomial
Property (A1) is obvious for a = b = 1. The proof
of (A2) for {a = 2,b = 2} is presented in the core of
the article. For general a and b, the proof is made by
induction. Assumingthattheinequalities(A1)and(A2)
6 Weuseherethefactthat,onthelattice, Gisboundedbysome
arefulfilled for{a≤m,b≤n}\{a=m,b=n},we have
positiveconstant(G(r)<D). Consequently, wecanfurtheruse
to prove that these properties are also valid for a = m theboundG2n(r)≤CG2(r)wherenisapositiveintegerandC
and b=n. apositiveconstant.
9
expansion and the positivity of (connected and discon- tionfunctions,onewithevenandonewithoddpowersof
nected) correlation functions to obtain the following in- the fields, or the product of a correlation function with
equality: odd powers of the fields and a positive constant. In all
cases, the correlationfunctions that appear in the right-
hϕm(x)ϕn(y)ic ≤ hand side fulfill the conditions of our hypothesis. We
m−1 n therefore conclude that hϕm(x)ϕn(y)ic satisfies property
×
(A1) for m and n odd (see Footnote 6). This concludes
(cid:18) a (cid:19)(cid:18)b(cid:19)
a∈{0,2X,···,m−1}b∈{1,3X,···,n−1} the proof of the induction hypothesis.
hϕa(0)ihϕm−a−1(0)ihϕ(x)ϕb(y)i hϕn−b(0)i (A8)
c
m−1 n Using the fact that the property (A1) is valid for a =
+ ×
(cid:18) a (cid:19)(cid:18)b(cid:19) b = 1, it is easy to check, by applying several times the
a∈{1,3X,···,m−2}b∈{0X,2,···,n}
induction property, that (A1) and (A2) are valid for any
hϕa+1(x)ϕb(y)ihϕm−1−a(x)ϕn−b(y)i a and b.
The firstterminvolvesthe productofacorrelationfunc-
tionwithoddpowersofthefieldsandapositiveconstant.
The secondsuminvolveseither a productoftwocorrela-
[1] J. M. Maldacena, Int. J. Theor. Phys. 38, 1113 (1999) [21] O. J. Rosten, arXiv:1411.2603 [hep-th].
[Adv.Theor. Math. Phys.2, 231 (1998)]. [22] J. Polchinski, Nucl. Phys. B 231, 269 (1984).
[2] S. El-Showk, M. F. Paulos, D. Poland, S. Rychkov, [23] C. Wetterich,Phys. Lett. B 301, 90 (1993).
D. Simmons-Duffin and A. Vichi, Phys. Rev. D 86, [24] U. Ellwanger, Z. Phys.C 58, 619 (1993).
025022 (2012). [25] T. R.Morris, Int.J. Mod. Phys.A 9, 2411 (1994).
[3] F.Gliozzi andA.Rago,J.HighEnergyPhys. 10(2014) [26] J.Berges,N.TetradisandC.Wetterich,Phys.Rept.363,
042. 223 (2002).
[4] S. El-Showk, M. F. Paulos, D. Poland, S. Rychkov, [27] T. Machado and N. Dupuis, Phys. Rev. E 82, 041128
D.Simmons-DuffinandA.Vichi,J.Stat.Phys.157,869 (2010).
(2014). [28] C. Becchi, hep-th/9607188.
[5] F. Kos, D. Poland and D. Simmons-Duffin, J. High En- [29] U. Ellwanger, Phys.Lett. B 335, 364 (1994).
ergy Phys. 06 (2014) 091. [30] J.M.Pawlowski,AnnalsPhys.(N.Y.) 322,2831(2007).
[6] S. El-Showk, M. Paulos, D. Poland, S. Rychkov, [31] J. M. Kosterlitz and D. J. Thouless, J. Phys. C 6, 1181
D. Simmons-Duffin and A. Vichi, Phys. Rev. Lett. 112, (1973).
141601 (2014). [32] A.B.Zamolodchikov,Sov.J.Nucl.Phys.44,529(1986)
[7] Y. Nakayama and T. Ohtsuki, Phys. Rev. D 91, [Yad. Fiz. 44, 821 (1986)].
021901(R) (2015). [33] T. R.Morris, Phys.Lett. B 345, 139 (1995)
[8] M. S. Virasoro, Phys.Rev. D 1, 2933 (1970). [34] T. R. Morris and M. D. Turner,Nucl. Phys. B 509, 637
[9] A. M. Polyakov, JETP Lett. 12, 381 (1970) [Pisma Zh. (1998).
Eksp.Teor. Fiz. 12, 538 (1970)]. [35] T. R.Morris, Phys.Rev.Lett. 77, 1658 (1996).
[10] A.A. Migdal, Phys. Lett.B 37, 386 (1971). [36] T. R.Morris, Nucl.Phys. B 495, 477 (1997).
[11] A.A.Belavin,A.M.PolyakovandA.B.Zamolodchikov, [37] V.Rivaand J. L.Cardy,Phys.Lett. B622, 339 (2005).
Nucl.Phys. B 241, 333 (1984). [38] S. El-Showk,Y. Nakayamaand S.Rychkov,Nucl.Phys.
[12] A.B.Zamolodchikov,JETPLett.43,730(1986)[Pisma B 848, 578 (2011)
Zh.Eksp. Teor. Fiz. 43, 565 (1986)]. [39] A. Abdesselam and V. Rivasseau, private communica-
[13] J. Polchinski, Nucl.Phys. B 303, 226 (1988). tion.
[14] P.DiFrancesco,P.MathieuandD.S´en´echal,Conformal [40] J.L. Lebowitz, Commun. Math. Phys. 35, 87 (1974).
field theory (Springer,New York,1997), p.890. [41] J. Glimm and A.Jaffe, Quantum Physics: A Functional
[15] K.G.WilsonandJ.B.Kogut,Phys.Rept.12,75(1974). Integral Point of View, 2nd ed. (Springer-Verlag, New
[16] I.Jack and H.Osborn, Nucl. Phys.B 343, 647 (1990). York,1987).
[17] M.A.Luty,J.PolchinskiandR.Rattazzi,J.HighEnergy [42] J. Zinn-Justin,Int.Ser. Monogr. Phys. 113, 1 (2002).
Phys.01 (2013) 152. [43] K. Pohlmeyer, Commun. Math. Phys. 12, 204 (1969).
[18] A. Dymarsky, Z. Komargodski, A. Schwimmer and [44] R. B. Griffiths, J. Math. Phys. 8, 478 (1967); 8, 484
S.Theisen, arXiv:1309.2921 [hep-th]. (1967).
[19] A. Dymarsky, K. Farnsworth, Z. Komargodski, [45] D. G. Kelly and S. Sherman, J. Math. Phys. 9, 466
M. A.Luty and V. Prilepina, arXiv:1402.6322 [hep-th]. (1968).
[20] L. Schafer, J. Phys. A 9, 377 (1976).