Conformal symmetry of the critical 3D Ising model inside a sphere

Journal of High Energy Physics, Aug 2015

We perform Monte-Carlo simulations of the three-dimensional 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.

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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 , 4169-007 Porto , Portugal 1 Catarina Cosme 2 Centro de F ́ısica, Universidade do Minho 3 P-4710-057, Braga , Portugal We perform Monte-Carlo simulations of the three-dimensional 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 Monte-Carlo 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 Monte-Carlo 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 Monte-Carlo 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 spin-flip 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 spin-flip 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 Z2-even 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, Monte-Carlo simulation, high-temperature 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 one-point 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 y-coordinates (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 y-coordinates is mapped to the point original flat metric becomes − r2)2 (R4 + 2R2y · r + r2y2)2 dyμdyμ ≡ Ω2(y)dyμdyμ , in the y-coordinates, 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 two-point 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 Monte-Carlo simulation In order to perform a Monte-Carlo 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 Monte-Carlo 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 one-point 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 non-trivial prediction of conformal invariance. Our second test is related to the two-point 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 non-trivial 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 Monte-Carlo 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 two-point 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 three-point correlation functions. We plan to verify this prediction with Monte-Carlo 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 [FP7-People-2010-IRSES] and [FP7/2007-2013] 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. NORTE-07-0124-FEDER- 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. 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Catarina Cosme, J. M. Viana Parente Lopes, João Penedones. Conformal symmetry of the critical 3D Ising model inside a sphere, Journal of High Energy Physics, 2015, 22, DOI: 10.1007/JHEP08(2015)022