PPrepared for submission to JHEP preprint
Chiral Lattice Fermions From Staggered Fields
Simon Catterall a a Department of Physics, Syracuse University, Syracuse, NY 13244, USA
E-mail: [email protected]
Abstract:
We construct lattice theories from reduced staggered fermions in four dimen-sions that we argue are capable of producing chiral theories in the continuum limit. Theconstruction employs Yukawa interactions of Fidkowski-Kitaev type which depend on thelattice site parity. We propose that such interactions are capable of generating cut-off scalemasses for half of the lattice fermions while preserving all symmetries. We argue that theremaining lattice fermions organize themselves in the continuum limit into a set of sixteenmassless Majorana or equivalently Weyl fermions. We show in the context of the latticemodel how this fermion content is needed to cancel off a discrete anomaly arising in thecontinuum limit. a r X i v : . [ h e p - l a t ] D ec ontents It has long been a goal of lattice field theory to be able to describe continuum chiral gaugetheories. All of the standard local lattice fermion prescriptions; Wilson, staggered, domainwall and overlap are only capable of describing vector-like theories. The reason is wellknown - the Nielsen-Ninomiya theorem asserts that a wide class of fermion discretizationswith exact chiral symmetry will necessarily contain equal numbers of left and right handedfields [1]. One natural way to evade this theorem is to start from a vector-like lattice theory andintroduce interactions that are capable of generating large masses for modes of a singlechirality leaving behind a low energy theory containing fermions of the opposite chirality.Perhaps the earliest example of such a proposal was made by Eichten and Preskill in theearly days of lattice gauge theory [2]. The idea was to introduce four fermion terms into theaction which, it was hoped, were capable of gapping the unwanted mirror fermions. Theearly numerical work to test this idea made use of Wilson and staggered lattice fermions[3–7] and appeared to invalidate the approach; to generate large mirror masses large fourfermion or Yukawa couplings were needed and typically this resulted in the formation ofsymmetry breaking condensates coupling left and right handed states via Dirac mass terms[8]. More recently this approach was revived for lattice fermion actions with superior chiralproperties - in a series of papers Poppitz et al. have investigated models using overlapfermions [9–11] while a gauge invariant path integral measure for overlap chiral fermionsin SO (10) was constructed in [12]. Complementary to this work Grabowska and Kaplanproposed a lattice regulator for chiral gauge theories using domain wall fermions [13]. Anearlier proposal combining domain wall fermions and appropriate four fermion interactions While domain wall and overlap formulations have been very successful at describing systems withapproximate global axial symmetries it has not been possible so far to generalize these to true chiral latticegauge theories. – 1 –as made by Creutz et al in [14]. However, again, the overall conclusion of this work wasthat it was difficult, if not impossible, to decouple the chiralities in the continuum limit. However, in recent years, a series of developments in condensed matter physics haveprovided new insights into the problem. One of the key new ingredients has been thediscovery of models in which fermions can acquire masses without generating symmetrybreaking fermion condensates. This field was launched by the seminal paper of Kitaev andFidkowski (FK) [16] who showed that it was possible to design a four fermion interactionthat was capable of generating masses for precisely eight zero dimensional Majorana modeswithout breaking symmetries. Subsequent work generalized this to higher dimensions find-ing that eight Majorana fermions are needed also in two dimensions and sixteen Majoranafermions in three and four (spacetime) dimensions [17–20] . It is now understood thatthese magic numbers of fermions are tied to the cancellation of certain discrete anomaliesin these theories [21, 22]. Indeed, one way to understand the observation of symmetrybroken phases in some lattice four fermion theories is that they arise as a consequence ofsuch discrete anomalies - since the vector-like lattice theory is anomaly free when the fourfermion coupling is zero the anomaly must cancel in the I.R even for strong coupling. Theformation of a bilinear condensate coupling light and heavy modes is one simple way forthis to happen.The phenomenon of symmetric mass generation can also be seen in vector-like latticetheories and a number of recent numerical studies have provided good evidence for theexistence of massive symmetric phases in dimensions from two to four [23–30]. Whilesuch phases had been observed in early lattice studies, they were typically separated fromthe weak coupling regime by regions where the symmetries were spontaneously broken,and the massive symmetric phases were interpreted as lattice artifacts. The commonelement in the new numerical work was that the models employed a variant of the usualstaggered fermion prescription containing half as many fermions - a formalism termedreduced staggered fermions. This reduced formalism, which can be thought of as imposinga Majorana condition on the staggered fermions, will form a key ingredient in our proposalfor constructing chiral lattice theories.It was also realized by Xu, Wen, and others that this new method for symmetric massgeneration might allow one to construct anomaly free chiral lattice gauge theories andseveral proposals have been made [19, 31–33]. However, it is fair to say that none of theseproposals show, in a concrete way, both how to achieve the required mass decoupling forsome subset of lattice fields and how the remaining light fields naturally lead to Majoranaor Weyl fermions in the continuum limit. Our construction aims to bridge this gap byfurnishing an explicit lattice theory in which certain single component relativistic latticefermions can be gapped by a FK type interaction while simultaneously producing a lowenergy theory which is chiral in the continuum limit.We start our discussion with a quick review of reduced staggered fermions and howthey may be given a mass without breaking symmetries using a four fermion interaction. The exception to this was L¨uscher’s formal construction of a path integral for U (1) chiral gauge theoryin [15] – 2 –e show that this construction naturally yields a vector-like continuum limit. To target achiral theory requires a very specific modification of the structure and symmetries of thequartic interactions and this is discussed in detail. We argue that the symmetric massgeneration mechanism must necessarily fail if the continuum limit does not correspondto sixteen massless Majorana or Weyl fermions in agreement with recent work on discreteanomalies. We show how to understand these anomalies from within the reduced staggeredfermion construction. We summarize our conclusions and discuss open questions in the finalsection of the paper. The usual staggered fermion action in D dimensions is easily arrived at by spin diagonal-izing the naive fermion action on a hypercubic lattice and takes the form [34] S = (cid:88) x,µ η µ ( x ) χ ( x ) D Sµ χ ( x ) + (cid:88) x mχ ( x ) χ ( x ) (2.1)where η µ ( x ) = ( − (cid:80) µ − i =0 x i are the usual staggered fermion phases and the symmetricdifference is given by D Sµ χ a ( x ) = 12 ( χ a ( x + µ ) − χ a ( x − µ )) (2.2)If m = 0 a further reduction is possible by keeping only one (single component) fermion ateach lattice site. Explicitly we introduce the projectors P ± defined by P ± = 12 (1 ± (cid:15) ( x )) (2.3)where the site parity is given by (cid:15) ( x ) = ( − (cid:80) D − µ =0 x µ . The lattice action decomposes into S = (cid:88) x,µ η µ ( x ) (cid:0) χ + D Sµ χ − + χ − D Sµ χ + (cid:1) (2.4)where P + χ = χ + etc. The reduction corresponds to, for example, retaining only the fields P + χ and P − χ . This results in the reduced staggered fermion action whose continuum limitcorresponds to 2 D/ − Dirac fermions or equivalently 2 D/ Majorana fermions [35]. S = (cid:88) x,µ χ a ( x ) η µ ( x ) D Sµ χ a ( x ) (2.5)where we have relabeled χ → χ on odd parity lattice sites. This action is invariant undera staggered shift symmetry: χ ( x ) → ξ µ ( x ) ξ ν ( x + µ ) χ ( x + µ + ν ) (2.6)where ξ µ = ( − (cid:80) Di = µ +1 x i . The reduced staggered action is also invariant under a U (1)symmetry: χ → e iα(cid:15) ( x ) χ (2.7)– 3 –hile the single flavor theory does not allow for a mass term this can be remedied bycoupling 2 N flavors of reduced staggered field. However, since the goal here is to prohibitfermion bilinears, we focus instead on possible four fermion terms. The very simplest modelrequires four reduced staggered fields transforming under an SO (4) global symmetry : S = (cid:88) x,µ χ a ( x ) η µ ( x ) D Sµ χ a ( x ) + G (cid:88) x (cid:15) abcd χ a ( x ) χ b ( x ) χ c ( x ) χ d ( x ) (2.8)Notice that these symmetries forbid any fermion bilinear operator from appearing in thequantum effective action.It is often convenient to replace the four fermion term by a Yukawa interaction withan auxiliary scalar field. The scalar action that was employed in [25, 27] is given by S = G (cid:88) x χ a χ b σ + ab + 12 (cid:0) σ + ab (cid:1) (2.9)where σ + ab is a self-dual scalar transforming in a fundamental representation of an SO (3)subgroup of the original SO (4) symmetry. σ + ab = P + σ ab = 12 (cid:18) δ ac δ bd + 12 (cid:15) abcd (cid:19) σ cd (2.10)One should note that in the Yukawa formulation U (1) symmetry is broken down to adiscrete Z symmetry of the form: ψ a ( x ) → ω n ψ a (2.11) σ + ab → ω − n σ + ab n = 0 . . . ω = e i(cid:15) ( x ) π . At first sight the appearanceof Z rather than Z is surprising. It requires that the scalar σ + transform under a Z whichincludes the transformation σ + → iσ + . This is equivalent to sending G → − G . Whenformulating the Yukawa interaction we made the choice to use one of the two SO (3)’s thatmake up SO (4). We could just as easily have chosen the other which would have exchanged σ + → σ − . This generates precisely the same change in the sign of G . Thus we infer thatthe partition function is invariant under a change G → iG or equivalently of Z phasetransformations of the scalar.One intuitive way to understand how a fermion mass can arise in these models is torewrite the four fermion operator as (cid:15) abcd χ a ( x ) χ b ( x ) χ c ( x ) χ d ( x ) = Ω a ( x ) χ a ( x ) (2.13)corresponding to a bilinear mass term formed by pairing an elementary fermion with acomposite fermion Ω a ( x ) = (cid:15) abcd χ b ( x ) χ c ( x ) χ d ( x ). It is easy to see that a condensate ofthis four fermion operator arises in the strong coupling limit G → ∞ which can hence be We impose a reality condition on the fermions which allows for only a SO (4) rather than SU (4)symmetry. This is compatible with the even-odd site reduction. – 4 – σ + - / GL=8 κ =-0.05 κ =0.0 κ =0.05 Figure 1 . Four fermion condensate vs G interpreted as generating a fermion mass. It is also clear that one expects massless fermionsat G = 0. Thus one expects at least one phase transition to separate the massless andmassive symmetric regimes. Evidence of a continuous transition separating these phaseshas been seen in both three and four dimensions [23–28].To underline these conclusions weinclude a plot in fig. 1 of the four fermion condensate in the four dimensional model takenfrom our earlier paper [27]. The rapid increase of the condensate close to G ∼ κ that appears is the coefficient ofa scalar kinetic term which must be tuned in four dimensions to see this direct transitionbetween massless and massive symmetric phases. These earlier results encourage the beliefthat a continuum limit may exist in these lattice models in which fermions can acquiremasses without breaking symmetries.Since reduced staggered fields yield Dirac fermions in the continuum limit the mecha-nism described above is only capable of generating mass in vector-like theories. To targeta chiral theory we need to modify the quartic interactions in a manner that subjects only asubset of the reduced staggered field to strong Yukawa interactions. In a staggered theorythere is a natural way to do this using the lattice site parity. The next section will describethe needed four fermion interactions and the nature of the continuum limit. We will alsosee the modified model will in general exhibit an anomaly in the continuum limit. This isdiscussed in section 4 where it is shown that cancellation of this anomaly is only possiblefor eight flavors of reduced staggered field. The exception to this is two dimensions where the four fermion coupling is asymptotically free and onesees [29] a single symmetric gapped phase for all non-zero lattice couplings. – 5 – χ s t ag G κ =0.05 L=6L=8L=12 Figure 2 . Fermion susceptibility vs G At this point we will focus on four dimensions and modify the Yukawa interactions to treatthe even and odd parity sites in an asymmetric fashion. The action is given by S = (cid:88) x,µ η µ ( x ) χ T ( x ) D Sµ χ ( x ) + (cid:88) x ( GP + + gP − ) (cid:2) χ ( x ) T Λ χ ( x ) (cid:3) + 12 σ A ( x ) (3.1)where several flavors of reduced staggered fermion are coupled through an hermitian, anti-symmetric matrix Λ( x ) = σ A ( x )Γ A . In practice we will follow FK and take Γ A to be Diracmatrices appropriate to the real, eight dimensional spinor representation of Spin(7) [16].Explicitly, Γ A = (cid:0) σ , σ , σ , σ , σ , σ , σ (cid:1) (3.2)where the notation indicates that the Dirac matrices Γ A are built from tensor products ofPauli matrices. The significance of this choice will be discussed later. We will also take G to be large to drive symmetric mass generation in the even site parity sector while g iskept small and serves merely to regulate a zero mode that appears in the odd parity sectorat g = 0 corresponding to the shift symmetry χ − ( x ) → χ − ( x ) + α − (3.3)To understand the continuum limit and why this lattice model may allow us to decouplethe different chiralities that appear in the continuum limit we can assemble the staggeredfields in a unit hypercube into a continuum-like matrix field Ψ labeled by both spinor andflavor indices [36]. We suppress the Spin(7) indices for simplicity. This matrix fermion In the lattice literature this is termed the spin-taste basis. It is equivalent to the K¨ahler-Dirac repre-sentation used in lattice susy constructions [37]. – 6 –esides on a lattice with twice the lattice spacing.Ψ = (cid:88) { n µ =0 , } χ ( x + n µ ) γ n µ (3.4)where γ n µ = γ n γ n γ n γ n . In a chiral basis it is easy to see that Ψ has the block structureΨ = (cid:32) E O (cid:48)
O E (cid:48) (cid:33) (3.5)where the 2 × E, E (cid:48) and
O, O (cid:48) contain only even and odd lattice sitestaggered fields. In terms of the matrix Ψ it is easy to see that the restriction to evenand odd parity lattice sites can be expressed via the matrix projectorsΨ ± = 12 (Ψ ± γ Ψ γ ) (3.6)This in turn dictates that the internal flavor symmetry inherited by Ψ from the reducedstaggered field is Spin(4) corresponding to the generators [ γ µ , γ ν ] that commute with γ .Indeed, in the continuum limit Ψ transforms under independent Spin(4) Lorentz and flavorsymmetries which act by left and right multiplication on the matrix [34, 36]. Furthermore,in this limit, the use of a real staggered field χ implies that Ψ obeys the reality condition:Ψ = γ Ψ ∗ γ (3.7)This in turn ensures that E (cid:48) = − σ E ∗ σ and O (cid:48) = σ O ∗ σ . The following table showsthe different representations for the blocks under the Lorentz SU (2) × SU (2) and (inter-nal) flavor SU (2) × SU (2) symmetries in the continuum theory: The restoration of these2 × E ( , ) ( , ) O ( , ) ( , ) E (cid:48) ( , ) ( , ) O (cid:48) ( , ) ( , ) Table 1 . Representations of the blocks entering into the continuum matrix field Ψ continuum symmetries is tied to the existence of exact shift symmetries of the staggeredfermion action given by χ ( x ) → ξ µ ( x ) ξ ν ( x + µ ) χ ( x + µ + ν ) (3.8) χ ( x ) → η µ ( x ) η ν ( x + µ ) χ ( x + µ + ν ) (3.9)Using these representations it is clear that the continuum limit comprises two independentsets of Majorana fermions consisting of the block pairs ( O, O c ) and ( E, E c ). These twoblock are coupled to the scalar field σ A via independent Yukawa couplings. If we assume Notice that the structure of the resultant continuum Euclidean action corresponds to that in [38]. – 7 –hat the even parity fields acquire a symmetric mass because of the large coupling G these will decouple from the low energy spectrum. Conversely, if g is small we expect theassociated four fermion coupling to be an irrelevant operator and the odd parity fieldsshould yield a pair of massless Majorana fermions in the continuum limit. But masslessMajorana fermions are equivalent to Weyl fermions for real representations and hence itshould be possible to describe the continuum limit in terms of a pair of (say) left handedWeyl fermions corresponding to the two columns of the block O . These transform in the( , , ) representation of the global Spin(7) × SU (2) × SU (2) flavor symmetry.The appearance of sixteen Weyl fermions in the continuum from our reduced staggeredfermion theory is in accordance with the vanishing of a discrete spin- Z anomaly for systemsof Weyl fermions [20, 22]. This anomaly arises in connection to the symmetry: ψ L → − iψ L (3.10) ψ R → + iψ R (3.11)and takes the form ν = n + − n − mod 16 (3.12)A similar symmetry in two dimensions - chiral fermion parity - is responsible for ananalogous anomaly cancellation condition which requires eight Majorana fermions. Thiscan also be seen in our staggered fermion model in which the gapped theory would yieldeight light Majorana fermions as g → g growing in the I.R. This would yield a finite, albeit exponentially small, symmetricmass for the light Majorana modes in the continuum limit and prevent an interpretationof the low energy spectrum in terms of chiral fermions.This Yukawa interaction can be reduced to the subgroups Spin( N ) for 2 ≤ N < A to run from 1 . . . N as described in [19]. The resultant theories withreduced symmetry should still be capable of generating symmetric masses for the sixteenright handed mirrors and produce theories with a low energy spectrum containing sixteenleft-handed Weyl fermions in four dimensions. In the condensed matter literature the keyfeature which makes this gapping mechanism possible is the non-degeneracy of the groundstate for this interaction which continues to hold even in the reduced symmetry cases.Systems with a unique ground state cannot undergo spontaneous symmetry breaking andhence offer the possibility of symmetric mass generation.As we have stressed anomaly cancellation is a necessary condition for symmetric massgeneration. With this in mind in the next section we show how this discrete anomaly ariseswithin the context of the lattice model. In this section we show how the discrete anomaly cancellation condition arises in the contextof this reduced staggered fermion model. The fermion operator we are interested in takesthe form M ( σ ) = η µ ( x )∆ µ δ ab + Λ ab ( GP + + gP − ) (4.1)– 8 –ith Λ = σ A Γ A . To expose the anomaly we will generalize the interaction slightly andallow the dimension of the matrix Λ be 2 N . Let us consider the eigenvalue equation: M [ σ ] φ n = λ n φ n (4.2)Acting on the left with an element of the Z transformation eqn. 2.12 yields M [ ω σ ] ω − φ n = (cid:0) λ n ω (cid:1) ω − φ n (4.3)Thus the non-zero eigenvalues of M [ ω σ ] are just those of M ( σ ) up to a phase ω . Thisnaively implies that Pf( M [ ω σ ]) = Pf( M [ σ ]) (4.4)This argument can be trivially repeated for the other possible phase rotations of σ . Howeverwe need to be careful in the presence of zero modes which can arise as g, G → x ) is some fixed matrix corresponding to a constant σ field of unit length. Theeigenvalues of Λ are then just ±
1. We can then effect the phase rotation σ → ω σ byrotating the couplings ( g, G ) → ω ( g, G ). Within the zero mode subspace we can nowperform an SO (2 N ) rotation to bring Λ to the canonical 2 × J = diag( λ iτ ⊕ λ iτ ⊕ . . . ⊕ λ N iτ ) (4.5)with λ i = gP − + GP + depending on the site parity. If the couplings are now rotated bythe phase ω the Pfaffian changes by the factor (cid:89) (cid:15) ( x )= ± e i(cid:15) ( x ) N (4.6)where the two factors arise from the even and odd parity zero modes. Hence in a theorywhere both χ + and χ − have zero modes (a theory whose continuum limit is vector-like)there is no net phase and the Pfaffian is indeed invariant. However if G → ∞ and thesystem generates a symmetric mass of order the cut-off for the χ + modes only the light χ − fermions contribute to this phase. Invariance of the Pfaffian then dictates that thenumber of reduced staggered fields 2 N must be a multiple of eight to cancel this globalanomaly. This illustrates why the FK interaction plays such a special role in symmetricmass generation – it corresponds to a symmetry group Spin(7) which possesses a real eightdimensional representation - the minimal number of fermions needed to cancel the anomaly.In the continuum limit we have argued previously that each gapped reduced staggered fieldyields two Weyl fermions. Thus we learn that one can only cancel the anomaly in thecontinuum if it contains sixteen Weyl fermions in agreement with arguments based on theDai-Freed theorem [22].There is another consequence of the cancellation of the anomaly - the theory should beindependent of the sign of the fermion bilinear squared ( χ T Λ χ ) . But this is just the state-ment that there is no energetic reason for the bilinear to condense thereby spontaneouslybreaking the Spin(7) symmetry. This is the essence of the condensed matter arguments that– 9 –re used to design interactions for symmetric mass generation and we see explicitly herethat it is tied to the cancellation of a discrete anomaly as has been pointed out previouslyin the literature eg. [21]. It should be remarked that the arguments for the appearance ofthis anomaly are quite analogous to those used to demonstrate Witten’s global anomalyfor odd numbers of Weyl fermions in SU (2) gauge theory [39]. We have argued that it is possible to generate masses without symmetry breaking for latticetheories built from reduced staggered fermions. The reduced staggered fermion possesses acontinuum limit in D dimensions that is naturally described by 2 D/ Majorana rather thanDirac fermions. To gap the fermions without breaking symmetries one employs four fermioninteractions that couple many flavors of lattice field. In the simplest case the structure ofthe interactions is independent of the lattice site parity and the resulting theory is vector-like in the continuum limit. In this case there is good evidence from numerical simulationthat massive symmetric phases are indeed realized.In this paper we have shown how to generalize these models to construct lattice theorieswhich attempt to gap only lattice fields associated with sites of fixed site parity. Potentiallythis can yield a continuum limit with half the number of degrees of freedom – 2 D/ − Majorana fermions. If these remaining Majorana fields become massless in the continuumthey can be traded for an equivalent number of Weyl fermions of fixed chirality. In thisway we will have constructed a lattice model capable of generating a chiral theory in thecontinuum limit.To accomplish this goal we employ a Yukawa interaction whose structure is borrowedfrom condensed matter physics where it has been shown to generate mass without breakingsymmetries in low dimensional systems with specific numbers of Majorana fermions. Thisis now understood as a consequence of the cancellation of a certain discrete anomaly. Wehave shown how this anomaly is manifested in the context of reduced staggered fermionsand argue that after cancellation of the anomaly the continuum limit of the lattice modelin four dimensions yields sixteen Weyl fermions that transform under a global Spin(7) × SU (2) × SU (2) flavor symmetry.While we have focused on four dimensions in this paper the same arguments can beused in two or three dimensions. In two dimensions the dynamics of the theory will likelyproduce a non-zero mass for the light Majorana fields arising in the continuum limit ofthe lattice model and hence will not allow for a description in terms of chiral fermions.In fact the eight Majoranas that would be produced by such a lattice construction areconsistent with the cancellation of another discrete anomaly connected to chiral fermionparity [17, 18, 22, 40]. It also seems possible that a similar construction may work in threedimensions. In this case the unit cube on the lattice contains eight reduced staggeredfields. Gapping say the even site parity fields would leave four light fields which can beassembled into two Majorana fermions. The continuum limit would then contain sixteenMajorana fermions again consistent with the expectation from continuum discrete anomalycancellation. – 10 –t is important to note that in the absence of anomalies the Spin(7) symmetry canbe straightforwardly gauged to yield a chiral lattice gauge theory in the continuum limit.This is indeed an exciting prospect.Some caveats are in order. We have assumed Lorentz invariance is restored in thecontinuum limit in order to identify the chirality and flavor representation of the vari-ous continuum fermions arising from the underlying staggered lattice fields. Conventionalarguments suggest that the presence of exact lattice shift symmetries should make this au-tomatic but its possible that non-perturbative physics may interfere with those arguments.Furthermore, while symmetric mass generation in the vector-like case has been checkedwith Monte Carlo simulation, this may be problematic in the models described here be-cause of sign problems. Also, as for Dirac fermions, it is likely that a full Higgs-Yukawamodel will be needed to have a hope of obtaining a continuum limit in four dimensions.Finally, it is possible that lattice bilinear operators coupling fermions at different sites inthe hypercube may condense. Such spontaneous breaking is not prohibited by the FKinteraction which is a single site operator. Condensates of this form would however spon-taneously break the flavor shift symmetries. Such condensates have not been seen in anyof the vector-like model simulations but it is possible that these would be chiral modelsbehave differently. More work is needed to decide this issue. Acknowledgments
This work was supported by the US Department of Energy (DOE), Office of Science, Officeof High Energy Physics under Award Number DE-SC0009998. The author would like tothank Nouman Butt and Goksu can Toga for discussions and David Tong for patientlyeducating me about discrete anomalies.
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