Conformal symmetry of the critical 3D Ising model inside a sphere
Received: April
Conformal symmetry of the critical 3D Ising model
0 Rua do Campo Alegre 687 , 4169007 Porto , Portugal
1 Catarina Cosme
2 Centro de F ́ısica, Universidade do Minho
3 P4710057, Braga , Portugal
We perform MonteCarlo simulations of the threedimensional Ising model at the critical temperature and zero magnetic field. We simulate the system in a ball with free boundary conditions on the two dimensional spherical boundary. Our results for one and two point functions in this geometry are consistent with the predictions from the conjectured conformal symmetry of the critical Ising model.
Boundary Quantum Field Theory; Conformal and W Symmetry; Lattice

inside a sphere
1 Introduction
2 Ising model and Conformal Field Theory CFT inside a sphere Results from MonteCarlo simulation Conclusion
It is a long standing conjecture that continuous phase transitions are described by conformal
invariant field theories [1]. Under reasonable assumptions this conjecture has been proven
in two dimensions [2, 3] and, recently, in four dimensions [4, 5]. A general proof in three
dimensions has not yet been found. However, assuming the validity of this conjecture it has
been possible to formulate conformal bootstrap equations and find approximate solutions
that predict the Ising critical exponents with high accuracy [6–10]. This success can also
be viewed as strong evidence for conformal invariance of the 3D Ising model at the critical
Conformal invariance of the critical 3D Ising model can also be tested directly with
lattice MonteCarlo simulations. In particular, in this work, we test the predictions of
conformal symmetry for the critical Ising model in a ball with free boundary conditions on
the two dimensional spherical boundary. In two dimensions, the analogous geometry (disk)
was analyzed in [11]. There have been other MonteCarlo studies of conformal invariance in
the 3D Ising model. Using the standard cubic lattice hamiltonian, [12] showed that some
two point functions in the presence of a line defect have the functional form predicted
by conformal invariance. In [13], the authors used an anisotropic hamiltonian (with a
continuous direction) to simulate the 3D Ising model in several cylindrical geometries and
measured correlation functions compatible with conformal invariance. See also [14, 15] for
an alternative implementation of a 3D cylindrical geometry.
Ising model and Conformal Field Theory
The Ising hamiltonian is
H[{s}] = −
neighbours in a cubic lattice. We are interested in correlation functions of local operators
at the critical temperature
hO1(x1) . . . On(xn)i =
1 X e−βcH[{s}]O1(x1) . . . On(xn) ,
where the partition function is
Z =
The local operators in the Ising model can be classified by their quantum numbers with
respect to the Z2 spinflip symmetry and the point group symmetry of the cubic lattice.
The simplest local lattice operator that is invariant under the lattice symmetries that
preserve the point x and is odd under spin flip is the local spin field s(x). In the sector of
operators invariant under the spinflip symmetry and the lattice symmetries the simplest
local operators are the identity I and the energy density
e(x) =
1 X s(x)s(x + δ)
At the critical temperature, the Ising model has infinite correlation length and its
want to test is that we can define a Conformal Field Theory (CFT) that describes the
the local lattice operators can be written in terms of the operators of the Ising CFT that
have the same symmetry properties. In particular, the spin field can be expanded in terms
s(x) = bsσ aΔσ σ(x) + bsσ0 aΔσ0 σ0(x) + . . . ,
O is the scaling dimension of the operator O and the b’s are dimensionless constants
that depend on the normalization convention of the CFT operators. Similarly, the local
energy density operator can be written as
e(x) = beI I + be a
where I is the identity operator and , 0, . . . are the lowest dimension scalar primary
operators in the Z2even sector. The best estimates for these scaling dimensions are [7–10]
These have been determined by a variety of methods,2 like direct experimental
measurements, MonteCarlo simulation, hightemperature expansions, expansion and, more
recently, conformal bootstrap techniques.
1In general this expansion also includes descendant scalar operators, which we did not write to avoid
2See [16] for a review.
We normalize the CFT operators imposing the following correlation functions in the
infinite system without boundaries
hOi(x)Oj (y)i =
This can be used to fix the coefficients in (2.5) and (2.6). The recent Monte Carlo
simulations of [17] found
beI = 0.330213(12) ,
be = 0.237(3) .
hO(x)i = 0 , ∀ O 6= I .
CFT inside a sphere
Let us consider the three dimensional critical Ising model in a ball with free boundary
conditions on the two dimensional spherical boundary. We would like to test if, in the
continuum limit, this system is described by a Boundary Conformal Field Theory (BCFT).
In order to do this, we start by discussing what are the implications of conformal symmetry
for correlation functions in this geometry.
Consider first the onepoint function hO(r)i of a scalar operator placed at distance
r from the centre of a sphere of radius R. Without conformal invariance, this one point
function would take the general form
for an arbitrary function f . Imposing conformal symmetry leads to
for some constant a .
In order to see how this comes about, we define the ball
hO(r)i =
f (z) =
(x1)2 + (x2)2 + (x3)2 < R2 ,
xμ = (R2 − r2) R4 + 2R2y · r + r2y2 + rμ ,
has several nice properties. The first is that it preserves the spherical boundary. In other
words, the ball (3.3) corresponds exactly to the same region in the ycoordinates
(y1)2 + (y2)2 + (y3)2 < R2 ,
3It might be helpful to understand this change of coordinates as a sequence of simpler steps. Start by
doing a translation to bring the point x
− r )
The second nice property is that the origin in the ycoordinates is mapped to the point
original flat metric becomes
− r2)2
(R4 + 2R2y · r + r2y2)2 dyμdyμ ≡ Ω2(y)dyμdyμ ,
in the ycoordinates, i.e. it is a conformal transformation.
Correlation functions inside a sphere with a flat metric are equal to correlation
func
Notice that this is true in any theory because it follows just from a relabelling of points
without changing the physical geometry. Remarkably, correlation functions of scalar
primary operators in CFTs also satisfy
In other words, CFT correlation functions transform in a simple way under Weyl
transformations (or local rescalings) of the metric. Equations (3.7) and (3.8) together lead to4
hO(x = r)i = Ω−Δ(0)hO(y = 0)i =
R2 − r2
as anticipated in (3.2).
it follows that
sphere is given by
hO(0)O(y)ic
In this way we relate a generic two point function inside the sphere to a two point function
where one of the points is at the centre of the sphere. From spherical and scaling symmetry,
hO(0)O(y)ic =
R2 − y
Therefore, we conclude that the two point function of scalar primary operators inside a
hO(x1)O(x2)ic =
(R2 − x12)Δ(R2 − x22)Δ fOO(ζ)
4From now on, we drop the subscript indicating the metric when it is the standard flat Cartesian metric.
where we considered a flat boundary, and used coordinates ~z along the boundary and the
distance to the boundary z. Normalizing the boundary operators Oe to have unit two point
where Oe is the boundary operator with lowest dimension that appears in the boundary
OPE of O (excluding the identity).
obtained5
We will consider the Ising model with free boundary conditions which is known to be
described by a BCFT usually called the ordinary transition [18, 22]. This BCFT can be
a = −0.751(4) .
The lowest dimension boundary operator in the Z2 even sector is the displacement
operafunctions obey Ward identities. In particular,
system. This gives
The two point function of a bulk and a boundary operator is fixed by conformal symmetry.
D is the normalization of the twopoint function of the displacement operator
e
dw~ hDe (w~ )O(z, ~z)i = − ∂z hO(z, ~z)i .
hDe (w~ ) (z, ~z)i = a De CDe [z2 + (~z − w~ )2]3
(R2 − x21)(R2
R2(x1 − x2)2
and it is controlled by the same singularity as the two point function (2.8) of the infinite
is controlled the boundary Operator Product Expansion (OPE) [18–21]
O(z, ~z) =
D
e
L = 8
L = 16
L = 32
L = 64
L = 128
L = 256
L = 512
L = 1024
Results from MonteCarlo simulation
In order to perform a MonteCarlo simulation of the critical Ising model, we need to know
the critical temperature with high precision.6 We used Wolff’s cluster algorithm [28] to
reduce critical slowing down and used the value
of the critical temperature from [29]. To check that this is a good estimate of the critical
temperature we measured the Binder cumulant
UB = 1 − 3hm2i2
of spins in a system with periodic boundary conditions. In figure 1, we plot the Binder
cumulant for several system sizes.
We also used these simulations with periodic boundary conditions to determine the
fit them to the theoretical expectation from equation (2.6)7
he(x)iL = beI + c
than the size of the system.
description of the lattice model.
d = −0.084(3), in agreement with (2.9) and [30].
6We need to be sufficient close to the critical temperature so that the correlation length is much larger
7The correction proportional to d comes from the leading irrelevant operator in the effective action
the 3D Ising model with periodic boundary conditions for several linear sizes L of the system. The
bounds (in light grey).
We are now ready to compare our results from the MonteCarlo simulation with the
predictions from conformal invariance.
We consider the critical Ising model (2.3) in a
three dimensional cubic lattice excluding all spins outside a sphere of radius R as shown
in figure 3. The interaction bounds connecting spins inside the sphere with spins outside
the sphere are also dropped.
In figure 4 we show the onepoint function of the energy density operator inside the
tion (3.1)–(3.2) from conformal invariance, we conclude that
he(r)iR − beI =
(r $
)a !
RR==12172a7a
R = 63a
RR==316a3a
RR==153a1a
RR==7a15a
R = 7a
1 − (r/R)2
for several values of the radial coordinate r and of the sphere radius R. The collapse of all points
into a single straight line confirms the prediction (4.4) of conformal symmetry. There are deviations
due to finite size effects and due to statistical uncertainty, specially in the larger systems.
R = 127a
RR==311a5a
R = 15a
RR==7a7a
1 − (r/R)2 "#%
the same value of r. The statistical error bars are smaller than the size of the dots. The black
where the dots stand for terms that vanish in the continuum limit a/R
fixed. The plot in figure 5 confirms this prediction and the values of be and a given in 2.9
and 3.19. In figures 4 and 5 one can notice deviations from spherical symmetry due to the
underlying cubic lattice, specially for points close to the spherical boundary. For large R we
R =R12=7a127a
R =R63=a 63a
R =R31=a 31a
R =R15=a 15a
R =R7a= 7a
should fall into a single curve up to statistical uncertainties and finite size effects.
have bigger statistical error due to the smaller number of independent samples harvested
first direct verification of a nontrivial prediction of conformal invariance. Our second test
is related to the twopoint function inside the sphere.
From (2.5) and (3.14), we obtain
vanish in the continuum limit. In other words, conformal invariance predicts that the
dimensionless function
Fss(x1, x2) =
hs(x1)s(x2)iR
a2Δσ (1 − x12/R2)−Δσ (1 − x22/R2)−Δσ
points collapse in a single smooth curve up to the statistical error bars and finite system
size effects. Moreover, using
ali→m0 Fss(x1, x2) = bs2σfσσ(ζ)
R =R12=7a127a
R =R63=a 63a
R =R31=a 31a
R =R15=a 15a
R =R7a= 7a
This reduces the statistical errors and shows better convergence to a single curve. The two straight
lines are fits to the asymptotic behaviour using the values (2.7), (2.9) and (3.19).
We also preformed a similar analysis for the two point function of the energy density
operator inside the sphere. In figures 8 and 9, we plot
Fee(x1, x2) =
R2Δ he(x1)e(x2)iR − he(x1)iRhe(x2)iR
we can fit the asymptotic behaviour of the curve to conclude that C
value is a very crude estimate because Fee has very large statistical uncertainty in the
is very small and it takes a long time to simulate the large systems required to explore the
De ≈ 0.012. The last
Conclusion
We gave strong evidence confirming the nontrivial predictions of conformal symmetry
for correlation functions of the critical Ising model in a ball geometry.
We hope our
work strengths the confidence in the conformal bootstrap methods that assume conformal
symmetry from the start.
It would be nice to obtain more precise measurements of scaling dimensions and OPE
coefficients of several boundary operators. However, the cubic lattice discretization of the
ball geometry we used is not ideal for this purpose because it introduces large finite size
and boundary effects. It would also be interesting to study other BCFT of the critical
Ising model, like the special and the extraordinary transition (see [31] for a MonteCarlo
R = 127a
R = 63a
R = 1R27=a 31a
R = 6R3a= 15a
R = 3R1a= 7a
R = 15a
R = 7a
The combination (4.8) involving the connected twopoint function of the energy
invariance predicts that all points should fall into a single curve up to statistical uncertainties and
finite size effects. The statistical uncertainty looks biased because we are using a logarithmic scale
and therefore we can not plot the points with Fee < 0.
study of the special transition). These can be implemented introducing another coupling
between the boundary spins. Unfortunately, it is not obvious how to do this in an elegant
fashion in our ball geometry.
In the absence of boundaries, a fundamental prediction of conformal symmetry is the
functional form of threepoint correlation functions. We plan to verify this prediction with
MonteCarlo simulations, in the same spirit of this paper. Such study would also be able
to check several conformal bootstrap predictions for OPE coefficients of the Ising CFT. It
is curious that in two dimensions this was done 20 years ago [32].
Acknowledgments
We are grateful to Slava Rychkov for useful discussions and for suggesting this work. The
research leading to these results has received funding from the [European Union] Seventh
Framework Programme [FP7People2010IRSES] and [FP7/20072013] under grant
agreements No 269217, 317089 and No 247252, and from the grant CERN/FP/123599/2011.
Centro de F´ısica do Porto is partially funded by the Foundation for Science and
Technology of Portugal (FCT). J.V.P.L. acknowledges funding from projecto Operacional Regional
do Norte, within Quadro de Referˆencia Estrat´egico Nacional (QREN) and through Fundo
Europeu de Desenvolvimento Regional (FEDER), ref. NORTE070124FEDER 000037.
R = 127a
R = 63a
R = 1R27=a 31a
R = 6R3a= 15a
R = 3R1a= 7a
R = 15a
R = 7a
size. This reduces the statistical errors and shows better convergence to a single curve, although
behaviour using the values (2.7), (2.9), (3.19) and C
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
theory, JETP Lett. 43 (1986) 730 [INSPIRE].
(1988) 226 [INSPIRE].
[2] A.B. Zamolodchikov, Irreversibility of the flux of the renormalization group in a 2D field
invariance in four dimensions, arXiv:1309.2921 [INSPIRE].
invariance, conformality and generalized free fields, arXiv:1402.6322 [INSPIRE].
4D CFT, JHEP 12 (2008) 031 [arXiv:0807.0004] [INSPIRE].
(2012) 025022 [arXiv:1203.6064] [INSPIRE].
Rev. E 67 (2003) 036107.
[9] F. Kos, D. Poland and D. SimmonsDuffin, Bootstrapping mixed correlators in the 3D Ising
model, JHEP 11 (2014) 109 [arXiv:1406.4858] [INSPIRE].
[10] D. SimmonsDuffin, A semidefinite program solver for the conformal bootstrap, JHEP 06
[11] Y. Deng and H.W.J. Bl¨ote, Conformal invariance and the Ising model on a spheroid, Phys.
arXiv:1407.7597 [INSPIRE].
Rept. 368 (2002) 549 [condmat/0012164] [INSPIRE].
[13] Y.J. Deng and H.W.J. Bl¨ote, Bulk and surface critical behavior of the threedimensional
Ising model and conformal invariance, Phys. Rev. E 67 (2003) 066116 [INSPIRE].
Lett. B 721 (2013) 299 [arXiv:1212.6190] [INSPIRE].
operatorproductexpansion coefficients in the 3D Ising model from offcritical correlators,
Phys. Rev. D 91 (2015) 061901 [arXiv:1501.04065] [INSPIRE].
514 [INSPIRE].
[19] D.M. McAvity and H. Osborn, Conformal field theories near a boundary in general
dimensions, Nucl. Phys. B 455 (1995) 522 [condmat/9505127] [INSPIRE].
3503 [condmat/9610143] [INSPIRE].
07 (2013) 113 [arXiv:1210.4258] [INSPIRE].
Cambridge U.K. (1996).
[21] P. Liendo, L. Rastelli and B.C. van Rees, The bootstrap program for boundary CFTd, JHEP
[1] A.M. Polyakov , Conformal symmetry of critical fluctuations , JETP Lett . 12 ( 1970 ) 381 [3] J. Polchinski , Scale and conformal invariance in quantum field theory, Nucl . Phys . B 303 [4] A. Dymarsky , Z. Komargodski , A. Schwimmer and S. Theisen , On scale and conformal [5] A. Dymarsky , K. Farnsworth , Z. Komargodski , M.A. Luty and V. Prilepina , Scale [6] R. Rattazzi , V.S. Rychkov , E. Tonni and A. Vichi , Bounding scalar operator dimensions in [7] S. ElShowk et al., Solving the 3D Ising model with the conformal bootstrap , Phys. Rev. D 86 [8] S. ElShowk et al., Solving the 3D Ising model with the conformal bootstrap II.
Rev . B 85 ( 2012 ) 174421.
Rev . B 84 ( 2011 ) 134405. [31] M. Hasenbusch , Monte carlo study of surface critical phenomena: the special point , Phys. [32] G.T. Barkema and J. McCabe , Monte Carlo simulations of conformal theory predictions for the three state Potts and Ising models , J. Statist. Phys . 84 ( 1996 ) 1067 [heplat/9510050]