# Transition in the spectral gap of the massless overlap Dirac operator coupled to abelian fields in three dimensions

TTransition in the spectral gap of the massless overlap Diracoperator coupled to abelian ﬁelds in three dimensions

Rajamani Narayanan ∗ Department of Physics, Florida International University, Miami, FL 33199 (Dated: February 24, 2021)

Abstract

The low lying spectrum of the massless overlap Dirac operator coupled to abelian ﬁelds in threedimensions with three diﬀerent measures are shown to exhibit two phases: a strong coupling gappedphase and a weak coupling gapless phase. The vanishing of the gap from the strong coupling sidewith a Maxwell and a conformal measure is governed by a Gaussian exponent. Contrary to thisresult, the vanishing of the gap from the strong coupling side with a compact Thirring measure isnot consistent with a Gaussian exponent. The low lying spectrum with a non-compact Thirringmeasure does not exhibit a simple non-monotonic behavior as a function of the lattice size on theweak coupling side. Our combined analysis suggests exploring the possibility of a strongly coupledcontinuum theory starting from a compact lattice Thirring model where a compact U(1) gauge ﬁeldwith a single link action is coupled to even number of ﬂavors of massless overlap Dirac fermions. ∗ Electronic address: rajamani.narayanan@ﬁu.edu a r X i v : . [ h e p - l a t ] F e b . INTRODUCTION Strongly coupled theory of massless fermions interacting with an abelian gauge ﬁeld inthree dimensions (two space and one Euclidean time) has been studied over several decadesboth analytically and numerically [1–17] and has attracted recent attention due to its rele-vance in condensed matter physics and three dimensional duality [18–26]. Another possibilityfor a theory of strongly interacting fermions in three dimensions is the Thirring model whichis shown to be renormalizable in the limit of large number of ﬂavors [27–31] and a quest toﬁnd a strongly interacting continuum theory away from the limit of large number of ﬂavorshas been underway for several decades [32–47]. Unlike QED , the Thirring model after oneconverts the four fermi interaction into a bilinear with an auxiliary vector ﬁeld is not gaugeinvariant. The interaction between fermions and the vector ﬁeld is formally gauge invariantbefore regularization and one could perform an integral over all gauge transformations ofthe action for the vector ﬁeld, namely (cid:0)(cid:82) d x A k ( x ) (cid:1) , and enforce gauge invariance of theaction by stating that only gauge invariant observables will be computed. Such a conditionis usually not imposed when one converts the continuum action to a lattice regulated actionfor numerical analysis. At the outset, all recent numerical analysis to date [41–47] assumethat fermions couple to a non-compact vector ﬁeld with two diﬀerent approaches treatingthe non-compact vector ﬁeld on a site [41, 42, 46] or a link [43–45, 47]. Recent numericalanalysis of QED has shown that theory remains scale invariant for all even number of ﬂa-vors [12, 13] as long as monopoles remain irrelevant. There is evidence for a critical numberof ﬂavors in the Thirring model [44, 47] if one uses domain wall fermions [48, 49] to regulatethe fermions on the lattice and place the vector ﬁeld on links. On the other hand, there is noevidence for symmetry breaking for any even number of ﬂavors if one uses SLAC fermionsto regulate the fermions on the lattice and place the vector ﬁeld on sites [42, 46]. Currentconservation was used to show that all divergences in the N ﬂavor Thirring model up toorder N can be removed by standard renormalizations there by rendering it ﬁnite [31]. Thismotivated us to see the eﬀects of lattice regularization that preserves gauge invariance ofthe fermionic determinant. Toward that end we only focus on fermionic observables in apure gauge measure, otherwise referred to as the quenched limit. The quenched limit ofthe Thirring model has been studied before [44] where evidence for symmetry breaking isfound at suﬃciently strong coupling but a possible transition to an unbroken phase at weak2oupling was not fully explored since this limit is not physically interesting. Nevertheless, itis interesting from the view point of the studying the eﬀects of diﬀerent approach to latticeregularization. We will diﬀer from [44] is one key aspect – the lattice fermion operator willcouple to the compact vector ﬁeld thereby making it a gauge covariant interaction. We willuse the overlap Dirac operator [13, 50] which is just a limit of the domain wall operator usedin [44].Consider the spectrum of the three dimensional Euclidean Dirac operator D/ = σ k ( ∂ k + iA k ); σ = ; σ = − ii ; σ = − , (1)averaged over some measure, P ( A k ). Since the spectrum is invariant under the gauge trans-formation, A k → A k + ∂ k χ, (2)we can assume that the measure is also gauge invariant. We will also assume that themeasure is invariant under parity, A k ( x ) → − A k ( − x ), and we will not concern ourselveswith the parity anomaly. The following three choices of the measure are of relevance in thecontext of QED and Thirring model: • The Maxwell measure given by P M ( A ) = exp (cid:20) − β (cid:90) d xF jk (cid:21) ; F jk = ∂ j A k − ∂ k A j . (3)The spectrum is relevant in the quenched limit of QED where the spectral density, ρ ( λ ) , attains a non-zero value, ρ (0), in the continuum limit, β → ∞ [14]. • The conformal measure given by P C ( A ) = exp (cid:20) − β (cid:90) d xF jk √− ∂ F jk (cid:21) . (4)The spectrum at various values of β ( N ) can be matched with that of continuum parityinvariant QED with 2 N ﬂavors [17]. • The Thirring measure given by P T ( A ) = exp (cid:20) − β (cid:90) d xA k (cid:21) gauge averaging ⇒ exp (cid:20) − β (cid:90) d xF jk − ∂ F jk (cid:21) . (5)Analysis of the spectrum as a function of β will shed some light into the existence ofa strongly coupled theory in the continuum, albeit, in the quenched limit.3ur primary aim in this paper is to compare the spectrum of the overlap-Dirac operatoron the lattice for the above three measures suitably discretized on the lattice. On the onehand, the Maxwell measure is reasonably well understood due to the existence of a welldeﬁned continuum theory in the β → ∞ limit. In addition, the behavior of the conformaltheory with the conformal measure has been matched with the behavior of QED withvarying number of ﬂavors [17]. A comparison of the behavior with the Thirring measurewith the other two measures can be used to explore the existence of a strongly interactingcontinuum theory at a ﬁnite value of β .Denoting the lattice link variables on a site of length a by θ k ( x ) = aA k ( x ), the latticeoverlap-Dirac operator will couple to the compact variable, U k ( x ) = e iθ k ( x ) . It is natural touse a non-compact measure on the lattice for the Maxwell and the conformal case in orderto suppress the presence of monopoles in the continuum theory. But, we will consider theabove Thirring measure and a variant since we are interested in eventually studying a ﬁeldtheory deﬁned at a ﬁnite value of β . Noting that fermions couple to U k ( x ), we rewrite thenormalized measure for each θ k ( x ) as [51] (cid:114) β π (cid:90) ∞−∞ dθ k ( x ) e − β θ k ( x ) = (cid:114) β π (cid:90) π − π dθ k ( x ) (cid:34) ∞ (cid:88) n = −∞ e − β ( θ k ( x )+2 nπ ) (cid:35) = (cid:90) π − π dθ k ( x ) ∞ (cid:88) n = −∞ e − n β π U nk ( x ) , (6)which is nothing but a Villain type action . An alternative is to use a Wilson type actionfor the link variables, namely, e β ( U k ( x )+ U ∗ k ( x ) )2 πI ( β ) = e β cos θ k ( x ) πI ( β ) = ∞ (cid:88) n = −∞ I n ( β )2 πI ( β ) U nk ( x ) . (7)The coeﬃcients in the character expansion for the two choices reach the same limit as β → ∞ but our interest is to look for a critical point at a ﬁnite value of β , possibly close to zero.Close to β = 0, I n ( β ) = β | n | | n | ( | n | )! , and suppression of Wilson loop with a given size will bestronger in the Villain type action compared to the

Wilson type action. The author would like to thank Simon Hands for bringing this to his attention. I. MASSLESS FERMION SPECTRUM ON THE LATTICE

The massless overlap-Dirac operator, D o , and the associated anti-Hermitian propagator, A , are [13] D o = 1 + V V = X √ X † X ; X = B + D ; D = 12 (cid:88) k =1 σ k ( T k − T † k ); B = 12 (cid:88) k =1 (2 − T k − T † k ) − m w ;( T k φ )( x ) = e iθ k ( x ) φ ( x + ˆ k ); A = 1 − V V . (8)The Wilson mass parameter, m w , can be taken anywhere in the range (0 ,

2) to realize asingle massless fermion. Changing the value of m w will result in diﬀerent values for thelattice spacing eﬀects and we will set it to m w = 1 for all computations in this paper. Wewill assume anti-periodic boundary conditions for fermions unless otherwise speciﬁed. Letthe spectrum of A − be deﬁned by1 + V − V ψ j = i Λ j ψ j ; j = ± , ± , · · · , · · · < Λ − < Λ − < < Λ < Λ < · · · . (9)In the weak coupling limit, β = ∞ , all gauge actions discussed in Section I will resultin gauge ﬁelds, θ k ( x ), that are gauge equivalent to zero and the spectrum will be that ofmassless free fermions as expected. In the strong coupling limit, β = 0, all gauge actionsdiscussed in Section I will result in gauge ﬁelds, θ k ( x ), that are uniformly and independentlydistributed in [ − π, π ]. The spectrum of the lattice Dirac operator will show a gap, namely,lim L →∞ (cid:104) Λ ± (cid:105) will be consistent with unity and this corresponds to an eigenvalue of V being ± i . It might be possible to obtain this result analytically using resolvents but it will besuﬃcient for our purposes to have shown it numerically by computing the spectrum on afew L lattices and extrapolating to L → ∞ as shown in Figure 1. To emphasize that thespectral gap is an eﬀect at strong coupling, one could have repeated the calculation at β = 0but by coupling the fermions to a smeared link [52, 53]. Smearing takes a distribution oflinks that is uniform into one that favors θ µ ( x ) = 0. The eﬀect on the gap will depend on thesmearing parameters and the initial range of θ k ( x ) which could be any real number insteadof restricting it to − [ π, π ]. 5 L FIG. 1: A plot of (cid:104) Λ (cid:105) = (cid:104) Λ − (cid:105) is shown as a function of L in the strong coupling limit, β = 0.The data is ﬁtted to 1 + a L + a L and shown to be consistent. A. Spectral gap for a Maxwell measure

The Maxwell measure given in Eq. (3) translates to [17] (cid:89) n exp (cid:34) − L βf ( n )2 (cid:88) j =1 (cid:12)(cid:12)(cid:12) ˜ θ j ⊥ ( n ) (cid:12)(cid:12)(cid:12) (cid:35) (10)on a L lattice where θ k ( x ) = (cid:88) n (cid:88) j =1 θ j ⊥ ( n ) v jk ( n ) exp (cid:20) πin · xL (cid:21) (11)and f j ( n ) = e πinjL − f ( n ) = (cid:118)(cid:117)(cid:117)(cid:116) (cid:88) j =1 | f j ( n ) | ;6 b < L , , , > L=6L=8L=10L=12L=14L=16 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 b < L L > L=6L=8L=10L=12L=14L=16 b g FIG. 2: The top left panel shows the low lying spectrum of the overlap Dirac operator with theMaxwell measure. The lowest scaled eigenvalue is shown on the top right panel and the presenceof a critical coupling is indicated by the point where the curves at diﬀerent values of L cross eachother. The bottom panel show the square of the gap extrapolated to L = ∞ for β ≤ .

25 alongwith a simple linear regression. (cid:88) k =1 v ∗ jk ( n ) f k ( n ) = 0; j = 1 , (cid:88) m =1 v ∗ jm ( n ) v km ( n ) = δ jk . (12)The zero mode ( n = 0) is not included in the measure. The low lying spectrum of theoverlap Dirac operator, Λ i ; i = 1 , , , L = 6 , , , , , β = 0 .

05 to β = 1 . β = 0 .

05 on 100 independentconﬁgurations at each L and β . The results are plotted in the top left panel of Figure 2.The spectrum shows the discrete nature at ﬁnite volume and the eigenvalues themselves godown as L increases. To show the presence of a transition from strong coupling to weakcoupling, we plot the scaled lowest eigenvalue, Λ L , in the top right panel of Figure 2. Thereis clear evidence for a critical coupling, β c ≈ .

28. The spectrum has a gap in the strongcoupling side ( β < β c ). The spectrum is gapless in the weak coupling side ( β > β c ) sincethe scaled eigenvalue itself is going down as L increases. The scaled eigenvalue on the weakcoupling side falls faster away from β → ∞ and this is consistent with the presence of acondensate that properly scales with β . A careful matching with random matrix theoryshould result in a condensate Σ( β ) for β > β c . We do not pursue this direction here since avalue of the condensate has already been numerically computed [14]. In order to emphasizethe presence of a gap on the strong coupling side, we extrapolated the lowest eigenvalue, Λ ,as a function of L to a gap at L = ∞ using a ﬁt of the form g + a L + a L . The square of thegap, g , so obtained with errors obtained by single elimination jackknife has been plotted for β ≤ .

25 in the bottom panel of Figure 2. The square of the gap at β = 0 . , . , . , .

25 ﬁta simple linear regression quite well with an estimate of the critical coupling consistent with β c ≈ .

28. If this simple minded analysis survives further scrutiny, the transition from thestrong coupling (gapped side) to weak coupling (gapless side) is a second order transitionwith Gaussian exponents.

B. Spectral gap for a conformal measure

The conformal measure given in Eq. (4) amounts to changing the weight of each modein the Maxwell measure (c.f. Eq. (10)) to (cid:89) n exp (cid:34) − L βf ( n )2 (cid:88) j =1 (cid:12)(cid:12)(cid:12) ˜ θ j ⊥ ( n ) (cid:12)(cid:12)(cid:12) (cid:35) . (13)8 .1 1 10 100 b < L , , , > L=6L=8L=10L=12 0.01 0.1 1 10 100 b < L , , , > L=6L=8L=10L=12L=14L=160.01 0.1 1 10 100 b < L L > L=6L=8L=10L=12L=14L=16 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 b g FIG. 3: The top left panel shows the low lying spectrum of the overlap Dirac operator coupled tounsmeared gauge ﬁelds generated with the conformal measure. The top right panel shows the lowlying spectrum of the overlap Dirac operator coupled to smeared gauge ﬁelds generated with theconformal measure.The lowest scaled eigenvalue is shown on the bottom panel and the presenceof a critical coupling is indicated by the point where the curves at diﬀerent values of L cross eachother. The bottom right panel show the square of the gap extrapolated to L = ∞ for β ≤ . This measure is of relevance in computing fermionic observables and matching them toobservables in QED coupled to N f ﬂavors (assumed to be even to preserve parity) of twocomponent massless fermions [17]. The coupling β becomes equal to N f when N f → ∞ but one can match β ( N f ) for the entire range of N f ≥

2. To obtain the matching function, β ( N f ), the fermions were coupled to smeared gauge ﬁelds in [17] and we will study the9ﬀect of smearing on the spectral gap in this section. The low lying spectrum of the overlapDirac operator coupled directly to the unsmeared gauge ﬁelds generated by Eq. (13), Λ i ; i =1 , , , L = 6 , , ,

12 for a range of coupling from β = 0 . β = 512 on 100 independent conﬁgurations at each L and β . The results are plottedin the top left panel of Figure 3. Like with the Maxwell measure, the spectrum shows thediscrete nature at ﬁnite volume and the eigenvalues themselves go down as L increases. Agap develops close to β = 1 and this might hinder the matching, β ( N f ), over the entirerange of N f . If the gauge ﬁelds appearing in the overlap Dirac operator are smeared, thegap shifts to smaller β and the results for Λ i ; i = 1 , , , L = 6 , , , , , β = 0 . β = 512 on 100 independent conﬁgurations ateach L and β are shown in the top right panel of Figure 3. To show the presence of atransition from strong coupling to weak coupling, we plot the scaled lowest eigenvalue, Λ L ,with smeared gauge ﬁelds in the bottom left panel of Figure 2. There is clear evidence fora critical coupling, β c ≈ .

05. The spectrum has a gap in the strong coupling side ( β < β c ).The spectrum is gapless in the weak coupling side ( β > β c ) since the scaled eigenvalueitself is going down as L increases. A continuum limit can be obtained at every value of β > β c with this measure and the scaling of the eigenvalue with L at a ﬁxed β gives theanomalous dimension at that β . The anomalous dimension goes down as β increases andthis is consistent with the trend in the dependence of the eigenvalues with L for β > . β > . in [17]. A careful analysisof the anomalous dimension for the entire range of β > β c might be interesting and mightnot even be a monotonic function of β . In order to emphasize the presence of a gap on thestrong coupling side, we extrapolated the lowest eigenvalue, Λ , as a function of L to a gapat L = ∞ using a ﬁt of the form g + a L + a L . The square of the gap, g , so obtained witherrors obtained by single elimination jackknife has been plotted for β ≤ .

05 in the bottomright panel of Figure 3. The square of the gap for β ∈ [0 . , . β ≈ . .2 1 5 25 b < L , , , > L=6L=8L=10L=12L=14L=16 0.2 1 5 25 b < L L > L=6L=8L=10L=12L=14L=160 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 b g b E rr o r i n f it g + a /L + a /L a /L + a /L + a /L FIG. 4: The top left panel shows the low lying spectrum of the overlap Dirac operator coupled togauge ﬁelds generated with the compact Thirring measure. The lowest scaled eigenvalue is shownon the top right panel and the presence of a critical coupling is indicated by the point where thecurves at diﬀerent values of L cross each other. The bottom left panel show the gap extrapolatedto L = ∞ for β ≤ β ∈ [1 . , g (cid:54) = 0 and g = 0. C. Spectral gap for a compact Thirring measure

We now move on to the Thirring measure which is the main new point of this paper. Ouraim is to compare the behavior of the low lying spectrum of the overlap Dirac operator withthe Thirring measure to the ones from the Maxwell and the conformal measure. To thisend, we ﬁrst focus on the compact Thirring measure given in Eq. (7). We generated θ k ( x )with a measure exp[ β (cos( θ k ( x )) − k and x on a L lattice. The11ehavior of the four lowest eigenvalues as a function of β are shown in the top left panel ofFigure 4 for six diﬀerent values of L and the qualitative similarity with the correspondingbehaviors in Figure 2 with the Maxwell measure and Figure 3 with the conformal measureis evident. Like with the Maxwell and the conformal measures, we have plotted the scaledlowest eigenvalue, Λ L as a function of β in the top-right panel of Figure 4. There is evidencefor a critical value around β c ≈

4. Whereas the increase with increasing L below β c is aspronounced as in the Maxwell and conformal measures, the behavior above β c seems to beessentially independent of L and the value of Λ i L is close to π indicating free ﬁeld behavior.Contrary to this behavior, the free ﬁeld behavior with the Maxwell measure has not set iteven at β = 1 . L to go below π as L is increased. Thefree ﬁeld behavior in Λ L does seem to set in with the conformal measure for large enough β but the values stay well below π for a wide range of β along with a variation in the behavioras a function of β . Like in the Maxwell and conformal measures, we extrapolated Λ toobtain a gap using the ﬁt g + a L + a L . The result with errors obtained by single eliminationjackknife is shown in the bottom left panel of Figure 4. There are qualitative diﬀerenceswhen one compares the gap with the Thirring measure to the ones from the Maxwell andconformal measures. The square of the gap with the Maxwell and conformal measures ﬁta simple minded linear regression quite well for g close to 1 and close to 0. On the otherhand, we have plotted g as a function of β for the Thirring model and we see a region inthe gap, namely, [0 . , . β c ≈ . L as a function of β for diﬀerent values of L in the top right panel of Figure 4. To furtherunderstand this discrepancy, we also ﬁtted Λ to a L + a L + a L which assumes there is nogap. The ﬁt errors, namely the value of the least square function, from the two diﬀerent ﬁtsare shown in the bottom right panel of Figure 4. It is clear that a non-zero gap is favoredfor β <

2. It is also reasonably clear that a ﬁt with no gap is favored for β >

8. Theregion, 2 < β <

7, is murky and it is diﬃcult to conclude the presence or the absence ofa gap with the data that is currently available. Combining this with a possible behaviorof g ∼ ( β − .

1) for β ∈ [1 . , b < L , , , > L=6L=8L=10L=12L=14L=16 0 0.5 1 1.5 2 2.5 3 b < L L > L=6L=8L=10L=12L=14L=160 0.5 1 1.5 2 2.5 3 b < L , , , > L=6L=8L=10L=12L=14L=16 0 0.5 1 1.5 2 2.5 3 b < L L > L=6L=8L=10L=12L=14L=16

FIG. 5: The top and bottom left panels shows the low lying spectrum of the overlap Dirac operatorcoupled to gauge ﬁelds generated with the non-compact Thirring measure. The lowest scaledeigenvalue is shown on the top and bottom right panels. The fermions obey anti-periodic boundaryconditions in the top panels whereas they obey periodic boundary conditions in the bottom panels.

III. WHY USE A COMPACT THIRRING MEASURE?

The Thirring measure that arises from converting a four fermi interaction to a bilinearwith an auxillary vector ﬁeld is given by Eq. (5). We generated θ k ( x ) using a measureexp[ − β θ k ( x ))] independently for each k and x on a L lattice and studied the behavior of thelow lying spectrum of the overlap Dirac operator. In this case, we computed the spectrumwith both anti-periodic and periodic boundary conditions for fermions. The results withanti-periodic and periodic boundary conditions are shown in the top and bottom panelsrespectively. The monotonic behavior of the eigenvalues as a function of L as a ﬁxed β seen13n the strong coupling side is absent in the weak coupling side. The presence of a singlecritical coupling is not evident in the plots of the scaled lowest eigenvalue in the plots onthe right panels. Since the fermion always sees the compact gauge ﬁeld, the diﬀerence in thebehavior between the non-compact gauge action (c.f. Eq. (6)) and the compact gauge action(c.f. Eq. (7)) is in suppression of larger Wilson loops in the non-compact action comparedto the compact gauge action. This diﬀerence in the actions plays a strong qualitative rolein the behavior of the spectrum at intermediate coupling which is the region of interest inthe Thirring model. IV. DISCUSSION

We numerically studied the low lying spectrum of the overlap Dirac operator coupled toa compact Abelian gauge ﬁeld. We investigated four diﬀerent measures for the gauge ﬁeld.All of them clearly show a lattice strong coupling phase where the spectrum is gapped:There are no eigenvalues in the range i [ − g, g ]; g >

0. The natural interpretation of thegapped phase is the eﬀect of a random gauge potential that persists up to a certain valueof the coupling constant (see [54] for example). One can change the location of the criticalcoupling where the gap closes by coupling the overlap Dirac operator to a smeared gaugeﬁeld; the transition from the gapped phase to a gapless phase is a lattice transition. Thelocation of the lattice transition will also depend on the choice of the Wilson mass parameter, m w , used in the overlap Dirac operator. One expects to realize a continuum theory in thegapless phase. In order to see if a continuum theory can be realized in the three dimensionalThirring model, we compared the fermion spectrum with a compact Thirring measure to aMaxwell measure and a conformal measure. The Maxwell measure is relevant for QED andthe conformal measure becomes relevant for QED due to its scale invariant behavior. Allthree measures show a clear separation of the two phases. There is reasonable evidence thatthe gap with the Maxwell and conformal measures go to zero as √ β c − β from the strongcoupling side as one approaches the critical coupling. This Gaussian like behavior is notseen with the Thirring measure where there is some indication that the gap behaves closerto ( β c − β ) from the strong coupling side. But the location of the transition is itself not welldetermined based on the data used in this paper and further work has to be done to explorethe intermediate region of coupling and perform a careful ﬁnite volume analysis. We also14tudied the spectrum with the non-compact Thirring measure and the behavior of the lowlying eigenvalues in the potentially gapless phase is non-monotonic in the size of the lattice.This eﬀect is a consequence of suppressing Wilson loops of larger size and can be avoidedby using the compact Thirring action.Our analysis suggests that it would be interesting to study the 2 N ﬂavor compact Thirringmodel with the lattice action including fermionic sources deﬁned by S = β (cid:88) k,x cos θ k ( x ) + N (cid:88) j =1 (cid:0) ¯ ψ j D o ψ j + ¯ ψ j D † o ψ j (cid:1) + N (cid:88) j =1 ¯ η j Aη j . (14)A lattice model similar to this one with staggered fermions and an explicit integrationover all gauge transformations of θ k ( x ) was already proposed in [55]. In order to make aconnection between the above action and the one for QED with 2 N ﬂavors of masslessfermions, consider the generalized continuum action deﬁned by S = 2 N g (cid:90) d x ( ∂ k A j − ∂ j A k ) (cid:20) − ∂ (cid:21) p ( ∂ k A j − ∂ j A k ) + (cid:90) d x ¯ ψ j σ k ( ∂ k + iA k ) ψ j , (15)with the coupling constant, g , having a mass dimension of (1 − p ). The p = 0 theorycorresponding to QED shows scale invariant infra-red behavior for all even number ofﬂavors [12, 13]. Furthermore the induced action for the gauge ﬁelds is well described by p = conformal theory with no dynamical fermions [17]. Theories with p > are notrenormalizable by simple power counting but are shown to be well deﬁned in the large N limit [27–31]. Strictly in the large N limit, the beta function is exactly equal to β ( ¯ α ) = (2 p −

1) ¯ α (1 − ¯ α ) , (16)where ¯ α ( p ) is the dimensionless running coupling constant [4]. This suggests that the infra-red and ultra-violet behavior are ﬂipped when a theory with p is compared to (1 − p ). Itwould be interesting to see if one can realize a continuum theory with p = 1 using Eq. (14)by ﬁrst ﬁnding the location in the lattice coupling where the gap in the fermion spectrumcloses and studying the scaling limit from the weak coupling side. Acknowledgments

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