Bootstrapping critical Ising model on three-dimensional real projective space

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by  Yu Nakayama

Bootstrapping critical Ising model on three-dimensional real projective space
Yu Nakayama1
1Kavli Institute for the Physics and Mathematics of the Universe (WPI),
University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8583, Japan
(Dated: April20, 2016)
GivenaconformaldataonaflatEuclideanspace,weusecrosscapconformalbootstrapequations
to numerically solve the Lee-Yang model as well as the critical Ising model on a three-dimensional
real projective space. We checktherapid convergence of our bootstrap program in two-dimensions
from theexactsolutionsavailable. Based on thecomparison, weestimate thatoursystematicerror
onthenumericallysolvedone-pointfunctionsofthecriticalIsingmodelonathree-dimensionalreal
projectivespaceislessthanonepercent. Ourmethodopensupanovelwaytosolveconformalfield
6 theories on non-trivial geometries.
1
0
2 The numerical conformal bootstrap in higher dimen-
r sional conformal field theories (CFTs) has provided us
p with the most efficient way to derive critical exponents
A
in yet-to-be analytically solved theories such as the crit-
ical Ising model [1][2][3][4] in three dimensions. It also
9
1 judges (non)-existence of controversial CFTs in various
areas of theoretical physics (e.g. [5][6][7]).

] Almost allof these numericalstudies havebeen aimed
h
atderivinga conformaldata,i.e. operatorspectrumand
t
- operator product expansion (OPE) coefficients, on a flat
p
Euclidean space (see, however, [8][9][10] for numerical
e
h bootstrap with defects or boundaries). Studying CFTs FIG. 1. Involutions used in two-dimensional real projec-
[ on space-time with non-trivial geometry should give us tive spaces: The left panel describes the involution (t,Ω~) →
more information about the properties of the CFT. For (−t,−Ω~)ontheLorentziancylinderwhiletherightpanelde-
1v2 emxeanmsipolnes,,twhhereerehtahveemboedeunlanruimnvearroiuasncsetupduitesssitnrotnwgocodni-- fsucrnidbaems tehnetailndvoolmutaiionnc~xan→be−c~x~xh2osoenntth≥eE0uocrlidre2a=n~xsp2a≥ce1..The
5 straints on the spectrum.
8 Inthispaper,weusetheconformalbootstrapprogram
6 tosolveCFTsond-dimensionalrealprojectivespaces. A In two dimensions, CFTs on a projective space have
0 real projective space is defined by an involution importantapplicationstounorientedstringtheory. They
1. give a worldsheet realization of orientifold planes. In ra-
0 ~x tionalCFTs,onemayusetheconformalbootstrapequa-
~x (1)
6 →−~x2 tion on the projective space or modular bootstrap equa-
1 tion on the Klein bottle and Mo¨bius strip to solve the
on the flat Euclidean space R parametrizedby a vector
v: ~x, and it preserves the SO(1,dd) subgroup of the original crosscap states. This is possible with the help of the
i enhanced Virasoro symmetry with null vectors [16].
X (Euclidean) conformal symmetry SO(1,d + 1). Alter-
So far, there has been no rigorous way to use an ana-
natively, one may conformally map it to the Lorentzian
r logueof modularbootstrapin higherdimensionalCFTs.
a cylinder R S , where the involution is given by the
t d 1
CPT transf×orma−tion (t,Ω~) ( t, Ω~), where Ω~ is a In this paper, we, then, resort to conformal bootstrap
→ − − equations on real projective spaces to derive one-point
unit vector on R parametrizing S . When d is even,
d d 1 functions. Our input parameters are conformal data on
the real projected space is not orien−table. See fig 1 for
aflatEuclideanspace. We study the criticalIsingmodel
illustration.
andtheLee-Yangmodelasourexamples,butourmethod
has wider applicability to the other non-trivial CFTs as
long as the conformal data on the flat Euclidean space
More recently, we found that CFTs on real projec- are sufficiently well-known.
tive spaces have a surprising application to construct- Three dimensionalCFTs are notoriouslyhardto solve
ing bulk local operators in the context of holography analytically, but the recent development in numerical
[11][12][13]. There the building block of CFTs on real conformal bootstrap allows us to obtain conformal data
projective spaces, so-called Ishibashi states [15] can be with sufficiently high precision. We use these data to
mapped to bulk local operators in the dual AdS space- numerically solve the conformal bootstrap equations on
time. Thisobservationgivesageometricalinterpretation realprojectivespaces. Wechecktherapidconvergenceof
of the crosscap conformal blocks in terms of correlation ourbootstrapprogramintwo-dimensionsfromtheexact
functions in AdS space-time [13]. solutions available, and we estimate that our systematic
2
erroronthenumericallysolvedone-pointfunctionsofthe numericallytodetermineA withagivenconformaldata
i
criticalIsingmodelonathree-dimensionalrealprojective ∆ and C on the flat Euclidean space.
i 12i
space is less than one percent. As far as we are aware, One way to numerically solve the crosscap bootstrap
our result is the first prediction of the one-point func- equation is to truncate the sum over i and demand that
tions of the critical Ising model on a three dimensional (5) hold around the symmetric point η = 1/2. While
real projective space. there are many options how to determine A approxi-
i
Let us considera CFT ona d-dimensionalrealprojec- matelyatthispoint,weusethesimilarderivativeexpan-
tive space. One-point functions of scalar primary opera- sionsstudiedin[17][18][10]: werequireup-to(2M+1)th
tories O with conformal dimension ∆ are given by order derivatives at η = 1/2 exactly vanish (while even
i i
number of derivatives automatically vanish). Although
A
O (~x) = i . (2) the method has been sometimes called “a severe trunca-
h i i (1+~x2)∆i tion” in the literature [22], we will see that the conver-
genceisremarkablyrapidandjustafewordertruncation
From the rotationalinvariance, higher spin operators do
shows a reasonably accurate estimate of the real projec-
not possess non-zero one-point functions. Once A are
i tivespaceone-pointfunctionsinexactlysolvedexamples
known in addition to conformal data on the flat Eu-
as we will see below. We note that in some cases like
clidean space, the CFT on real projective spaces is com-
a free boson CFT, the sum is actually over a finite set
pletely solved in principle.
in the crosscapbootstrap equation and the truncation is
Two-point functions can be parameterized as
then exact. This is in sharp contrast with the conformal
bootstrap equations on a flat Euclidean space in which
O (~x )O (~x )
1 1 2 2
h i the infinite sum is always necessary.
=(1+~x21)−∆12+∆2(1+~x22)−∆22+∆1 G (η) , (3) On a theoretical basis of this truncation approach,
(~x1 ~x2)2(∆1+2∆2) 12 we note the following. If the coefficients C12iAi in (4)
− wereallpositive, whichis indeed the caseforthe exactly
where η = (1+(~x~x121−)(~x12+)2~x22) is the crosscap crossratio. By spolelvmaebnletatlwmoa-dteimriaeln)s,ioonnealcocuriltdicuasleItshinegHmaroddye-lLi(tsteleewsouopd-
using the conformal invariance and OPE O O =
1 2 type asymptotic analysis to show that the convergence
×
C O in the ~x ~x limit (i.e. η 0 or A B in
i 12i i 1 → 2 → → of the conformalpartial wavedecomposition(4) is expo-
fiPg 1), one can derive the conformal partial wave decom- nentially fast in line with the studies of [23][24][25] for
position [14]:
four-pointfunctions on a Euclidean space. In particular,
only including the operators ∆ <∆ in the summation
G12(η)= C12iAiη∆i/2 would cause the exponentiallyisuppr∗essed error of order
Xi ∆∗
2− 2 in G12(η = 1/2) as ∆ . While we unfortu-
2F1 ∆1−∆2+∆i,∆2−∆1+∆i;∆i+1 d;η . nately do not know the pos∗iti→vit∞y of C12iAi for generic
(cid:18) 2 2 − 2 (cid:19) CFTs,therapidconvergencewewillneverthelessseemay
(4) imply that no dangerous cancellation, which could ruin
the asymptotic analysis, is happening.
We are going to fix the overall normalization such that In order to test the efficiency of our numerical con-
for the identity operator A = 1, and C = δ . Note formal bootstrap program, we begin our studies in two-
1 ij1 ij
thatthesumistakenonlyoverscalarprimaryoperators. dimensions. Rational CFTs on a two-dimensional real
The similar expression for disk two-point functions was projectivespacecanbeexactlysolved,andtheLee-Yang
derived in [19]. In relation to the holography, each term model and the critical Ising model that we examine are
inthesumcorrespondstothebulk-boundarythree-point suchexamples. The formeris anexample ofnon-unitary
functions in AdS space-time [20][21][14]. CFTs, but in our bootstrap program, unitarity does not
Now we may also expand the two-point functions by play a crucial role.
uAsingBtheinmfiigrro1r).chAasnsunmelinOgPtEha~xt1O→t−ra~~xxn222sf(oi.rem.sηtr→ivi1alolyr spiWneopfiresrtatsoturdσyianttwhoe-pLoeien-tYfaunngctmioondheσl(o~xn1)aσr(e~xa2l)iproofjtehce-
′ i
un→der the involution(i.e. O (~x) Γ O ( x~ ) with unit tive space. In terms of the global conformal block, the
Γ ), we have the crossing eqiuat∼ion ij j −~x2 scalar OPE is given by
ij
∆1+∆2 ∆1+∆2 σ σ =σ+1+ǫ′+ǫ′′+ǫ3+ (6)
1 η 6 η 6 × ···
− G (η)= G (1 η) .
(cid:18) η2 (cid:19) 12 (cid:18)(1 η)2(cid:19) 12 − with the conformal dimensions ∆σ = 0.4 and ∆i =
− (5) 0.4,0,4,7.6,11.6,12 . −
− ···
In the crosscap bootstrap equation, we truncate the
This will be called the crosscap bootstrap equation. A sum over the first (n + 1)th scalar exchange (e.g. for
generalization to the case in which O transforms non- n = 2, we include σ,1, and ǫ) and demand that up to
i ′
trivially under the involution is straightforward, but we (2n+1)-thderivativesofthecrosscapbootstrapequation
will not use it in this paper. Our programis to solve (5) atη =1/2vanish. Theresultingapproximatesolutionof
3
the crosscap conformal bootstrap equation for C A Exact (1,2) (2,1) (2,2) (3,1)
σσσ σ
and Cσσǫ′Aǫ′ are summarized in table I. We see that al- CσσǫAǫ 0.20711 0.21499 0.20653 0.20696 0.20712
ready at n=2, the error in C A is one percent, and Cσσǫ′Aǫ′ 0.01563 0.01248 0.01593 0.01571 0.01558
the convergence to the exact sσoσlσutioσn (see supplemental spectrum ∆ǫ′ =4.4 ∆ǫ′′ =9.1 ∆ǫ′′ =8.4 ∆ǫ3 =10.8
predictions ∆ǫ′′ =11.1 ∆ǫ3 =12.2
material) is quite fast.
TABLE III. Solutions of crosscap bootstrap equation with
Exact (2,0) (3,0) (4,0) (5,0) additional self-consistent predictions of the spectrum in two-
CσσσAσ −3.9338 −3.9101 −3.9364 −3.9345 −3.9342 dimensionalcriticalIsingmodel. Withthenotation(n,m)we
Cσσǫ′Aǫ′ −0.01818 −0.02012 −0.01795 −0.01811 −0.01814 truncate the sum up to n+1 scalar operators including the
identity operator, and predict m additional conformal data.
TABLEI.Solutionsofcrosscap bootstrap equationwithvar-
ious truncations in two-dimensional Lee-Yang model. With
the notation (n,0), we truncate the sum up to n+1 scalar Let us begin with the Lee-Yang model in three di-
operators including theidentity operator. mensions. The conformal data, in particular the dimen-
sions of irrelevant operators, is much less known in this
Similarly, we can study a two-point function model, but we use the recent result obtained from the
σ(~x )σ(~x ) ofthe spinoperatorinthe two-dimensional self-consistent numerical bootstrap for four-point func-
1 2
chritical Isingimodel on a real projective space. In terms tions on a Euclidean space in [18]:
oftheglobalconformalblock,thescalarOPEisgivenby
∆σ =0.235 , ∆ǫ′ =4.75 . (8)
σ σ =1+ǫ+ǫ′+ǫ′′+ǫ3+ (7) In table IV we show the numerical solutions of the
× ···
truncated crosscap bootstrap equation. Since we have
with the conformal dimensions ∆ = 0.125 and ∆ =
σ i much less conformal data than in two-dimensions, we
0,1,4,8,9,12, . The approximate solutions of the
··· have also tried to improve our estimate by including the
crosscap bootstrap equation are summarized in table II.
self-consistent predictions of higher dimensional opera-
Again, we see the convergence is remarkably rapid with
tors as shown in table IV. As in two-dimensions, we do
increasing n.
not expect that the self-consistent prediction gives the
precise conformal dimensions, but we expect the one-
Exact (2,0) (3,0) (4,0) (5,0)
point functions become better approximated.
CσσǫAǫ 0.20711 0.20407 0.20757 0.20693 0.20710
Cσσǫ′Aǫ′ 0.01563 0.01702 0.01539 0.01572 0.01563 (2,0) (1,1) (1,2) (2,1) (2,2)
CσσσAσ −1.369 −1.352 −1.3587 −1.3656 −1.3650
TABLEII.Solutionsofcrosscapbootstrapequationwithvar- Cσσǫ′Aǫ′ −0.00613 −0.00409 −0.00485 −0.00561 −0.00551
ious truncations in two-dimensional critical Ising model. spectrum ∆ǫ′ =5.5 ∆ǫ′ =5.1 ∆ǫ′′ =9.6 ∆ǫ′′ =8.9
predictions ∆ǫ′′ =11.3 ∆ǫ3 =12.4
So far, we have used the conformal data on a flat Eu- TABLE IV. Solutions of crosscap bootstrap equation with
clidean space as a given input. However, in harmony additionalself-consistentpredictionsofthespectruminthree-
with a spirit of [10], one may determine the conformal dimensional Lee-Yangmodel.
data self-consistently from the crosscap bootstrap equa-
tion. Thisisdonebyrequiringthatupto(2(n+2m)+1)- Finally, we numerically solve the crosscap bootstrap
th derivative vanish given n input conformal data. The equation in the critical Ising model in three dimen-
condition determines m additional conformal data. sions. As for the conformal data on the flat Euclidean
In table III, we summarize our attempt to deter- space in particular for higher dimensional scalar oper-
mineconformaldataofthetwo-dimensionalcriticalIsing ators, we use two sets of parameters computed by the
model on the flat Euclidean space self-consistently from self-consistent truncated bootstrap solutions of bound-
the crosscap bootstrap equation. We typically see that ary two-point functions [10] and the extremal functional
the predicted conformal dimensions are larger than the method [26] in numerical bootstrap of the four-point
known exact values, but this may be expected because functions on the flat Euclidean space.
theyrepresentcontributionsofallthehigherdimensional The one from the truncated boundary bootstrap is
operatorsthanthe truncatedones. We alsoobservethat
Set A: ∆ =0.518154, ∆ =1.41267,
irrespectiveoftheprecisionofthepredictedconformaldi- σ ǫ
mensions, the obtained one-point functions become bet- ∆ǫ′ =3.8303, ∆ǫ′′ =7.27, ∆ǫ3 =12.90. (9)
ter approximated in all examples we have studied.
The one from the extremal functional method imple-
Given the success of our numerical conformal boot-
mented by Juliboots [27] with n =21, m =1 is
strap program on a real projective space in two- max max
dimensions, we now apply the same idea to three- Set B: ∆ =0.518151, ∆ =1.412658,
σ ǫ
dimensional CFTs. This will result in the first predic-
tions of one-point functions of interacting CFTs on a ∆ǫ′ =3.83032,∆ǫ′′ =6.9905, ∆ǫ3 =10.83,
∆ =15.22, ∆ =21.07 . (10)
three-dimensional real projective space. ǫ4 ǫ5
4
Here, the extremalfunctional includes order n2max opera- for the three-dimensional critical Ising model on a real
2
tors, but the dimensions of scalar operators ∆ > 10 are projective space. From the numerical convergence and
sensitive to n used. This will be the one of the main the comparison with the experience in two-dimensional
max
sources of the systematic error in our program. We have cases, we estimate that our systematic error is less than
also checked the sensitivity with the implementation by one percent.
SDPB [4], but the results do not improve much.
Inthispaper,wehaveprovedthattheconformalboot-
Based on these conformal data, we compute the ap-
strapisusefulnotonlyonflatEuclideanspaces,butalso
proximatesolutionsofthecrosscapbootstrapequationas
on real projective spaces. We would like to conclude the
shownin table V.For comparison,we havealsoincluded
paper with future directions to be pursued.
our self-consistent predictions of the operator spectrum
with n = 2 and m = 2. The result is ∆ǫ′′ = 7.60 and First of all, we have studied the simplest models i.e.
∆ =10.7,whichmaybe comparedwiththe aboveself-
ǫ3 theLee-YangmodelandthecriticalIsingmodel,butour
consistent truncated solutions from the boundary boot-
methodcanbeeasilygeneralizedtoothermodelssuchas
strap.
critical O(n) models. Assuming the simplest involution
that commutes with O(n) symmetry, the crosscap boot-
(2,0) (4,0)A (4,0)B (6,0)B (2,2) strap equations for σ (~x )σ (~x ) take the same form
i 1 j 2
h i
CσσǫAǫ 0.690 0.7015 0.7022 0.70197 0.7006 sinceonlytheO(n)singletscalaroperatorcontributesto
Cσσǫ′Aǫ′ 0.054 0.0475 0.0470 0.04714 0.0480 thesum,soonemaysolvethe equationsinthesameway
as we have done in this paper. The only difference is the
input conformal data, which can be obtained along the
TABLE V. Solutions of crosscap conformal bootstrap equa-
tion for the critical Ising mode in three dimensions. line of [28][29] from the conformal bootstrap.
Another interesting future direction is to study other
In order to check the stability of the truncation and
correlation functions. Indeed, since we have deter-
estimate the systematic error with respect to the less
mined the projective space one-point function only from
reliable spectrum of higher dimensional operators, we
σ(~x )σ(~x ) ,thestudyoftheothercorrelationfunctions
1 2
have repeated our analysis by artificially changing the h i
yields strong consistency checks. More ambitiously we
values of ∆ and ∆ in set B, ranging between (14,16)
ǫ4 ǫ5 may imagine that we would be able to predict confor-
and (18,25)respectively. The predictions of C A and
σσǫ ǫ mal data on the flat Euclidean space such as C /C
σσǫ ǫǫǫ
Cσσǫ′Aǫ′ then fluctuate around 0.70197± 0.00008 and fromtheconsistencyconditionsofmulti-correlatorcross-
0.04714 0.00007. As in [10], we may regardthe fluctu-
± cap bootstrap equations.
ation as a rough estimate of the systematic error of our
approach, which is small and comparable to the trunca- Finally, it would be interesting to check our predic-
tion error we saw in the two-dimensional critical Ising tions from the direct studies of the correlation functions
model. on real projective spaces e.g. from epsilon expansions,
Combining the known OPE coefficients (see e.g. [2]: Hamiltonian truncations, and Monte-Carlo simulations.
the number we use below is obtained from the output of On computers, there is no obstruction to study statisti-
Juliboots with nmax =21, mmax =1) cal models on real projective spaces while experimental
tests may require some ingenuity.
Cσσǫ =1.0518 , Cσσǫ′ =0.0530 , (11)
The authorwouldliketo thank T.Ohtsuki,H. Ooguri
we finally predict and S. Rychkov for discussions and suggestions. This
work is supported by the World Premier International
Aǫ =0.667(2) , Aǫ′ =0.896(5) (12) Research Center Initiative (WPI Initiative), MEXT.
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19
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3 1 1
JHEP 1511, 106 (2015) [arXiv:1504.07997 [hep-th]]. G (η)=(1 η)3/8 F ( , ; ;η)
σσ 2 1
− 4 4 2
3 5 3
+c η1/2(1 η)3/8 F ( , ; ;η), (A6)
I 2 1
Appendix A: Exact bootstrap solutions in − 4 4 2
two-dimensions
where c = √2 1 = 0.20711. This number was also ob-
I 2−
Inthisappendix,wewillpresenttheexactsolutionsof tained from the modular bootstrap in [16].
the crosscapbootstrapequationswith the Virasorosym- Note that the two-point function can be expressed in
metry in two-dimensional Lee-Yang model and the criti- terms of more elementary functions by using
cal Ising model. In rational CFTs, we only need a finite
number of Virasoro conformal blocks to solve the cross- 3 5 3 √2 1 √1 η
F ( , ; ;η)= − −
capbootstrapequations. Furthermoreduetonullvectors 2 1 4 4 2 p√1 η√η
in the Virasoroalgebra,one may derive explicit forms of −
3 1 1 1+√1 η
the Virasoro crosscap conformal block in the minimal
F ( , ; ;η)= − . (A7)
models. Then the crosscap bootstrap equations are re- 2 1 4 4 2 p√2√1 η
−
duced to algebraic equations that can be easily solved.
In the Lee-Yang model, we have the Virasoro OPE In terms of the global conformal block, we have the
decomposition
[σ] [σ]=[1]+[σ] . (A1)
×
η2 9η4
Accordingly, the two-point function of the spin operator G (η)=1+ F (2,2;4;η)+ F (4,4;8;η)+
σσ 2 1 2 1
64 40960 ···
σ on the real projective space can be decomposed into
two Virasoro blocks: 1 1 1 η29 9 9
+cI(η22F1( , ;1;η)+ 2F1( , ;9;η)+ )
2 2 16384 2 2 ···
G (η)=(1 η)1/5 F (2,3;6;η) (A8)
σσ 2 1
− 5 5 5
+c η 1/5(1 η)1/5 F (1,2;4;η) , (A2) Thecoefficientsinfrontofeachblockwerequotedasthe
LY − 2 1
− 5 5 5 exact results in table II.
6
In the above solutions of the crosscapbootstrapequa- may be useful.
tions, Euler’s transformation
F (a,b;c;η)=(1 η)c a b F (c a,c b;c;η)
2 1 − − 2 1
− − −
Γ(c)Γ(c a b)
= − − F (a,b;a+b+1 c;1 η)
2 1
Γ(c a)Γ(c b) − −
− −
Γ(c)Γ(a+b c)
+ − (1 η)c−a−b
Γ(a)Γ(b) −
F (c a,c b;1+c a b;1 η) (A9)
2 1
× − − − − −