Dispersion relation and spectral range of Karsten-Wilczek and Borici-Creutz fermions
DDispersion relation and spectral range of
Karsten-Wilczek and Borici-Creutz fermions
Stephan D¨urr a,b and
Johannes H. Weber c a Department of Physics, University of Wuppertal, 42119 Wuppertal, Germany b J¨ulich Supercomputing Centre, Forschungszentrum J¨ulich, 52425 J¨ulich, Germany c Department of Computational Mathematics, Science and Engineering and Department ofPhysics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA
Abstract
We investigate some properties of Karsten-Wilczek and Borici-Creutz fermions, whichare the best known varieties in the class of minimally doubled lattice fermion actions.Our focus is on the dispersion relation and the distribution of eigenvalues in the free-fieldtheory. We consider the situation in two and four space-time dimensions, and we discusshow properties vary as a function of the Wilson-like lifting parameter r . The choice of any lattice fermion action is a bit of a compromise. Ideally, one would wantto realize ultra-locality, chiral symmetry, and absence of doublers (in addition to a correctcontinuum limit, of course). But these are precisely the ingredients which, according to theNielsen-Ninomiya theorem, cannot possibly coexist [1, 2]. Furthermore, in real life computa-tional expedience is an important criterion. It follows that the choice of a lattice action whichis well-suited to the specific needs of a planned numerical investigation is an important decisionwhich impacts the subsequent analysis of the lattice data in a profound manner.Staggered fermions and Wilson fermions represent two popular choices in this context.Staggered fermions put a focus on ultra-locality and chiral symmetry, at the expense of having4 species in the continuum [3]. Wilson fermions, on the other hand, prioritize ultra-locality andabsence of doublers, at the expense of breaking chiral symmetry [4].Staggered fermions seem ideally suited to study theories with four degenerate fermions(or a multiple thereof) in the continuum. The details of taste-breaking [5–9] and the non-commutativity of the continuum limit ( a →
0) and the chiral limit ( m →
0) [10] imposepractical difficulties, but to the best of our knowledge there is no concern about the theoreticalsoundness of this formulation of QCD with N f ∈ N dynamical flavors.Things are different, if one desires to study QCD with fewer than 4 degenerate fermions, suchas real-life QCD where m d , m u , m s , m c are pairwise different. Given the continuum argumentthat N f degenerate fermions would raise the functional determinant of a single species to the N f -th power, Marinari, Parisi and Rebbi suggested by means of “reverse engineering” that onewould take the square-root of the staggered determinant to simulate 2 degenerate fermions orthe quarter-root for a non-degenerate fermion species [11].1 a r X i v : . [ h e p - l a t ] A ug n practice, it seems the approach of “rooting” the staggered determinant yields convincingnumerical results for real-life QCD, with small statistical error bars even at physical values ofthe quark masses m d , m u , m s , m c [12–14], and with some field-theoretic underpinning throughrooted staggered chiral perturbation theory [15–17]. Nevertheless, this approach has beencriticized [18, 19] on the grounds of the argument that – in the presence of the cut-off – no localtheory can be constructed (or has been constructed) that would implement exactly (i.e. down tomachine precision) the square-root of the staggered determinant as the determinant of a valid 2species formulation. There have been several reviews of the issue at lattice conferences [20–24]which essentially collected pieces of evidence in favor of the approach. But it holds true that nostrict mathematical proof has been found, and no side has been able to convince the opponent.Given this situation it is natural to ask whether a local formulation with chiral symmetryand just 2 species (the minimum required by the Nielsen-Ninomiya theorem) would shed somelight on the issue. Since staggered fermions emerge from the naive formulation by an ingeniousprocedure of “thinning out” the degrees of freedom by a factor 2 d/ in d space-time dimensions,one might dream of a similarly ingenious second step that would reduce the degeneracy from 4to 2 in d = 4 dimensions. However, the eigenvalue spectrum of staggered fermions on interactingbackgrounds shows a 4-fold near-degeneracy (i.e. no exact degeneracy) [25, 26], and this meansthat such a second reduction step cannot possibly take place.However, there is a better approach. It is based on adding an extra term (of mass-dimensionfive) to the naive action which lifts 14 of the 16 species in d = 4 dimensions, albeit withthe important difference to the Wilson term that it does not break chiral symmetry. Such“minimally doubled fermions” have been proposed by Karsten [27] and Wilczek [28], and laterby Creutz [29] and Borici [30]. More recently, yet another variety with “twisted ordering”has been proposed by Creutz and Misumi [31]. Also the proposal of augmenting the Karsten-Wilczek action by a “flavored chemical potential term” has been made [32, 33]. From theviewpoint of computational efficiency, all these formulations augur well, since they are ultra-local with on-axis links only. In the literature issues of mixing with lower-dimensional operatorshave been addressed, and how one can defeat them with appropriate tuning strategies [34–42].Also the consistency of these formulations with the index theorem has been verified [43, 44].Furthermore, some minimally doubled actions have been shown to possess an extra “mirrorfermion” symmetry [45], and it has been demonstrated that the Karsten-Wilczek determinantis invariant under all of the discrete symmetries [46].Still, some basic features of minimally doubled fermion actions have hardly been explored,for instance the respective quark-level free-field dispersion relations (including the cut-off effectson a heavy quark mass) and spectral bounds. In this article we try to fill some of thesegaps for Karsten-Wilczek (KW) and Borici-Creutz (BC) fermions. To understand how theseformulations differ from the naive one we think it is useful to introduce a lifting parameter r ,similar to what is done for Wilson fermions. Hence for r = 0 we start with the naive action, andwe expect to see a cascade of species reductions as r increases, eventually realizing 2 species at r = 1. Since chiral symmetry holds throughout, the Nielsen-Ninomiya theorem demands thatthe reduction proceeds in steps of (integer multiples of) 2.The remainder of this article is organized as follows. In Sec. 2 we give a brief review ofthe situation with naive and Wilson fermions, mostly to specify our notation. The results forKW fermions are presented in Sec. 3, and those for BC fermions in Sec. 4. We summarize ourfindings in Sec. 5, and more lengthy calculations are arranged in appendices A-E.2 Naive and Wilson fermions
Throughout this article ∂ µ and ∂ ∗ µ denote the discrete forward an backward derivative, respec-tively, and ∇ µ = ( ∂ µ + ∂ ∗ µ ) / ∇ µ ψ ( x ) = 12 (cid:104) U µ ( x ) ψ ( x + ˆ µ ) − U † µ ( x − ˆ µ ) ψ ( x − ˆ µ ) (cid:105) (1)where U µ ( x ) is the parallel transporter from x + ˆ µ to x , and ˆ µ denotes a times the unit-vectorin direction µ . Similarly, (cid:52) µ = ∂ ∗ µ ∂ µ = ∂ µ ∂ ∗ µ denotes the second discrete derivative, that is (cid:52) µ ψ ( x ) = U µ ( x ) ψ ( x + ˆ µ ) − ψ ( x ) + U † µ ( x − ˆ µ ) ψ ( x − ˆ µ ) (2)in the presence of a gauge field U µ ( x ).With this notation the naive Dirac operator is defined as D nai ( x, y ) = (cid:88) µ γ µ ∇ µ ( x, y ) + mδ x,y (3)where the anti-hermitean behavior ∇ † µ = −∇ µ and the anti-commutation property { γ µ , γ } = 0of the hermitean γ -matrices imply that the naive Dirac operator is γ -hermitean, i.e. γ D nai γ = D † nai . In the free-field limit this operator assumes a diagonal form in momentum space, D nai ( p ) = i (cid:88) µ γ µ a sin( ap µ ) + m = i (cid:88) µ γ µ ¯ p µ + m with ¯ p µ = 1 a sin( ap µ ) (4)which again highlights the anti-hermitean nature of the derivative (momentum) term.The Wilson Dirac operator follows by adding a hermitean and positive semi-definite termof dimension 5 to the naive Dirac operator D W ( x, y ) = (cid:88) µ γ µ ∇ µ ( x, y ) − ar (cid:88) µ (cid:52) µ ( x, y ) + mδ x,y . (5)The hermitean behavior (cid:52) † µ = (cid:52) µ together with the properties used in the naive case implythat the Wilson (W) operator is γ -hermitean, i.e. γ D W γ = D † W . An unpleasant feature isthat the term (cid:80) µ (cid:52) µ ( x, y ) mixes (on interacting gauge backgrounds) with the identity. As aresult, the bare mass m in (5) gets renormalized, and chiral symmetry is broken [47]. In thefree-field limit the Wilson operator assumes a diagonal form in momentum space, D W ( p ) = i (cid:88) µ γ µ a sin( ap µ ) + ar (cid:88) µ { − cos( ap µ ) } + m = i (cid:88) µ γ µ ¯ p µ + ar (cid:88) µ ˆ p µ + m with ˆ p µ = 2 a sin( ap µ r = 1 the 15 unwanted species do not propagate in anyof the on-axis directions [47]. 3 Naive operator, L/a=64, am=0
Naive operator, L/a=64, am=0.5
Figure 1: Free-field dispersion relation aE versus a | (cid:126)p | of the naive Dirac operator at am = 0and am = 0 .
5. The dashed curves give the continuum dispersion relations, and the vertical linesshow the end of the Brillouin zone in the (1 , , , , , ,
1) directions, respectively.4
Wilson operator, L/a=64, am=0, r=1
Wilson operator, L/a=64, am=0.5, r=1
Figure 2: Same as Fig. 1, but for the Wilson operator at r = 1.5 Wilson, L/a=32, r=1
Figure 3: Free-field Wilson eigenvalue spectrum at r = 1 in the complex plane.With these expressions in hand, we are in a position to study the quark-level free-fielddispersion relations. Both for naive and Wilson fermions, the energy aE can be given as ananalytic function of the spatial momentum a(cid:126)p , see App. A for a brief account of this standardcalculation. The results are shown in Fig. 1 for the naive formulation and in Fig. 2 for theWilson action at r = 1. In either figure the situation at am = 0 and at am = 0 . a | (cid:126)p | = π, √ π, √ π the naive action features well for small enough momenta. In particularat a | (cid:126)p | = 0 it features better than the Wilson action, since the gap to the continuum curve issmaller. This can be understood on analytical grounds, too. The energy at zero momentumis nothing but the heavy quark mass. As detailed in App. B, it is known to be afflicted withcut-off effects O (( am ) ) in the naive case, but O ( am ) in the Wilson case.From expression (4) or (6) one finds the eigenvalues of the free-field operators. Since each γ µ ( µ = 1 , ..,
4) has eigenvalues ±
1, it follows that the upper end of the massless naive eigenvaluespectrum is realized for ap µ along the hyperdiagonal (1 , , , | Im( λ nai ) | ≤ ≤ Re( λ W ) ≤ r , | Im( λ W ) | ≤ . (8)The complex eigenvalue spectrum of the Wilson operator at am = 0 is shown in Fig. 3. Thesymmetry about the real axis reflects the pairing property imposed by the γ -hermiticity. As6s well known, the five branches in the Wilson eigenvalue spectrum correspond to species withmultiplicities 1 , , , ,
1, and chiralities + , − , + , − , +, respectively [47]. In total 8 species thushave correct chirality, and 8 have opposing chirality. The free-field eigenvalue spectrum of thenaive (or staggered) operator follows by horizontally projecting the λ W onto the imaginaryaxis (and reducing degeneracies by a factor 4 in the staggered case). Under this operation theseparation between right-chirality and opposite-chirality species gets lost, and this feature willcarry on to minimally doubled actions. The Karsten-Wilczek proposal is to restrict the Wilson term in (5) to the spatial components D KW ( x, y ) = (cid:88) µ γ µ ∇ µ ( x, y ) − i ar γ (cid:88) i =1 (cid:52) i ( x, y ) + mδ x,y (9)with an extra factor i γ to make it anti-hermitean and anti-commuting with γ [27, 28]. As aresult the Karsten-Wilczek (KW) operator is γ -hermitean, i.e. γ D KW γ = D † KW . An issuediscussed in the literature is that γ (cid:80) i (cid:52) i ( x, y ) mixes (on interacting backgrounds) with γ [34–42]. In the free-field limit the KW operator assumes a diagonal form in momentum space, D KW ( p ) = i (cid:88) µ γ µ a sin( ap µ ) + i arγ (cid:88) i =1 { − cos( ap i ) } + m = i (cid:88) µ γ µ ¯ p µ + i ar γ (cid:88) i =1 ˆ p i + m (10)which again highlights the anti-hermitean nature of either term. This formulation was shownto have 2 species for r = 1 in the original works [27, 28], but how this number decreasesfrom 16, at r = 0, to the minimally doubled value has, to the best of our knowledge, not beeninvestigated. We find that the number of species is reduced in three steps (in d = 4 dimensions).At r = 1 / , / , / , ,
6, respectively, so the species chainis 16 → → →
2. See App. C for details, e.g. the situation with d (cid:54) = 4.Starting from eqn. (10) one can work out the free-field dispersion relation of KW fermions,see App. A for details. For a given momentum configuration a(cid:126)p the Euclidean energy aE is, ingeneral, complex valued, and its real part is plotted in Fig. 4 for r = 1. Again, two values of thequark mass are used, am = 0 and am = 0 .
5. In either case the KW dispersion relation followsthe continuum curve faithfully, out to momentum values a | (cid:126)p | (cid:39)
1. In particular at a | (cid:126)p | = 0it features much better than the Wilson action, reminiscent of the naive action. This is nota coincidence, since the rest energy has the same functional dependence on am as the naiveaction, see App. B for details. In other words, cut-off effects on this quantity start at O (( am ) )only, unlike the O ( am ) signature of Wilson fermions.From eqn. (10) one finds the eigenvalues of the free-field KW operator. The result for r = 1and am = 0 is shown in Fig. 5. On a 32 lattice one finds 4 · purely imaginary and γ -pairedeigenvalues, as expected. Plotting the eigenvalues against the index means that the inverseslope encodes for the density of the λ KW / i on the imaginary axis.It is instructive to repeat this for a series of r values; the result is shown in Fig. 6, withvertical lines marking the abscissa values r = 1 / , / , / Karsten-Wilczek, L/a=64, am=0, r=1
Karsten-Wilczek, L/a=64, am=0.5, r=1
Figure 4: Same as Fig. 1, but for the Karsten-Wilczek (KW) operator at r = 1.8 -6-4-20246 Karsten-Wilczek, L/a=32, r=1
Figure 5: Free-field eigenvalue spectrum of the KW operator at r = 1. The imaginary part isplotted against the index.The spectral range is seen to increase with growing r . In addition, the low-energy end of theeigenvalue spectrum seems unstable for small r , but stable in a broad range around r = 1.Given Fig. 6, one may wonder about the existence of an analytic function which describesthe upper end as a function of r . In App. E we derive the free-field spectral bound | Im( λ KW ) | ≤ (cid:40) (cid:113) (4 + 6 r ) / (1 − r ) ( r ≤ / r ( r ≥ /
3) (11)in d = 4 dimensions. Hence at r = 1 the imaginary parts λ KW / i cover the range [ − , − ,
2] for naive and staggered fermions. On the other hand, the smallest non-zeroKW eigenvalue is found in essentially the same place as the smallest staggered eigenvalue, seeFig. 7. This amounts to an enhancement of the condition number of D † D , compared to thestaggered formulation at the same am , by a factor up to 3 . = 12 .
25 (in the chiral limit). In Fig. 7 one finds the small (in absolute magnitude) KW eigenvalues by projecting the black dots ontothe y -axis, and the staggered counterparts by projecting them onto the x -axis. Hence, min( | λ KW | ) (cid:39) .
2, andmin( | λ stag | ) (cid:39) . box, if we disregard the non-topological zero-modes. In large boxes the spectral gapdecreases as 1 /L , so we anticipate min( | λ | ) (cid:39) . box for both KW and staggered fermions. r . The vertical linesat r = 1 / , / , / Karsten-Wilczek versus staggered, L/a=32, r=1
Figure 7: Sorted free-field eigenvalues (with a 2-fold degeneracy removed) of the KW operatorat r = 1 plotted versus sorted staggered eigenvalues (with a 4-fold degeneracy removed). Inboth cases the imaginary part Im( λ ) = λ/ i at am = 0 is used, and plenty of degeneraciesremain. The dotted line shows the identity for comparison. The basis for Borici-Creutz fermions in d space-time dimensions is the idempotent operatorΓ = 1 √ d (cid:88) µ γ µ with Γ = 12 d { (cid:88) α γ α , (cid:88) β γ β } = 2 d d = 1 (12)and { Γ , γ µ } = √ d and { Γ , γ } = 0. This suggests to define the set of dual gamma-matrices γ (cid:48) µ = Γ γ µ Γ = (cid:16) √ d − γ µ Γ (cid:17) Γ = 2 √ d Γ − γ µ (13)which are hermitean and satisfy the Dirac-Clifford algebra, since (13) implies { γ (cid:48) µ , γ (cid:48) ν } = 2 δ µν and { Γ , γ (cid:48) µ } = √ d . Furthermore, one finds { γ µ , γ (cid:48) ν } = γ µ ( 2 √ d Γ − γ ν ) + ( 2 √ d Γ − γ ν ) γ µ = 2 √ d { γ µ , Γ } − { γ µ , γ ν } = 4 d − δ µν (14) { γ (cid:48) µ , γ ν } = ( 2 √ d Γ − γ µ ) γ ν + γ ν ( 2 √ d Γ − γ µ ) = 2 √ d { Γ , γ ν } − { γ µ , γ ν } = 4 d − δ µν . (15)The Borici-Creutz (BC) proposal is to dress the Wilson term in (5) with i times (13), i.e. D BC ( x, y ) = (cid:88) µ γ µ ∇ µ ( x, y ) − i ar (cid:88) µ γ (cid:48) µ (cid:52) µ ( x, y ) + mδ x,y (16)11here our second term differs in sign from the original proposal [30]. Note that the secondterm is anti-hermitean and anti-commutes with γ , since γ (cid:48) µ γ = Γ γ µ Γ γ = − Γ γ µ γ Γ = Γ γ γ µ Γ = − γ Γ γ µ Γ = − γ γ (cid:48) µ (17)and this renders the BC operator γ -hermitean, i.e. γ D BC γ = D † BC . An issue discussed inthe literature is whether (cid:80) µ γ (cid:48) µ (cid:52) µ mixes (on interacting backgrounds) with Γ [34–42]. In thefree-field limit the BC operator assumes a diagonal form in momentum space, D BC ( p ) = i (cid:88) µ γ µ ¯ p µ + i ar (cid:88) µ γ (cid:48) µ { − cos( ap µ ) } + m = i (cid:88) µ γ µ ¯ p µ + i ar (cid:88) µ γ (cid:48) µ ˆ p µ + m (18)in which the bracket { − cos( ap µ ) } may be split and the sum over γ (cid:48) µ performed by means of (cid:88) µ γ (cid:48) µ = 2 √ d Γ − (cid:88) µ γ µ = 2 √ d Γ − √ d Γ = √ d Γ . (19)Furthermore, the free-field form (18) highlights the invariance under any permutation of the d axes. In App. C we discuss, for d = 4, how the number of species is reduced by 6 at r = 1 / √ r = 1 / √
2; so the species chain is 16 → →
2. Of course, the number of speciesis unchanged by a sign flip of r .In two (Euclidean) space-time dimensions it is customary to use γ = σ , γ = σ . Similarly,the chirality operator is defined as γ = − i γ γ = − i σ σ = σ . Upon using the simplificationsΓ = 1 √ (cid:16) σ + σ (cid:17) = 1 √ (cid:18) − i1 + i 0 (cid:19) = (cid:18) e − i π/ e +i π/ (cid:19) (20) σ (cid:48) = Γ σ Γ = 12 ( σ + σ ) σ ( σ + σ ) = 12 ( σ + σ + σ + σ σ σ ) = σ (21) σ (cid:48) = Γ σ Γ = 12 ( σ + σ ) σ ( σ + σ ) = 12 ( σ σ σ + σ + σ + σ ) = σ (22)the operators (9) and (16) are seen to take the simple form D KW ( x, y ) = (cid:88) µ σ µ ∇ µ ( x, y ) − i ar σ (cid:52) ( x, y ) + mδ x,y (23) D BC ( x, y ) = (cid:88) µ σ µ ∇ µ ( x, y ) − i ar σ (cid:52) ( x, y ) − i ar σ (cid:52) ( x, y ) + mδ x,y (24)which shows that the BC operator is not a symmetrized form of the KW operator; it has anextra term. This explains why we deviate, in the sign of the mass-dimension 5 term in eqns. (16,18), from the literature. With our convention the joint terms in eqns. (23, 24) have like sign.Starting from eqn. (18) one can work out the free-field dispersion relation of BC fermions,see App. A for details. For a given momentum configuration a(cid:126)p the Euclidean energy aE is,in general, complex valued, and its real part is plotted in Fig. 8 for r = 1. Again, two valuesof the quark mass are used, am = 0 and am = 0 .
5. In either case the BC dispersion relationfollows the continuum curve reasonably well, out to momentum values a | (cid:126)p | (cid:39) , a range slightlynarrower than what was found in the KW case. Specifically at a | (cid:126)p | = 0 it features much betterthan the Wilson action, though a little worse than the KW action. In App. B the rest energy12 Borici-Creutz operator, L/a=64, am=0, r=1
Borici-Creutz operator, L/a=64, am=0.5, r=1
Figure 8: Same as Fig. 1, but for the Borici-Creutz (BC) operator at r = 1.13 -4-3-2-101234 Borici-Creutz, L/a=32, r=1
Figure 9: Free-field eigenvalue spectrum of the BC operator at r = 1. The imaginary part isplotted against the index.of a heavy BC fermion is found to have a contribution ∝ ( am ) , nearly as good as ∝ ( am ) of the KW action (both values are for r = 1 in d = 4 dimensions). Unlike the KW energy, theBC energy has an imaginary contribution ∝ am , which is not a desirable property. What isparticularly disconcerting in the free-field dispersion relation of a BC fermion at heavy quarkmass is that the global minimum is not necessarily at a | (cid:126)p | = 0. For am = 0 . r = 1and am = 0 is shown in Fig. 9. Similar to the KW case, eigenvalues come in γ -pairs and arepurely imaginary. The only difference to Fig. 5 is that the BC range is slightly narrower.It is instructive to repeat this for a series of r values; the result is shown in Fig. 10, withvertical lines marking the abscissa values r = 1 / √ , / √ r . In addition, the low-energy end of theeigenvalue spectrum seems far less stable than in the KW case, the canonical choice r = 1seems to represent a small island of stability.Given Fig. 10, one may wonder about the existence of an analytic function which describesthe upper end as a function of r . In App. E we derive the free-field spectral bound | Im( λ BC ) | ≤ r + √ r ) (25)in d = 4 dimensions. Hence at r = 1 the imaginary parts λ BC / i cover the symmetric range[ − . , √ − ,
2] for naive and staggered fermions. For r = 1 the14igure 10: Imaginary part of the free eigenvalues of the BC operator in linear and logarithmicrepresentation (for the upper half-spectrum) versus the lifting parameter r . The vertical linesat r = 1 / √ , / √ Borici-Creutz versus staggered, L/a=32, r=1
Figure 11: Sorted free-field eigenvalues (with a 2-fold degeneracy removed) of the BC operatorat r = 1 plotted versus sorted staggered eigenvalues (with a 4-fold degeneracy removed). Inboth cases the imaginary part Im( λ ) = λ/ i at am = 0 is used, and plenty of degeneraciesremain. The dotted line shows the identity for comparison.smallest non-zero BC eigenvalue is found in essentially the same place as the smallest staggeredeigenvalue, see Fig. 11. This amounts to an enhancement of the condition number of D † D ,compared to the staggered formulation at the same am , by a factor up to (1 + √ = 5 . In this paper we tried to fill some of the most obvious gaps in the knowledge about the two mostpopular minimally doubled fermion actions, namely the formulations due to Karsten-Wilczek(KW) and Borici-Creutz (BC), respectively. The gaps concern the eigenvalue spectra and thedispersion relations (including the leading cut-off effects on the heavy fermion mass) in thefree-field limit. We studied these issues as a function of the lifting parameter r , in order to seehow the number of species gets reduced from 16 (at r = 0) to 2 (at r = 1). Our investigationwas limited to KW and BC fermions, but there are two more approaches, “twisted ordering” In Fig. 11 one finds the small (in absolute magnitude) BC eigenvalues by projecting the black dots ontothe y -axis, and the staggered counterparts by projecting them onto the x -axis. Hence, min( | λ BC | ) (cid:39) .
2, andmin( | λ stag | ) (cid:39) . box, if we disregard the non-topological zero-modes. In large boxes the spectral gapdecreases as 1 /L , so we anticipate min( | λ | ) (cid:39) . box for both BC and staggered fermions. r can be introduced tostudy how the number of species gets reduced [31–33].Regarding the eigenvalue spectra we find an extension, relative to the staggered/naive one,by a factor 3 . √ (cid:39) . r = 1 andvanishing quark mass). This leads to an enhancement of the condition number of D † D (asrelevant for generating dynamical ensembles) by a factor up to 12 .
25, or 5 . | λ | ) = Im( λ ) + ( am ) , with Im( λ ) given by (11, 25), respectively.In addition, we studied the dispersion relations. On the one hand, we find that the KWoperator features very well in this respect. It follows the continuum dispersion relation moreclosely than the Wilson operator. In particular at a(cid:126)p = (cid:126) O (( am ) ), just like the naive/staggered action, not at O ( am ) like theWilson operator. On the other hand, the dispersion relation of the BC operator in d = 4dimensions shows some more problematic features, including a funny behavior at small a | (cid:126)p | and an imaginary part of the heavy quark rest mass which starts at O ( am ).Obviously, there remain many unexplored issues with these fermion formulations. We thinkit would be interesting to study the behavior of small eigenvalues on interacting backgrounds(especially some with non-zero topological charge), and how they implement the constraintsimposed by the Nielsen-Ninomiya theorem. Our Figs. 7 and 11 are inspired by Figs. 6 and 7of Ref. [26], and we hope to see the “fingerprint property” of low-energy fermion eigenvalues confirmed with the KW and BC formulations, too. Also some more light on the mixing patternwith lower-dimensional operators (beyond what was found in [34–42]) might prove useful. Over-all, we feel a collaboration aiming for exploratory large-scale production runs with minimallydoubled fermions would be well advised to give first priority to the KW formulation. Acknowledgements : This work was supported by the German DFG through the collaborativeresearch grant SFB-TRR-55. This work was supported by the U.S. Department of Energy, Officeof Science, Office of Nuclear Physics and Office of Advanced Scientific Computing Researchwithin the framework of Scientific Discovery through Advance Computing (SciDAC) awardComputing the Properties of Matter with Leadership Computing Resources. By this we mean that the pattern of low-energy Dirac operator eigenvalues is characteristic of the gaugebackground and nearly independent of the fermion formulation, at least at small enough lattice spacings. Dispersion relations
A.1 Naive fermions
The naive operator and its Green’s function take the form D nai = (cid:88) µ γ µ ∇ µ + m = i (cid:88) µ γ µ ¯ p µ + m (26) G nai = − i (cid:80) µ γ µ ¯ p µ + m (i (cid:80) ρ γ ρ ¯ p ρ + m )( − i (cid:80) σ γ σ ¯ p σ + m ) = − i (cid:80) µ γ µ ¯ p µ + m ¯ p + m . (27)The dispersion relation follows from searching for zeros of the denominator with p → i E , so0 = (cid:88) i sin ( ap i ) − sinh ( aE ) + ( am ) (28)means that the physical solution is given by the positive root aE = (cid:115) asinh (cid:16) (cid:88) i sin ( ap i ) + ( am ) (cid:17) . (29) A.2 Wilson fermions
The Wilson operator and its Green’s function take the form D W = (cid:88) µ γ µ ∇ µ − ar (cid:52) + m = i (cid:88) µ γ µ ¯ p µ + ar p + m (30) G W = − i (cid:80) µ γ µ ¯ p µ + ar ˆ p + m (i (cid:80) ρ γ ρ ¯ p ρ + ar ˆ p + m )( − i (cid:80) σ γ σ ¯ p σ + ar ˆ p + m ) = − i (cid:80) µ γ µ ¯ p µ + ar ˆ p + m ¯ p + ( ar ˆ p + m ) (31)with ¯ p µ = a sin( ap µ ) and ˆ p µ = a sin( ap µ ). It follows thatˆ p = (cid:88) µ ˆ p µ = 4 a (cid:88) µ sin ( ap µ da − a (cid:88) µ cos( ap µ ) (32)or ar ˆ p = dra − ra (cid:80) µ cos( ap µ ), and searching for a zero of the denominator with p → i E yieldssinh ( aE ) = (cid:88) i sin ( ap i ) + (cid:16) dr − r cosh( aE ) − r (cid:88) i cos( ap i ) + am (cid:17) (33)= (cid:88) i sin ( ap i ) + r cosh ( aE ) − r cosh( aE ) (cid:104) dr + am − r (cid:88) i cos( ap i ) (cid:105) + (cid:104) ... (cid:105) . For r = 1 the identity cosh − sinh = 1 turns this into a linear equation in cosh( aE )2 cosh( aE ) (cid:104) d + am − (cid:88) i cos( ap i ) (cid:105) = 1 + (cid:88) i sin ( ap i ) + (cid:104) d + am − (cid:88) i cos( ap i ) (cid:105) (34)which one solves for aE > x ) = ln( x + √ x −
1) for x >
1. For r (cid:54) = 1 onestays with a quadratic equation in cosh( aE )0 = 1 + (cid:88) i sin ( ap i ) + ( r −
1) cosh ( aE ) − r cosh( aE ) (cid:104) dr + am − r (cid:88) i cos( ap i ) (cid:105) + (cid:104) ... (cid:105) (35)which one addresses by first solving for a real positive cosh( aE ) and then inverting the cosh.18 .3 Karsten-Wilczek fermions The KW operator and its Green’s function take the form D KW = (cid:88) µ γ µ ∇ µ − i ar γ d d − (cid:88) i =1 (cid:52) i + m = i (cid:88) µ γ µ ¯ p µ + i ar γ d d − (cid:88) i =1 ˆ p i + m (36) G KW = − i (cid:80) µ γ µ ¯ p µ − i ar γ d (cid:80) d − i =1 ˆ p i + m (i (cid:80) ρ γ ρ ¯ p ρ + i ar γ d (cid:80) d − i =1 ˆ p i + m )( − i (cid:80) σ γ σ ¯ p σ − i ar γ d (cid:80) d − j =1 ˆ p j + m )= − i (cid:80) µ γ µ ¯ p µ − i ar γ d (cid:80) d − i =1 ˆ p i + m ( (cid:80) d − i =1 γ i ¯ p i + γ d ¯ p d + ar γ d (cid:80) d − i =1 ˆ p i ) + m = − i (cid:80) µ γ µ ¯ p µ − i ar γ d (cid:80) d − i =1 ˆ p i + m (cid:80) d − i =1 ¯ p i + (¯ p d + ar (cid:80) d − i =1 ˆ p i ) + m (37)where in the last step specific properties of the Dirac-Clifford algebra were used. Searching fora zero of the denominator with ar (cid:80) i ˆ p i = ra (cid:80) i { − cos( ap i ) } and p → i E yields0 = d − (cid:88) i =1 sin ( ap i ) + (cid:16) i sinh( aE ) + r d − (cid:88) i =1 { − cos( ap i ) } (cid:17) + ( am ) (38)which does not necessarily yield a real solution for E . In such a situation one should go for acomplex E , and treat its real part as the “energy” of the respective mode. In other wordssinh( aE ) = i r d − (cid:88) i =1 { − cos( ap i ) } ± (cid:118)(cid:117)(cid:117)(cid:116) d − (cid:88) i =1 sin ( ap i ) + ( am ) (39)yields a complex sinh( aE ), and through the asinh function the definition of a complex aE isobtained, whose positive real part is plotted against (cid:113)(cid:80) d − i =1 p i . A.4 Borici-Creutz fermions
The BC operator and its Green’s function take the form D BC = (cid:88) µ γ µ ∇ µ − i ar (cid:88) µ γ (cid:48) µ (cid:52) µ + m = i (cid:88) µ γ µ ¯ p µ + i ar (cid:88) µ γ (cid:48) µ ˆ p µ + m (40) G BC = − i (cid:80) µ γ µ ¯ p µ − i ar (cid:80) µ γ (cid:48) µ ˆ p µ + m (i (cid:80) ρ γ ρ ¯ p ρ + i ar (cid:80) ρ γ (cid:48) ρ ˆ p ρ + m )( − i (cid:80) σ γ σ ¯ p σ − i ar (cid:80) σ γ (cid:48) σ ˆ p σ + m ) (41)= − i (cid:80) µ γ µ ¯ p µ − i ar (cid:80) µ γ (cid:48) µ ˆ p µ + m (cid:80) ρ,σ γ ρ γ σ ¯ p ρ ¯ p σ + ar (cid:80) ρ,σ γ ρ γ (cid:48) σ ¯ p ρ ˆ p σ + ar (cid:80) ρ,σ γ (cid:48) ρ γ σ ˆ p ρ ¯ p σ + a r (cid:80) ρ,σ γ (cid:48) ρ γ (cid:48) σ ˆ p ρ ˆ p σ + m and our task is to further simplify the denominator. The first term is symmetric in ¯ p ρ ↔ ¯ p σ ;it may be rewritten as (cid:80) ρ,σ { γ ρ , γ σ } ¯ p ρ ¯ p σ = (cid:80) λ ¯ p λ , where the Dirac-Clifford property of the γ -matrices has been used. For exactly the same reason the fourth term may be rewritten as a r (cid:80) ρ,σ { γ (cid:48) ρ , γ (cid:48) σ } ˆ p ρ ˆ p σ = a r (cid:80) λ ˆ p λ , where the Dirac-Clifford property of the γ (cid:48) -matrices hasbeen used. The two cross-terms are a bit trickier to deal with. It proves useful to notice thatthe second term can be inflated to look like ar (cid:80) ρ,σ γ ρ γ (cid:48) σ ¯ p ρ ˆ p σ + ar (cid:80) ρ,σ γ σ γ (cid:48) ρ ¯ p σ ˆ p ρ . Similarly, thethird term can be brought into the form ar (cid:80) ρ,σ γ (cid:48) ρ γ σ ˆ p ρ ¯ p σ + ar (cid:80) ρ,σ γ (cid:48) σ γ ρ ˆ p σ ¯ p ρ . Accordingly, the19econd and third terms can be combined into ar (cid:80) ρ,σ { γ ρ , γ (cid:48) σ } ¯ p ρ ˆ p σ + ar (cid:80) ρ,σ { γ (cid:48) ρ , γ σ } ˆ p ρ ¯ p σ , andthe relations (14, 15) suggest replacing the latter expression by ard (cid:80) ρ,σ ¯ p ρ ˆ p σ − ar (cid:80) λ ¯ p λ ˆ p λ + ard (cid:80) ρ,σ ˆ p ρ ¯ p σ − ar (cid:80) λ ˆ p λ ¯ p λ . Putting everything together we thus arrive at G BC = − i (cid:80) µ γ µ ¯ p µ − i ar (cid:80) µ γ (cid:48) µ ˆ p µ + m (cid:80) λ ¯ p λ − ar (cid:80) λ ¯ p λ ˆ p λ + a r (cid:80) λ ˆ p λ + ard (cid:80) ρ,σ ¯ p ρ ˆ p σ + m (42)and our task is to search for a zero of the denominator, i.e. to solve0 = (cid:88) λ ¯ p λ − ar (cid:88) λ ¯ p λ ˆ p λ + a r (cid:88) λ ˆ p λ + 2 ard (cid:88) ρ,σ ¯ p ρ ˆ p σ + m = (cid:88) λ (cid:104) ¯ p λ − ar p λ (cid:105) + 2 ard (cid:88) ρ,σ ¯ p ρ ˆ p σ + m (43)with the substitution p → i E for aE . Using ¯ p ρ = a sin( ap ρ ) and ˆ p σ = a { − cos( ap σ ) } yields0 = (cid:88) λ (cid:104) sin( ap λ ) − r { − cos( ap λ ) } (cid:105) + 4 rd (cid:88) ρ,σ sin( ap ρ ) { − cos( ap σ ) } + ( am ) (44)which the substitution then brings into the form (with i, j running from 1 to d − (cid:88) i (cid:104) sin( ap i ) − r { − cos( ap i ) } (cid:105) + (cid:104) i sinh( aE ) − r { − cosh( aE ) } (cid:105) + 4 rd (cid:88) i,j sin( ap i ) { − cos( ap j ) } + 4i rd sinh( aE ) (cid:88) j { − cos( ap j ) } + 4 rd (cid:88) i sin( ap i ) { − cosh( aE ) } + 4i rd sinh( aE ) { − cosh( aE ) } + ( am ) . (45)In d = 2 space-time dimensions this expression simplifies to (each sum contains a single term)0 = (cid:88) i (cid:104) sin( ap i ) − r { − cos( ap i ) } (cid:105) − sinh ( aE ) + r { − cosh( aE ) } + 2 r (cid:88) i,j sin( ap i ) { − cos( ap j ) } + 2i r sinh( aE ) (cid:88) j { − cos( ap j ) } + 2 r (cid:88) i sin( ap i ) { − cosh( aE ) } + ( am ) (46)while in d = 4 space-time dimensions one finds0 = (cid:88) i (cid:104) sin( ap i ) − r { − cos( ap i ) } (cid:105) − sinh ( aE ) + r { − cosh( aE ) } + r (cid:88) i,j sin( ap i ) { − cos( ap j ) } + i r sinh( aE ) (cid:88) j { − cos( ap j ) } + r (cid:88) i sin( ap i ) { − cosh( aE ) } − i r sinh( aE ) { − cosh( aE ) } + ( am ) . (47)In the special case r = 1 the d = 2 version simplifies to0 = (cid:88) i (cid:104) sin( ap i ) − { − cos( ap i ) } (cid:105) + (cid:104) (cid:88) i sin( ap i ) (cid:105) { − cosh( aE ) } + 2 (cid:88) i,j sin( ap i ) { − cos( ap j ) } + 2i sinh( aE ) (cid:88) j { − cos( ap j ) } + ( am ) (48)20hile the d = 4 version takes the form0 = (cid:88) i (cid:104) sin( ap i ) − { − cos( ap i ) } (cid:105) + (cid:104) (cid:88) i sin( ap i ) − i sinh( aE ) (cid:105) { − cosh( aE ) } + (cid:88) i,j sin( ap i ) { − cos( ap j ) } + i sinh( aE ) (cid:88) j { − cos( ap j ) } + ( am ) . (49)These equations look complicated, and this is why we shall work our way backwards, from thesimplest case to the more complicated case.A peculiar feature of the d = 2 , r = 1 case is that the equation is linear in sinh( aE ) andcosh( aE ). This suggests multiplying eqn. (48) with exp( aE ) to obtain0 = (cid:88) i (cid:104) sin( ap i ) − { − cos( ap i ) } (cid:105) e aE + (cid:104) (cid:88) i sin( ap i ) (cid:105) { e aE − e aE − } + 2 (cid:88) i,j sin( ap i ) { − cos( ap j ) } e aE + i[ e aE − (cid:88) j { − cos( ap j ) } + ( am ) e aE (50)which is a quadratic equation in e aE . Evidently, this means that we should go for the twocomplex e aE as function of ap , to obtain a complex aE whose positive real part is plottedagainst | p | . By contrast, the d = 4 , r = 1 case has a mixed term in sinh( aE ) cosh( aE ).The hyperbolic semi-angle substitution t = tanh( aE/ aE ) = 2 t/ (1 − t ),cosh( aE ) = (1 + t ) / (1 − t ) and 1 − cosh( aE ) = − t / (1 − t ), turns eqn. (49) into0 = (cid:88) i (cid:104) sin( ap i ) − { − cos( ap i ) } (cid:105) − (cid:104) (cid:88) i sin( ap i ) − t − t (cid:105) t − t + (cid:88) i,j sin( ap i ) { − cos( ap j ) } + 2i t − t (cid:88) j { − cos( ap j ) } + ( am ) (51)and upon multiplying this equation with (1 − t ) one finds the (possibly modified) condition0 = (cid:88) i (cid:104) sin( ap i ) − { − cos( ap i ) } (cid:105) (1 − t ) − (cid:104) (cid:88) i sin( ap i ) (cid:105) t (1 − t ) + 4i t + (cid:88) i,j sin( ap i ) { − cos( ap j ) } (1 − t ) + 2i (cid:88) j { − cos( ap j ) } t (1 − t ) + ( am ) (1 − t ) (52)which amounts to a fourth-order polynomial in t .For generic r we resort to the hyperbolic semi-angle substitution, regardless of the space-time dimension. For d = 2 we obtain the relation0 = (cid:88) i (cid:104) sin( ap i ) − r { − cos( ap i ) } (cid:105) + 4[ r − t (1 − t ) + 2 r (cid:88) i,j sin( ap i ) { − cos( ap j ) } + 2i r t − t (cid:88) j { − cos( ap j ) }− r (cid:88) i sin( ap i ) 2 t − t + ( am ) (53)and upon multiplying this equation with (1 − t ) one finds the (possibly modified) condition0 = (cid:88) i (cid:104) sin( ap i ) − r { − cos( ap i ) } (cid:105) (1 − t ) + 4[ r − t + 2 r (cid:88) i,j sin( ap i ) { − cos( ap j ) } (1 − t ) + 4i r (cid:88) j { − cos( ap j ) } t (1 − t ) − r (cid:88) i sin( ap i ) t (1 − t ) + ( am ) (1 − t ) (54)21hich amounts to a fourth-order polynomial in t . Note that for r = 1 the second term ineqn. (53) vanishes. It is then sufficient to multiply the equation with 1 − t , and one ends upwith a quadratic polynomial in t (equivalent to the procedure used above). In other words,after setting r = 1 and dropping a factor 1 − t eqn. (54) is equivalent to eqn. (50). For d = 4the same semi-angle substitution yields0 = (cid:88) i (cid:104) sin( ap i ) − r { − cos( ap i ) } (cid:105) − t (1 − t ) + 4 r t (1 − t ) + r (cid:88) i,j sin( ap i ) { − cos( ap j ) } + i r (cid:88) j { − cos( ap j ) } t − t − r (cid:88) i sin( ap i ) 2 t − t + i r t − t t − t + ( am ) (55)and upon multiplying this equation with (1 − t ) one finds the (possibly modified) condition0 = (cid:88) i (cid:104) sin( ap i ) − r { − cos( ap i ) } (cid:105) (1 − t ) − t + 4 r t + r (cid:88) i,j sin( ap i ) { − cos( ap j ) } (1 − t ) + 2i r (cid:88) j { − cos( ap j ) } t (1 − t ) − r (cid:88) i sin( ap i ) t (1 − t ) + 4i rt + ( am ) (1 − t ) (56)which amounts to a fourth-order polynomial in t . Upon setting r = 1 eqn. (56) simplifies toeqn. (52) without further ado.Using the built-in capabilities of a computer algebra program or a numerical package suchas matlab/octave, it is straight-forward to find all (in general complex-valued) solutions to afourth-order polynomial with given numerical coefficients. In this spirit we evaluate, for a given( p , p , p ) configuration, the four solutions t and apply aE = 2 atanh( t ) to obtain the energies.The one with the smallest positive real part is interpreted as the energy of the fermion in thatmomentum configuration, and its imaginary part gives the damping of the pertinent mode.This is the numerical basis of all dispersion relations shown in this article. On the analyticalside, one may proceed one step further upon expanding the physical solution in powers of am .This yields results relevant to assess the suitability of these actions for heavy-quark physics, asdiscussed in the main part of the article and App. B. B Suitability for heavy-quark physics
B.1 Naive fermions At a(cid:126)p = (cid:126) aE ) = am (57)and this means that the series expansion in powers of am takes the form aE = am (cid:26) −
16 ( am ) + 340 ( am ) + O (( am ) ) (cid:27) . (58)Hence, the rest-mass of a fermion in the naive discretization has cut-off effects O (( am ) ).22 .2 Wilson fermions At a(cid:126)p = (cid:126) d and r = 1 simplifies tocosh( aE ) = 12(1 + am ) + 1 + am aE ) = 1 + am , that is for aE = log(1 + am ). The series expansion aE = am (cid:26) − am + 13 ( am ) −
14 ( am ) + O (( am ) ) (cid:27) (60)shows that such cut-off effects scale as O ( am ). For arbitrary d and generic r one starts fromthe quadratic equation ( r −
1) cosh ( aE ) − r ( r + am ) cosh( aE ) + 1 + ( r + am ) = 0 whereuponcosh( aE ) = r ( r + am ) ± (cid:113) ram + ( am ) r − r →
1, with the solution found in this special case. This yields the expansion aE = am (cid:26) − r am + 3 r −
16 ( am ) − [5 r − r am ) + O (( am ) ) (cid:27) (62)which, again, in the special case r = 1 is found to agree with the previous expansion. Thelesson is that cut-off effects of Wilson fermions are linear in am . It is impossible to get rid ofthis undesirable term through a clever choice of r , since for r = 0 we are back to 2 d species. B.3 Karsten-Wilczek fermions At a(cid:126)p = (cid:126) − sinh ( aE ) + ( am ) and thus to theform (57) of the naive action. Accordingly, the expansion of the rest energy of a static KWfermion in powers of am agrees with (58). Hence, the KW action yields a 2 species formulationwhich maintains the desirable heavy-quark features of the naive discretization. B.4 Borici-Creutz fermions At a(cid:126)p = (cid:126) d space-time dimensions takes the form0 = (cid:104) i sinh( aE ) − r { − cosh( aE ) } (cid:105) + 4i rd sinh( aE ) { − cosh( aE ) } + ( am ) (63)which for d = 2 simplifies to 0 = − sinh ( aE ) + r { − cosh( aE ) } + ( am ) , while for d = 4 ittakes the form 0 = − sinh ( aE ) + r { − cosh( aE ) } − i r sinh( aE ) { − cosh( aE ) } + ( am ) .It seems instructive to first consider the case r = 1. In this case the d = 2 version assumesthe compact form 0 = 2 − aE ) + ( am ) , while the d = 4 version can be rewritten as0 = [2 − i sinh( aE )][1 − cosh( aE )] + ( am ) . In d = 2 dimensions the solution at r = 1 iscosh( aE ) = 1 + 12 ( am ) [ d = 2 , r = 1] (64)23hich expands as aE = am (cid:26) −
124 ( am ) + 3640 ( am ) + O (( am ) ) (cid:27) [ d = 2 , r = 1] . In d = 4 dimensions even at r = 1 the solution can only be given as the logarithm of the rootsof the polynomial i z − (4 + 2i) z + (8 + 4( am ) ) z − (4 − z − i, and a power expansion yields aE = am (cid:26) am − am ) − i8 ( am ) + 92310240 ( am ) + O (( am ) ) (cid:27) [ d = 4 , r = 1] . (65)For r (cid:54) = 1 and d = 2 we notice that eqn. (63) is quadratic in cosh( aE ), whereuponcosh( aE ) = − r ± (cid:113) − r )( am ) − r [ d = 2] (66)but only the first solution (with positive sign) is physical, since it is the one which agrees, inthe limit r → aE = am (cid:26) r −
424 ( am ) + 35 r − r + 48640 ( am ) + O (( am ) ) (cid:27) [ d = 2] (67)and a quick check reveals that each coefficient in the r = 1 expansion is recovered in that limit.For r (cid:54) = 1 and d = 4 the solution of eqn. (63) can only be given as the logarithm of the roots ofthe polynomial (i r + r − z + ( − r − r ) z + (4 m + 6 r + 2) z + (2i r − r ) z − − i r + r = 0,and a power expansion yields aE = am (cid:26) r am − r + 1696 ( am ) + i[ r − r ]16 ( am ) − r − r − am ) + O (( am ) ) (cid:27) [ d = 4] (68)which, for r →
1, would indeed simplify to (65).In short, we find that in d = 2 dimensions the rest-mass of a BC fermion has discretizationeffects O (( am ) ) for generic r . For r = 4 / O (( am ) ). By contrast,in d = 4 dimensions the rest mass of a BC fermion has O ( am ) cut-off effects, but this orderaffects only the imaginary part. Quite generally, it seems that in d = 4 dimensions the realpart of E/m is even in r and am , while the imaginary part is odd in r and am .Another way to see the difference between the cases d = 2 and d = 4 is to apply thehyperbolic semi-angle substitution to eqn. (63). Multiplying it with (1 − t ) yields0 = − t + 8i rt + 4 r t − rd t + ( am ) (1 − t ) where t = tanh( aE/ d = 2 the troublesome cubic term is gone0 = − t + 4 r t + ( am ) (1 − t ) [ d = 2] (69)and the equation is bi-quadratic, while for d = 4 one ends up with0 = − t + 4i rt + 4 r t + ( am ) (1 − t ) [ d = 4] (70)which is a genuine fourth-order equation in t . 24 Check of zero location in Green functions
C.1 Naive fermions
The denominator of G nai at am = 0 is a ¯ p = (cid:80) µ sin ( ap µ ). It has 16 zeros in the Brillouinzone, one at ap µ ∈ { , π } for each µ , if the range in each direction is taken to be ] − π , π ]. C.2 Wilson fermions
The denominator of G W at am = 0 is a ¯ p + ( a r ˆ p ) ; evidently it is only zero if a ¯ p = 0 and a ˆ p = 0 hold simultaneously. The first term has 16 zeros in the Brillouin zone, the second oneonly one, at (0 , , , r/a , 4 r/a , 6 r/a and 8 r/a , with degeneracies 4 , ,
4, and 1, respectively.
C.3 Karsten-Wilczek fermions
The denominator of G KW at am = 0 is zero if (cid:80) d − i =1 ¯ p i + (¯ p d + ar (cid:80) d − i =1 ˆ p i ) = 0. This holds if0 = (cid:88) i sin ( ap i ) ∧ ap d ) + 2 r (cid:88) i sin ( ap i ap i ∈ { , π } for each i , if the range is taken tobe ] − π , π ]. Hence we need to evaluate the second requirement for the 2 d − spatial momentumconfigurations, e.g. (0 , , , , π ), (0 , π, π, , , π, π ), ( π, , π ), ( π, π, π, π, π ) for d = 4. For (0 , ,
0) the second requirement reads 0 = sin( ap d ) + 0, and this implies ap d ∈ { , π } .For each of (0 , , π ), (0 , π, π, ,
0) the second requirement reads 0 = sin( ap d ) + 2 r , whichhas two solutions (in ap d ) for | r | < that merge into one at | r | = , hence the numberof flavors changes here by 6. For each of (0 , π, π ), ( π, , π ), ( π, π,
0) the second requirementreads 0 = sin( ap d ) + 4 r , which has two solutions for | r | < that merge into one at | r | = ,hence the number of flavors changes here by 6. For ( π, π, π ) the second requirement reads0 = sin( ap d ) + 6 r , which has two solutions for | r | < that merge into one at | r | = . Insummary, | r | = marks the watershed (for d = 4) between a deformed naive fermion and a 14species formulation, | r | = marks the transition to 8 species, and | r | = marks the transitionto a minimally doubled lattice fermion with poles at (0 , , ,
0) and (0 , , , π ).In view of a similar discussion below for BC fermions, it is perhaps useful to illustratethe solutions to the system (71) in the ( r, ap ) plane, see Fig. 12. The degeneracies andmultiplicities of the modes are given in the legend and the caption. The main mode (0 , , , , +)”, since it is non-degenerate with correct chirality. The doubler mode(0 , , , ± π ) is labeled “survivor (1 , − )”, since it is non-degenerate with opposite chirality.A contour plot for KW fermions in d = 2 dimensions is shown in Fig. 13. The momentumrange is ] − π, π [ for both ap and ap . At r = 0 one starts with the naive action. At infinites-imally small r the poles at ( − π,
0) and ( − π, − π ) [which have opposite chiralities] start movingtowards each other. At r = 1 / − π, − π/
2) and annihilate. The two remainingpoles are located at (0 , , − π ), with opposite chirality. Their position is independent of r .25 r<1/2 (3,+)r<1/2 (3,-)r<1/4 (3,-)r<1/4 (3,+)r<1/6 (1,+)r<1/6 (1,-)survivor (1,-)survivor (1,+) Figure 12: Illustration of the free-field pole structure of the Karsten-Wilczek operator in d = 4dimensions. The momentum ap is always plotted as a function of the parameter r . The 3-folddegenerate solution that emerges from (0 , , π ), (0 , π, π, , ap = ± π ,has correct chirality. It annihilates, at r = 1 /
2, with the 3-fold degenerate counterpart thatemerges from ap = 0 with opposite chirality. The 3-fold degenerate solution that emergesfrom (0 , π, π ), ( π, , π ), ( π, π, ap = ± π , has opposite chirality. It annihilates,at r = 1 /
4, with the 3-fold degenerate counterpart that emerges from ap = 0 with correctchirality. The non-degenerate solution that emerges from ( π, π, π ), together with ap = ± π , hascorrect chirality. It annihilates, at r = 1 /
6, with the non-degenerate counterpart that emergesfrom ap = 0 with opposite chirality. The non-degenerate solution that emerges from (0 , , ap = ± π has opposite chirality and lives for any r . The non-degenerate solution stemmingfrom the same spatial momentum, but with ap = 0, has correct chirality and lives for any r . C.4 Borici-Creutz fermions
For BC fermions in d = 2 dimensions, eqn. (44) at am = 0 simplifies to0 = (cid:104) sin( ap ) − r { − cos( ap ) } (cid:105) + (cid:104) sin( ap ) − r { − cos( ap ) } (cid:105) + 2 r (cid:104) sin( ap ) + sin( ap ) (cid:105)(cid:104) − cos( ap ) − cos( ap ) (cid:105) . (72)26 W in 2D, log(denominator), r=0.001 -3 -2 -1 0 1 2-3-2-1012 -10-9-8-7-6-5-4-3-2-10
KW in 2D, log(denominator), r=0.2 -3 -2 -1 0 1 2-3-2-1012 -10-9-8-7-6-5-4-3-2-10
KW in 2D, log(denominator), r=0.4 -3 -2 -1 0 1 2-3-2-1012 -10-9-8-7-6-5-4-3-2-10
KW in 2D, log(denominator), r=0.6 -3 -2 -1 0 1 2-3-2-1012 -10-9-8-7-6-5-4-3-2-10
KW in 2D, log(denominator), r=1 -3 -2 -1 0 1 2-3-2-1012 -10-8-6-4-20
KW in 2D, log(denominator), r=3 -3 -2 -1 0 1 2-3-2-1012 -10-8-6-4-202
Figure 13: Contour plots of the denominator of the KW propagator in d = 2 space-timedimensions for r ∈ { . , . , . , . , , } . Two poles annihilate at r = 1 / C in 2D, log(denominator), r=0.001 -3 -2 -1 0 1 2-3-2-1012 -10-9-8-7-6-5-4-3-2-10
BC in 2D, log(denominator), r=0.2 -3 -2 -1 0 1 2-3-2-1012 -10-9-8-7-6-5-4-3-2-10
BC in 2D, log(denominator), r=0.4 -3 -2 -1 0 1 2-3-2-1012 -10-8-6-4-20
BC in 2D, log(denominator), r=0.6 -3 -2 -1 0 1 2-3-2-1012 -10-8-6-4-20
BC in 2D, log(denominator), r=1 -3 -2 -1 0 1 2-3-2-1012 -10-8-6-4-20
BC in 2D, log(denominator), r=3 -3 -2 -1 0 1 2-3-2-1012 -10-8-6-4-202
Figure 14: Contour plots of the denominator of the BC propagator in d = 2 space-timedimensions for r ∈ { . , . , . , . , , } . Three poles merge into one at r = 1 / √ (cid:39) . p = p ≡ p . In this case we have0 = (cid:104) cos( ap ) − (cid:105)(cid:104) ( r −
1) cos( ap ) − r sin( ap ) − ( r + 1) (cid:105) (73)and thus one solution, ap = 0, is independent of r . To the second square bracket we apply thetrigonometric semi-angle substitution t = tan( ap/
2) with sin( ap ) = 2 t/ (1 + t ) and cos( t ) =(1 − t ) / (1 + t ). Upon multiplying the result with 1 + t , the second factor becomes0 = ( r − − t ) − r t − ( r + 1)(1 + t ) = − rt + 1) (74)and this yields the 2-fold zero t = − /r , hence ap = − /r ).For the non-symmetric modes it is useful to notice that (72) is the sum of two squares0 = (cid:104) sin( ap ) + r { − cos( ap ) } (cid:105) + (cid:104) sin( ap ) + r { − cos( ap ) } (cid:105) (75)and one can thus reformulate the condition as a system of two coupled equations0 = sin( ap ) + r { − cos( ap ) } ∧ ap ) + r { − cos( ap ) } . (76)The aforementioned trigonometric semi-angle substitution turns this into0 = t t + rt t ∧ t t + rt t (77)which, after multiplication by (1 + t )(1 + t ), leads to the conditions0 = t (1 + t ) + rt (1 + t ) ∧ t (1 + t ) + rt (1 + t ) . (78)There are four real solutions, { t = 0 , t = 0 } , { t = − /r, t = − /r } , { t = r − √ − r − r r , t = r − − √ − r − r r }{ t = r − − √ − r − r r , t = r − √ − r − r r } (79)where the first two are again symmetric in p ↔ p , and the last two interchange under t ↔ t .For the square-root in (79) to be real, one needs 1 − r − r ≥
0, and this means r ≤ /
3. At r = 1 / √ t = t = −√
3, and thus coincide with the symmetric solution, t = − /r = −√ < r < / √ / √ < r the BC action in d = 2dimensions encodes for 2 species (which live on the diagonal of the Brillouin zone).A contour plot for BC fermions in d = 2 dimensions is shown in Fig. 14. The momentumrange is ] − π, π [ for both ap and ap . At r = 0 one starts with the naive action. Forinfinitesimally small r the poles in the ( ap , ap ) plane at ( − π,
0) and (0 , − π ) start movingtowards the diagonal, and the pole at ( − π, − π ) moves along the diagonal, while the pole at(0 ,
0) stays invariant. At r = 1 / √ r > / √ , r → ∞ the two surviving polepositions are arbitrarily close to each other. 29or BC fermions in d = 4 dimensions, eqn. (44) at am = 0 simplifies to0 = (cid:104) sin( ap ) − r { − cos( ap ) } (cid:105) + ... + (cid:104) sin( ap ) − r { − cos( ap ) } (cid:105) + r (cid:104) sin( ap ) + ... + sin( ap ) (cid:105)(cid:104) − cos( ap ) − ... − cos( ap ) (cid:105) . (80)Let us first focus on a symmetric mode, i.e. put p = p = p = p ≡ p . In this case we have0 = (cid:104) cos( ap ) − (cid:105)(cid:104) ( r −
1) cos( ap ) − r sin( ap ) − ( r + 1) (cid:105) (81)exactly as in d = 2 dimensions, and the solution is again given by ap = 0 or ap = − /r ).For the asymmetric modes it is useful to notice that (80) is the sum of four squares0 = (cid:104) sin( ap ) + r { ap ) − cos( ap ) − cos( ap ) − cos( ap ) } (cid:105) + (cid:104) sin( ap ) + r { − cos( ap ) + cos( ap ) − cos( ap ) − cos( ap ) } (cid:105) + (cid:104) sin( ap ) + r { − cos( ap ) − cos( ap ) + cos( ap ) − cos( ap ) } (cid:105) + (cid:104) sin( ap ) + r { − cos( ap ) − cos( ap ) − cos( ap ) + cos( ap ) } (cid:105) (82)and one can thus reformulate the condition as a set of four coupled equations0 = sin( ap ) + r { ap ) − cos( ap ) − cos( ap ) − cos( ap ) } ap ) + r { − cos( ap ) + cos( ap ) − cos( ap ) − cos( ap ) } ap ) + r { − cos( ap ) − cos( ap ) + cos( ap ) − cos( ap ) } ap ) + r { − cos( ap ) − cos( ap ) − cos( ap ) + cos( ap ) } . (83)By adding two successive equations, this system may be reformulated as0 = sin( ap ) + sin( ap ) + r { − cos( ap ) − cos( ap ) } ap ) + sin( ap ) + r { − cos( ap ) − cos( ap ) } ap ) + sin( ap ) + r { − cos( ap ) − cos( ap ) } ap ) + sin( ap ) + r { − cos( ap ) − cos( ap ) } (84)or one might add three and subtract one out of the four equations to obtain0 = − sin( ap ) + sin( ap ) + sin( ap ) + sin( ap ) + 2 r { − cos( ap ) } ap ) − sin( ap ) + sin( ap ) + sin( ap ) + 2 r { − cos( ap ) } ap ) + sin( ap ) − sin( ap ) + sin( ap ) + 2 r { − cos( ap ) } ap ) + sin( ap ) + sin( ap ) − sin( ap ) + 2 r { − cos( ap ) } . (85)Finally, one might add all four equations to obtain0 = sin( ap ) + ... + sin( ap ) + r { − cos( ap ) − ... − cos( ap ) } (86)and an obvious question is which one of the four equivalent systems (83), (84), (85), or (86)would be most useful for finding actual solutions.30he last version is useful for finding the symmetric mode. With p = ... = p ≡ p eqn. (86)simplifies to 0 = sin( ap ) + r { − cos( ap ) } , or 0 = sin( ap )[cos( ap ) + r sin( ap )]. This meanssin( ap ) = 0 or cos( ap ) = − r sin( ap ). Hence ap ∈ { , − /r ) } , as was found previously.The last but one version is useful for solutions with 3-to-1 momentum pairing. Without lossof generality we assume p = p = p ≡ p , p ≡ q , so eqn. (85) takes the form0 = 1 sin( ap ) + sin( aq ) + 2 r { − cos( ap ) } ap ) − sin( aq ) + 2 r { − cos( aq ) } (87)and the trigonometric semi-angle substitution t = tan( ap/ , u = tan( aq/
2) turns this into0 = t t + u u + r { − − t t } t t − u u + r { − − u u } . (88)After multiplication by (1 + t )(1 + u ) one ends up with0 = 1 t (1 + u ) + u (1 + t ) + 2 rt { u } t (1 + u ) − u (1 + t ) + 2 ru { t } (89)and the real-valued solutions include t = u = 0 and t = u = − r (which are the previouslyfound symmetric solutions) as well as two non-trivial solutions for | r | ≤ / √
2, namely { t = r (1 − s )2 r + r + 2 s − , u = 2 r + s − r (2 r − }{ t = r (1 + s )2 r + r − s − , u = 2 r − s − r (2 r − } (90)with s ≡ √− r − r + 1. For r → / √ { t → − √ , u → −∞ }{ t → − √ , u → + ∞ } (91)meaning ap → − / √
2) and aq → ∓ π . We also determine the values which thesolutions (90) assume at r = 1 / √
3; we find { t → −√ , u → −√ }{ t → − √ , u → √ } (92)which means that only the first one of these two solutions matches onto the symmetric solutionat r = 1 / √
3. With respect to the general solution (90) let us recall that the choice p ≡ q wasone out of four possibilities, hence we have eight rather than two non-trivial solutions.The second version is useful for solutions with 2-to-2 momentum pairing. Without loss ofgenerality we assume p = p ≡ p , p = p ≡ q , so eqn. (84) takes the form0 = sin( ap ) + r { − cos( aq ) } aq ) + r { − cos( ap ) } (93)31ut this is identical to the system (76) for BC fermions in d = 2 dimensions. It follows that { t = 0 , u = 0 } and { t = − /r, u = − /r } are the symmetric solutions, and { t = r − √ − r − r r , u = r − − √ − r − r r }{ t = r − − √ − r − r r , u = r − √ − r − r r } (94)are the non-symmetric ones. Evidently, the second solution emerges from the first one byinterchanging t ↔ u . The square-root is real for | r | ≤ / √
3, and at this point the non-symmetric solutions take the form t = − /r = −√ , u = − /r = −√
3, which means that theymerge into the symmetric solution. We recall that the choice p = p ≡ p was one out of threepossibilities, hence we have six rather than two non-symmetric solutions.The first version of the system was not used at all. It seems (83) would be most useful forfinding a totally unsymmetric mode, i.e. one with pairwise unequal p , p , p , p . Apart fromhaving already found 2 + 8 + 6 = 16 solutions (for r small enough), permutations demanded byinvariance under exchange of any axes would beef up a totally unsymmetric solution to 4! = 24solutions, and that is too many of them.Overall, we thus arrive at the following picture for BC fermions in d = 4 dimensions. Forinfinitesimally small r there is an invariant solution, ap = (0 , , , ap = − arctan(1 /r )(2 , , , r = 1 / √ r slightly above 1 / √ r = 1 / √ r the BC action is minimally doubled, i.e. has one species of each chirality.Following a similar attempt in the KW case, we try to illustrate the various modes in the( r, ap ) or ( r, aq ) plane in Fig. 15. At r = 1 / √ r = 1 / √ , , ,
0) has correct chirality, symmetric solution has correct chirality for r < / √ r > / √ d = 4 dimensions it is more difficult to visualize the moving of the various poles as afunction of r than in d = 2 dimensions. While it seems impossible to visualize the originalsystem (80), we can visualize each one of the successor relations (83), (84), (85), and (86)under the assumption of the associate momentum pairing. Eqn. (83) would be most usefulwithout any pairing, but we just learned that this cannot yield a solution. Eqn. (84) is mostuseful with 2-to-2 pairing, and the reduced form, eqn. (93), can be visualized as a contour plot of[sin( ap )+ r { − cos( aq ) } ] +[sin( aq )+ r { − cos( ap ) } ] . But this is identical to the functional thatwas visualized in the d = 2 case, so the figure would look like Fig. 14, with the axes indicatingthe joint momenta p and q , respectively. Eqn. (85) is most useful with 3-to-1 pairing, and32 t _31_upper(2,-/+)u_31_upper(2,-/+)t _31_lower (2,-)u_31_lower (2,-)t _22_upper(3,+)u_22_upper(3,+)symm.(1,+/-)trivial (1,+) Figure 15: Illustration of the free-field pole structure of the Borici-Creutz operator in d =4 dimensions. Throughout, the momentum ap or aq is plotted as a function of the liftingparameter r . The full/dashed lines give 2 arctan( t ) , u ) with the 3-to-1 solutions t, u defined in the upper/lower line of (90). The dash-dotted lines give 2 arctan( t ) , u )with the 2-to-2 solutions t, u defined in the upper line of (94). The lower line of that systeminterchanges t ↔ u , and would give the same graph. The fat-dotted lines indicate the symmetricsolution − /r ) and the trivial solution. The horizontal dotted lines are at latticemomentum − π/ − π/
3, and − π , respectively. The vertical dotted lines are at r = 1 / √ r = 1 / √
3, respectively. In d = 2 dimensions all 3-to-1 paired solutions would be absent,the dash-dotted curve would refer to (79), and the fat-dotted curves would be unchanged.the reduced form, eqn. (87), can be visualized as a contour plot of [sin( ap ) + sin( aq ) + 2 r { − cos( ap ) } ] + [3 sin( ap ) − sin( aq ) + 2 r { − cos( aq ) } ] . Here p is the 3-fold momentum, and q isthe single momentum. The pertinent contours, with momentum range ] − π, π [ for both ap and aq , are shown in Fig. 16. For small r one sees the trivial solution t = u = 0, the symmetricsolution t = u = − r , as well as the upper line of (90). At r = 1 / √ r = 1 / √ C in 4D, log(3-to-1 pairing), r=0.001 -3 -2 -1 0 1 2-3-2-1012 -8-6-4-202
BC in 4D, log(3-to-1 pairing), r=0.3 -3 -2 -1 0 1 2-3-2-1012 -8-6-4-202
BC in 4D, log(3-to-1 pairing), r=0.65 -3 -2 -1 0 1 2-3-2-1012 -8-6-4-202
BC in 4D, log(3-to-1 pairing), r=0.75 -3 -2 -1 0 1 2-3-2-1012 -8-6-4-202
BC in 4D, log(3-to-1 pairing), r=1 -3 -2 -1 0 1 2-3-2-1012 -8-6-4-202
BC in 4D, log(3-to-1 pairing), r=3 -3 -2 -1 0 1 2-3-2-1012 -8-6-4-2024
Figure 16: Contour plots for the solutions to (87) for r ∈ { . , . , . , . , , } . Thereare touch- and endpoints at r = 1 / √ (cid:39) . r = 1 / √ . Hyperdiagonal propagation of Borici-Creutz fermions
Recall that the substitution p → i E would bring eqn. (44) to the form (45). If we let the fermionpropagate along the hyperdiagonal direction, we should use the substitution i E = ( p + p ) / √ d = 2 dimensions, and i E = ( p + p + p + p ) / d = 4 dimensions. The “spatial”momenta should be orthogonal to this direction, hence q ≡ ( p − p ) / √ d = 2 dimensions,and q ≡ ( − p + p + p − p ) / , q ≡ ( p − p + p − p ) / , q ≡ ( p + p − p − p ) / d = 4dimensions, since this definition establishes the orthogonality relation q ⊥ q ⊥ q ⊥ q .In d = 2 dimensions, eqn. (44) simplifies to0 = (cid:104) sin( ap ) − r { − cos( ap ) } (cid:105) + (cid:104) sin( ap ) − r { − cos( ap ) } (cid:105) + 2 r (cid:104) sin( ap ) + sin( ap ) (cid:105)(cid:104) − cos( ap ) − cos( ap ) (cid:105) + ( am ) (95)and with p = ( q + i E ) / √ p = ( − q + i E ) / √ − r sinh( √ aE ) + ( r −
1) cos( √ aq ) cosh( √ aE )+ 4 cos( aq √ − r cosh( aE √ r sinh( aE √ r + ( am ) (96)which is a quartic equation in e aE/ √ . Specifically at q = 0 it simplifies to0 = − r sinh( √ aE ) + ( r −
1) cosh( √ aE )+ 4( − r cosh( aE √ r sinh( aE √ r + ( am ) (97)and upon setting r = 1 it further simplifies to0 = 4 (cid:104) − cosh (cid:16) aE √ (cid:17)(cid:105)(cid:104) (cid:16) aE √ (cid:17)(cid:105) + ( am ) . (98)This equation has formally four solutions aE = √ (cid:16) RootOf( − i + i z + (2 − z − (4 + m ) z + (2 + 2i) z ) (cid:17) (99)out of which the physical one expands as aE = am (cid:26) − √ am −
13 ( am ) + √ am ) + 1740 ( am ) + O (( am ) ) (cid:27) [ r = 1] . (100)For generic r the expanded solution reads aE = am (cid:26) − √ r am − ( r am ) + √ r (5 r + 3)32 ( am ) +( 7 r
32 + 3 r
16 + 3160 )( am ) + O (( am ) ) (cid:27) (101)which, in the limit r →
1, is seen to coincide with the previous expansion.As an aside we mention that choosing the propagation direction orthogonal to the hyper-diagonal axis, i.e. p = ( q + i E ) / √ p = ( q − i E ) / √
2, yields0 = (4( r −
1) cos ( q/ √ − r − ( E/ √ − r cos( q/ √
2) + 4 r sin( q/ √ E/ √ − r + 2) cos ( q/ √ − r sin( q/ √
2) cos( q/ √
2) + 4 r + m (102)35hich is a quadratic equation in cosh( aE/ √ q = 0 it simplifies to0 = 2( r −
1) cosh (cid:16) aE √ (cid:17) − r cosh (cid:16) aE √ (cid:17) + 2 r + 2 + ( am ) (103)and upon setting r = 1 it further simplifies to 0 = − aE/ √
2) + 4 + ( am ) . This equationis linear in cosh( aE/ √
2) and yields cosh( aE/ √
2) = 1 + ( am ) / aE = am (cid:26) −
148 ( am ) + 32560 ( am ) + O (( am ) ) (cid:27) [ r = 1] . (104)The quadratic equation (103) has the unique physical solutioncosh (cid:16) aE √ (cid:17) = 2 r − (cid:113) − r − am ) r −
1) (105)since the other mathematical solution does not match onto the r = 1 case, and expands as aE = am (cid:26) r −
448 ( am ) + 35 r − r + 482560 ( am ) + O (( am ) ) (cid:27) . (106)In this peculiar case choosing r = 4 / aE to O (( am ) ).In d = 4 dimensions, eqn. (44) simplifies to0 = (cid:104) sin( ap ) − r { − cos( ap ) } (cid:105) + ... + (cid:104) sin( ap ) − r { − cos( ap ) } (cid:105) + r (cid:104) sin( ap ) + ... + sin( ap ) (cid:105)(cid:104) − cos( ap ) − ... − cos( ap ) (cid:105) + ( am ) (107)and with p = ( − q + q + q + i E ) / p = ( q − q + q + i E ) / p = ( q + q − q + i E ) / p = ( − q − q − q + i E ) / aq aq aq (cid:104) i r sinh( aE r cosh( aE (cid:105) + 8 cos( aq aq aq (cid:104) − r cosh( aE r sinh( aE (cid:105) − r −
1) sin( aq ) sin( aq ) sin( aq ) sinh( aE )+ 2( r −
1) cos( aq ) cos( aq ) cos( aq ) cosh( aE )+ 2i r (cid:104) cos( aq ) cos( aq ) cos( aq ) − cos( aq ) − cos( aq ) − cos( aq ) (cid:105) sinh( aE )+ 2 r sin( aq ) sin( aq ) sin( aq ) cosh( aE ) + 6 r + 2 + ( am ) (108)which is a bi-quadratic equation in cosh( aE/
2) and sinh( aE/ q = q = q = 0, it simplifies to0 = 4( r −
1) cosh ( aE r + i r sinh( aE − cosh( aE − r −
1) + ( am ) (109)and upon setting r = 1 it further simplifies to0 = 8 (cid:104) − cosh( aE (cid:105)(cid:104) aE (cid:105) + ( am ) . (110)36his equation has formally four solutions, but only aE = 2 log (cid:16) − i4 (cid:104)
2i + am + (cid:113) am + ( am ) (cid:105)(cid:17) (111)is physical, since it expands as aE = am (cid:26) − i4 am −
16 ( am ) + i8 ( am ) + 17160 ( am ) + O (( am ) ) (cid:27) (112)while the remaining ones have a constant imaginary part and/or start with a negative slope in am . The quartic equation (109) has the unique physical solution aE = 2 log (cid:16) − i r r ) (cid:104) r + am + (cid:113) ram + ( am ) (cid:105)(cid:17) (113)since the remaining ones do not match onto the r = 1 case, and it expands as aE = am (cid:26) − i r am − r + 124 ( am ) + i r (5 r + 3)64 ( am ) + 35 r + 30 r + 3640 ( am ) + O (( am ) ) (cid:27) (114)which, in the limit r →
1, is found to reproduce the previous result.In short, for BC fermions with propagation in the hyperdiagonal direction we find similarproperties than with the standard propagation along the d -th axis. In d = 2 and d = 4dimensions the rest mass of a BC fermion with diagonal propagation direction has O ( am ) cut-off effects, but this order affects only the imaginary part. The coefficient of the O (( am ) ) cut-offeffects is − [3 r + 1] /
12 in d = 2 dimensions and − [3 r + 1] /
24 in d = 4 dimensions. Again, itseems that the real part of E/m is even in r and am , while the imaginary part is odd in r and am . Overall, we do not see any compelling advantage of the (++) or (++++) propagationdirection over the standard propagation in the d -th direction. A peculiarity of d = 2 space-timedimensions is that the propagation direction can be chosen orthogonal to the hyperdiagonaldirection, and in this case the O ( am ) cut-off effects disappear, and for r = 4 / O (( am ) ). E Spectral bounds
E.1 Karsten-Wilczek operator
Plugging in the momenta in eqn. (10) at m = 0 yields aD KW / i = d − (cid:88) i =1 γ i sin( ap i ) + γ d (cid:110) sin( ap d ) + 2 r (cid:88) i sin ( ap i (cid:111) (115)and with ω KW ≡ max( λ KW / i) it follows that the symmetry among the spatial axes implies ω = ( d −
1) sin ( ap ) + (cid:110) sin( ap d ) + 2( d − r sin ( ap (cid:111) (116)for some appropriately chosen momentum configuration. Obviously, setting ap d = π/ ( ap/
2) = 1 − cos( ap ) we thus need to maximize ω = ( d −
1) sin ( ap ) + (cid:110) d − r [1 − cos( ap )] (cid:111) (117)37ver p ∈ [ − π, π ]. We need to keep in mind that at either endpoint we have the value ω = (cid:110) d − r (cid:111) = (cid:40) (1 + 6 r ) [ d = 4](1 + 2 r ) [ d = 2] . (118)Taking the derivative with respect to ap and setting it to zero yields0 = cos( ap ) + (cid:110) d − r [1 − cos( ap )] (cid:111) r (119)and this leads to the solutioncos( ap ) = ( d − r + r ( d − r − (cid:40) (3 r + r ) / (3 r −
1) [ d = 4] r/ ( r −
1) [ d = 2] . (120)Plugging the d = 4 result into the general expression yields ω = 3[1 − (3 r + r ) (3 r − ] + (cid:110) r [1 − r + r r − (cid:111) = 4 + 6 r − r (121)and equating this with the endpoint value shows that the switching beween the two solutionshappens at r = 1 /
3. Plugging the d = 2 result into the general expression yields ω = 1 − r ( r − + (cid:110) r [1 − rr − (cid:111) = 21 − r (122)and equality with the endpoint value is reached at r = 1 / d = 4 dimensions we find the spectral bound (11). In d = 2 dimensions | Im( λ KW ) | ≤ (cid:40) (cid:113) / (1 − r ) r ≤ /
21 + 2 r r ≥ / r generalizes to 1 + 2( d − r in d dimensions. For r → √ d , which is the upper bound of the staggered free-field eigenvalue spectrum.The upper envelope of the numerical data in Fig. 6 is well consistent with the bound (11).For r < / r > / E.2 Borici-Creutz operator
Plugging in the momenta in eqn. (18) at m = 0, and using eqn. (13) yields aD BC / i = (cid:88) µ γ µ sin( ap µ ) + 2 r (cid:88) µ (cid:104) √ d Γ − γ µ (cid:105) sin ( ap µ /
2) (124)and with the definition (12) we obtain the expression aD BC / i = (cid:88) µ γ µ sin( ap µ ) + 4 rd (cid:88) ν γ ν · (cid:88) µ sin ( ap µ / − r (cid:88) µ γ µ sin ( ap µ /
2) (125)38here we may interchange the indices µ ↔ ν in the middle term. This yields aD BC / i = (cid:88) µ γ µ (cid:110) sin( ap µ ) + 4 rd (cid:88) ν sin ( ap ν / − r sin ( ap µ / (cid:111) (126)and with ω BC ≡ max( λ BC / i) it follows that the momentum symmetry implies ω = d (cid:110) sin( ap ) + 2 r sin ( ap/ (cid:111) . (127)Using 2 sin ( ap/
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