QED 3 -inspired three-dimensional conformal lattice gauge theory without fine-tuning
QQED -inspired three-dimensional conformal lattice gauge theory without fine-tuning Nikhil Karthik ∗ and Rajamani Narayanan † Physics Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA Department of Physics, Florida International University, Miami, FL 33199 (Dated: December 3, 2020)We construct a conformal lattice theory with only gauge degrees of freedom based on the inducednon-local gauge action in QED coupled to large number of flavors N of massless two-componentDirac fermions. This lattice system displays signatures of criticality in gauge observables, withoutany fine-tuning of couplings and can be studied without Monte Carlo critical slow-down. By couplingexactly massless fermion sources to the lattice gauge model, we demonstrate that non-trivial anoma-lous dimensions are induced in fermion bilinears depending on the dimensionless electric charge ofthe fermion. We present a proof-of-principle lattice computation of the Wilson-coefficients of variousfermion bilinear three-point functions. Finally, by mapping the charge q of fermion in the modelto a flavor N in massless QED , we point to an universality in low-lying Dirac spectrum and anevidence of self-duality of N = 2 QED . Introduction. –
Extraction of conformal field theory(CFT) data plays an important role in our understand-ing of critical phenomena. An important set of confor-mal data are the scaling dimensions of operators thatclassify the relevant and irrelevant operators in a CFT.This data can be used to abstract the source of dynami-cal scale-breaking in the long-distance limit of quantumfield theories in terms of few symmetry-breaking opera-tors that turn relevant. The operator product expansion(OPE) coefficients in the CFT correlation functions areanother set of highly constrained conformal data. Theformal structure of CFT and its data has been exploredover decades and one can refer [1] for a survey of the sub-ject; [2] for a discussion not restricted to two dimensionsand [3–5] for recent developments in dimensions greaterthan two. Monte Carlo (MC) studies of strongly inter-acting CFTs are difficult owing to a combined effect ofthe required precise tuning of couplings, an increase inMC auto-correlation time closer to a critical point andthe need for large system sizes. Notwithstanding suchdifficulties, the CFT data in many bosonic spin systemshave been extracted from traditional MC (e.g., [6, 7]for recent determinations in 3d O( N ) models) as wellas using radial lattice quantization [8–10]. At present,however, three-dimensional fermionic CFTs have been ofgreat interest, particularly owing to recent works relatedto dualities [11–13], and therefore, MC based search forthree-dimensional fermionic CFTs (such as, [14–19]) is ofparamount importance.One such three-dimensional interacting fermionic CFTis approached in the infrared limit of the parity-invariantnoncompact quantum electrodynamics (QED ) with N (even) flavors of massless two-component Dirac fermionsin the limit of large- N ; to leading order, the effect offermion is to convert the p − Maxwell photon propaga-tor into a conformal 16(
N g p ) − photon propagator [20] ∗ [email protected] † rajamani.narayanan@fiu.edu in the limit of small momentum p , where g is the di-mensionful Maxwell coupling. This suggests replacingthe usual Maxwell action for the gauge field A µ by aconformal gauge action [21] S g = 1 q (cid:90) d p (2 π ) A µ ( p ) (cid:18) p δ µν − p µ p ν p (cid:19) A ν ( − p ) , (1)with a dimensionless coupling q ( N ) = 32 /N for large- N ,thereby obtaining results consistent with an interactingconformal field theory in a N expansion. The conformalnature of the above quadratic action can be seen in thetensorial structure of n -point functions of field strength F µν that is consistent with conformal symmetry [21, 22].Since the dimension of F µν is fixed by gauge-invariance,it is only for the 1 /p kernel of the above quadratic action,the coupling becomes dimensionless in three-dimensions.Both [20, 21] approaches are consistent with a scale in-variant field theory only if N is above some critical valuebut recent numerical analyses [23, 24] of QED haveshown that the theory likely remains scale- (or conformal-) invariant all the way down to the minimum N = 2. Thissuggests that the induced gauge action from the fermionis conformal for any non-zero N , and it might be possibleto capture many aspects of the infrared physics of QED by appropriately modeling this induced non-local action— we do so by using the quadratic conformal gauge ac-tion Eq. (1), however with an otherwise unknown q - N relation, q ( N ), which for general N needs to be deter-mined from first principles, and assuming that effect ofterms in the induced-action which are higher-order ingauge field are negligible. This motivated us to considerthe action in Eq. (1) in its own right as an interactingCFT for any q obtained without tuning any couplings,and probed by massless spectator fermions. It is the pri-mary aim of this letter to use a lattice regularization ofEq. (1) and show that this CFT induces non-trivial con-formal data in fermionic observables depending on thevalue of q , thereby making it a powerful model systemfor lattice studies of fermion CFTs. Finally, we will closethe loop and demonstrate numerically that this confor-mal gauge theory for arbitrary q probed by spectator a r X i v : . [ h e p - l a t ] D ec fermions can describe universal features in a correspond-ing N -flavor QED . The model and signatures of its criticality in pure-gauge observables –
The noncompact U(1) lattice gaugemodel we consider is the regularized version of Eq. (1)on L periodic lattice, given by Z = (cid:32)(cid:89) x,µ (cid:90) ∞−∞ dθ µ ( x ) (cid:33) e − S g ( θ ) , with S g = 12 (cid:88) µ,ν =1 (cid:88) x,y F µν ( x ) (cid:3) − / ( x, y ) F µν ( y ) , (2)where θ µ ( x ) are real-valued gauge fields that reside onthe links connecting site x to x + ˆ µ , with a field strength F µν = ∆ µ θ ν ( x ) − ∆ ν θ µ ( x ) where ∆ µ is the discrete for-ward derivative. The three-dimensional discrete Lapla-cian is (cid:3) = (cid:80) µ ∆ µ ∆ † µ . The model lacks any tunabledimensionful parameter at the cost of being non-local,which is not a hindrance for a numerical study; a MCsampling of the gauge fields weighted by Eq. (2) becomessimple in the Fourier basis where the Laplacian is diag-onalized and the modes are decoupled. We absorbed thefundamental real-valued charge q in Eq. (1) in a redef-inition of gauge fields when defining the parameterlesslattice model, and hence the observables will couple togauge fields as qθ µ ( x ), or integer multiples thereof. Wediscuss further details of the model and the algorithm inthe Supplementary Material, which includes Refs. [25–30].The absence of tunable parameters in the lattice ac-tion by itself is not an indication of it being critical. Astrong evidence of the scale invariant behavior was seenin the sole dependence on aspect-ratio ζ = l/t of all l × t Wilson loops, W ( qθ ), after a simple perimeter term isremoved. The asymptotic behavior [31] is characterizedby νζ as ζ → ∞ and νζ for ζ → ν = − . q that should be universal for all theoriesapproaching this CFT, such as QED (refer Supplemen-tary Material). Another interesting pure-gauge observ-able is the topological current, V top µ ≡ q π (cid:80) νρ (cid:15) µνρ F νρ ,which is trivially conserved in this noncompact U(1) the-ory. We also checked that its two point function for1 (cid:28) | x | (cid:28) L/ (cid:80) µ (cid:10) V top µ (0) V top µ ( x ) (cid:11) = C top V | x | − , with the coefficient C top V = q π as expected from the continuum regulatedcalculation [32–34]. The trivial q dependence of confor-mal data in pure-gauge observables becomes nontrivialin gauge invariant observables formed out of spectatormassless fermions. Conformal data in fermionic observables. –
The lat-tice model per se does not have dynamical fermions.But, one can couple spectator massless fermion sourcesto the model in order to construct a variety of gauge-invariant hadronic correlation functions. Formally, thesource term for a pair of parity-conjugate Dirac fermionsis ¯ ψ + q G q ψ + q − ¯ ψ − q G q ψ − q , where G q is the exactly masslessoverlap lattice fermion propagator [24, 35, 36] coupled − − . − − . − − .
501 2 3 4 00 . .
52 2 . . . . . l og h ˜ Λ ( q ) i log( L )Charge qq = 0 . q = 1 . q = 1 . q = 2 . q = 2 . q = 3 . l og h ˜ G ( ) S + S − ( L / ) i log( L ) FIG. 1. Mass anomalous dimension as computed at differentcharges q . Left: The dependence of smallest Dirac eigenvalue˜Λ ( q ), normalized by free theory value, on L . The curves arethe fits to extract the leading L − γ S dependence. Right: Thefinite size-scaling of the scalar two-point function ˜ G ( | x | ) atseparations | x | = L/
4. The lines are the expected asymp-totic dependence ˜ G ( | x | = L/ ∼ L γ S at different q , with γ S determined from ˜Λ n . to the gauge-fields through the gauge-links e iqθ µ ( x ) (seeSupplementary Material for the implementation of over-lap Dirac operator, which includes Refs. [37, 38]). Theflavor-triplet fermion bilinears are defined by taking ap-propriate derivatives O ± ( x ; q ) = (cid:18) ∂∂ ¯ ψ ± q Γ ∂∂ψ ∓ q (cid:19) ( x ); O ( x ; q ) = 1 √ (cid:18) ∂∂ ¯ ψ + q Γ ∂∂ψ + q + ∂∂ ¯ ψ − q Γ ∂∂ψ − q (cid:19) ( x ) , (3)of the effective action; Γ = 1 for scalar bilinear, S ± , ,and Pauli matrices Γ = σ µ for the conserved vector bi-linears, V ± , µ . Practically, this procedure is equivalentto a prescription of replacing fermion lines with mass-less fermion propagators to form gauge-invariant observ-ables. We also imposed anti-periodic boundary condi-tions on fermion sources in all three directions which issymmetric under both lattice rotation and charge con-jugation while removing the issue of trivial Dirac zeromodes present even in the free field q = 0 limit. We willdenote the n point functions formed out of these fermionbilinears by G ( n ) ( x ij ; q ) and the dependence on the x ij ,the separation between the location of the i th and j th bilinears should match the structure deduced from con-formal symmetry. Since we are only interested in changesto observables from free-field theory, we form the ratios˜ G ( n ) ( x ij ; q ) = G ( n ) ( x ij ; q ) /G ( n ) ( x ij ; 0), which we hence-forth refer to as reduced n -point functions; this also helpsdecrease any finite-size and short-distance lattice effectsthat are already present in the free-field case.We define scaling dimensions ∆ i = 2 − γ i govern-ing the scaling ˜ G (2) O i O i ( x ) = C i | x | γ i for distanceslarger than few lattice spacings. The scaling dimen-sion ∆ S ( q ) = 2 − γ S ( q ) of S ± , is an example of non-trivial conformal data that is induced in this model. . . . . . . .
10 5 10 15 20 25 30˜ C V +3 V − V ˜ C V +1 V − V ˜ C S + S − V q = 1 . ˜ C i j k ( z ; z ) z FIG. 2. Left: A configuration of collinearly placed operators.Right: The effective OPE coefficients ˜ C ijk ( z , z ; q ) of threedifferent collinear three-point functions (distinguished by col-ors and slightly displaced) are shown as a function of z atthree different fixed z =6 (open triangles), 8 (open circles),10 (filled circles). The q -dependent non-zero γ S can be obtained from thefinite-size scaling (FSS) of the scalar two-point function,˜ G (2) S + S − ( | x | = ρL ) = L γ S ( g ( ρ ) + O (1 /L )) at fixed ρ .The data for log (cid:104) ˜ G (2) S + S − (cid:105) at ρ = 1 / L ) using values of q ranging from q = 0 . L ) dependence (which is 2 γ S ) increasesmonotonically from 0 when q is increased. Better esti-mates of γ S ( q ) were obtained by studying the FSS of thelow-lying discrete overlap-Dirac eigenvalues Λ j ( L ; q ), sat-isfying G − q v j = − Λ j v j ; the FSS, Λ j ( L ; q ) ∝ L − − γ S ( q ) , is a consequence of the FSS of the scalar susceptibility.In the left panel of Fig. 1, we show the reduced eigen-values, ˜Λ j ( L ; q ) ≡ Λ j ( L ; q ) / Λ j ( L ; 0) for j = 1 as a func-tion of L along with curves from combined fits using afunctional form ˜Λ j ( L ; q ) = a j L − γ S (1 + (cid:80) k b jk L − k ) tofirst five ˜Λ j using data from L = 6 up to L = 36 (re-fer Supplementary Material, which includes Ref. [39]).Such a functional form with leading scaling behavior andsubleading scaling corrections nicely describes the dataand leads to precise estimates of γ S ( q ) that increasescontinuously from γ S = 0 to O (1) in the vicinity of q ≈
2; this dependence is captured to a good accuracy by γ S ( q ) = 0 . q + 0 . q + O ( q ), over this en-tire range of q . For some charge q = q c ≈ .
9, the valueof γ S becomes greater than 1.5, which is the unitaritybound on scalars in a three-dimensional CFTs (c.f., [4]);therefore, within the framework of constructing fermionicobservables in this pure gauge theory, we need to restrictourselves to values of q < q c to be consistent with beingan observable in a CFT. Unlike the scalar bilinear, V aµ isconserved current and hence, does not acquire an anoma-lous dimension. Therefore, the only non-trivial confor-mal data is the two-point function amplitude, C V ( q ) = (cid:80) µ =1 ˜ G (2) V aµ V aµ ( | x | ; q ) that we were able to obtain from theplateau in the reduced vector two-point correlator as afunction of separations, 0 (cid:28) | x | (cid:28) L/ q -dependence can be parameterizedas 4 π C V ( q ) = 1 − . q + 0 . q + O ( q ). In order to demonstrate further the efficacy ofthe model as a CFT with non-trivial conformaldata in the massless spectator fermion observablesthat is tractable numerically on the lattice, we alsopresent a proof-of-principle computation of the OPEcoefficients ˜ C ijk ( q ) of the reduced three-point func-tions ˜ G (3) O O O ( x , x , x ; q ) when three operators liecollinearly, that is, x = (0 , , x = (0 , , z ) and x = (0 , , z + z ) as described in the left panel of Fig. 2.We looked at three distinct three-point functions, chosenso as to reduce finite size effects, and whose dependencesare fixed by conformal invariance [2] to be˜ G (3) V + µ V − µ V ( z , z ) = ˜ C V + µ V − µ V ; µ = µ ⊥ (= 1 ,
2) or 3 , ˜ G (3) S + S − V ( z , z ) = ˜ C S + S − V z γ S , (4)when 0 (cid:28) z , z , z + z (cid:28) L/ z and z dependent OPE coefficientswhich will display a plateau as a function of z , z pro-vided the theory is a CFT. In the right part of Fig. 2,we show the three effective OPE coefficients as a func-tion of z at three different fixed z (= 6 , ,
10) as deter-mined on 64 lattice using q = 1 .
5. The plot demon-strates the independence of the three coefficients on z by a plateau over a wide range of z that is not too smallor too large. It also demonstrates their independence on z since the data from three different intermediate val-ues of z are consistent, with this being quite non-trivialespecially for ˜ C S + S − V as it comes from a cancellationwith a factor z γ S . The conformal symmetry in generalallows non-degenerate OPE coefficients ˜ C V +3 V − V = a + bb and ˜ C V + µ ⊥ V − µ ⊥ V = bb , with a = 0 , b = b in free theory.From Fig. 2, it is evident that a (cid:54) = 0 and b (cid:54) = b , clearlyindicating that the result is for an interacting CFT. Relevance of the model to QED . – We will show a cor-respondence between the behavior of the CFT at one par-ticular q and QED with N flavors of massless two com-ponent fermions. Our surprising observation for whichwe will present empirical evidences is that, for any finite N , as long as QED flows to an infrared fixed point, thedominant effect of fermion determinant in QED path-integral is to induce a non-local quadratic conformal ac-tion for the gauge fields with a coupling q = Q ( N ) forsome function Q that has to be determined ab initio ,with the only condition being Q ( N ) ∼ (cid:112) /N for large values of N . That is, if the map Q ( N ) is known for all N , then one can study universal features of the N -flavorQED by studying the same properties in the confor-mal lattice model at the corresponding q = Q ( N ) withnon-dynamical massless fermion sources, whose purposeis simply to aid the construction of fermionic n -pointfunctions. In order to find Q ( N ), we propose to mapvalues of q in the lattice model to N in QED suchthat the values of scalar anomalous dimensions γ S , deter-mined non-perturbatively in both theories, are the same.Such an identification of q and N is made in the bot- . . . . . .
750 0 . . . N = 2 N = 4 N = 6 N = 800 . . .
030 0 . . . C V C top V γ S qq FIG. 3. Bottom panel: Mass anomalous dimension γ S isshown as a function of charge q . The filled circles are numeri-cal determinations in the lattice model and the black band isthe resulting spline interpolation of the data. The expectedregion corresponding to N = 2 , , , are shownby the rectangular boxes, so as to match the values of γ S . Thedashed line is the unitarity bound on γ S . Top panel: The C V in the lattice model and C top V = q / (4 π ) are shown as a func-tion of q . The two intersect in the region of q correspondingto N = 2 QED , as inferred from the bottom panel. tom panel of Fig. 3, where we have plotted γ S ( q ) asa function of q , and determined expected 1- σ rangesof q that corresponding to N = 2 , , , based on estimates of γ S from our previous lattice stud-ies of QED [23, 24] ; namely, we find the expectedranges q ∈ [2 . , . , [1 . , . , [1 . , . , [1 . , . N = 2 , , , V aµ and V top µ behave as | x | − with amplitudes C V ( q )having a non-trivial dependence on q and C top V ( q ) beingquadratic in q . In the top-panel of Fig. 3, we have shownthese q -dependences of the two amplitudes, wherein onefinds C top V increases as q / (4 π ) whereas C V decreasesfrom the free field value 1 / (4 π ) as a function of q , andthe two curves intersect around q = 2 .
6; at this intersect-ing point, ( V + µ , V µ , V − µ , V top µ ) form an enlarged set of de-generate conserved vector currents in the lattice model.It is fascinating that this value of q ≈ . N = 2 QED , where such adegeneracy is expected from a conjectured self-duality of N = 2 QED [40–42] (conditional to the theory beingconformal), and the q - N mapping presented here sug-gests that such a degeneracy could occur in N = 2 QED (and also numerically observed in [43]).Quite strikingly, we also find evidence for microscopicmatching between QED and the conformal model stud-ied in this paper. The probability distribution P ( z i )of the scaled low-lying discrete Dirac eigenvalues z i =Λ i / (cid:104) Λ i (cid:105) are universal to QED in the infrared limit andthe lattice model at the matched point Q ( N ). In the . . . . . . P ( z ) z model: q = 2 . L = 32model: q = 2 . L = 28model: q = 2 . L = 24QED N = 2; ‘ = 200QED N = 2; ‘ = 144RMT P ( z ) z P ( z ) z . . . . . . P ( z ) z model: q = 2 . L = 32model: q = 2 . L = 28model: q = 2 . L = 24QED N = 8; ‘ = 200QED N = 8; ‘ = 144RMT P ( z ) z P ( z ) z FIG. 4. Distribution of scaled eigenvalues z i = Λ i (cid:104) Λ i (cid:105) forthe three lowest eigenvalues (left to right) from the conformallattice model at q = 2 . q = 2 . N = 2 and N = 8 QED . For thelattice model, results from L = 24 , ,
32 are shown, whereas for QED , results from two large box sizes (cid:96) (measured inunits of coupling g ) are shown. top panels of Fig. 4, we show the nice agreement be-tween P ( z i ) for the lowest three eigenvalues from N = 2QED at two different large box sizes (cid:96) (measured inunits of Maxwell coupling g ) [23, 24] which are in the in-frared regime, and the distributions P ( z i ) from the latticemodel discussed here at q = 2 . q for N = 2. Such an agreement is again seenbetween P ( z i ) in the lattice model at q = 2 . q for N = 8)and in N = 8 QED shown in the bottom panels. To con-trast, such universality in low-lying eigenvalue distribu-tion has previously been studied only between fermionictheories with a condensate and random matrix theories(RMT) with same global symmetries [44]. The results for P ( z i ) from non-chiral RMT [44] corresponding to N = 2and 8 flavor theories are also shown for comparison intop and bottom panels of Fig. 4, using analytical resultsin [45, 46]; the observed disagreement between P ( z i ) in N ≥ and the corresponding RMTs is an evidencefor the absence of condensate in parity-invariant QED with any non-zero number of massless fermions (as pre-viously observed by us in [23]), and instead, the strikingcompatibility of the QED distributions with those froma CFT studied here is a remarkable counterpoint. Discussion. –
We have presented a three dimensionalinteracting conformal field theory where one can computeconformal data by a lattice regularization without finetuning. We showed that by probing this CFT with mass-less spectator fermions, one is able to obtain a more elab-orate set of conformal data that is tunable based on thecharge of the fermions. For the sake of demonstration, weonly computed two and three point functions of fermionbilinear that have the same charge. A simple extensionfor the near future is a computation of n -point functionsof four-fermi operators ¯ ψ n q ¯ ψ n q ψ n q ψ ( n + n − n ) q that isgauge-invariant nontrivially and has only connected di-agrams. We demonstrated a direct correspondence be-tween the model with charge- q fermions and an N -flavorQED ; by tuning q so as to match a scaling exponent(we chose γ S ), one is able to observe many other univer-sal features between the two corresponding theories. Westress that we did not perform an all-order calculationin 1 /N for QED [32, 47, 48] via a lattice simulation ofthe model; rather, the lattice calculation is an all-ordercomputation in charge- q which might or might-not beexpandable in 1 /N via a mapping q = Q ( N ) that wedetermined by a non-perturbative matching condition. However, a lattice perturbation theory approach to theresults presented here would be interesting. It would alsobe interesting to use this model to test for robust pre-dictions of infrared fermion-fermion dualities [12, 13] bytuning the value of q = Q ( N ) and adding required level- k lattice Chern-Simons term det[(1 − G ) / (1 + G )] k [49]. ACKNOWLEDGMENTS
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In this appendix, we write down a general U(1) gauge theory, of which the non-compact model considered in thispaper is a specific case. To avoid confusion, the terminology compact and non-compact in the lattice field theorylanguage means that they are U(1) theories with and without monopoles respectively [23, 25]. The U(1) model, thatin general has monopole defects, can be defined using a Villain-type [26] action: Z ≡ (cid:32)(cid:89) x,µ (cid:90) ∞−∞ dθ µ ( x ) (cid:33) (cid:88) { N } e − S g ( N ) (5)where S g ( N ) = 12 (cid:88) µ,ν =1 (cid:88) x,y (cid:20) F µν ( x ) − πq N µν ( x ) (cid:21) (cid:104) (cid:3) − / (cid:105) ( x, y ) (cid:20) F µν ( y ) − πq N µν ( y ) (cid:21) ; F µν ( x ) = ∆ µ θ ν ( x ) − ∆ ν θ µ ( x ) , (6)for integer valued fluxes N µν defined over plaquettes, and q is the real valued dimensionless charge. The theory hasthe U(1) gauge symmetry θ µ ( x ) → θ µ ( x ) + ∆ µ χ ( x ) as well as a symmetry θ µ ( x ) → θ µ ( x ) + πq m µ ( x ) for integers m µ . Fermions sources ψ nq in this model couple to θ via compact link variables e inqθ µ ( x ) . Monopoles of integer valuedmagnetic charges q mon at a cube at site x is given by12 (cid:88) ρ,µ,ν =1 (cid:15) ρµν ∆ ρ N µν ( x ) = 2 πq mon ( x ) . (7)The non-compact U(1) theory is a specific case obtained by the restriction that the number of monopoles at any site x is zero, i.e., q mon ( x ) = 0. This gives the condition that the integer valued fluxed N µν ( x ) be writable as a curl ofinteger valued links: N µν ( x ) = ∆ µ m ν ( x ) − ∆ ν m µ ( x ) . (8)Under such a condition, the explicitly U(1) symmetric partition function in Eq. (5) can be equivalently written as thenon-compact action we study in this paper, Z ≡ (cid:32)(cid:89) x,µ (cid:90) ∞−∞ dθ µ ( x ) (cid:33) e − S g ( N =0) , (9)by appropriately redefining θ µ ( x ) → θ µ ( x ) − πq m µ ( x ) in the original action. Such a connection also means that theobservables O ( θ ) be restricted to those invariant under θ µ ( x ) → θ µ ( x ) + πq m µ ( x ) for the equivalence of two ways ofwriting the U(1) theory without monopoles. We only studied the non-compact action above in this paper.A future study of the compact model with monopole degrees of freedom will be very interesting for the followingreason. In the weak-coupling limit of q →
0, the monopoles will get suppressed energetically, and hence be irrelevant,and we expect the theory would remain conformal as the noncompact theory. This irrelevance of monopoles mightcontinue up to some critical q = q c beyond which monopoles could become relevant (their scaling dimension becomesmaller than 3) [27], and the theory could be confining like the pure gauge compact Maxwell theory [28]. This studywill be feasible using the approaches presented in [29, 30]. II. MONTE-CARLO ALGORITHM IN FOURIER SPACE
The lattice action in real space is non-local, but it is diagonal in momentum space. In this appendix, we describethe Monte-Carlo algorithm in momentum space to generate independent gauge field configurations. Our conventionfor Fourier transform χ → ˜ χ on the lattice is χ ( x ) = (cid:88) (cid:48) L − n ,n ,n =0 ˜ χ ( n ) e πin · xL ; ˜ χ ( n ) = 1 L (cid:88) L − x ,x ,x =0 ˜ χ ( x ) e − πin · xL . (10)here the prime over the sum denotes that the zero momentum mode n = 0 is excluded. The reality of a function χ ( x ) implies ˜ χ ∗ ( n ) = ˜ χ (¯ n ) where ¯ n i = − n i mod L . with the lattice momentum given by f µ ( n ) = e πinµL −
1, thelattice action for the model in Eq. (2) can be written as S g = L (cid:88) n (cid:88) µ,ν (cid:112) f ( n ) (cid:104) f µ (¯ n )˜ θ ν (¯ n ) − f ν (¯ n )˜ θ µ (¯ n ) (cid:105) (cid:104) f µ ( n )˜ θ ν ( n ) − f ν ( n )˜ θ µ ( n ) (cid:105) ; f ( n ) ≡ (cid:88) µ =1 f µ ( n ) f ∗ µ ( n ) . (11)Assuming we will only be interested in computing observables that are gauge invariant, we will generate the twophysical degrees of freedom per momentum that are perpendicular to the zero mode,˜ θ (cid:107) = f ( n ) f ( n ) f ( n ) . (12)We are free to pick the two directions perpendicular to the zero mode due to the degeneracy in this plane. When( n + n ) (cid:54) = 0, we choose the normalized eigenvectors˜ θ ⊥ = 1 (cid:112) f ∗ ( n ) f ( n ) + f ∗ ( n ) f ( n ) f ∗ ( n ) − f ∗ ( n )0 ;˜ θ ⊥ = 1 f ( n ) (cid:112) f ∗ ( n ) f ( n ) + f ∗ ( n ) f ( n ) f ∗ ( n ) f ( n ) f ∗ ( n ) f ( n ) − f ∗ ( n ) f ( n ) − f ∗ ( n ) f ( n ) , (13)and when n = n = 0, we choose ˜ θ ⊥ = ; ˜ θ ⊥ = . (14)With these choice, the Monte Carlo algorithm is simple;1. Pick random numbers c ,µ ( n ) , c ,µ ( n ) ∼ N (cid:104) µ = 0 , σ = 1 / ( L (cid:112) f ( n )) (cid:105) .2. Construct ˜ θ µ ( n ) = c ,µ ( n )˜ θ ⊥ ,µ ( n ) + c ,µ ( n )˜ θ ⊥ ,µ ( n ).3. Construct the gauge fields in real space as θ µ ( x ) = (cid:80) (cid:48) n ˜ θ µ ( n ) e πin · xL .Just as a similar algorithm for pure gauge Maxwell theory, the Monte-Carlo algorithm for this conformal action isfree of auto-correlation by construction. The expense of the anti-Fourier transform in the last step can be drasticallyreduced by using a standard Fast Fourier Transform algorithm. III. TOPOLOGICAL CURRENT CORRELATOR
The topological current is V top µ ( x ) = q π (cid:88) ν,ρ =1 (cid:15) µνρ F νρ , (15)which is conserved on the lattice. To compute the two-point function, the source for V top µ ( x ) is added as S J = q π (cid:88) µνρ L − (cid:88) x ,x ,x =0 (cid:15) µνρ J µ ( x )∆ ν θ ρ ( x ) = qL π (cid:88) µνρ (cid:88) (cid:48) L − n ,n ,n =0 ˜ θ ∗ µ ( n ) (cid:16) (cid:15) µνρ f ∗ ν ( n ) ˜ J ρ ( n ) (cid:17) + cc , (16)and only couples to ˜ θ ⊥ j as expected. Thenln Z ( k, J ) Z (0) = 12 (cid:88) x,y J µ ( x ) G µν ( x − y ) J ν ( y ) , (17) − − − − − − − − − L = 256 l og (cid:16) q G ( ) V t o p (cid:17) log( | x | ) V top lattice π | x | FIG. 5. Topological current correlator, scaled by 1 /q , is plotted as a function of current-current separation | x | . The data iscompared with the continuum expectation 1 / (cid:0) π | x | (cid:1) . where G µν ( x ) = (cid:104) V top µ ( x ) V top ν (0) (cid:105) = q π L (cid:88) (cid:48) L − n ,n ,n =0 δ µν f ( n ) − f ∗ µ f ν (cid:112) f ( n ) e πin · xL . (18)The two-point function traced over the directions becomes G (2) V top ( x ) = (cid:88) µ =1 G µµ ( x ) = q π L (cid:88) (cid:48) L − n ,n ,n =0 (cid:112) f ( n ) e πin · xL L →∞ −−−−→ q π (cid:90) π − π d p e ip · x (cid:113)(cid:80) µ sin p µ . (19)In Fig. 5, we plot q − G (2) V top ( x ) as a function of | x | for x = (0 , , z ) as determined using the above expression on L = 256 lattice to show the effect of lattice regularization. For comparison, the continuum result [32–34] q − G (2) V top ( x ) = π | x | is also plotted as the black line. It is clear for intermediate 1 (cid:28) | x | (cid:28) L/
2, the value of C top V = q / (4 π ) isreproduced by the lattice regularization. This intermediate range of | x | indeed increases as one keeps increasing L . IV. WILSON-LOOP
We consider l × t rectangular Wilson loop defined as W q ( l, t ) ≡ − log (cid:42) exp (cid:32) iq (cid:88) x ∈ l × t F µν ( x ) (cid:33)(cid:43) . (20)We compute its expectation value by coupling a source J ( x ) = iq l − (cid:88) y =0 [ δ ( x , z + y ) , δ ( x , z ) δ ( x , z ) − δ ( x , z + y ) , δ ( x , z + t ) δ ( x , z )] ,J ( x ) = iq t − (cid:88) y =0 [ δ ( x , z + l ) , δ ( x , z + y ) δ ( x , z ) − δ ( x , z ) , δ ( x , z + y ) δ ( x , z )] ,J ( x ) = 0 , (21) − . − − . − − . − − . − − .
500 10 20 30 40 50 60 ∆ W ( ζ ) ζL = 512 data L = 256 data L = 128 data L = 64 data FIG. 6. The data for the perimeter subtracted Wilson-loop ∆ W ( l, t ) from multiple ( l, t ) have been plotted together as afunction of ζ = l/t . The near data collapse shows the dependence only on ζ . where q denotes the charge. Upon a Fourier transform the non-zero vectors are,˜ J ( n ) = iqL e − πin · zL (cid:16) − e − πin lL (cid:17) (cid:16) − e − πin tL (cid:17) − f ∗ ( n )1 f ∗ ( n ) , n , n (cid:54) = 0; iqL e − πin · zL (cid:16) − e − πin tL (cid:17) l , n = 0 , n (cid:54) = 0; iqL e − πin · zL (cid:16) − e − πin lL (cid:17) − t , n (cid:54) = 0 , n = 0; (22)and we note that θ †(cid:107) ˜ J ( n ) = 0, implying that the Wilson loop operator only couples to the physical degrees of freedom.The logarithm of the expectation value of the Wilson loop is proportional to q and its expression after factoring outthe q is W ( l, t ) = 4 L (cid:88) (cid:48) L − n =1 ,n =1 ,n =0 (cid:112) f ( n ) sin πn lL sin πn tL (cid:20) | f | + 1 | f | (cid:21) + 1 L (cid:88) (cid:48) L − n =1 ,n =0 l (cid:112) f ( n ; n = 0) sin πn tL + 1 L (cid:88) (cid:48) L − n =1 ,n =0 t (cid:112) f ( n ; n = 0) sin πn lL (23)In the limit L → ∞ , we can write the above expression as an integral W ( l, t ) = 12 π (cid:90) π − π d p sin p l sin p t (cid:113)(cid:80) k =1 sin p k (cid:34) p + 1sin p (cid:35) . (24) A. Conformal behavior of Wilson loop
The integral in Eq. (24) results in a non-trivial dependence on l and t which includes a perimeter term. We showthat it is possible to extract the conformal behavior by evaluating the lattice sum in Eq. (23). The semi-analyticexpression above by itself is hard to understand; hence we numerically evaluated the expressions for different L lattices to determine the behavior of l × t rectangular Wilson loops as a function of l, t . Since the Wilson line dependson charge as a simple q , we divide the results by 1 /q and present the results here (we will drop the index q below.) − − − − − − − W ( l , t ) t l = 8 l = 16 l = 32 l = 64 0 . . . . . . . . . . .
33 10 20 30 40 50 60 70 80 90 100 V ( l ) lL = 512 L = 256 L = 128 L = 640 . − . l FIG. 7. Extraction of static quark potential V ( l ) = lim t →∞ t − W ( l, t ). Left panel shows −W ( l, t ) as a function of t at differentfixed l on L = 256 lattice. The straight lines are fits to −W ( t ; l ) = − A ( l ) − t V ( l ) starting from t = 30. The right panel showsthe extracted static fermion potential V ( l ) as function of l , and data from different L are shown. The data for l < L/ V ( l ) = 0 . − . /l . The 1 /l is expected in conformal gauge theories in any dimension. In a gauge theory which is critical, one expects W ( l, t ) to depend only on the aspect ratio of the loop ζ = lt up tolinear corrections from the perimeter of the loop, p = l + t . In Fig. 6, we show the ζ -dependence for the difference∆ W ( l, t ) ≡ W ( l, t ) − W ( p/ , p/ , (25)constructed such that any perimeter term gets canceled. For a given L , Wilson loops of various possible l and t havebeen put together in the plot. We have shown the results using L = 64 , ,
256 and 512. One can see that the resultsfrom various l × t loops fall on a universal curve to a good accuracy that depends only on ζ . This clearly demonstratesthe underlying gauge theory is conformal. At a fixed ζ , one sees a little scatter of points around a central value; thisis because the lattice corrections increase when the size of a Wilson-loop at a given ζ is comparable to the latticespacing itself. This can be justified by observing that as L is increased towards 512, the scatter of points at given ζ becomes lesser, due to the possibility of having larger loops with the same ζ . For large ζ , one finds a linear tendencyof ∆ W ( ζ ) originating from the 1 /t static potential as we discuss below.We extract the static fermion potential V ( l ) by looking for the asymptotic behavior W ( l, t ) = A ( l ) + t V ( l ) (26)for larger t at fixed l . For this, we fitted the above form to W ( l, t ) for 25 < t < L/ −
10, and obtained V ( l ), using L = 64 , ,
256 and 512. This is demonstrated in the left panel of Fig. 7 where −W is plotted as a function of t for different fixed l = 8 , , ,
64 on L = 256 lattice. The fits to the above form are the straight lines. In the rightpanel of Fig. 7, we plot the extracted potential V ( l ) as a function of l . We have shown the potential as extracted from L = 64 ,
128 and 256 as the different colored symbols. For 1 < l (cid:28) L/
2, the data is nicely described by the form V ( l ) = 0 . − . l . (27)It is important to remember that this functional form is not the Coulomb potential in three dimensions (whichis instead logarithmic in 3d), and instead, this functional form is dictated by the conformal invariance in gaugetheories [31]. The coefficient ν ≈ . q dependence, for Wilson loop of charge q , the coefficient will be ν ( q ) = 0 . q .) By changing fit ranges,we find about 1% variation in our estimates of ν ; Therefore, we quote an estimate with a systematic uncertainty, ν ( q ) = 0 . q . V. OVERLAP FERMION PROPAGATOR
The details on the overlap formalism in three dimensions to study exactly massless fermions on the lattice canbe found in [24]. Here, we recall the important aspects of the implementation of the overlap Dirac operator. Themassless overlap propagator G q for a two-component Dirac fermion of charge q is given by G q = V ( qθ ) − V ( qθ ) + 1 , (28)here V ( qθ ) is a unitary 2 L × L matrix. The matrix V is constructed using Wilson-Dirac operator kernel as V ( qθ ) ≡ (cid:112) X ( qθ ) X † ( qθ ) X ( qθ ) . (29) X is the Wilson-Dirac operator with mass − m w , X ( qθ ) = /D ( qθ ) + B ( qθ ) − m w , (30)where /D and B are the naive lattice Dirac operator and the Wilson mass term respectively, /D ( qθ ) = 12 (cid:88) µ =1 σ µ (cid:2) T µ ( qθ ) − T † µ ( qθ ) (cid:3) , B = 12 (cid:88) µ =1 (cid:2) − T µ ( qθ ) − T † µ ( qθ ) (cid:3) , (31)in terms of the covariant forward shift operator, [ T µ f ]( x ) = e iqθ µ ( x ) f ( x + ˆ µ ). The three Pauli matrices are denoted as σ µ .We improved the overlap operator by using 1-HYP smeared [23, 38] fields θ sµ ( x ) instead of θ µ ( x ) in the aboveconstruction, which suppresses gauge field fluctuations of the order of lattice spacing and in particular, reduces thenumber of few lattice-spacing separated monopole-antimonopole pairs which are artifacts in a noncompact theory [24].We implemented ( XX † ) − / by using Zolotarev expansion [37] up to 21st order, which was found sufficient in [24].We used m w = 1 in the Wilson-Dirac kernel. VI. EXTRACTION OF MASS ANOMALOUS DIMENSION FROM DIRAC EIGENVALUES
We determined the low-lying Dirac eigenvalues Λ i with, 0 ≤ Λ ≤ Λ ≤ . . . , using the anti-Hermitian inverse overlapfermion propagator G − q v i = − Λ i v i , (32)where v i are the eigenvectors. It is easier determined equivalently using14 ( V ( qθ ) + 1)( V † ( qθ ) + 1) v i = Λ i i v i , (33)using the Kalkreuter-Simma algorithm [39]. We determined the smallest eight eigenvalues Λ j this way, and used only j ≤ q = 0 . , . , . , . , . , . , . L lattices with L = 4 , , , , , , , , , , , ,
36. For each of those L in that order, we used the following number of configurations; 680,680,680,680,680,680,680,680,680,278,210,153configurations respectively. We formed the ratio ˜Λ j = Λ j ( L ; q )Λ j ( L ; 0) , (34)to study the effect of non-zero q and reduce any finite- L corrections already present in free theory.We used the finite-size scaling of the low-lying Dirac eigenvalues ˜Λ i ( L ) ∝ L − γ S to determine the scalar anomalousdimension γ S . One way to see it is that the scalar susceptibility χ q = (cid:82) d x (cid:104) S ± , ( x ) S ∓ , (0) (cid:105) = L − (cid:80) j Λ − j scales as L − γ S , which implies that Λ j ∝ L − − γ S for all j in the large- L limit. In Fig. 8, we have shown the dependence of˜Λ i for i = 1 to 5 as a function of L in a log-log scale; the different panels correspond to q ranging from 0.25 to 3.0.One can see that for larger q , one does not a see a perfect log( L ) scaling dependence and the subleading correctionsget larger in the range of L used. Therefore, we used the following ansatz to capture the leading L − γ S scaling alongwith sub-leading corrections which we model to be 1 /L k corrections for integer k :˜Λ j ( L ) = a j L − γ S (cid:32) N max (cid:88) k =1 b j,k L − k (cid:33) . (35)We performed a combined fit of the above ansatz to the L -dependence of ˜Λ j for j = 1 to 5. Using N max = 4, we wereable to fit the data at all q ranging from L = 6 to 36 with χ / dof < .
2. The error-bands from such fits are shownalong with the data in Fig. 8. By reducing N max = 2, we were able to fit data ranging from L = 14 to 36, and there ispossibly a systematic effect to slightly increase the estimated γ S , but such changes were within error-bars. Therefore,we take our estimates that fit the data over wider range using N max = 4 as our best estimate in this paper. Thedeterminations of γ S from different ranges of L and goodness-of-fit are summarized in Table I. . − . − . − . − . − . l og ( ˜ Λ n ) log( L ) q = 0 . n = 1 n = 2 n = 3 n = 4 n = 5 − . − . − . − . l og ( ˜ Λ n ) log( L ) q = 0 . n = 1 n = 2 n = 3 n = 4 n = 5 − . − . − . − . − . − . − . − . − . l og ( ˜ Λ n ) log( L ) q = 1 . n = 1 n = 2 n = 3 n = 4 n = 5 − . − . − . − . − . − . − .
101 2 3 4 l og ( ˜ Λ n ) log( L ) q = 1 . n = 1 n = 2 n = 3 n = 4 n = 5 − − . − . − . − .
201 2 3 4 l og ( ˜ Λ n ) log( L ) q = 2 . n = 1 n = 2 n = 3 n = 4 n = 5 − . − . − . − . − − . − . − . − . .
21 2 3 4 l og ( ˜ Λ n ) log( L ) q = 2 . n = 1 n = 2 n = 3 n = 4 n = 5 − . − − . − − . .
51 2 3 4 l og ( ˜ Λ n ) log( L ) q = 3 . n = 1 n = 2 n = 3 n = 4 n = 5 FIG. 8. Plot shows log(Λ n ( L ; q ) / Λ n ( L ; q = 0) j ) versus log( L ) for different charges q . In each panel, the different coloredsymbols are the first five eigenvalues, n = 1 to 5. The eigenvalues are ordered by magnitude. The curves are the fits asdescribed in the text. VII. TWO-POINT FUNCTIONS
We computed the two-point functions by coupling fermion sources of charge q to the gauge fields as described inEq. (3) in the main text. The expressions for two-point functions in terms of the fermion propagators are G (2) S ± , S ∓ , ( | x − y | ) = (cid:42) (cid:88) α =1 |G ααq ( x, y ) | (cid:43) ; G (2) V ± , µ V ∓ , µ ( | x − y | ) = − (cid:42) (cid:88) α,β,γ,δ =1 σ αβµ G βγ ( x, y ) σ γδµ G δα ( y, x ) (cid:43) . (36)Since all the propagators G are determined for the same value of charge q , we have suppressed the index for q . Wedetermined the correlator in a standard fashion by using a point source vector v α (cid:48) ( x (cid:48) ) = δ x (cid:48) ,x δ α (cid:48) ,α using terms such L range N max γ S χ / dof0.25 6-36 4 0.011(11) 26.0/3414-36 2 0.022(10) 13.1/240.50 6-36 4 0.036(21) 26.5/3414-36 2 0.058(15) 13.9/241.00 6-36 4 0.112(40) 28.9/3414-36 2 0.156(28) 17.7/241.50 6-36 4 0.242(54) 28.0/3414-36 2 0.299(41) 17.2/242.00 6-36 4 0.459(68) 27.8/3414-36 2 0.522(56) 17.6/242.50 6-36 4 0.888(64) 32.8/3414-36 2 0.922(63) 22.9/243.00 6-36 4 1.657(56) 39.1/3414-36 2 1.619(66) 27.2/24TABLE I. The estimates of scalar dimensions γ S as estimated using the FSS of Dirac eigenvalues Λ j are tabulated. Here, q isthe charge used in the Dirac operator, L range is the range of lattice size from which the eigenvalues are used in the fits, N max is the number of 1 /L k corrections used in Eq. (35), and γ S is the estimated scaling dimension from the fits to Eq. (35). Theminimum χ per degree of freedom of the fits as a measure of goodness-of-fit are also tabulated. − − − − − − − − − .
75 1 1 .
25 1 . .
75 2 2 .
25 2 . .
75 3 L = 40 l og ( G ( ) S ( | x | )) log( | x | )Scalar q = 1 . q = 1 . q = 2 . q = 2 . ∝ | x | − − − − − − − − − − .
75 1 1 .
25 1 . .
75 2 2 .
25 2 . .
75 3 L = 40 l og ( G ( ) V ( | x | )) log( | x | )Vector q = 1 . q = 1 . q = 2 . q = 2 . ∝ | x | − FIG. 9. The flavor triplet scalar and vector correlators from different q are compared. L = 40 lattice was used. The conservedvector correlator is unchanged up to slight change in normalization C V that seems to decrease with q . For the scalar, both theamplitude as well as the exponent γ S changes; the decrease in γ S is also apparent from the plot. as G β,αq ( y, x ) = (cid:20) ( V − V † ) · (cid:2) (1 + V )(1 + V † ) (cid:3) − · v (cid:21) β ( y ) , (37)and the identity G α,β ( x, y ) = −G β,α ( y, x ) to compute backward propagators. We used conjugate gradient (CG) todetermine (cid:2) (1 + V )(1 + V † ) (cid:3) − · v , using a stopping criterion 3 · − . For the inner-CG to determine V = ( XX † ) − / X ,we used a stopping criterion of 3 · − . We chose x = (0 , ,
0) and y = (0 , , z ) along the lattice axes, and hence | y − x | = z .In Fig. 9, we show the vector and scalar two-point functions as a function of | x | as determined on L = 40 lattice.For each of them, we have shown the correlators from q = 1 . , . , . C V . From the plot, one can see a decreasing tendency in C V , which will . . . . .
910 5 10 15 20 25 30 q = 0 . q = 0 . q = 1 . q = 1 . q = 2 . q = 2 . C e ff V ( | x | ) | x | FIG. 10. The effective amplitude ˜ C eff V ( | x | ), defined in Eq. (39), is shown as a function of operator separation | x | . The differentcolored symbols are from different q specified in right-side of the plot. For each q , data from L = 28 (triangle) and L = 48(circle) are shown. The value of the vector two-point function amplitude ˜ C V is estimated from the plateau region | x | ∈ [8 , analyze in detail in the later part. For the scalar, there is both a decrease in scaling dimension due to the non-zero γ S and a decrease in amplitude.The correlators on a periodic lattice have three parts; 1) a small distance part consisting | x | of the order of fewlattice spacing where the operators have contributions from the primary scaling operators as well as of secondaryscaling operators of higher scaling dimensions. 2) an intermediate | x | that is larger than few lattice spacings andalso smaller than L/ | x | of the order of L/ | x | − dependence with an appropriately chosen amplitude C V . One can see that there is only a short intermediate regionin | x | on the typical lattices L ∼
40 used, where there is a | x | − behavior, and hence fitting such a functional formto the correlator to extract the scaling dimension and the amplitude is not a good way. Instead, in order to obtainthe scaling dimension from the correlator, it is best to use the finite size scaling; for a critical theory, the two pointfunction G (2) ( | x | , L ) should have a scaling form L − ( g ( | x | /L ) + O (1 /L )) and hence by keeping | x | = ρL for fixed ρ ,one can extract ∆ from the FSS (for example,refer [6]). We chose ρ = 1 / C V , we found it optimal to use the reduced two point function˜ G (2) ( | x | , q ; L ) = G (2) ( | x | , q ; L ) G (2) ( | x | , q = 0; L ) , (38)which removes finite lattice spacing and finite-volume effects that are already present in free theory; for the vector,which is where we are interested in the amplitude the most, this was optimal since the behavior of correlators fornon-zero q and zero q were more of less the same and hence we can find ˜ C V ( q ) ≡ C V ( q ) /C V ( q = 0) very well. Forthis, we define an effective | x | -dependent ˜ C V ( | x | ) as˜ C eff V ( | x | , q ; L ) ≡ (cid:88) µ =1 ˜ G (2) V + µ V − µ ( | x | , q ; L ) . (39)If there were perfect | x | − scaling in both q (cid:54) = 0 and q = 0 vector two-point functions, the effective C eff V will exhibita plateau at all distances | x | . Instead in the actual case, one can expect a plateau only over an intermediate | x | . InFig. 10, we show ˜ C V ( | x | , q ; L ) as a function of | x | ; the different colors correspond to different q and for each q , wehave shown the results using L = 28 and 48 lattices as the open triangles and filled circles respectively. One findsthat at fixed | x | , the values of ˜ C V ( | x | , q ; L ) for these ranges of L above 20 are consistent within errors and hence havereached their thermodynamic limits within statistical errors. For | x | ∈ [8 ,
12] which is larger than few lattice spacingsand at the same time much smaller than L/ L used, one finds a plateau and we estimate C V byaveraging over these values of | x | . Such estimates are shown as the bands in Fig. 10. We take the determination of C V on the largest L = 48 we computed to be our estimate. In order to compute C V ( q ), we use the continuum valueof C V ( q = 0) = 1 / (4 π ) [32]. III. THREE-POINT FUNCTIONS
In a CFT, the conformal invariance dictates the form of three-point functions of primary operators. In the latticemodel, the local operators we construct in general are not the scaling operators, and hence, we expect to observescaling only when the distances | x ij | between any pair of operators are large, but at the same time, smaller than L/ G V + i V − i V , G V + i V − i V and G S + i S − i V in the main text as further evidenceto the conformal nature of the lattice theory and also to demonstrate that the system is a very good model systemfor furthering the lattice framework to study fermionic CFTs.We constructed the three-point functions in terms of the fermion propagators as G (3) S + S − V ρ ( x , x , x ) = −√ (cid:20) (cid:42) (cid:88) α,β,γ,δ =1 G α,β ( x , x ) G β,γ ( x , x ) σ γ,δρ G δ,α ( x , x ) (cid:43) (cid:21) , (40)and G (3) V + µ V − ν V ρ ( x , x , x ) = −√ (cid:20) (cid:42) (cid:88) α,β,γ,δ =1 σ αβµ G βγ ( x , x ) σ γδν G δρ ( x , x ) σ ρλρ G λα ( x , x ) (cid:43) (cid:21) . (41)The contractions above require two overlap inversions per choice of x at fixed x . From the three-point functions,we constructed the reduced three-point function˜ G (3) O + i O − j O k ≡ G (3) O + i O − j O k ( x , x , x ; q, L ) G (3) O + i O − j O k ( x , x , x ; q = 0 , L ) . (42)By construction, in free field theory, this ratio is normalized to 1 at all separations and hence we expect this ratio toremove both short distance lattice corrections as well as large distance finite volume corrections. We specialized tocollinear three-point function [2] in order to greatly simplify the x ij dependence. That is, we used x = (0 , , , x =(0 , , z ) , x = (0 , , z + z ). In this configuration of operators, We expect the above ratio to behave as˜ G (3) O + i O − j O k ( z , z ) = ˜ C O + i O − j O k z γ Oi + γ Oj − γ Ok z γ Oj + γ Ok − γ Oi ( z + z ) γ Ok + γ Oi − γ Oj , (43)where the scaling dimension of the operator is ∆ O = 2 − γ O , and ˜ C O + i O − j O k is the ratio of OPE coefficients˜ C O + i O − j O k ( q ) = C O + i O − j O k ( q ) /C O + i O − j O k ( q = 0). For the vector γ V = 0. Therefore, the expressions above sim-plify for our three-point functions. In general, if the operators only had overlap with the primary scaling operator,one would expect, ˜ G (3) V + i V − i V ( z , z ) = ˜ C V + i V − i V , ˜ G (3) V +3 V − V ( z , z ) = ˜ C V +3 V − V , ˜ G (3) S + S − V ( z , z ) = ˜ C S + S − V z γ S , (44)at all distances. However, on the lattice, our choice of operators generically overlap with both primary as well assecondary operators, and hence, one can expect to see the above behavior of three-point functions with smallestscaling dimensions as that of primaries, only when0 (cid:28) z , z , z + z (cid:28) L/ . (45)Therefore, we turn the expression above around, and define effective OPE coefficients C eff O + i O − j O k ( z , z ) as˜ C eff V + i V − i V ( z , z ) ≡ ˜ G (3) V + i V − i V ( z , z ) , ˜ C eff V +3 V − V ( z , z ) ≡ ˜ G (3) V +3 V − V ( z , z ) , ˜ C eff S + S − V ( z , z ) ≡ ˜ G (3) S + S − V ( z , z ) z − γ S . (46)We presented the results in these effective OPE coefficient in Fig. 2 in the main text. First, we observe a plateaufor intermediate distances which is a nice demonstration of the spectator fermion observables satisfying the CFTonditions. We can extract the OPE coefficient from the value of C eff in the plateau region. The condition in Eq.(45), also tells us the optimal ordering of three-point functions to look at. For example, we could have constructed athree-point function ˜ G (3) S + V − S ( z , z ) which behaves as ˜ C S + V − S ( z + z ) γ S . Given a finite L = 64 lattice we use, ifwe used z , z ≈ ∼ . L so as to be in a scaling region, then z + z ≈ ∼ . L , which might suffer from finitesize effect. Hence the usage of infinite-volume factor ( z + z ) γ S might not be correct. This is the reason, we usedthe ordering of operators given in Eq. (43) and Eq. (44).Since the three-point function computation is for a small-scale computational project we undertook, we used onlya single, large lattice extent L = 64 and one value of q = 0 .
15 using 850 statistically independent configurations. Wepresented the results from this computation in Fig. 2.
IX. EIGENVALUE DISTRIBUTIONS
In this section, we compare distributions P ( z i ) of scaled Dirac eigenvalues z i ≡ Λ i (cid:104) Λ i (cid:105) , (47)as determined in N -flavor QED with that in the model using charge q bilinears. Here, q is chosen to be in the vicinity ofthe expected uncertainty range where we expect γ S for the charge q bilinear will be equal to that in the N flavor QED .As we mentioned in the main text, we expect these ranges to be q ∈ [2 . , . , [1 . , . , [1 . , . , [1 . , . P ( z i ) by histograms of the z i as sampled in the Monte Carlo. For QED , we used differentphysical boxes of volume (cid:96) (measured in units of Maxwell coupling g ) which we expect to be in the infrared regime ofQED . For QED , we used the eigenvalues data in L = 20 , ,
28 (which determines the lattice spacing, the continuumlimit of QED is in the limit L → ∞ ) from our study using massless Wilson-Dirac fermions [23]. Since the results fromthe three different L for QED gave similar result, we only show the results for L = 28 in the histograms below. Wealso checked that using eigenvalues from our later study [24] of N = 2 QED using exactly massless overlap fermionsgave results consistent with the histograms for N = 2 QED using Wilson-Dirac fermions shown here. In Fig. 11,we have compared P ( z i ) from N = 2 , , , q = 2 . , . , . , . N, q . The agreement isalmost perfect and supports the claim that the observed agreement cannot be a mere coincidence.The N -flavor non-chiral Gaussian Unitary Ensemble random matrix theories (nonchiral RMT) [44] is given by thepartition function Z RMT ( N ) = (cid:90) [ dH ] det ( H ) N e − tr( H ) , (48)where H are M × M Hermitian matrices, in the limit of M → ∞ . The eigenvalues λ i in the RMT are the eigenvaluesof H (ordered according to their absolute values). We compare analytical results [45, 46] for the distributions of λ i / (cid:104) λ i (cid:105) from the N -flavor RMTs in the different panels of Fig. 11. For a theory with a condensate, the Diraceigenvalue distribution must agree with the one from the corresponding RMT. One can see that the Dirac eigenvaluedistributions disagree with those from the RMT, and instead, they agree with the distributions from the conformalgauge theories for tuned values of q studied in this paper. X. COMPARISON WITH LEADING /N RESULTS
Since any remnant q -dependent ambiguity in normalization factors that convert the operators in the model tooperators in QED cannot affect the scaling exponents, we expect the coefficient of the leading q dependence of γ S to be the same as the coefficient of 32 /N in large- N expansion. Indeed, our determination of the leading coefficient0 . q is consistent with the large- N expectation [48] of 2 / (3 π ) ≈ .
068 within errors. Taking an example q - N relation of the form 32 /N = q + bq + . . . , shows that coefficients of orders q and higher might have contributionsfrom all orders in 32 /N , and hence we do not expect such universality in coefficients beyond leading order in q .A similar comparison for two-point function amplitudes, for example C V , cannot be made due to possible q -dependent conversion factors, Z ( q ) = 1 + q + . . . , between operators in the model and in QED . In fact, we differin the leading q contribution to [ C V ( q ) /C V (0) − q coefficient to be − . /N in large- N expansion [32] for C V in QED is ≈ . q with that in the full N -flavor QED , which is not a veryuseful statement as such. However, one can study renormalization group invariant concepts such as the degeneracyof current correlators, as presented in this paper. . . . . . . P ( z ) z model: q = 2 . L = 32model: q = 2 . L = 28model: q = 2 . L = 24QED N = 2; ‘ = 200QED N = 2; ‘ = 144RMT P ( z ) z P ( z ) z . . . . . . P ( z ) z model: q = 2 . L = 28QED N = 4; ‘ = 200QED N = 4; ‘ = 144RMT P ( z ) z P ( z ) z . . . . . . P ( z ) z model: q = 2 . L = 28QED N = 6; ‘ = 200QED N = 6; ‘ = 144RMT P ( z ) z P ( z ) z . . . . . . P ( z ) z model: q = 2 . L = 32model: q = 2 . L = 28model: q = 2 . L = 24QED N = 8; ‘ = 200QED N = 8; ‘ = 144RMT P ( z ) z P ( z ) z FIG. 11. Distribution of scaled eigenvalues z i = Λ i (cid:104) Λ i (cid:105) for the three lowest eigenvalues (left to right) from the conformal latticemodel at q = 2 . , , , . , . N = 2 , , , respectively. For thelattice model, results from L = 24 , ,
32 are shown for N = 2 ,
8, and L = 28 for N = 4 ,
6. For QED , results from two largebox sizes (cid:96) (measured in units of coupling g2