PHYSICAL REVIEW D 98, 014502 (2018)
Lattice ϕ4 field theory on Riemann manifolds: Numerical tests for the 2D Ising CFT on S2
† ‡ Richard C. Brower,* Michael Cheng, and Evan S. Weinberg Boston University, Boston, Massachusetts 02215, USA ∥ George T. Fleming§ and Andrew D. Gasbarro Yale University, Sloane Laboratory, New Haven, Connecticut 06520, USA
Timothy G. Raben¶ Brown University, Providence, Rhode Island 02912, USA and University of Kansas, Lawrence, Kansas 66045, USA
Chung-I Tan** Brown University, Providence, Rhode Island 02912, USA
(Received 7 June 2018; published 6 July 2018)
We present a method for defining a lattice realization of the ϕ4 quantum field theory on a simplicial complex in order to enable numerical computation on a general Riemann manifold. The procedure begins with adopting methods from traditional Regge calculus (RC) and finite element methods (FEM) plus the addition of ultraviolet counterterms required to reach the renormalized field theory in the continuum limit. The construction is tested numerically for the two-dimensional ϕ4 scalar field theory on the Riemann two- sphere, S2, in comparison with the exact solutions to the two-dimensional Ising conformal field theory (CFT). Numerical results for the Binder cumulants (up to 12th order) and the two- and four-point correlation functions are in agreement with the exact c ¼ 1=2 CFT solutions.
DOI: 10.1103/PhysRevD.98.014502
I. INTRODUCTION Weyl transform from flat Euclidean space Rd to a compact Sd Lattice field theory (LFT) has proven to be a powerful spherical manifold ,orinRadial Quantization [2],a nonperturbative approach to quantum field theory [1]. Weyl transformation to the cylindrical boundary, R Sd−1 dþ1 However the lattice regulator has generally been restricted × ,ofAdS . Other applications that could benefit to flat Euclidean space, Rd, discretized on hypercubic from extending LFT to curved manifolds include two- Λ π dimensional condensed matter systems such as graphene lattices with a uniform ultraviolet (UV) cutoff UV ¼ =a in terms of the lattice spacing a.Hereweproposeanew sheets [3], four-dimensional gauge theories for beyond approach to enable nonperturbative studies for a range of the standard model (BSM) strong dynamics [4,5],and problems on curved Riemann manifolds. There are many perhaps even quantum effects in a space-time near applications that benefit from this. In the study of massive systems such as black holes. conformal field theory (CFT), it is useful to make a Before attempting a nonperturbative lattice construction, one should ask if a particular renormalizable field theory in flat space is even perturbatively renormalizable on a general *[email protected] smooth Riemann manifold. Fortunately, this question was † [email protected] – ‡ addressed with an avalanche of important research [6 9] [email protected] in the 1970s and 1980s. A rough summary is that any UV §[email protected] ∥ complete field theory in flat space is also perturbatively [email protected] ¶[email protected] renormalizable on any smooth Riemann manifold with **[email protected] diffeomorphism invariant counterterms corresponding to those in flat space [9]. Taking this as given, in spite of our Published by the American Physical Society under the terms of limited focus on ϕ4 theory, we hope this paper is the the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to beginning of a more general non-perturbation lattice the author(s) and the published article’s title, journal citation, formulation for these UV complete theories on any smooth and DOI. Funded by SCOAP3. Euclidean Riemann manifolds.
2470-0010=2018=98(1)=014502(26) 014502-1 Published by the American Physical Society RICHARD C. BROWER et al. PHYS. REV. D 98, 014502 (2018)
The basic challenge in constructing a lattice on a sphere cylinder to zero, both cases are conveniently presented —or indeed on any nontrivial Riemann manifold—is the together in Appendix A. lack of an infinite sequence of finer lattices with uniformly The organization of the article is as follows. In Sec. II we decreasing lattice spacing [2,10,11]. For example, unlike review the basic Regge calculus/finite element method the hypercubic lattice with toroidal boundary condition, the framework as a discrete form of the exterior calculus and largest discrete subgroup of the isometries of a sphere is the show its failure for a quantum field theory beyond the icosahedron in 2d and the hexacosichoron, or 600 cell, in classical limit due to UV divergences. The reader is referred 3d. This greatly complicates constructing a suitable bare to the literature [13–16] for more details and to Ref. [19] for lattice Lagrangian that smoothly approaches the continuum the extension to Dirac fermions. In Sec. III we address the 4 limit of the renormalizable quantum field theory when the central issue of counterterms in the interacting ϕ theory UV cut-off is removed. Here we propose a new formulation required to restore the isometries on S2 in the continuum of LFT on a sequence of simplicial lattices converging to a limit. Sec. IV compares our Monte Carlo simulation for general smooth Riemann manifold. Our strategy is to fourth and sixth order Binder cumulants with the exactly represent the geometry of the discrete simplicial manifold solvable c ¼ 1=2 CFT. In Sec. V we extend this analysis to using Regge Calculus [12] (RC) and the matter fields using the two-point and four-point correlation functions. We fit the finite element method (FEM) [13] and discrete exterior the operator product expansion (OPE) as a test case of how calculus (DEC) [14–17]. Together these methods define a to extract the central charge, OPE couplings, and operator lattice Lagrangian which we conjecture is convergent in the dimensions. classical (or tree) approximation. However, the conver- gence fails at the quantum level due to ultraviolet diver- II. CLASSICAL LIMIT FOR SIMPLICIAL LATTICE gences in the continuum limit. To remove this quantum FIELD THEORY obstruction, we compute counterterms that cancel the ultraviolet defect order by order in perturbation theory. The scalar ϕ4 theory provides the simplest example. On a We will refer to the resultant lattice construction as the smooth Riemann manifold, ðM;gÞ, the action, quantum finite element (QFE) method and give a first Z numerical test in 2d on S2 at the Wilson-Fisher CFT fixed ffiffiffi 1 d p μν∂ ϕ ∂ ϕ point. S ¼ d x g 2 g μ ðxÞ ν ðxÞ M While our current development of a QFE Lagrangian and 1 numerical tests are carried out for the simple case of a 2-d 2 ξ˜ ϕ2 λϕ4 ϕ þ 2 ðm þ 0RicÞ ðxÞþ ðxÞþh ðxÞ ; ð3Þ scalar ϕ4 theory projected on the Riemann sphere we attempt a more general framework. Since the map to the 0 ≡ Riemann sphere, R2 → S2, is a Weyl transform, the CFT is is manifestly invariant under diffeomorphisms: x ¼ fðxÞ 1 2 x0ðxÞ. We include the coupling to the scalar Ricci curvature guaranteed [18] to be exactly equivalent to the c ¼ = ˜ ˜ (ξ0Ric), where ξ0 ¼ðd − 2Þ=ð4ðd − 1ÞÞ and an external Ising CFT in flat space and therefore presents a convenient 2 and rigorous test of convergence to the continuum theory. constant (scalar) field h. Ric ¼ðd − 1Þðd − 2Þ=R on the More general examples can and will be pursued mapping sphere Sd of radius R. The field, ϕðxÞ,isanabsolute scalar conformal field theories in flat Euclidean space Rdþ1 either or in the language of differential calculus a 0-form with a to R × Sd, appropriate for radial quantization, fixed value at each point P in the manifold, independent of co-ordinate system: ϕ0ðx0Þ¼ϕðxðx0ÞÞ for x0 ¼ fðxÞ.For Xdþ1 future reference to the discussion in Sec. IIffiffiffi B on the discreteffiffiffi 2 μ μ 2t 2 Ω2 p d p 1 dsflat ¼ dx dx ¼ e ðdt þ d dÞ exterior calculus, we identify vold ¼ gd x ¼ gdx ∧ μ¼1 ∧ dxd, as the volume d-form. 4 Weyl 2 2 The scalar ϕ theory in d 2, 3 has a (super-)renorma- !ðdt þ dΩ Þ; ð1Þ ¼ d lizable UV weak coupling fixed point and a strong coupling Wilson-Fisher conformal fixed point in the infrared (IR), with t ¼ logðrÞ or to the sphere, Sd, as illustrated in the phase diagram in Fig. 1. In passing, we Xd also note its similarity to 4-d Yang Mills theory with a Weyl 2 μ μ σðxÞ Ω2→ Ω2 sufficient number of massless fermions to be in the dsflat ¼ dx dx ¼ e d d d d: ð2Þ μ¼1 conformal window [5], which is a central motivation for this research. Both theories are UV complete, perturba- Current tests of the QFE method for the 3-d Ising model in tively renormalizable at weak coupling, and have a strong radial quantization, R3 → R × S2, are underway, so we coupling conformal fixed point in the IR. The mass terms 2 2 take the opportunity here to give a brief introduction to both (m0ϕ or m0ψψ¯ , respectively) must be tuned either to or geometries. By considering the sphere Sd as a dimensional near the critical surface to reach the conformal or mass reduction of the cylinder R × Sd by taking the length of the deformed theory.
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(iii) QFE: Quantum corrections: Quantum counterterms are added to the discrete lattice action to cancel defects at the UV cutoff (see Sec. III B). 0.6 A. Geometry and Regge calculus The Regge calculus approach to constructing a discrete 0.4 approximation to a Riemann manifold, ðM;gÞ, proceeds as follows. First the manifold, M, is replaced by a simplicial complex, Mσ, composed of elementary simplicies: trian- gles in 2d as illustrated in Fig. 2, tetrahedrons in 3d, etc. 0.2 This graph defines the topology of the manifold in the language of category theory [20]. Next a discrete metric is introduced as a set of edge lengths on the graph: 0.0 gμνðxÞ → gσ ¼flijg. Assuming a piecewise flat interpola- tion into the interior of each simplex, we now have the Regge representation of a Riemann manifold ðMσ;gσÞ that is continuous but not differentiable. The curvature is given -0.2 by a singular distribution at the boundary of the each -0.2 0.0 0.2 0.4 0.6 simplex; in 2d, concentrated at the vertices, and in higher − 2 “ ” 4 dimensions at the d -dimensional hinges. The inte- FIG. 1. The phase plane of ϕ for d<4, depicting the grated curvature over the defect is easily computed by renormalization flow from the repulsive weak coupling ultra- parallel transport of the tangent vector around each defect. μ2 λ 0 0 violet (UV) fixed point at ð 0; 0Þ¼ð ; Þ to the infrared (IR) Each simplex is parameterized by using d þ 1 local μ2 λ Wilson Fisher fixed point at ð ; Þ, parametrized in terms of the i 2 2 barycentric coordinates, 0 ≤ ξ ≤ 1. Using the constraint, bare parameters, λ0 ∼ λ and μ ∼−m . 0 ξ0 þ ξ1 þ þξd ¼ 1 to eliminate ξ0, our ϕ4 action in Eq. (3) on this piecewise flat Regge manifold is given as a sum over each simplex, To formulate a lattice action, we introduce a simplicial Z complex, or triangulation in 2d, as illustrated in Fig. 2, and X pffiffiffiffiffiffi 1 → dξ ij∂ ϕ ξ ∂ ϕ ξ a discrete action, S Sσ ¼ d Gσ 2 Gσ i ð Þ j ð Þ σ∈M σ σ X ϕ − ϕ 2 1 1 ð i jÞ 2 ξ˜ ϕ2 ξ λϕ4 ξ S ¼ K þ 2 ðm þ 0RicÞ ð Þþ ð Þ : ð5Þ 2 ij l2 hiji ij X X We may choose an isometric embedding into a sufficiently pffiffiffiffi 1 pffiffiffiffi þ g m2ϕ2 þ λ ϕ4 þ h g ϕ ð4Þ high dimensional flat Euclidean space with vertices at i 2 i i i i i i ⃗ ⃗ i i y ¼ rn so a point in the interior of each simplex and its induced metric are given by where i labels all vertices and the sum hiji runs over all links with proper lengthffiffiffiffi lij. The technical requirement is to fix the p 2 λ weights, (Kij, gi, mi , i) as functions of bare couplings and the target manifold, on a sequence of increasingly fine tessellations so that in the limit of vanishing lattice spacing, a ¼ OðlijÞ, the quantum path integral converges to the continuum renormalized quantum theory with a minimal set of fine tuning parameters. For the ϕ4 theory, there is one parameter to tune in the approach to the critical surface: the μ2 → μ2 λ relevant mass parameter 0 ð 0Þ, illustrated in Fig. 1. The construction of our QFE simplicial lattice action can be broken into three steps: (i) Simplicial geometry: The smooth Riemann mani- fold, ðM;gÞ, is replaced by simplex complex, FIG. 2. A 2-d simplicial complex with points (σ0), edges (σ1) M ð σ;gσÞ with piecewise flat cells (see Sec. II A). and triangles (σ2) (illustrated in yellow). At each vertex σ0 there is ϕ (ii) Classical lattice action: The continuum field ðxÞ is a dual polytope, σ0 (illustrated in red), and at each link, σ1, there replaced by a discrete sum over FEM basis elements, is a dual link σ1 and its associated hybrid cell σ1 ∧ σ1 (illustrated i ϕðxÞ → W ðxÞϕi (see Sec. II B). in blue).
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d d X X the hybrid cell, Aij ¼ lijjσ1ðijÞj=2 ¼jσ1ðijÞ ∧ σ1ðijÞj,as ⃗ ξn⃗ ξi⃗ ⃗ y ¼ rn ¼ li0 þ r0 ð6Þ illustrated in Fig. 2. n¼0 i¼1 For higher dimensions, a natural generalization of the kinetic term is ∂y⃗ ∂y⃗ ⃗ ⃗ Gij ¼ · ≡ li0 · lj0: ð7Þ ∂ξi ∂ξj X ϕ − ϕ 2 X 1 ð i jÞ 1 pffiffiffiffi 2 Sσ ϕ V m g ϕ ; pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ ¼2 ij 2 þ 2 i i ð11Þ lij This defines the volume element, Gσ ¼ det½Gij and hi;ji i ij inverse metric, Gσ , as well. σ ∧ σ Since we are not considering dynamical gravity, this where Vij ¼j 1ðijÞ 1ðijÞj ¼ lijSij=d is the product of − 1 piecewise flat metric is frozen (i.e., quenched) and chosen the length of the link (lij) times the volume of the d - “ ” σ to conform as closely as possible to our target manifold for dimensional surface, Sij ¼j 1ðijÞj, of the dual polytope normal to the link hi; ji. A mass term has also been included the quantum field theory. For example this can be achieved pffiffiffiffi weighted by the dual lattice volume g ¼jσ ðiÞj.This by computing the edge length lij, from the proper distance i 0 between points on the original manifold xi, xj or from the elegant form was recommended in the seminal papers on random lattices by Christ, Friedberg and Lee [21–23] and Euclidean distance jr⃗i − r⃗jj in the isometric embedding space, r⃗ ¼ y⃗ðx Þ to first order relative to the local curvature subsequently in the FEM literature by the application of the i i ’ of the target manifold. At this stage, the dynamical simplicial Stokes theorem for discrete exterior calculus quantum field ϕðxÞ is still a continuum function on the (DEC). However, we note this generalization is not equiv- 2 piecewise flat Regge manifold. alent to linear FEM for d> (See Ref. [19]). To appreciate the DEC approach [14–16,24], it is useful to expand a little on the geometry of a simplicial complex, B. Hilbert space and discrete exterior calculus S, and its Voronorï dual, S . A pure simplicial complex S The second step is the approximation of the matter field consists of a set of d-dimensional simplices (designated by ϕðxÞ as an expansion, σd) “glued” together at shared faces (boundaries) consisting of d − 1-dimensional simplices (σ −1), iteratively giving a 0 1 d d ϕðxÞ → ϕσðξÞ¼EσðξÞϕ0 þ EσðξÞϕ1 þ þEσðξÞϕd; sequence: σd → σd−1 → σ1 → σ0. This hierarchy is ð8Þ specified by the boundary operator, into a finite element basis on each simplex σ . The simplest Xn d ∂σ … −1 kσ …ˆ … form is a piecewise linear function, EiðξÞ¼ξi, on each nði0i1 inÞ¼ ð Þ n−1ði0i1 ik inÞ; ð12Þ simplex. In this case we are using essentially the same k¼0 linear approximation for both the metric, gμνðxÞ, and where iˆ means to exclude this site. Each simplex matter, ϕðxÞ, fields. To evaluate the FEM action, we simply k σ ði0i1…i Þ is an antisymmetric function of its arguments. plug the expansion in Eq. (8) into Eq. (5) and perform the n n The signs in Eq. (12) keep track of the orientation of each integration. For the kinetic term, this is particularly simple simplex. It is trivial to check that the boundary operator is because the gradients of the barycentric coordinates are closed: ∂2σ ¼ 0. On a finite simplicial lattice ∂ is a matrix constant. For 2d, this gives the well-known form on each n and its transpose, ∂T, is the coboundary operator. The triangle, circumcenter Voronorï dual lattice, S , is composed of σ ← σ ← ← σ σ 2 2 2 polytopes, 0 1 d, where n has dimension l31 þ l23 − l12 2 Iσ ¼ ðϕ1 − ϕ2Þ þð23Þþð31Þ d − n as illustrated in Fig. 2. A crucial property of this 8A123 circumcenter duality is orthogonality. Each simplicial 1 ð3Þ 2 element σ ∈ S is orthogonal to its dual polytope σ ∈ S . ¼ A12 ðϕ1 − ϕ2Þ þð23Þþð31Þð9Þ n n 2 As a consequence, the volume, ð3Þ ! − ! where A12 is the area of the triangle formed by the sites 1, n ðd nÞ jσn ∧ σnj¼ jσnjjσnj; ð13Þ 2, and the circumcenter, σ2ð123Þ. d! The free scalar action on the entire simplicial complex is σ ∧ σ now found by summing over triangles, of the hybrid cell, n n, is a simple product. Hybrid cells, constructed from simplices σn in S and their X 1 X ϕ − ϕ 2 i j orthogonal dual σn in S , give a proper tiling of the Iσ½ϕ ¼ A : ð10Þ 2 ij l discrete d-dimensional manifold with the special case σ ij ij ffiffiffiffi h i p jσ0ðiÞj ¼ 1 and jσ0ðiÞj ¼ gi. This is a first modest step Each link hiji receives two contributions—one from each into discrete homology and De Rham cohomology on a triangle that it borders—resulting in the total area for simplicial complex.
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The discrete analogue of differential forms, ωk,isa pairing or map,
hωk; σki ≡ ωkðσkÞ¼ωkði0i1…ikÞ; ð14Þ to the numerical value of a field from sites in the case of 0-forms, from links in the case of 1-forms, and from k-simplices in the case of k-forms. Of course this is familiar to lattice field theory, associating scalars (ω0 ∼ ϕx), gauge μ μ ν fields (ω1 ∼ Aμdx ) and field strengths (ω2 ∼ Fμνdx ∧ dx ) with sites, links and plaquettes respectively. This enables us to define the discrete analogue of the exterior derivative dωk of a k-form by replacing the continuum Stokes’ theorem by the discrete map or DEC Stokes’ theorem: Z Z dωðxÞ¼ wðxÞ σ ∂σ kþ1 kþ1
→ hdω ; σ 1i¼hω ; ∂σ 1ið15Þ k kþ k kþ FIG. 3. The discrete Laplacian at a site i is given by the sum on i; j ϕ − ϕ =l The discrete exterior derivative is automatically closed all links h i (in red) weighed by gradients ð i jÞ ij multiplied by the surface S ¼ 2V =l (in black) and normal- (dd ¼ 0) because the boundary operator is closed (∂∂ ¼ 0). ij ij ij ized by the dual volume jσ ðiÞj ¼ V (in yellow). Applying Eq. (15) to the discrete exterior derivative of a o i scalar (0-form) field trivially gives, ω σ uniquely fixes the dual field values, h k; ki and the action hdϕ; σ1ðijÞi ¼ hϕ; ∂σ1ðijÞi ¼ ðϕi − ϕjÞ; ð16Þ of a discrete codifferential, δ ¼ d through Stokes theo- rem in Eq. (15) on S . the standard finite difference approximation on each link. Putting this all together, we consider the DEC Next we need to define the discrete analogue of the Laplace-Beltrami operator on scalar fields, ðδ þ dÞ2ϕ ¼ Hodge star ( ). This is the first time an explicit dependence ðdδ þ δdÞϕ ¼ d dϕ,is on the metric is introduced. In the continuum, a k-form is an −1 μ1 μ2 σ X σ antisymmetric tensor, ω x k! ωμ μ …μ dx ∧dx ∧ j 0ðiÞj j 1ðijÞj kð Þ¼ð Þ 1; 2 k d dϕ ϕ − ϕ μ ðiÞ¼ ð i jÞð19Þ ∧ k σ σ dx , in a orthogonal basis of 1-forms dual to tangent j 0ðiÞj ∈ j 1ðijÞj μ μ j hi;ji vectors: dx ð∂νÞ¼δν. The Hodge star takes a k-dimen- X sional basis into its orthogonal complement, 1 Vij ϕi − ϕj ¼ pffiffiffiffi ; ð20Þ μ μ μ gi lij lij ðdx 1 ∧ dx 2 ∧ ∧ dx k Þ j∈hi;ji μ μ μ ¼ dx kþ1 ∧ dx kþ2 ∧ ∧ dx n ð17Þ which corresponds to the action in Eq. (11). For 2d this is illustrated in Fig. 3, as the sum of fluxes through the where μ1; μ2; ; μ is an even permutation of n boundaries ∂σ0ðiÞ with surface area, Sij=ðd − 1Þ! ¼ ð1; 2; ;nÞ. However the wedge product (or equivalently Vij=lij ¼jσ1ðijÞ ∧ σ1ðijÞj=lij. This is identical to linear the Levi-Civita symbol) is not a tensor but a weight 1 tensor finite elements in 2d. Besides providing an alternate density [25]. The true volume k-form (or tensor), volk ¼ pffiffiffiffiffi μ μ pffiffiffiffiffi approach to linear finite elements for constructing the 1 ∧ ∧ k gkdx dx , requires a factor of gk which discrete Hilbert space for scalar fields in d>2 dimensions, under the Hodge star operation in Eq. (17) gives the pffiffiffiffiffiffiffiffiffi pffiffiffiffiffi it provides a useful geometric framework for fields with identity, g − ðvol Þ¼ g vol − , where we have used n k k pffiffiffi k pnffiffiffiffiffikpffiffiffiffiffiffiffiffiffi spin. We note however that additional considerations are orthogonality to factor g ¼ gk gd−k, that is, between still needed for non-Abelian gauge fields [22], Dirac the plane and its dual. Consequently on the simplicial fermions [19] and Chern-Simons terms [26]. The best complex, it is reasonable that the proper definition of the geometrization of simplicial field theories and the error discrete Hodge star, estimates thereof are an active research topic. To complete the simplicial action, we need to add the ω σ σ ω σ σ h k; kij kj¼h k; kij kj; ð18Þ potential term. This may be constructed in a variety of pffiffiffiffiffi pffiffiffiffiffiffiffiffiffi ways. If one follows strictly the linear finite element replaces these factors gk, gn−k by finite volumes jσkj, prescription, the expression for the local polynomial σ j kj respectively. The Hodge star identity in Eq. (18) potential (e.g., mass and quartic terms) will not be local
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σ but rather point split on eachP simplex d. For example, in ϕ ξ d ξiϕ 2d, after expanding ð Þ¼ i¼0 i and evaluating the integral over the linear elements, the contribution for the quadratic term on a single triangle, σ2ð123Þ,is Z pffiffiffiffiffiffi 2 2 d ξ GσϕσðξÞ σ A123 ϕ2 ϕ2 ϕ2 ϕ ϕ ϕ ϕ ϕ ϕ ¼ 6 ð 1 þ 2 þ 3 þ 1 2 þ 2 3 þ 3 1Þ: ð21Þ The general expression for a homogeneous polynomial over a d-simplex with volume Vd ¼jσdj is given as as sum over distinct partitions of n: Z V d!n! X dV ϕ ξ n d ϕk0 ϕk1 …ϕkd : dð ð ÞÞ ¼ ! P 0 1 d ð22Þ σd ðn þ dÞ ð ki¼nÞ i Nonetheless in the spirit of dropping higher dimensional FIG. 4. The L ¼ 3 refinement of the icosahedron with V ¼ operators in a derivative expansion, we choose local terms 2 þ 10L2 ¼ 92 vertices or sites. The icosahedron on the top left is 2 9 approximatingffiffiffiffi the potential at each vertex weighted by the refined in the top right with L ¼ equilateral triangles on each p face, and then on the bottom the new vertices are projected onto volume, gi ¼jσ0ðiÞj, of the dual simplex σ0ðiÞ. This approximation leads to our complete simplicial the unit sphere. The resulting simplicial complex preserves the icosahedral symmetries. action, combining the DEC Laplace-Beltrami operator for the kinetic term and the local approximation for the boundary, faces, edges, and vertices are F ¼ 20L2, E ¼ 30L2 and N ¼ 2 þ 10L2, respectively, satisfying the Euler identity 1 X V X ffiffiffiffi μ2 2 ij ϕ − ϕ 2 λ p ϕ2 − 0 F − E þ N ¼ 2. S ¼ 2 ð i jÞ þ 0 gj i 2 l 2λ0 hiji ij i The images of the vertices on the sphere are then X ffiffiffiffi connected by new links consistent with the unique p ϕ þ h gi i; ð23Þ Delaunay triangulation [27]. In the triangulation, each i vertex is connected to five or six neighboring vertices by 2 2 where μ0 ¼ −m0=2. We will use this action for our edges hx; yi. The lengths are set to the secant lengths lxy ¼ 2 ˆ − ˆ discussion of S . jrx ryj in the embedding space between the vertices Finally we should acknowledge that there are many other on the sphere, which approximates the geodesic length 2 R S2 alternatives in the FEM literature worthy of consideration to OðlxyÞ. The extension to a lattice for the × cylinder which may offer improved convergence and faster restora- introduces a uniform (periodic) lattice perpendicular to the tion of the continuum symmetries. Our goal here is to find spheres at t ¼ 0; 1; ;Lt − 1. The sequence of refine- the simplest discrete action on simplicial lattice capable of ments as L → ∞ divides the total curvature into vanish- reaching the correct continuum theory with no more fine ingly small defects at each vertex. As an alternative tuning than is required on the hypercubic lattice. formulation, we could replace the flat triangles with spherical triangles [28], introducing spherical areas, geo- C. Spectral fidelity on S2 desic lengths, moving the curvature uniformly into the 2 3 interior of each triangle. In this formulation, each triangle We represent S embedded in R parametrized by 3-d would have a vanishing deficit angle in the continuum unit vectors: limit. Both discretizations are equivalent at Oða2Þ, and as 3 2 2 2 rˆ ¼ðrx;ry;rzÞ ∈ R ;rx þ ry þ rz ¼ 1: ð24Þ such we prefer flat triangles due to their relative simplicity. The first test of our construction is to look at the In order to preserve the largest available discrete subgroup spectrum of the free theory, of Oð3Þ, we start with the regular icosahedron illustrated on 1 A 2 ffiffiffiffiffi the left in Fig. 4 and subsequently divide each of the 20 ϕ ϕ xy ϕ − ϕ 2 m0 p ϕ2 2 S0 ¼ 2 xMx;y y ¼ 2 ð x yÞ þ 2 gx x; ð25Þ equilateral triangles into L smaller equilateral triangles. 2lxy Then we project the vertices radially outwards onto the surface of the circumscribing sphere, dilating and distorting where x; y ¼ 1; ;N enumerates the lattice sites on a each triangle from its equilateral form, but preserving finite simplicial lattice. The spectrum is computed from the exactly the icosahedral symmetries. The total numbers of generalized eigenvalue condition
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pffiffiffiffiffi ð0Þ 2 pffiffiffiffiffi MxyϕnðxÞ¼En gxϕnðxÞ¼ðEn þ m0Þ gxϕnðxÞ; ð26Þ
ð0Þ where the eigenvalues at zero mass are En . Each distinct eigenvalue En has right/left generalized eigenvectors, ϕnðxÞ=ϕnðxÞ, with the orthogonality and completeness relations, X ffiffiffiffiffi p gx ϕnðxÞϕmðxÞ¼δnm; ð27Þ x X ffiffiffiffiffi p ϕnðyÞϕnðxÞ¼δxy= gx: ð28Þ n The spectral decomposition for the free propagator is X 1 ϕ ðxÞϕ ðyÞ G m2 ≡ n n 29 xyð 0Þ ¼ ð0Þ 2 ð Þ M xy n En þ m0
Alternatively we may define a Hermitian form, M˜ ¼ g−1=4Mg−1=4, with complex conjugate right and left 1=4 eigenvectors, gx ϕnðxÞ¼hxjni and ϕnðxÞ¼hnjxi, respectively. In Dirac notationP the completenessP and 1 1 orthogonality are given by ¼ xjxihxj, ¼ njnihnj and δx;y ¼hxjyi, δn;m ¼hnjmi respectively. Returning to the example of the FEM sphere, S2,we have verified that the generalized eigenvalues which lie well below the cut-off are well fitted by the continuum 2 1 ∈ − spectrum, E ¼ lðl þ 1Þ, with the 2l þ 1 degeneracy FIG. 5. Top: The l þ spectral values for m ½ l; l are lm 8 m ¼ −l; …l. Indeed, any finite eigenvalue approaches its plotted against l for L ¼ . Bottom: The spectral values averaged over m fitted to l 1.00002l2 − 1.27455 × 10−5l3 − 5.58246 × continuum value as 1=L2 in the limit of infinite refinement þ 10−6l4 for L ¼ 128 and l ≤ 32. L → ∞. In addition, we note that the right eigenvectors are well approximated by the continuum spherical harmonics, YlmðrˆxÞ, evaluated at the lattice sites rˆx. This is illustrated in It is useful to compare the low spectrum of our FEM Fig. 5, where the eigenvalues are estimated by computing operator on S2, shown in Fig. 5, to the corresponding diagonal matrix elements, spectrum on the hypercubic refinement of the torus T 2, shown in Fig. 6. For a finite Ld toroidal lattice, the exact ˆ ˆ YlmðrxÞMxyYlmðryÞ hypercubic spectra is given by E ¼ P pffiffiffiffiffi ; ð30Þ l;m g Y ðrˆ ÞY ðrˆ Þ X x x lm x lm x 2 2 En ¼ 4sin ðkμ=2Þþm0 2 1 μ against l. On the left of Fig. 5, the lack of l þ degenerate multiplets on a coarse lattice with L ¼ 8, as we approach X 1 ≃ 2 2 − 4 the cut-off at large l ∼ OðLÞ, shows the breakdown of m0 þ kμ 12 kμ þ ; ð31Þ rotational symmetry. On the right of Fig. 5, the degeneracy μ of the spherical representation holds to high accuracy and where the discrete eigenvalues are enumerated by kμ ¼ the average of the 2l þ 1 eigenvalues at each l level gives 2πnμ=L for integer nμ ∈ ½−L=2;L=2 − 1 . The eigenvec- the correct continuum dispersion relation, lðl þ 1Þ,to tors can be found by a Fourier analysis, and are given by Oð10−5Þ for l ≪ L ¼ 128. 1 1 We will refer to the exact convergence of fixed eigen- ffiffiffiffi −ik·x ffiffiffiffi ik·x ϕnðxÞ¼p e ; ϕnðxÞ¼p e : ð32Þ values and their associated eigenvectors to the continuum N N as the cut-off is removed as spectral fidelity. This is a theoretical consequence of FEM convergence theorems for Converting to dimensionful variables (m, p) and holding a shape regular linear elements as the diameter goes uni- physical mass fixed (mP≡ m0=a), the dispersion relation is 2 2 2 − 1 2 4 formly to zero [13]. We do not provide a proof but will a En ¼ m þ p 12 μa pμ þ . The Lorentz break- assume that this property holds for our implementation if ing term vanishes as a2 ¼ Oð1=L2Þ. Again spectral fidelity we apply the DEC [15] to the Laplace-Beltrami operator. holds for any fixed spectral value En as the lattice
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D. Obstruction to nonlinear quantum path integral Having demonstrated the spectral fidelity of our con- struction on S2 at the Gaussian level, we turn next to the more difficult problem of an interacting quantum theory, starting with the FEM action given in Eq. (23).Wehave performed extensive Monte Carlo simulations for the path integral given the FEM action in Eq. (23) on S2 with a conclusive result: the FEM action does not converge to a spherically symmetric theory as you approach the con- tinuum and fails to have a well defined critical surface. Attempting to locate the critical surface, we monitor the Binder cumulant. The results are given on the left-hand side of Fig. 7. As we increase the cut-off (or L → ∞) while 4 2 2 2 FIG. 6. The zero mass spectrum En ¼ ðsin ðkx= Þþ μ 2 nμπ tuning the only relevant coupling, 0, the fourth Binder sin ðk =2ÞÞ, kμ ¼ , nμ ∈ ½−L=2;L=2 − 1 for the 2-d lattice y L cumulant should stabilize to the known, exact value U ¼ 32 32 4 Laplacianqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi on a regular × square lattice plotted against 0.8510207ð63Þ for the Ising CFT (see Sec. IV for details). 2 2 nx þ ny. Instead we see that there is an alarming instability at large L ¼ Oð100Þ, which we believe is due to the lack of well- defined critical surface with a second order phase transition converges to the continuum. However, as we approach the required to define a continuum limit. A more graphic cutoff, it is increasingly distorted. indication of this failure is present when we consider the Clearly the spectral fidelity on the hypercubic torus is average value of hϕ2i as a function of position on the sphere 2 i comparable to the FEM spectra on S . This is not as shown on the right-hand side of Fig. 7. Due to a spatial surprising; indeed the square lattice can also be viewed variation in the UV cutoff, this is not uniform, with as a FEM realization. One simply divides each square into variation that can be seen by comparing the regions close two right angle triangles and notices that the formula in to and far form the 12 exceptional five-fold vertices of the Eq. (11) implies a zero contribution on the diagonal links. original icosahedron. As we will show, this failure is also This generalizes to higher dimensional hypercubic lattices evident in a lattice perturbation expansion. At small λ0 a when using the DEC form. Of course, the major difference spherically asymmetric contribution to hϕ2i is given by a 2 i between the square lattice on R and the simplicial sphere UV divergent one-loop diagram. S2 is the former breaks one rotational isometry of the In conclusion, in the application of FEM to quantum continuum but conserves two discrete subgroups of trans- field theory, we have encountered a fundamentally new lations, whereas the later breaks all three isometries of problem. The FEM methodology has been developed to Oð3Þ down to a fixed finite subgroup independent of the give a discretization for nonlinear PDEs that converges to refinement. the correct continuum solution as the simplicial complex
ϕ2 FIG. 7. On the left the Binder cumulants for the FEM Lagrangian with no QFE counterterm. On the right the amplitude of h i i in simulations with the unrenormalized FEM Lagrangian.
014502-8 LATTICE ϕ4 FIELD THEORY ON RIEMANN … PHYS. REV. D 98, 014502 (2018) is refined. In the context of a Lagrangian system for a quantum field, this implies a properly implemented FEM should therefore guarantee convergence to all smooth classical solutions of the EOM as the cutoff is removed. However, quantum field theory is more demanding. The path integral for an interacting quantum field theory is sensitive to arbitrarily large fluctuations—even in pertur- bation theory—on all distance scales, down to the lattice spacing, due to ultraviolet divergences. This amplifies local UV cut-off effects on our FEM simplicial action. Our solution is to introduce a new quantum finite element (QFE) lattice action that includes explicit counterterms to regain the correct renormalized perturbation theory. We conjecture that, for any ultraviolet complete theory, if the QFE lattice Lagrangian is proven to converge to the continuum UV theory to all orders in perturbation theory, this is sufficient to define its nonperturbative extension to the IR. We believe this is a plausible conjecture consistent with our experience on hypercubic lattices for field theories in Rd, but it is far from obvious. It needs careful theoretical and numerical support to determine its validity. In Secs. IVand V we give extensive numerical test for ϕ4 on S2 on the critical surface. In particular the new Monte Carlo simulation of the Binder cumulant U4 on the right-hand side of Fig. 12 including the quantum counter term gives a critical value FIG. 8. The one-loop diagram is logarithmically divergent in 2d 0 85020 58 90 of the Binder cumulant, U4;cr ¼ . ð Þð Þ, in agree- and linearly divergent in 3d, whereas the two-loop is finite and logarithmically divergent for 2d and 3d, respectively. The three- ment with the continuum value, U4 ¼ 0.8510, to about one part in 103. While this is promising, the limitations due to loop diagram is finite in both 2d and 3d. statistics and the restriction of our studies to the simplest Σ −12λ δ − 96λ2 3 scalar 2-d CFT is duly acknowledged. ðx; yÞ¼ 0G0ðx; yÞ ðx yÞþ 0G0ðx; yÞ; ð33Þ as illustrated in the first two panels in Fig. 8. In both 2d and III. ULTRAVIOLET COUNTERTERMS ON 3d the three-loop diagram, and all higher-order diagrams, THE SIMPLICIAL LATTICE are UV finite. In 4d there is a logarithmically divergent contribution to the four-point function which contributes to To remove the quantum obstruction to criticality for the the quartic coupling λ0; however the complete nonpertur- FEM simplicial action, we begin by asking if we can add bative theory is the trivial free theory with no interesting IR counterterms to the action Eq. (23) to reproduce the physics. renormalized perturbation expansion order by order in the continuum limit at the UV weak coupling fixed point. 4 A. Lattice perturbation expansion On a hypercubic lattice, it has been proven for ϕ theory 4 [29] that this can be achieved by taking the lattice UV cut- The perturbation expansion for the ϕ theory on our off to infinity, holding the renormalized mass and coupling FEM lattice (or indeed any lattice) starts with the partition fixed. Here we will suggest how this can be achieved on a function, Z simplicial lattice for 2d and 3d. While we do not attempt a −1ϕ ϕ −λ ϕ4 λ Dϕ 2 iMi;j j i i proof, the similarity with the hypercubic example strongly Zðm0; 0Þ¼ ie ; ð34Þ suggests that a proof could be found at the expense of increased technical difficulty. by expanding in the quartic term. In our FEM representa- The ϕ4 theory is super-renormalizable in 2d and 3d, with tion, the action is one-loop and two-loop divergent diagrams, respectively, 1 ϕ ϕ ϕ λ ϕ4 only contributing to the two-point function. The one-loop S½ i ¼2 iMi;j j þ i i diagram is logarithmically divergent in 2d and linearly 1 divergent in 3d. The two-loop diagram is UV finite in 2d, ϕ ϕ 2ϕ2 λ ϕ4 ¼ 2 ½ iKi;j j þ mi i þ i i : ð35Þ but logarithmically divergent in 3d. The divergent contri- 2 pffiffiffiffi 2 butions renormalize the mass via the one-particle irreduc- For convenience, both bare parameters m ≡ g m0 and pffiffiffiffi i i pffiffiffiffi ible (1PI) contribution to the self-energy, λi ≡ giλ0 include the factor of the local dual volume gi.
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The quadratic form (Mi;j) includes both the DEC Laplace- Beltrami operator (Ki;j) and the bare mass (m0). Following the standard Feynman rules for a perturbative λ ϕ ϕ −1 expansionR in 0, we compute the propagator, h i ji¼Z −S Dϕϕiϕje , to second order: ϕ ϕ −1 h i ji¼½M i;j −1 −12λ −1 −1 þ½M i;i1 ð i1 ½M i1;i1 Þ½M i1;j −1 96λ λ −1 3 −1 þ½M i;i1 ð i1 i2 ½M i1;i2 Þ½M i2;j: ð36Þ After amputating the external lines, we find the inverse ˜ propagator Mijðm0; λ0Þ¼Mi;j þ Σijðm0; λ0Þ, where Σ λ −12λ −1 δ 96λ λ −1 3 λ3 FIG. 9. The numerical fit of the lattice one loop log divergent ijðm0; 0Þ¼ i½M ii ij þ i j½M i;j þ Oð 0Þ: determining coefficient Q compared to the continuum value: pffiffiffi ð37Þ Q ¼ 3=ð8πÞ¼0.0689161…. pffiffiffi is the 1PI simplicial self-energy. 3 1 −1 ≈ ½M xx 8π log 2 2 m ax B. One loop counterterm pffiffiffi pffiffiffi 3 3 2 Since there is no analytic spectral representation of the a 1 ¼ 8π logðNÞþ 8π log 2 þ Oð =NÞ: ð41Þ free FEM Green’s function, we compute the one loop ax diagram in coordinate space by numerical evaluation of the 2 2 The logarithmic divergence is regulated by the IR mass, m . propagator. To be concrete, for 2d on S and for 3d on 2 Of course on the lattice, at fixed dimensionless bare mass, R × S , the Gaussian term is 2 2 2 m0 ¼ a m and bare coupling, λ0, there is no actual UV ϕ ϕ divergence. The renormalized perturbation expansion is x;t1 Mx;t1;y;t2 y;t2 pffiffiffiffiffi 2 2pffiffiffiffiffi 2 defined by fixing the physical mass (m), which in the ϕ K ϕ g ϕ − ϕ 1 m g ϕ ; ¼ x;t1 x;y y;t1 þ xð x;t x;t Þ þ 0 x x;t continuum limit (N → ∞) corresponds to a vanishing 2 2 ð38Þ effective lattice spacing, a m ¼ Oð1=NÞ. On a hypercubic lattice, there is a universal cut-off (π=ax ¼ π=a), whereas where the sites are now labeled by i ¼ðx; tÞ. The 2-d 2 on a simplicial lattice the one-loop term separates into geometry, S , can be viewed as a special case with a single two terms on the right side of Eq. (10): (1) a position sphere at t ¼ 0. The integer t labels each sphere along the independent divergence and (2) a finite (scheme dependent) cylinder, and x indexes the sites on each sphere. Kx;y is constant for each site x. non-zero on only nearest neighbor links hx; yi. To take the We have checked numerically, to high accuracy, that the continuum limit we need to define the normalization first divergent term is independent of position. This is a convention of our lattice constants: crucial observation which we believe is a general conse- XN−1 X pffiffiffiffiffi quence of the FEM prescription. Spatial dependence for a gx ¼ N and Kx;y ¼ð2=3ÞE; ð39Þ divergent term would imply the need for a divergent x¼0 hx;yi counterterm to restore spherical symmetry. This departs where in this specific context E refers to the number of edges from the standard cutoff procedure on regular lattices and on the simplicial graph. The extra factor of 2=3 is introduced would be very difficult if not impossible to implement. to compensate, on average, for the six nearest neighbors per Moreover, we show in Fig.p9ffiffiffithat the fit coefficient is site on the sphere relative to a conventional square lattice accurately determined to be 3=ð8πÞ, which is the exact with four nearest neighbors. In flat 2-d space this factor of continuum value. 2=3 for the triangular lattice gives the same dispersion A heuristic argument for the value of this constant follows 2 2 4 S2 relation, E ¼ m0 þ k þ Oðk Þ, as the square lattice. from our tessellation as a nearly regular triangular lattice. In lattice perturbation theory on a 2-d FEM simplicial In flat space, relative to a squareP lattice, theR density of states −1 → 2 ρ lattice, we expect the logarithmically divergent one-loop on a triangularffiffiffi lattice is N n ffiffiffi d k TðkÞ, where p 2 p term to give a site-dependent mass shift, ρT ¼ 3=ð8π Þ. The extra factor of 3=2 is due to the area 2 2 2 jσ j of the dual lattice hexagon relative to the dual lattice m → m þ Δm : ð40Þ 0 0 0 x square on a hypercubic lattice. However, since this constant The numerical computation of the one-loop diagram on S2 defines the local charge in the 2-d Coulomb law and our does indeed have the appropriate form, linear elements do not implement local flux conservation,
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pffiffiffiffiffi FIG. 10. One loop counterterm plotted against logð gxÞ. FIG. 11. The contribution of the two loop term in 2d. this is surprising result. Nonetheless in Sec. III C we will The QFE one loop counterterm is introduced to cancel prove this based on the FEM principle of spectral fidelity and the finite position dependence in Eq. (41) that violates the renormalization group for the mass anomalous dimen- rotational invariance to order λ0. To project out the spherical sion. To the best of our knowledge this is a novel extension component of a local scalar density, ρx, we average over the of FEM convergence theorems. rotation group R ∈ SOð3Þ, Next consider the finite spatial dependence term in Z Z 1 1 X ffiffiffiffiffi Eq. (41). As illustrated in Fig. 10, it is almost exactly a ρ ˆ Ωρ θ ϕ ≃ −1 p ρ pffiffiffiffiffi 3 dR ðRrÞ¼4π d ð ; Þ N gx x: linear function of log½ gx . Again there is a heuristic volOð Þ x explanation for this. An almost perfect analytic expression ð43Þ can be found by considering the dilatation incurred by the projection of our triangular tessellation of the icosahedron ρ −1 ’ Applied to x ¼½M xx, the subtracted lattice Green s onto the sphere. The icosahedron is a manifold with a flat function is metric except for 12 conical singularities at the vertices. XN Our choice of the simplicial complex began with flat 1 pffiffiffiffiffi δG ¼½M−1 − g ½M−1 : ð44Þ equilateral triangles on each of the 20 faces of the original xx xx N x xx icosahedron. The radial projection onto the sphere is a Weyl x¼1 transformation to constant curvature with conformal factor This removes completely the logarithmic divergence, (or Jacobian of the map), leaving the position dependent finite counterterm which pffiffiffiffiffiffiffiffiffi 2 2 2 3=2 adds a contribution to the FEM action, ðx1 þ x2 þ 1 − R Þ gðxÞ ¼ eσðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi c ; ð42Þ X ffiffiffiffiffi 2 → 6 p λ δ ϕ2 1 − Rc SFEM SQFE ¼ SFEM þ gx 0 Gxx x: ð45Þ x where R is the circumradius for one of the 20 icosahedral c ϕ4 faces and x1, x2 the flat coordinates on that face. This In 2d theory this is the only UV divergent term. We also formula gives a line coinciding with the linear data in computed the two-loop contribution to the 1PI simplicial Fig. 10. Although this is an extremely good approximation self energy Eq. (37) which in coordinate space is given by ’ −1 3 to the counter term, it is not perfect since it neglects the the third power of the Green s function, ½M xy, between conical singularities in the map near the 12 exceptional the vertex at x and y. The position dependence, vertices of the original icosahedron. These exceptional X XN 1 pffiffiffiffiffi points are visible in the upper left corner of Fig. 10. We find δ½G 3 ¼ ½M−1 3 − g ½M−1 3 ; ð46Þ x xy N x xy that the true numerical computation does give better results y x¼1 and, moreover, it is a general method not requiring our is found again by subtracting the average contribution on careful triangulation procedure. A more irregular procedure pffiffiffiffiffi should also be allowed. The lesson is that the simplicial the sphere. The result is plotted in Fig. 11, against log½ gx metric, gij ¼flijg, on a given Regge manifold not only which demonstrates that it vanishes rapidly in the con- defines the intrinsic geometry but also a co-ordinate system tinuum limit as 1=L2 ∼ 1=N consistent with naïve power breaking diffeomorphism explicitly. The set of lengths counting in continuum perturbation theory. While we do includes both a definition of curvature and the discrete not attempt a general analysis of the simplicial lattice coordinate system. The counterterm must compensate for perturbation expansion, it is plausible based on analogous this arbitrary choice. studies on the hypercubic lattice [30,31] that after canceling
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2 2 cut-off dependence in the one loop UV divergent term the followed by taking the IR regulator to zero: m =Λ0 → 0. entire expansion restores the renormalized perturbation This proves that the coefficient cx ¼ 1=ð4πÞ is independent series in the continuum limit. However our test on the of position and identical to the continuum value on R2. 2-d Riemann sphere simplicial lattice goes further. Namely Of course, this is not surprising for flat space. we give numerical evidence that this 2-d QFE lattice action, Now we can apply the same reasoning to the simplicial when tuned to the critical surface, approaches the non- sphere S2, beginning with the spectral representation of perturbative continuum theory in the IR for the Ising CFT the Green’s function, field theory. X ϕ ðxÞϕ ðxÞ G m2 n n : 50 xxð Þ¼ ð0Þ 2 ð Þ C. Universal logarithmic divergence n En þ m0 The success of our QFE action prescription depends on On the basis of FEM spectral fidelity, assume the low the coefficient of the logarithmic divergence being exact in eigenspectrum for l ≤ L0 is well approximated to Oð1=NÞ the continuum limit. On S2 or any maximally symmetric by spherical harmonics, space, this implies the coefficient is also independent of E ≃ a2R−2lðl þ 1Þþm2; ð51Þ position. At first, this observation appears surprising. After n pffiffiffiffiffiffi 0 4π all, the UV divergence is sensitive to the local, short ffiffiffiffi ˆ ϕnðxÞ ≃ p YlmðrxÞ; ð52Þ distance cut-off which is not uniform. We claim this is a N consequence of the observed spectral fidelity of the DEC where R=a is the radius of the sphere in units of the lattice Laplace-Beltrami operator and renormalization group (RG) spacing. Matching the area of the sphere with N triangles for a logarithmic divergence relating UV divergences to the pffiffiffi 4π 2 2 3 2 1 IR regulator. Adapting the argument to a general 2-d we have R ¼ a N = .ForR ¼ the spectrum is l l 1 , but with our convention of setting a 1, we find Riemann manifold is in principle straight forward, match- ð þpÞffiffiffi ¼ 3 8π ing the local divergence to the one loop renormalization of R ¼ N=ð Þ. Next,P weffiffiffiffiffi fix the eigenvector normaliza- p 2 4π the continuum theory [7] at each point x. tion by requiring x gx jY00j ¼ N= . Let us first apply this renormalization group (RG) By splitting the spectral sumP at the cut-off L0, and using 4π ˆ ⃗ 2 1 argument to the hypercubic lattice, where we already know the addition formula, mYlmðrxÞYlmðryÞ¼ð l þ Þ and understand the answer. Suppose that the one-loop Plðrx · ryÞ for l 014502-12 LATTICE ϕ4 FIELD THEORY ON RIEMANN … PHYS. REV. D 98, 014502 (2018) ∂ γ 2 − 2 The second special feature of the one-loop diagram in ðm0Þ¼ m0 Gxxðm0Þ ϕ4 ∂m0 2d in theory is that it is both local on the lattice and pffiffiffi Z Λ2 2 logarithmically divergent in the continuum. In 3d the 3 0 m ≃ dEð0Þ 0 situation changes. First the one-loop term in the continuum 4π ð0Þ 2 2 0 ðE þ m0Þ is linearly divergent. However on the lattice such a pffiffiffi 3 1 divergence is not manifest because it enters as a dimen- ¼ 2 2 : ð56Þ sional factor of 1=a, which in the bare lattice action is 4π 1 þ m =Λ 0 0 chosen implicitly to be 1=a ¼ Oð1Þ. Also all UV power divergent diagrams are finite in the IR so that there is no 2 Λ2 → 0 In the limit Oðm0= 0Þ , we isolate the coefficient in need for a dimensionless mass regulator m0 ¼ am.Wehave Eq. (47) and prove that the divergence is both independent computed the one-loop diagram on the simplicial cylinder of position on the sphere with its coefficient identical to the R × S2, and found numerically that the counterterm is continuum one loop diagram on the sphere. The essential almost identical to the 2-d case. This is easy to understand. assumption we have made is the spectral fidelity property The spectral decomposition of the 3-d Green’s function, of the discrete DEC Laplace-Beltrami operator. In ffiffiffi p L0 ω − 0 Appendix A we provide the exact continuum propagator X X 2 1 i kðt t Þ 2 3 ð l þ ÞPlðrx · ryÞe on the sphere, G 0 ðm Þ ≃ xt;yt 0 8π l l 1 ω2 m2 l¼0 k ð þ Þþ k þ 0 ∞ X XN X iω ðt−t0Þ 1 ð2l þ 1ÞPlðcosðθÞÞ ϕ ðxÞϕ ðyÞe k Gðθ; μÞ¼ þ n n ; ð58Þ 4π 1 2 2 μ2 ð0Þ 2 2 l 0 ðl þ = Þ þ 2 E þ ω þ m Z¼ n>ðL0þ1Þ k n k 0 2 ∞ cosðμtÞdt ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S2 4π 0 2 coshðtÞ − 2 cosðθÞ is a natural generalization to the case on given in Eq. (53). We see that the breaking of rotational invariance 1 32 ≃ − 14ζ 3 μ2 θ2 μ4 on R × S2 is very similar to the case of S2 for each mode 4π log 1 − θ ð Þ þ Oð ; Þ; cosð Þ ωn and as such the overall breaking term will also be very ð57Þ similar to the 2-d case. Next, in 3d, there is a new, logarithmically divergent two-loop term. Again, a comparison between 2d and 3d is and identify the coefficient of the logarithmic divergence 2 2 interesting. The one-loop term in Hamiltonian form can be by relating the angle to the lattice cutoff by θ ∼ 1=L ¼ 4 0 viewed simply as a problem of normal ordering: ϕ ðxÞ ≕ Oð1=NÞ. 4 2 2 ϕ ðxÞ∶ þ ϕ ðxÞhϕ ðxÞi0. The diagram has no unitarity cuts and can be simply dropped when defining a normal ordered D. Comments on structure of the counterterms perturbation expansion. In 3d, the two-loop term introduces In spite of our very limited example of the QFE a nonlocal contribution with essential unitarity cuts that counterterms on S2, we hazard a general interpretation must be preserved in a renormalized perturbation theory. which we hope will guide extensions to any smooth On the lattice, the continuum position space two-loop Riemann manifold. This is based in part on current study diagram is replaced by the element-wise third power of ’ of counterterms in 3-d ϕ4 theory on R × S2. Nonetheless, the Green s function, we acknowledge the fact that our examples so far may have 2 2 −1 3 λ G t1 − t2; cos θ → λ M : 59 very special features not shared more generally. 0 ð ð xyÞÞ 0½ x;t1;y;t2 ð Þ The first special feature is that the S2 manifold is an example of a maximally symmetric manifold. This property Again, we must subtract the rotationally invariant contribu- − ˆ − ˆ 2 2 − 2 θ is shared by flat space, spheres, and anti-de Sitter space tion at fixed t1 t2 and jrx ryj ¼ cosð xyÞ.Wehave with constant curvature. On a d-dimensional maximally found numerically that although the rotationally invariant symmetric manifold, there is a full set of dðd þ 1Þ=2 part has a power fall-off dictated by naïve dimensionality, the δμ2 ϕ ϕ isometries implying that all points have identical metric residual breaking term, xy;t2−t1 x;t1 y;t2 , falls off exponen- properties. A consequence is in these cases, the counter- tially in lattice units. Consequently, in physical units, it is terms can be easily defined by projecting out the average exponential in ðt2 − t1Þ=a and jrˆx − rˆyj=a and therefore to over the full set of isometries as in Eq. (43) above. While leading order may be replaced by a local counter term in these maximally symmetric manifolds are actually of the QFE action. We postpone further analysis of counter paramount interest, we are also confident that this restric- terms to future publications. tion is not necessary. On a general manifold, the procedure We are also considering alternative methods that avoid the would be to compute the continuum UV divergence at each difficulty of computing individual UV divergent perturbative site and subtract it before defining the counterterm. diagram. For example, as a proof of concept, we have 014502-13 RICHARD C. BROWER et al. PHYS. REV. D 98, 014502 (2018) implemented the Pauli-Villars (or Feynman-Stuekelberg) Gnðx1; ;xnÞ¼hϕðx1Þ ϕðxnÞi approach. It introduces an intermediate scale of Pauli- 1 δ δ ≪ π Villars mass, MPV =a separating the IR from the UV ¼ 0 δ δ Z½J jJ¼0; ð61Þ and protects the UV diagram from the dependence of the Z½ Jðx1Þ JðxnÞ variable cutoff of the simplicial lattice. This separation of for our scalar field theory as a derivative expansion of the scales is another way to exploit the spectral fidelity of the low R R partition function, Z½J ¼ ½Dϕ exp½−S½ϕ þ ddxJðxÞϕðxÞ , spectrum, but this time to all orders in perturbation theory. in the current JðxÞ. Likewise replacing the current by a The implementation amounts to the addition of a ghost field pffiffiffiffiffiffiffiffiffi giving a simplicial equivalent of the continuum propagator, constant magnetic field, JðxÞ¼h gðxÞ, the derivatives of the partition function give the magnetic moments, 1 1 1 1 → − ¼ : ð60Þ n p2 p2 p2 þ M2 p2 þ p4=M2 mn ¼hM i PV PV Z Dpffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi E 2 … 2 ϕ ϕ Now, reaching the continuum requires a double limit: a → 0 ¼ d x1 d xn gðx1Þ ðx1Þ gðxnÞ ðxnÞ → ∞ at fixed aMPV,followedbyaMPV .Wehaveimple- 2 ð62Þ mented this on S by adding a quadratic PV term, −2 pffiffiffiffi R −M ϕ K K ϕ = g , to the FEM action in Eq. (23), 2 pffiffiffi PV x xz zy y z where M ¼ d x gϕðxÞ. From these moments, defining seeing qualitatively that this does work. However, in addition 2n 2 n homogeneous quotients, Q2 ¼hM i=hM i , the first three to the cost of the double limit, in the context of our current ϕ4 n Binder cumulants are [35] simulations, the PV term has the technical disadvantage that it prevents the use of the very efficient cluster algorithm 3 1 1 − of Ref. [32]. U4 ¼ 2 3 Q4 We are also exploring extensions of the renormalization group approach in our demonstration of the one-loop 15 1 1 U6 ¼ 1 − Q4 þ Q6 logarithmic divergences in 2d in Sec. III C. For asymp- 8 2 30 totically free theories in 4d, such as the non-Abelian gauge 315 2 1 2 1 2 2 1 − 2 − theory, the Lagrangian, F =g , has a dimensionless cou- U8 ¼ 272 3 Q4 þ 18 Q4 þ 136 Q6 630 Q8 : ð63Þ pling with a logarithmic divergence. We anticipate the use of the RG approach and perhaps the scale setting properties These are just the connected moment expansion of the free of Wilson flow [33] to correct the scheme dependence of a energy, simplicial lattice. As demonstrated in the classic paper by A. Hasenfratz and P. Hasenfratz [34] for pure non-Abelian Z R ffiffi − ϕ d p ϕ gauge theory, the lattice scheme dependence is a one-loop F½h ¼log ½Dϕ e S½ þh d x g ðxÞ ; ð64Þ effect which on a simplicial lattice should be replaced by a 2 local site dependent multiplicative counter term for F . 2 divided by appropriate factor of hM ni. Therefore they vanish in the Gaussian limit. The overall normalization IV. NUMERICAL TESTS OF UV has been chosen so that the Binder cumulants are unity in S2 COMPETITION ON the ordered phase. We designate the exact value of Binder cumulants for the critical Ising CFT by U2n. Here we present our first test of our QFE simplicial 2 lattice construction for the nonperturbative study of quan- On the sphere S the exact value of the Binder cumulants tum field theories on curved manifolds. The Monte Carlo U2n can be computed from the solution for the conformal 2 simulation of the 2-d scalar ϕ4 theory on the Riemann n-point functions on R . This a consequence of the special sphere, S2, must agree within statistical and systematic property for conformal correlators under a Weyl trans- 2 2 uncertainties with the exact solution of the Ising or c ¼ formation of the flat metric: gμνðxÞ¼Ω ðxÞδμν on S .For 1=2 minimal CFT in the continuum limit. The first test is to example the CFT correlators of a primary operator ϕ with fine-tune the mass parameter to the critical surface illus- dimension Δ obey the identity [18] trated in Fig. 1 and to compare the Monte Carlo compu- ϕ …ϕ tation of bulk Binder cumulants with analytic values. In h ðx1Þ ðxnÞigμν Sec. V, we extend our tests to the detailed form of the 1 1 ϕ ϕ conformal two-point and four-point correlators. ¼ Δ Δ h ðx1Þ ðxnÞiflat: ð65Þ Ωðx1Þ ΩðxnÞ S2 A. Ising CFT on the Riemann As also pointed out in Ref [18], even the Weyl anomalies Let us begin by defining the Euclidean correlations will cancel in homogeneous ratios of CFT correlators. functions Consequently as noted in Ref. [36] homogeneous moments 014502-14 LATTICE ϕ4 FIELD THEORY ON RIEMANN … PHYS. REV. D 98, 014502 (2018) in the Binder cumulants are computed by integration over of m4 is naïvely an eight-dimensional integral, but after the n-point function on the compact manifold. utilizing rotational invariance the integration is reduced to a In our current application, the Weyl transformation is a five-dimensional integral. The numerical four-point inte- stereographic projection from R2 to the Riemann sphere, gral evaluation in Ref. [36] used 1000 independent S2, which can be explicitly parametrized by Monte Carlo estimates, yielding an estimate of m4 ¼ 1.19878ð2Þ, leading to a prediction of the critical iϕ w ¼ðrx þ iryÞ=ð1 þ rzÞ¼tanðθ=2Þe ; ð66Þ value for the fourth Binder cumulant U4 ¼ 0.8510061ð108Þ after error propagation. rˆ ¼ðsin θ; cos ϕ; sin θ sin ϕ; cos θÞ; ð67Þ We computed both the four- and six-point integrals numerically using the MonteCarlo method of where w is the complex coordinate w ¼ x þ iy in the R2 Mathematica’s Integrate[] function. For the four-point 3 AccuracyGoal → 4 plane and rˆ is unit vector in R in Eq. (24). The resulting integral, we set , which yields a metric on S2 is Monte Carlo distribution with standard deviation approx- imately 10−4. We repeated the Monte Carlo estimation of 2 2 2 2 the integral 100 times. The quoted error is the standard ds 2 ¼ dwdw¯ ¼ cos ðθ=2Þds 2 : ð68Þ S ð1 þ ww¯ Þ2 R error on the mean of this sample distribution. We find m4 ¼ 1.1987531ð116Þ and the corresponding Binder cumulant 2 2 0 8510207 63 with Ω ðθÞ¼cos ðθ=2Þ. After the Weyl rescaling the two- is U4 ¼ . ð Þ. We again used the Mathematica point function on S2 is MonteCarlo integrator to compute m6.Weset AccuracyGoal → 3, which yields a Monte Carlo dis- 1 tribution with standard deviation approximately 10−3.We ϕ ˆ ϕ ˆ h ðr1Þ ðr2Þi ¼ Δ ; ð69Þ ð2 − 2 cos θ12Þ repeated the Monte Carlo estimation 50 times. We find m6 ¼ 1.632851ð253Þ, and a corresponding critical Binder 0 7731441 213 where θ12 is the angle between the two radial vectors, cumulant value of U6 ¼ . ð Þ. rˆ1, rˆ2, on the surface of the sphere. Incidentally a pedestrian proof uses standard trigonometric identities to show that B. Finite scaling fitting methods 2 2 Ωðθ1Þjw1 − w2j Ωðθ2Þ¼jrˆ1 − rˆ2j ¼ 2ð1 − cos θ12Þ.We To compute these cumulants, we must obtain estimates have chosen the normalization of ϕ so that the numerator for the even moments of the average magnetization. On the is unity. Just as Poincare invariance on the plane implies lattice we define the moments of the field on the simplicial that correlators are a function of the length (or Euclidean complex by distance on the plane, jw1 − w2j), rotational invariance on the sphere fixes the correlator to be a function of the X ffiffiffiffiffi −1 p ϕ n geodesic distance, θ12. mn ¼hðN gx xÞ i On R2 the conformal n-point correlation functions of the x X 2-d Ising model can be constructed, in principle, to any 1 pffiffiffiffiffiffi pffiffiffiffiffiffi ¼ h g ϕ … g ϕ ið71Þ order [37–41] and in practice have been computed up to Nn x1 x1 xn xn x1; ;xn sixth order [37,38,42]. To normalize the ratios Q2n we use the analytic result, where h i denotes the ensemble average and the spatial Z Z Ω Ω average is shown explicitly by a summation. Numerically, d 1 d 2 ϕ Ω ϕ Ω m2 ¼ 4π 4π h ð 1Þ ð 1Þi these moments are easy to compute by a weighted sum of Z the field over the complex on each field configuration, with 1 1 pffiffiffiffiffi θ the weights gx defined to be the areas of the Voronoï cells ¼ 1=8 d cos 12 −1 2ð2 − 2 cos θ12Þ [43] as computed on the flat triangles and normalizedP ffiffiffiffiffi to 11=4 N p 2 one per site on average, i.e., our convention x¼1 gx ¼ ¼ 7 ; ð70Þ N in Eq. (39). In our Monte Carlo calculations, the finite size of our with Δ ¼ 1=8 for the Ising model. This allows us to find the simplicial complex will break conformal invariance. If the relationship between the normalization of the continuum bare lattice parameters are tuned sufficiently close to the field ϕðxÞ and the normalization of our discretized field ϕx critical point, finite-size scaling (FSS) relations can be used in a QFE calculation. Similarly, higher moments can be to extract CFT data by fitting the volume dependence of computed numerically as integrals over the n-point corre- moments or cumulants of average magnetization [44].We lators on S2. give the key details below, and in Appendix B we give The integral of the four-point function on the sphere was further details of the FSS analysis used to parametrize our performed by Deng and Blöte in Ref. [36]. The computation numerical data as we take the infinite volume limit. 014502-15 RICHARD C. BROWER et al. PHYS. REV. D 98, 014502 (2018) It begins by expanding the free energy in a finite lattice correction to scaling is not due to a conformal quasiprimary volume N around the critical point ðgσ;gϵ ;gωÞ.ByZ2 operator ω, but rather a breaking of conformal symmetry due symmetry the critical point for the Z2 odd operator is to a finite volume. We know of no other example of a CFT gσ ¼ 0. Of course, we cannot directly vary the renormal- where this occurs. ized couplings ðgσ;gϵ;gω; Þbut instead can vary our bare Finally, to compare our simplicial calculations to the 2 couplings ðh; μ0; λ0Þ defined in our cutoff theory. The precise analytic results described in Sec. IVA, we can fit expansion then for the Z2 odd operator takes the form our estimates of the magnetization moments and use the fitted parameters to estimate the critical Binder cumulants. 3 5 gσ ¼ hα1 þ h α3=3! þ h α3=5! ; ð72Þ Note that even though we chose to approach the critical surface along a line of constant λ0 and were therefore and without loss of generality, we can rescale h so that R α 1 unable to determine the linear coordinate transformation 1 ¼ . There is mixing between the two even operators so of Eq. (73) leading to some of that dependence being that absorbed into redefinitions of fit parameters, e.g., a¯ 21 vs. 2 2 a21, our estimates of the critical Binder cumulants U2 are gϵ − gϵ Rϵμ Rϵλ μ0 − μ n ¼ þ : ð73Þ free of this ambiguity. − λ − λ gω gω Rωμ Rωλ 0 Since our current simulations were performed at a single C. Monte Carlo results near the critical surface λ fixed 0, we cannot directly determine the coordinate For the Monte Carlo simulation, we use the embedded transformation R. Instead, we define the derived quantities. dynamics algorithm of Brower and Tamayo [32], mixing a ¯ number of embedded Wolff cluster updates [45] with local a¯ 1 ¼ a 1Rϵμ; b 1 ¼ b 1Rωμ; c¯ 1 ¼ c 1Rϵμ; k k k k k k updates consisting of one Rosenbluth-Teller sweep [46] ð74Þ and one over relaxation sweep [47–50]. Table I shows our empirically determined number of Wolff cluster updates per μ2 μ2 − Rϵλ λ − λ μ2 μ2 − Rωλ λ − λ a ¼ ð 0 Þ; b ¼ ð 0 Þ; local update such that on average OðNÞ spins are flipped Rϵμ Rωμ between local updates. We can approximately locate the ð75Þ critical surface of the Wilson-Fischer fixed point by computing the Binder cumulants at fixed λ0, varying the where a 1, b 1, and c 1 are defined in Eq. (B12) and 2 2 2 k k k volume N ¼ L ¼ 2 þ 10L and the bare mass μ0, search- Eq. (B13) in Appendix B. Future simulations varying the ing for the region of parameter space where U4 ≈ 0.851. bare coupling near to the Wilson Fisher fixed points will We can understand the approximate behavior of the Binder improve the finite volume parametrization. The result of the cumulants by substituting the FSS expressions in Sec. IV B finite volume and scaling expansion is to parametrize our and reexpanding [51], giving data by 2 ¯ 2 2 d−Δϵ U2 ðμ0Þ¼U2 þ A2 ðμ0 −μ ÞL −2Δσ 2 2 d−Δϵ n n n a m2 ¼ L ½a20 þ a¯ 21ðμ0 − μaÞL ¯ 2 2 d−Δω 2Δσ −d þ B2nðμ0 − μ ÞL þ C2nL þ : ð79Þ ¯ μ2 − μ2 Ld−Δω L−d b þ b21ð 0 bÞ þc20 þ ; ð76Þ Δ 1 ¯ −4Δσ 2 2 d−Δϵ Since ϵ ¼ , the A2n term will cause the cumulant to m4 ¼ L ½a40 þ a¯ 41ðμ0 − μaÞL 2 2 diverge from its critical value if μ0 is not tuned to μa as ¯ μ2 − μ2 Ld−Δω L−3d þ b41ð 0 bÞ þc40 L → ∞. Importantly, the direction of the divergence will 2 2 −2d 2 depend on the sign of μ − μ . At smaller L, the C2 term þ α3L m2 þ 3m2 þ ; ð77Þ ð 0 aÞ n will tend to dominate since Δσ ¼ 1=8 and Δω ¼ 4 as L−6Δσ ¯ μ2 − μ2 Ld−Δϵ m6 ¼ ½a60 þ a61ð 0 aÞ previously noted. It is our experience that Eq. (79) should ¯ μ2 − μ2 Ld−Δω L−5d þ b61ð 0 bÞ þc60 3 −2d 2 TABLE I. The number of Wolff cluster updates totaling OðNÞ þ 15m4m2 − 30m2 þ α3L ðm4 − 3m2Þ spins near criticality. −4d þ 3α5L m2 þ ; ð78Þ pffiffiffiffi Wolff/local update where we introduce the length parameter L ¼ N in accord Lattice size ratio with the FSS analysis summarized in Appendix B.The L<15 4 ellipses are a reminder that our expansion drops higher order 15 ≤ L ≤ 42 5 terms in the fitting expressions, leading to a systematic error 43 ≤ L ≤ 99 6 100 ≤ ≤ 200 in the included parameters. For the d ¼ 2 Ising model, Δσ ¼ L 7 ¯ −9=4 201 ≤ L ≤ 364 8 1=8 and Δω ¼ 4,sotheb21 term scales like L whereas the −2 L>364 9 c20 term scales like L . Interestingly, the leading irrelevant 014502-16 LATTICE ϕ4 FIELD THEORY ON RIEMANN … PHYS. REV. D 98, 014502 (2018) FIG. 12. The left panel shows Binder cumulants up to twelfth 2 order vs. μ0 for fixed simplicial complex size, L ¼ 200 and 2 2 λ0 ¼ 1. The vertical dashed line indicates μa ≈ μcr. The right panel shows the cumulant U4 vs. simplicial complex size L in the pseudocritical region holding λ0 ¼ 1 fixed. Each point represents 2 a calculation and connecting lines indicate constant μ0. Con- tinuum estimate: U4 ¼ 0.8510207ð63Þ. not be used to actually fit the Binder cumulant data to accurately determine U2n since there tend to be delicate cancellations that occur in the expansion of the ratio. Instead, the moments should be analyzed separately using the parameterization in Eq. (76). In the left panel of Fig. 12, we show the cumulants for a FIG. 13. A subset of the data used in a simultaneous fit to the 0 03 fixed size of the simplicial complex (L ¼ 200) as we sweep three lowest even moments for data selection parameter w ¼ . . 2 through the critical region by varying μ0 holding λ0 ¼ 1 fixed. Higher order cumulants are statistically noisier but We start by getting a rough fit to the data using some μ2 ∈ μ2 μ2 ∈ since they pass through the critical region with a steeper initial data selection window 0 ½ min; max and L slope [35] they have comparable statistical weight in ½Lmin;Lmax and then form the following quantities determining the critical coupling μ2 ≈ μ2 . On the right a cr 2 2 d−Δϵ a¯ 1ðμ0 − μ ÞL we show the cumulant U4 vs. simplicial complex size L for k a δak1 ¼ μ2 a 0 fixed values of 0 in the critical region. As expected, the k 2 divergence at large L changes sign as μ passes through the ¯ μ2 − μ2 Ld−Δω 0 δ bk1ð 0 bÞ critical region. bk1 ¼ ak0 We will now proceed to describe our data fitting and L2Δσ −d c20 analysis procedure. To estimate quantities like U4 as δc20 ¼ ð80Þ accurately as possible, we use an iterative procedure to a20 2 determine which ensembles, labeled by ðμ0; λ0;LÞ,to Subsequently we adjust the range of data to be fit for each include in the fits to various moments of magnetization. moment of magnetization mn enforcing the data cuts: δak1, 014502-17 RICHARD C. BROWER et al. PHYS. REV. D 98, 014502 (2018) TABLE II. Best fit values for the parameters described in We see Oð10−3Þ agreement within the error estimates. Sec. IV B and Appendix B for data selection window with the Although even more stringent tests can be made, with the ratios in Eq. 80 bounded by w ¼ 0.03. development of faster parallel code and a more extensive use of FSS analysis, we feel this is substantial support for w 0.03 1 2 3 the convergence of our QFE method to the exact c ¼ = α3 −22.9ð4.7Þ × 10 minimal CFT. In the next section we give further tests for α 12 4 3 3 1010 5 . ð . Þ × the two-point and four-point correlation functions. a20 0.47895(11) a¯ 21 0.1873(41) V. CONFORMAL CORRELATOR ON S2 a40 −0.39006ð20Þ a¯ 41 −0.357ð12Þ With our results for the Binder cumulants, we are a60 1.3570(11) confident that the QFE lattice action on S2 has the correct a¯ 61 1.876(71) ¯ critical limit. Our analysis of magnetization moments and b21 −18ð2106Þ ¯ cumulants involves global averages over n-point correla- b41 −23ð2629Þ ¯ tors. Here we turn to correlators to examine more closely b61 115(13187) −19 9 3 1 other consequences of conformal symmetry and to provide c20 . ð . Þ more stringent tests for our QFE lattice action. We begin μ2 2 a 1.82241324(70) with the two-point functions S , and explain how, with μ2 b 4(274) rotational invariance, the exact analytic result can best be χ2=dof 1.001 tested against our numerical result via a Legendre expan- dof 1195 sion. We next move to the exact four-point correlator, which takes on a relatively simple form as a sum of Virasoro blocks. Finally we use this as a toy example to δbk1, δc20 ≤ w, where w is the width of the window, for all learn how to compute CFT data (dimensions and cou- k ≤ n. It makes sense that lower moments of magnetization plings) from the conformal block operator product expan- should have larger data selection regions since they vary sion (OPE). In particular, we extract the central charge c more slowly in the critical region relative to the higher from the energy momentum tensor contribution to the moments, as indicated in Fig. 12. We then refit the new data scalar four-point function. selection and reselect the data for the same w using the new fit values and then iterate until the process converges to a A. Two-point correlation functions stable data set for that w. We expect that fit parameters like In the continuum CFT on the Riemann sphere S2, 2 ak0 will have systematic errors of Oðw Þ due to higher the conformal two-point function, already mentioned in order terms in the Taylor series expansion of the moments Eq. (69),is of the free energy which we did not include in our fit. So, we would like to take w to be as small as possible but also gðθ12Þ¼hϕðrˆ1Þϕðrˆ2Þi have enough data left to adequately constrain all of the 1 1 important fit parameters, particularly the ak0. ¼ 2Δ ¼ Δ ; ð83Þ jrˆ1 − rˆ2j ð2 − 2 cos θ12Þ In Fig. 13 we see a subset of the data selected with w ¼ 0.03 and curves with error bands computed from the best fit 3 where rˆ1, rˆ2 are unit vectors in the embedding space R values of parameters shown in Table II. Using the values θ 0 03 defined earlier in Eq. (24) and 12 is the angle between them. shown in the table for w ¼ . , we get U4;cr ¼ The projection of the two-point function onto Legendre 0.85020ð58Þð90Þ and U6;cr ¼ 0.77193ð37Þð90Þ where the functions, Plðcos θ12Þ, can be computed analytically, first error is statistical from the fit and the second is Z systematic. 1 2 1=8 cont In summary, the comparison of our Monte Carlo estimate cl ¼ dz PlðzÞ −1 1 − z on the QFE S2 with the exact solution are the following: 8 8 24 408 680 22440 ⇒ 7;105;805;24955;64883;3049501; ð84Þ Monte Carlo Values∶ integrating over z ¼ cos θ12 for l ¼ 0; 1; 2; . U4;cr ¼ 0.85020ð58Þð90Þ U6;cr ¼ 0.77193ð37Þð90Þ In Fig. 14, we compare the analytic value to the moments ð81Þ of our correlation data. The fit is excellent showing only slight deviations at high l due to cutoff effects. Our ∶ simplicial calculation of the two-point function is made Analytic CFT Values 2 2 at a fixed value of μ0 closest to the pseudocritical values μ 0 8510207 63 0 7731441 213 a U4 ¼ . ð Þ U6 ¼ . ð Þð82Þ in Table II. In Fig. 14, the normalization of the simplicial 014502-18 LATTICE ϕ4 FIELD THEORY ON RIEMANN … PHYS. REV. D 98, 014502 (2018) ϕ4 theory on S2, both methods agree with the exact value Δσ ¼ 0.125 at the percent level. From our analysis of the two-point function, we have a direct demonstration of how rotational symmetry is restored in the continuum limit. Because the two-point function can be represented as an expansion in Legendre functions, it is a function of only the angle between any two points on the sphere and therefore rotationally invariant. For any fixed finite l, no matter how large, the simplicial coefficient clðLÞ converges to the continuum one as L → ∞. B. Four point functions The c ¼ 1=2 minimal CFT has only three Virasoro primaries 1, σ, ϵ, with an OPE expansion, σ × σ ¼ 1 þ ϵ; ϵ × σ ¼ ϵ; ϵ × ϵ ¼ 1: ð86Þ In our earlier paper [19], we constructed a simplicial Wilson representation of Dirac fermions S2. The map to the holomorphic, ψðzÞ, and antiholomorphic, ψ¯ ðz¯Þ, com- ponents of free Majorana fields on all 2-d Riemann surfaces [52] allowed us to define ϵ ¼ ψψ¯ and therefore compute the four-point functions: hϵ1ϵ2ϵ3ϵ4i and hσ1ϵ1σ2ϵ1i without Monte Carlo simulations. Here we test our QFE ϕ4 2 construction on S to compute the hσ1σ2σ3σ4i correlator. In the continuum limit the leading behavior of the lattice FIG. 14. On the top, we show the simplicial two-point functions 0 field is the primary operator: ϕ ∼ σðxÞþOð1=LΔσ −Δσ Þ. clðLÞ projected into Legendre coefficients for various values of x L, with the naïve dependence on l scaled out for clarity of The four-point function hϕðrˆ1Þϕðrˆ2Þϕðrˆ3Þϕðrˆ4Þi is a func- presentation. On the bottom, we show the relative difference tion invariant under any of the 24 permutations of the four between the simplicial and continuum Legendre coefficients as positions. This permutation invariance is easily enforced we take the continuum limit for fixed l. even on a finite lattice. In the continuum, the eight real coordinates can be reduced to five real coordinates using Legendre coefficients clðLÞ are chosen so that c0ðLÞ¼ rotational invariance alone. cont c0 ¼ 8=7. In the left panel, we see that for any fixed value Conformal symmetry allows for a further reduction to of L the difference between the simplicial and continuum four real coordinates, the two angular variables θ13 and θ24 values is small at small l but increases with increasing l.Yet defined as in Eq. (83), plus two real conformal cross ratios as L increases, all simplicial coefficients converge towards u and v their continuum values. The right panel shows how the ˆ − ˆ 2 ˆ − ˆ 2 simplicial coefficients behave at fixed l as we approach the ≡ jr1 r2j jr3 r4j 2 u 2 2 ¼jzj ; ð87Þ continuum limit. The scaling curves shown assume a naïve jrˆ1 − rˆ3j jrˆ2 − rˆ4j 1=L scaling of simplicial artifacts, which turns out to be a ˆ − ˆ 2 ˆ − ˆ 2 good description of the data. ≡ jr1 r4j jr3 r2j 1 − 2 v 2 2 ¼j zj ; ð88Þ More generally in a CFT on the sphere, we will fit the jrˆ1 − rˆ3j jrˆ2 − rˆ4j two-point function to determine the operator dimension, or equivalently a single complex variable z. Δσ. This can be done directly in coordinate space by fitting to Eq. (83) up to an overall normalization, or numerically to R ðw1 − w2Þðw3 − w4Þ cont Δ 1 2 1 − Δσ the Legendre coefficients, cl ð σÞ¼ −1 dzð =ð zÞÞ z ¼ ð89Þ ðw1 − w3Þðw2 − w4Þ PlðzÞ, which obey the closed form recursion relation, l − 1 Δσ 1 ðw2 − w3Þðw1 − w4Þ cont Δ þ cont cont Δ ⇒ 1 − cl ð σÞ¼ cl−1 ;c0 ð σÞ¼ : z ¼ ð90Þ l þ 1 − Δσ 1 − Δσ ðw1 − w3Þðw2 − w4Þ ð85Þ where w is the complex variable for the R2 prior to the cont 2 The overall normalization c0 is of course proportional to stereographic project to S in Eq. (67). In these variables, m2 defined in Eq. (62). With our current simulations of the four-point function can be written 014502-19 RICHARD C. BROWER et al. PHYS. REV. D 98, 014502 (2018) hϕ1ϕ2ϕ3ϕ4i GðzÞ¼ : ð91Þ hϕ1ϕ3ihϕ2ϕ4i where the dependence on θ13 and θ24 cancels in the ratio. While GðzÞ has the exchange symmetries in the t-channel (1 ↔ 3, 2 ↔ 4), combining this properly with the Möbius map z → 1 − z, z → 1=z gives 24 copies, transforming the blue fundamental zone on the left-hand side of Fig. 15 to the entire complex z plane. In the specific case of the c ¼ 1=2 minimal model, the explicit form of GðzÞ is known in polar coordinates z ¼ reiθ as Gðu;vÞ¼vΔgðu;vÞ 1 pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi ¼ ½j1 þ 1 − zjþj1 − 1 −zj 2jzj1=4j1 − zj1=4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi ¼ 2 þ 2 ð1 − zÞð1 − z¯Þ þ 2 zz¯: 2jzj1=4j1 − zj1=4 ð92Þ The first form is a sum of the two Virasoro blocks and the second form, given in Ref. [36], is the explicitly crossing symmetric form. For visualization, we plot GðzÞ in Fig. 15. Given that independent field configurations on our sim- plicial complexes can be generated in nearly OðL2Þ time, the dominant cost of computing the four-point function would naïvely be computing the product of four fields which scales as OðL8Þ and would be intractable for large L close to the continuum limit. We can reduce the cost by only sampling a subset of values of the four-point function on each indepen- dent configuration we generate. This is achieved by drawing OðL2Þ quartets of random points on the simplicial complex. This balances the computational cost equally between generating independent field configurations and sampling the four-point function. We then determine the product of the four simplicial fields, the unique coordinates ðθ13; θ24;zÞ under the permutations of the positions, and finally an estimate of GðzÞ by dividing our FIG. 15. We visualize the conformal function GðzÞ in contour and surface renderings. The outermost contour starts at GðzÞ¼ product of simplicial fields by the continuum two-point 0.6 and increases by steps of 0.1. Two symmetry planes are function, Eq. (69), which is justified by our success of the apparent: ReðzÞ¼1=2 and ImðzÞ¼0. These are related to previous section in showing that the simplicial two-point certain permutations of the four-point function. The entire set function converges correctly in the continuum limit, of permeations maps the blue zone exactly to the entire complex plane. hϕ1ϕ2ϕ3ϕ4i hϕ1ϕ2ϕ3ϕ4i GðzÞ¼ → as L → ∞: ð93Þ gðθ13Þgðθ24Þ hϕ1ϕ3ihϕ2ϕ4i We present a subset of our results for the conformal For convenience when studying the four-point function, we portion of the four-point function, Eq. (92), in Fig. 16. ∈ 0 1 θ 0 bin the data by coordinate on the complex plane. We partition Here, we consider the fixed slice r ½ ; for ¼ .We the unit disk jzj ≤ 1 on the complex plane into a radial grid see that, even for moderately small values of L, the of N2 bins of equal area. The angular width of each bin is measured four-point function converges well to the ana- 0 lytic, continuum result. Δθ ¼ 2π=N0 and we choose N0 mod 4 ¼ 0 so that the bins are centered at θn ¼ 0, π=2, π, 3π=2. This arrangement is ideal for using standard discrete cosine transforms (DCT-I) C. Operator product expansion for computing Fourier coefficients. For this study we chose In Sec. VB, we calculated the simplicial approximation N0 ¼ 64. to the scalar four-point function which, after taking a ratio 014502-20 LATTICE ϕ4 FIELD THEORY ON RIEMANN … PHYS. REV. D 98, 014502 (2018) FIG. 16. Results for the conformal part of the QFE scalar four- point function of the Ising universality class on S2. Gðr; θÞ should only depend on a single complex coordinate z ¼ reiθ. We have computed the four-point function in the entire complex plane in ϕ4 theory and show the function for r ∈ ½0; 1 at fixed θ ¼ 0 as it converges to the continuum result. with a product of two-point functions respecting t-channel symmetry, is described by a conformal function GðzÞ invariant under z → 1 − z and z → z¯. We computed the function everywhere in the unit circle, which can then be extended to the whole complex plane utilizing the permu- tation symmetry of the full four-point function. Our FIG. 17. On the top, a fit to the m ¼ 0 Fourier component of the representation of the function GðzÞ is a tabulation of values s-channel symmetric conformal part of the four-point function. on an equal area grid in polar coordinates ðr; θÞ. On the bottom, a fit to the m ¼ 2 Fourier component. A In the conformal bootstrap the expansion around z ¼ 0 simultaneous fit to both components can be used to fix the is used based on the operator product expansion (OPE). It is normalization of the four-point function and determine the central interesting to ask how well our Monte Carlo simulation on charge. S2 can determine the data in an OPE expansion, With a conserved energy momentum tensor T, there are X hϕ1ϕ2ϕ3ϕ4i special terms in the OPE expansion of the scalar four-point Δσ λ2 GsðzÞ¼ ¼jzj GðzÞ¼ OgΔO;lðzÞ; ð94Þ 2 gðθ12Þgðθ34Þ function which have integer powers: 1 þ λ g 2ðr; θÞ. The ΔO;l T T; identity operator term is normalized to unity by convention. where GsðzÞ is s-channel symmetric, i.e., symmetric under The conformal block for the energy momentum tensor has a interchange 1 ↔ 2 and 3 ↔ 4, ΔO are the scaling dimen- closed form expression, O sions of conformal primary operators , and l labels the 1 z spin. The functions gΔ ðzÞ are called conformal blocks g 2ðzÞ¼−3 1 þ 1 − logð1 −zÞ þ c:c: O;l T; z 2 whose functional form is completely determined by con- 2 1 1 9 formal symmetry. In d ¼ , there is an explicit representa- ¼ r2 cosð2θÞþ r3 cosð3θÞþ r4 cosð4θÞþ ; tion of the conformal blocks in terms of hypergeometric 2 2 20 functions, ð97Þ 2 1 ¯ and the OPE coefficient λ is related to the central charge of h ;2 ; ¯h ¯ ¯;2¯;¯ T gΔ;lðzÞ¼2 ½z 2F1ðh; h h zÞ ½z 2F1ðh; h h zÞ ; ð95Þ the CFT, 2 2 Δþl ¯ Δ−l 2 Δσd 1 where h ¼ 2 , h ¼ 2 . The expansion in Fourier modes is λ → d 2; T ¼ − 1 2 16 for ¼ ð98Þ given by CTðd Þ CT X∞ X 2 ¯ 2 nþn0 where C ¼ 2c ¼ 2ð1=2Þ¼1 for the d ¼ 2, c ¼ 1=2 ðhÞ ðhÞ 0 r T θ θ Δ n n gΔ;lðr; Þ¼ cosðm Þ r ¯ 0 ; minimal model. ð2hÞ ð2hÞ 0 n!n ! m¼0 jn−n0þlj¼m n n We can extract an estimate of the central charge c from ð96Þ our simplicial calculation of the scalar four-point function by modeling the result with the first few terms in the OPE Γ Γ using the Pochhammer symbol ðaÞn ¼ ða þ nÞ= ðaÞ. expansion 014502-21 RICHARD C. BROWER et al. PHYS. REV. D 98, 014502 (2018) TABLE III. A summary of the simultaneous fits to the m ¼ 0, 1, 2, 3 Fourier components of the OPE expansion parameterized in Eq. (97) for the s-channel symmetric conformal four-point function. The normalization (norm) fixes the vacuum term to be 1, consistent in the continuum limit with the normalization in Eq. (83) of the two point function. 2 2 μ Lrmin ≤ r ≤ rmax Norm Δϵ λϵ c 1.82241 9 0.25 ≤ r ≤ 0.75 0.2900 1.075 0.2536 0.4668 1.82241 9 0.30 ≤ r ≤ 0.70 0.2901 1.075 0.2533 0.4704 1.82241 9 0.35 ≤ r ≤ 0.65 0.2902 1.077 0.2533 0.4738 1.82241 9 0.40 ≤ r ≤ 0.60 0.2902 1.016 0.2427 0.4747 1.82241 18 0.25 ≤ r ≤ 0.75 0.2051 1.068 0.2563 0.4866 1.82241 18 0.30 ≤ r ≤ 0.70 0.2051 1.056 0.2544 0.4878 1.82241 18 0.35 ≤ r ≤ 0.65 0.2051 1.050 0.2535 0.4904 1.82241 18 0.40 ≤ r ≤ 0.60 0.2051 1.046 0.2526 0.4884 1.82241 36 0.25 ≤ r ≤ 0.75 0.1457 1.031 0.2528 0.4926 1.82241 36 0.30 ≤ r ≤ 0.70 0.1458 1.026 0.2519 0.4932 1.82241 36 0.35 ≤ r ≤ 0.65 0.1458 1.018 0.2508 0.4931 1.82241 36 0.40 ≤ r ≤ 0.60 0.1458 1.007 0.2486 0.4933 θ ∝ 1 λ2 θ λ2 θ Gsðr; Þ þ ϵgϵ;0ðr; Þþ TgT;2ðr; Þ: ð99Þ to convergence to the classical limit of any renormalizable quantum field theory on a smooth Euclidean Riemann We used a DCT-I to Fourier transform the data, simulta- manifold [19]. A rather nontrivial extension of the classical neously fitting m ¼ 0, 2 to four parameters: the overall finite element/DEC method for the lattice Dirac fermion and λ2 λ2 Δ normalization, ϵ , T and ϵ.Wechooseasinglevalueof non-Abelian gauge fields on a simplicial Riemannian mani- μ2 ¼ 1.822410 which is close to the pseudocritical value, fold is provided in Ref. [19] and Ref. [22] respectively. and we restrict the fitting range in r around 0.5 where the The local spin and gauge invariance on each site is achieved data shows the least discretization error. A fit for L ¼ 36 and by compact spin connections and compact gauge links. This 0.25 ≤ r ≤ 0.75 is shown in Fig. 17.Asummaryofseveral we believe provides the basic tools for classical simplicial fits is shown in Table III. As expected, the fit values converge constructions. 2 towards the continuum CFT results ðΔϵ ¼ 1; λϵ ¼ 1=4;c¼ Second, and most importantly, we show that reaching the 1=2Þ as L increases toward the continuum limit and as the fit continuum for a quantum FEM Lagrangian requires a range in r is confined to a narrower region around r ¼ 0.5 modification of the action by counterterms to accommodate where the discretization artifacts appear smallest. UV effects. We conjecture that for a superrenormalizable theory with a finite number of divergent diagrams, a new VI. CONCLUSION QFE lattice action can be constructed by a natural gener- 4 We have set up the basic formalism for defining a scalar alization of ϕ theory example presented here that removes field theory on nontrivial Riemann manifolds and tested it the scheme dependence of the FEM simplicial lattice UV numerically in the limited context of the 2-d ϕ4 theory by cutoff. The detailed generalization of this conjecture needs comparing it against the c ¼ 1=2 Ising CFT at the Wilson- to be investigated with many more examples and most Fisher fixed point. This test is at a sufficient accuracy likely alternative approaches. compared with the exact result to encourage us that we are Third, once we have successfully constructed the QFE able to define this strongly coupled quantum field theory on action with UV counterterms consistent with the continuum a curved manifold, in this case S2. Faster parallel code limit to all orders in perturbation theory, we conjecture that it could push these tests to much higher precision. While this represents a correct formulation of the nonperturbative may be pursued in future works, our goal here was to sketch quantum field theory on a curved Riemann manifold. This a framework for lattice quantum field theories on a non- parallels the conventional wisdom for LFTs on hypercubic trivial Riemann manifold as a quantum extension of FEM lattices that when lattice symmetries are protected (Lorentz, which we refer to as quantum finite elements (QFE). chiral, SUSY) only a finite set of fine tuning parameters are Whether or not this has generally applicability remains, of need to reach the continuum theory. Further research is need course, to be proven. to establish the range of applicability of our QFE proposal. Our construction first relies on theoretical issues in the More examples are obviously need to support this proposal. application of FEM that, in spite of the vast literature, are not There are many more direct extensions of this formu- to our knowledge proven in enough generality. Nonetheless, lation that we are currently entertaining. The first is to apply the documented experience with FEM for PDEs does argue QFE to the 3-d ϕ4 theory in both radial quantization and the that this framework likely extends when properly formulated 3-d projective sphere, S3. Our current software relies on a 014502-22 LATTICE ϕ4 FIELD THEORY ON RIEMANN … PHYS. REV. D 98, 014502 (2018) serial code and the very efficient Brower-Tamayo [32] X∞ Xl ˆ ˆ θ μ Ylmðr1ÞYlmðr2Þ modification of the cluster algorithm. We are in the process Gð ; Þ¼ 2 lðl þ 1Þþ1=4 þ μ of developing parallel code that will allow for large lattices l¼0 m¼−l X∞ and higher precision, which will also be a necessity in 1 ð2l þ 1ÞP ðcos θÞ l ; going beyond 2d and scalar fields. Although new software ¼ 4π 1 2 2 μ2 ðA1Þ 0 ðl þ = Þ þ requires substantial effort, we believe that all of the l¼ fundamental data parallel concepts and advanced algo- where rˆ1 · rˆ2 ¼ cosðθ12Þ¼z. For generality we have added rithms utilized in lattice QCD will still be applicable here. 2 2 a dimensionless “mass” term m0 ¼ μ þ 1=4 to regulate We note there are advantages to studying CFTs on spherical the IR. The series can also be summed to get Sd simplicial lattices. There are no finite volume approx- imations. The entire Rd is mapped to the sphere. For radial 1 θ μ quantization the only finite volume effect is the one radial Gð ; Þ¼4π ½Q−1=2þiμðzÞþQ−1=2−iμðzÞ : ðA2Þ dimension along the R × Sd−1 cylinder. With periodic boundary conditions, the finite extent of the cylinder is in term of associated Legendre functions. This Green’s also an interesting parameter for the study of CFTs at finite function (A2) can also be written in compact integral temperature. Radial quantization on R × Sd−1 allows direct representation, access to the Dilatation operator to compute operator Z 1 ∞ dimensions and the OPE expansion. Even our 2-d example θ μ iμt θ Gð ; Þ¼4π dte Gðt; Þ is worth further investigation at higher precision, and with Z−∞ an emphasis on new studies enabled by a fully non- 1 ∞ cosðμtÞdt ϕ4 S2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ðA3Þ perturbative formulation of the theory on .For 2π 0 2 coshðtÞ − 2 cosðθÞ example, we can consider new ways to compute the renormalization flow of the central charge from the UV exhibiting clearly the logarithmic singularity as 1 − z → to the IR. Further research is on going to hopefully justify θ2=2. Expanding the integral we find, this optimism. 1 32 Gðθ; μÞ ≃ log − 14ζð3Þμ2 þ Oðθ2; μ4Þ; ACKNOWLEDGMENTS 4π 1 − cosðθÞ We would like to thank Steven Avery, Peter Boyle, ðA4Þ Norman Christ, Luigi Del Debbio, Martin Lüscher, Ami Katz, Zuhair Kandker and Matt Walters for valuable dis- in agreement with Eq. (57) in Sec. III C. cussions. R. C. B. and G. T. F. would like to thank the Aspen b. Conformal propagator on a cylinder: Next, we Center and Kavali Institutes for their hospitality and R. C. B. consider the 3-d conformal propagator for radial quantiza- thanks the Galileo Gallilei Institute’s CFT workshop and the tion on R × S2, Higgs Centre for their hospitality during the completion of this manuscript. G. T. F. and A. D. G. acknowledges support Gðt; θÞ¼hϕðx1Þϕðx2Þi under Contract No. DE-SC0014664 and thanks Lawrence Δ 1 Livermore National Laboratory and Lawrence Berkeley ¼ðr1r2Þ 2Δ jx1 − x2j National Laboratory for hospitality during the completion 1 of this work. R. C. B. and E. S. W. acknowledge support by ¼ Δ DOE Grant No. DE-SC0015845 and T. R. and C.-I T. in part ½r1=r2 þ r2=r1 − 2 cosðθÞ by the Department of Energy under Contract No. DE- 1 ¼ ; ðA5Þ SC0010010-Task-A. T. R. is also supported by the ½2 coshðtÞ − 2 cosðθÞ Δ University of Kansas Foundation Distinguished Professor Grant of Christophe Royon. where t ¼ t2 − t1 ¼ logðr2=r1Þ. Note by setting t ¼ 0, this is exactly the conformal two point function on the sphere Eq. (69) as expected by dimensional reduction. (In the rest APPENDIX A: FREE THEORY ON SPHERE of the Appendix, we also drop the factor of 1= 4π to AND CYLINDER ð Þ coincide with the normalization convention in Sec. V.) Here we collect a few analytic expressions for the free However, if we return to the 3-d free theory, setting scalar theory that are useful in establishing convergence of Δ ¼ðd − 2Þ=2 ¼ 1=2, the dimensional reduction by set- 2 the simplicial propagators to the exact continuum on the S ting t ¼ 0 in the 3-d conformal propagator fails to give the and R × S2 manifolds. 2-d propagator. To understand this we recall that in 2d, a. Free scalar on a sphere: The Green’s function on the the scalar field, ϕðxÞ is not a primary operator. As is well sphere is given by spectral sum, known in string theory, conformal primaries are given by 014502-23 RICHARD C. BROWER et al. PHYS. REV. D 98, 014502 (2018) vertex operators : exp½ikϕ ∶. Instead, to reach the free Fðgσ;gϵ; fgω; g;aÞ¼log Z; ðB1Þ Green’s function from the cylinder, we proceed as follows. Z R The multipole expansion of the Coulomb term is − d ϕ Z ¼ ½Dϕ e Sþh d x ðxÞ ðB2Þ 1 1 X lþ1=2 ¼ pffiffiffiffiffiffiffiffiffi ðr1=r2Þ P ðcos θÞðA6Þ − l μ2 λ jx1 x2j r1r2 l expressed not as a function of the bare couplings h, 0, 0 but rather the relevant couplings gσ, gλ and the irrelevant for r2 >r1 or r2 >r1, after interchanging r1 and r2. Thus couplings fgω; g of the Wilson-Fisher fixed point. For the Green’s function is the lattice spacing, we assume a ∼ 1=L. Under a change of X scale by a factor l, the free energy renormalizes −ðlþ1=2Þt ðlþ1=2Þt Gðt; θÞ¼ ½θðtÞe þ θð−tÞe Plðcos θÞ l Fðgσ;gϵ; fgω; g;aÞ −d yσ yϵ yω ðA7Þ ¼ l Fðl gσ;l gϵ; fl gω; g;laÞþGðgσ;gϵÞðB3Þ pffiffiffiffiffiffiffiffiffi rescaling by r1r2 as in Eq. (A5) above. In Fourier space where yO ≡ d − ΔO, F is the singular, or scaling, part of this becomes the free energy and G is the regular part. In what follows, Z Δ ∞ we will continue to use yO instead of O for compactness ˜ iωt Gðω; θÞ¼ dte Gðt; θÞ of notation. pffiffiffiffiffiffi −∞ L ≡ d Ω Ω Z If we choose d, where d is the volume of the ∞ X iωt −ðlþ1=2Þt d-dimensional manifold, and differentiate the free energy k ¼ dte e Plðcos θÞ 0 times with respect to gσ and then take the limit gσ → 0 we Z l 0 X get iωt ðlþ1=2Þt þ dte e Plðcos θÞ; ðA8Þ −∞ ∂kF l F ðkÞ 1 L k ¼ ðgϵ; fgω; g; = Þ ∂gσ so Lkyσ−d ðkÞ Lyϵ Lyω 1 ðkÞ ¼ F ð gϵ; f gω; g; ÞþG ðgϵÞ: X∞ 1 1 G˜ ðω; θÞ¼ þ P ðcos θÞ ðB4Þ l 1=2 iω l 1=2 − iω l l¼0 þ þ þ X We compute derivatives of the free energy with respect to ð2l þ 1ÞP ðcos θÞ ¼ l : ðA9Þ the bare parameter h and then take the limit h → 0, to get l 1=2 2 ω2 l ð þ Þ þ cumulants of the magnetization 2 2 By fixing the frequency ω ¼ μ we regain the free propa- 1 ∂kF κ gator on the sphere. We take the value μ ¼ 0 as our standard k ¼ lim −1 ; ðB5Þ h→0 Ωk ∂hk IR regulated propagator which has an appealing interpreta- d tion as constant line source in 3d. Of course in practice on and put them in the form of Binder cumulants, the lattice for our 3-d simulations, we will introduce a finite cylinder of length T ¼ aL , with periodic boundary con- κ t ∝ 2n ditions. This corresponds to the thermal propagators, with U2n lim n ; ðB6Þ h→0 κ2 periodic t ∈ ½0;T represented by a discrete sum, Z ω 1 X with the normalization to be determined as described in d −iωt ˜ −iω t ˜ ϕ → −ϕ Gðt; θÞ¼ e Gðω; θÞ → e n Gðω ; θÞ Sec. IVA. We note that the symmetry guarantees 2π T n κ 0 n that all odd cumulants vanish, 2nþ1 ¼ and that h is an ðA10Þ odd-function of gσ. To take advantage of the RG scaling properties of the free energy near the critical point, we must over frequencies, ω ¼ 2πn=T. rewrite the cumulants carefully using the chain rule. To n α make the following manageable, we define 2nþ1 and understand that all expressions are in the limit gσ, h → 0, APPENDIX B: FINITE SIZE SCALING FOR CUMULANTS AND MOMENTS ∂2nþ1 α ≡ gσ 2nþ1 lim 2n 1 ; ðB7Þ This follows the renormalization group arguments first h→0 ∂h þ presented in [44] and reviewed in [51]. We begin with the L2Δσ κ α2 ð2Þ L−dþ2Δσ ð2Þ free energy Eq. (64), 2 ¼ 1½F þ G ; ðB8Þ 014502-24 LATTICE ϕ4 FIELD THEORY ON RIEMANN … PHYS. REV. D 98, 014502 (2018) L4Δσ κ α4 ð4Þ L−3dþ4Δσ ð4Þ ðkÞ 4 ¼ 1½F þ G G ¼ ck0 þ ck1ðgϵ − gϵ Þþ ðB13Þ −2dþ2Δσ ð2Þ −3dþ4Δσ ð2Þ þ α1α3½L F þ L G ; ðB9Þ Substituting these expressions into those for the cumulants 6Δσ 6 ð6Þ −5dþ6Δσ ð6Þ defined in Eqs. (B8)–(B11) nearly gives us an expression L κ6 ¼ α1½F þ L G we can fit to computed data. Of course, we cannot directly α3α L−2dþ2Δσ ð4Þ L−5dþ6Δσ ð4Þ þ 1 3½ F þ G vary the renormalized couplings ðgσ;gϵ;gω; Þbut instead 2 2 −4dþ4Δσ ð2Þ −5dþ6Δσ ð2Þ μ λ þðα3 þ 3α1α5Þ½L F þ L G can vary our bare couplings ðh; 0; 0Þ defined in our cutoff theory. Near the Wilson-Fisher fixed point we can relate the ðB10Þ two sets of couplings by expanding L8Δσ κ α8 ð8Þ L−7dþ8Δσ ð8Þ 8 ¼ 1½F þ G − μ2 − μ2 gϵ gϵ Rϵμ Rϵλ 0 5 −2dþ2Δσ ð6Þ −7dþ8Δσ ð6Þ ¼ þ ðB14Þ þ56α1α3½L F þL G − λ − λ gω gω Rωμ Rωλ 0 2 2 3 −4dþ4Δσ ð4Þ −7dþ8Δσ ð4Þ þð280α1α3 þ56α1α5Þ½L F þL G So, substituting Eq. (73) first into Eq. (B12)–(B13) leads to 56α α 8α α L−6dþ6Δσ ð2Þ L−7dþ8Δσ ð2Þ þð 3 5 þ 1 7Þ½ F þ G expressions that can be used to model cumulant data ðB11Þ computed in a cutoff theory close to the Wilson-Fisher fixed point. At this point, the pattern is clear and the only difficulty Often it is more convenient in the cutoff theory to extending the calculation to higher cumulants is computing compute moments instead of cumulants. The expansion the associated chain rule factors which are relatively straight- of moments in terms of cumulants is well known so we only forward to compute using a program like Mathematica. write the first few even moments here For example, D[f[g[h]],{h,8}] will give all the terms 8 8 needed for ∂ F=∂h , remembering to set to zero odd m2 ¼ κ2 ðB15Þ derivatives of F and even derivatives of g. Before we continue, we point out an important physical 2 m4 ¼ κ4 þ 3κ2 ðB16Þ concept. At finite L, the difference between the lattice scalar ϕ σ operator and the conformal primary operator is indicated κ 15κ κ 15κ3 α m6 ¼ 6 þ 4 2 þ 2 ðB17Þ by the presence of higher derivative terms 2nþ1 in the cumulant expressions. But, in the scaling limit L → ∞ all 2 2 4 those terms vanish and the cumulants are dominated by the m8 ¼ κ8 þ 28κ6κ2 þ 35κ4 þ 210κ4κ2 þ 105κ2 ðB18Þ moments of the free energy with respect to the coupling gσ σ 2 of the conformal primary operator . In this sense, the m10 ¼ κ10 þ 45κ8κ2 þ 210κ6κ4 þ 630κ6κ2 simplicial operator ϕðxÞ becomes the primary operator 1575κ2κ 3150κ κ3 945κ5 σðxÞ in the scaling limit. þ 4 2 þ 4 2 þ 2 ðB19Þ We can now apply the important physical principle that 2 close to the critical point we can expand the derivatives of m12 ¼ κ12 þ 66κ10κ2 þ 495κ8κ4 þ 1485κ8κ2 the free energy in a Taylor series 2 3 þ 462κ6 þ 13860κ6κ4κ2 þ 13860κ6κ2 ðkÞ yϵ 3 2 2 4 6 F ¼ ak0 þ ak1ðgϵ − gϵÞL þ þ 5775κ4 þ 51975κ4κ2 þ 51975κ4κ2 þ 10395κ2: yω þ bk1ðgω − gωÞL þ : ðB12Þ ðB20Þ [1] T. Appelquist, R. Brower, S. Catterall, G. Fleming, J. Giedt, [4] T. Appelquist, R. C. Brower, G. T. Fleming, A. Hasenfratz, A. Hasenfratz, J. Kuti, E. Neil, and D. Schaich, arXiv:1309 X.-Y. Jin, J. Kiskis, E. 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