Classification of emergent Weyl spinors in multi-fermion systems
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Classification of emergent Weyl spinors in multi - fermion systems
M.A. Zubkov ∗ Physics Department, Ariel University, Ariel 40700, Israel
In the fermionic systems with topologically stable Fermi points the emergent two - componentWeyl fermions appear. We propose the topological classification of these fermions based on the twoinvariants composed of the two - component Green function. We define these invariants using Wigner- Weyl formalism also in case of essentially non - homogeneous systems. In the case when valuesof these invariants are minimal ( ±
1) we deal with emergent relativistic symmetry. The emergentgravity appears, and our classification of Weyl fermions gives rise to the classification of vierbein.Transformations between emergent relativistic Weyl fermions of different types correspond to parityconjugation and charge conjugation.
I. INTRODUCTION
Electrons in solids are described by multi - componentspinors carrying band index. However, at low energieswe may describe electrons by effective spinors with theessentially reduced number of components. Only thoseenergy bands are relevant that cross Fermi energy. InDirac/Weyl semimetals [3–7] with Fermi points the emer-gent spinors are two component if Fermi energy is closeto the position of the Fermi points. Due to the repulsionof energy levels the Fermi points are unstable unless theyare protected by topology [15]. Therefore, the effectivedescription in terms of the two - component spinors typi-cally survives in the case when the topological invariantsprotecting the Fermi points are nonzero [1]. The earlierdiscussion of these issues may be found in [16–18]. Forthe review of the topological invariants (in momentumspace) protecting Fermi points and Fermi surfaces see[2]. The well - known example is graphene with emer-gent Weyl fermions [8, 9]. It is also worth mentioningthat the topological invariants are responsible for gaplessedge modes of topological insulators [10–12]. The similarphenomena are observed also in superfuids [13, 14].The minimal value of the topological invariant N (composed of the Green functions [2]) responsible for thestability of the Weyl points is ±
1. Weyl points with thelarger values of N are able to split into several Weylpoints with N = ±
1. Action of Weyl fermions with N = ± N give rise to more ex-otic types of Weyl fermions, typically with the nonlineartouching points of positive and negative energy branches.An example of such exotic Weyl points is given by the2 + 1 D multilayer graphene with the ABC stacking [22]. ∗ On leave of absence from NRC ”Kurchatov Institute” - ITEP, B.Cheremushkinskaya 25, Moscow, 117259, Russia
Also such exotic Weyl fermions appear in effective gravi-tational theories with anisotropic scaling [23–26], see [27–31].It is worth mentioning that the classification of topo-logical insulators [32, 33] may be obtained by a certainreduction of the topological classification of Fermi sur-faces and emergent spinors incident on them [1], for thedetails see [2].In the present paper we are going to extend the men-tioned above classification of Weyl points into two direc-tions. First of all, we notice that in general case of in-teracting systems the two topological invariants N and N (3)3 may be introduced. Both are composed of the Greenfunctions, and both of them are reduced to the abovementioned N in the absence of interactions. However,they become different in general case. As a result in caseof the minimal values N , N (3)3 = ± N is defined as an integral in momentum space andis valid for the homogeneous systems only. Using theWigner - Weyl formalism we extend the expressions ofboth N and N (3)3 to the non - homogeneous case, whenthey are given by integrals in phase space. II. EMERGENT WEYL SPINORS IN THEMULTI - FERMION SYSTEMS
Here we closely follow [40]. We start from the consid-eration of equilibrium condensed matter system with the n - component spinors ψ at zero temperature. Its realtime partition function has the form: Z = Z DψD ¯ ψ exp (cid:16) i Z dt X x ¯ ψ x ( t )( i∂ t − ˆ He − iǫ ) ψ x ( t ) (cid:17) , (1)Hamiltonian H is a Hermitian matrix depending on mo-mentum ˆ p . Sum over points of coordinate space is tobe understood as an integral over d x for continuous co-ordinate space. However, we may also consider latticetight - binding models, in which case we have the sumover the lattice points. In the absence of dependence of H on x Hamiltonian H depends on p only. Factor e − iǫ is introduced here in order to point out how the poles inFeynmann diagrams are to be treated. ǫ is assumed tobe very small, its appearance may be explained, for ex-ample, via equilibrium limit of the Keldysh non - equilib-rium path integral in lattice regularization (of time axis).Typically the factor e − iǫ is omitted and is restored onlyif it is necessary to speak about the ways the singularitiesin Feynmann diagrams are treated.Several branches of spectrum of ˆ H repel each other.Therefore, the minimal number n reduced = 2 of branchesare able to cross each other. And this minimal numberis fixed by topology of momentum space.Let us consider the position p (0) of the crossing of 2branches of ˆ H . Transformation via a certain Hermitianmatrix Ω: ˜ H ( p ) = Ω + ˆ H Ω brings Hamiltonian to thediagonal form. Then the first 2 × H reduced corre-sponds to the two crossed branches. The remaining blockˆ H gapped corresponds to the branches of spectrum sepa-rated from the crossing point by a gap. We may representthe functional integral as the product of the functionalintegral over ”gapped” modes Θ and the integral over 2reduced fermion components Ψ. At low energies (closeto the Fermi level coinciding with the branch crossingpoint) the contribution to the physical observables of thereduced fermions dominates over the contribution of thegapped ones because Θ contribute the physical quantitieswith the fast oscillating factors. Therefore, we effectivelydescribe the system close to the Fermi point via the par-tition function Z = Z D Ψ D ¯Ψexp (cid:16) i Z dt X x ¯Ψ x ( t )( i∂ t − ˆ H reduced e − iǫ )Ψ x ( t ) (cid:17) (2)In the following we will omit the subscript ”reduced” ofˆ H for simplicity. The general form of Hermitian matrix 2 × Z = Z D Ψ D ¯Ψexp (cid:16) i Z dt X x ¯Ψ x ( t )( i∂ t − [ m Lk (ˆ p )ˆ σ k + m (ˆ p )] e − iǫ )Ψ x ( t ) (cid:17) (3)where functions m Lk , m are real - valued. The nontriv-ial topology appears when the topological invariant com-posed of m L ( p ) has the nontrivial value N = e ijk π Z σ dS i ˆ m L · (cid:18) ∂ ˆ m L ∂p j × ∂ ˆ m L ∂p k (cid:19) , ˆ m L = m L | m L | (4)where σ is the S surface surrounding the branch crossingpoint p (0) j .On the language of the Matsubara Green function G H ( ω, p ) = 1 iω − ˆ H ( p )the value of N may be written as ( ω = p ): N (3)3 = 13! 4 π Z Σ Tr h G H ( ω, p ) dG − H ( ω, p ) ∧ dG H ( ω, p ) ∧ dG − H ( ω, p ) i (5)Here Σ is the three - dimensional hypersurface surround-ing the Fermi point in 4 D momentum space (composedof 3 D momentum space and the axis of ω = p ).If N = ± p = p (0) the follow-ing expansion takes place m La ( p ) = f ja ( p j − p (0) j ) . (6)with sign det f ja = N , and m ( p ) ≈ f j ( p j − p (0) j )with j = 1 , ,
3. The coefficients f ja ( a = 0 , , ,
3) ofexpansion may be considered as the source of emergentgravity. As a result, Eq. (10) receives the form ( σ ≡ Z = Z D Ψ D ¯Ψexp (cid:16) i Z dt X x ¯Ψ x ( t )( i∂ t − f ja (ˆ p j − p (0) j )ˆ σ a e − iǫ )Ψ x ( t ) (cid:17) (7)The dispersion of quasiparticles close to the point p (0) re-ceives the form of the Dirac cone. Due to nonzero valuesof f k this cone is tilted. Moreover, if k f j [ f − ] aj k > σ a f ja ( p j − p (0) j ) = 0 , j = 1 , , a = 0 , , , k f j [ f − ] aj k = 1. In this case the Fermipoint (the couple of the Fermi pockets) is reduced actu-ally to the Fermi line. III. TOPOLOGICAL INVARIANTS ANDEMERGENT VIERBEIN IN HOMOGENEOUSSYSTEMS
In the presence of interactions the partition functionfor the (reduced) two - component fermions appearingclose to the Weyl points can be written as follows: Z = Z DψD ¯ ψD Φexp (cid:16) iR [Φ] + i Z dt X x ¯ ψ x ( t )( i∂ t − ˆ H (Φ) e − iǫ ) ψ x ( t ) (cid:17) (9)Here Φ is a set of fields that provide interactions. R isa certain effective action of Φ. Operator ˆ H also dependson Φ. In mean field approximation we substitute thevalues of Φ by their ”mean” values. Fluctuations aroundthese mean values will give us the partition function ofthe form Z = Z D Ψ D ¯Ψ D Φexp (cid:16) iR [Φ] (cid:17) exp (cid:16) i Z dt X x ¯Ψ x ( t )( i∂ t m ′ Φ ,a σ a +[ µm ′ Φ ,a σ a − m L Φ ,k (ˆ p )ˆ σ k − m Φ (ˆ p )] e − iǫ )Ψ x ( t ) (cid:17) (10)Here it is assumed that Ψ carries an index enumeratingthe Weyl points of the system.Let us consider the Green function of the original multi- fermion system G ( t − t , x − y ) = − iZ Z DψD ¯ ψD Φexp (cid:16) iR [Φ]+ i Z dt X x ¯ ψ x ( t )( i∂ t − ˆ H (Φ) e − iǫ ) ψ x ( t ) (cid:17) ψ x ( t ) ¯ ψ y ( t ) (11)In order to construct the topological invariants responsi-ble for the stability of Fermi points we may also use theGreen function of the reduced low energy theory G ( t − t , x − y ) = − iZ Z D Ψ D ¯Ψ D Φexp (cid:16) iR [Φ] (cid:17) exp (cid:16) i Z dt X x ¯Ψ x ( t )( i∂ t m ′ Φ ,a σ a +[ µm ′ Φ ,a σ a − m L Φ ,k (ˆ p )ˆ σ k − m Φ (ˆ p )] e − iǫ )Ψ x ( t ) (cid:17) Ψ x ( t ) ¯Ψ y ( t ) (12)Then we compose the Fourier transform G ( P , P , P , P )of the Green function either from Eq. (11) or from Eq. (12). We substitute to G ( P , P , P , P ) the values P = iω and P j = p j for j = 1 , , N = 13! 4 π Z Σ Tr h G ( ω, p ) dG − ( ω, p ) ∧ dG H ( ω, p ) ∧ dG − H ( ω, p ) i (13)Here Σ is the three - dimensional hypersurface surround-ing the Fermi point in 4 D momentum space (composedof 3 D momentum space and the axis of ω = p ).In addition, we define G H ( ω, p ) = 1 iω − G (0 , p ) − (14)(Again, we may use either G of Eq. (11) or Eq. (12).)The second topological invariant is defined as N (3)3 = 13! 4 π Z Σ Tr h G H ( ω, p ) dG − H ( ω, p ) ∧ dG H ( ω, p ) ∧ dG − H ( ω, p ) i (15)For the minimal values of N and N (3)3 at each Weylpoint we represent m ′ Φ ,a ( p ) ≈ e e a , m L Φ ,a ( p ) ≈ e e ja ( p j − B j ) ,m Φ ( p ) ≈ e e j ( p j − B j ) , i, j = 1 , , e we denotethe determinant of matrix inverse to the 4 matrix e ka .The appearance of the field B = − µ reflects, that inthe presence of interaction the value of Fermi energy atthe position of the crossing of several branches of spec-trum may differ from zero. Thus, µ is chemical potentialcounted from the crossing point.As a result, the partition function of the model maybe rewritten as: Z = Z D Ψ D ¯Ψ De ik DB k e iS [ e ja ,B j , ¯Ψ , Ψ] (17)with S = S [ e, B ]+ 12 (cid:16) i Z dt e X x ¯Ψ x ( t ) e ja ˆ σ a ˆ D j e − iǫ Ψ x ( t ) + ( h.c. ) (cid:17) , (18)where the sum is over a, j = 0 , , , σ ≡
1, and ˆ D is the covariant derivative that includes the U (1) gaugefield B : ˆ D = ∂ t e iǫ − iµ, ˆ D k = ∂ k + iB k S [ e, B ] is the part of the effective action that dependson e and B only.In the absence of an external source of inhomogeneityin the leading order the field e ja may be considered asindependent of coordinates. The same refers to B , whichthen may be gauged off. In this case in order to de-fine topological invariants N and N (3)3 we use the Greenfunction G ( ω, p ) = 1 e e ja ˆ σ a P j (19)Obviously, N (3)3 and N may differ from each other dueto the nontrivial components of the emergent vierbein e a .The direct calculation gives N = sign det × e ja (Here a, j = 0 , , ,
3, and the determinant is of the 4 × N (3)3 = sign det × e ja where a, j = 1 , ,
3, and the determinant is of the 3 × IV. T, C, P TRANSFORMATIONS AND N , N (3)3 . Let us consider the interplay of these topological invari-ants and T, C, P transformations. For the left - handedfermion (the one with N = N (3)3 = +1) the transforma-tions are: T : Ψ( t, x ) → σ ¯Ψ T ( − t, x ) , ¯Ψ( t, x ) → Ψ T ( − t, x ) σ P : Ψ( t, x ) → − i Ψ( t, − x ) , ¯Ψ( t, x ) → ¯Ψ( t, − x ) iC : Ψ( t, x ) → σ ¯Ψ T ( t, x ) , ¯Ψ( t, x ) → Ψ T ( t, x ) σ At the same time for the right - handed ones (those with N = N (3)3 = − T : Ψ( t, x ) → σ ¯Ψ T ( − t, x ) , ¯Ψ( t, x ) → Ψ T ( − t, x ) σ P : Ψ( t, x ) → − i Ψ( t, − x ) , ¯Ψ( t, x ) → ¯Ψ( t, − x ) iC : Ψ( t, x ) → − σ ¯Ψ T ( t, x ) , ¯Ψ( t, x ) → − Ψ T ( t, x ) σ Let us also introduce the additional transformation R(both for the right - handed and for the left - handedfermions): R : Ψ( t, x ) → − Ψ( t, x ) , ¯Ψ( t, x ) → Ψ( t, x )These transformations lead to the same form of the ac-tion for the transformed spinor fields, in which the vier-bein is transformed as follows: T : e → e , e a → − e a , e k → − e k , e ka → e ka ,P : e → e , e a → e a , e k → − e k , e ka → − e ka ,C : e → − e , e a → e a , e k → − e k , e ka → e ka ,a, k = 1 , , N , N (3)3 : T : N → N , N (3)3 → N (3)3 C : N → − N , N (3)3 → N (3)3 P : N → − N , N (3)3 → − N (3)3 CP : N → N , N (3)3 → − N (3)3 R : N → N , N (3)3 → − N (3)3 In the limit of small ǫ the operator Q = e ka iD k σ a e − iǫ is Hermitian. The same refers to operator Q standingin the fermionic action R d xdt ¯Ψ Q Ψ also in general case.When Q remains in the class of such (almost) Hermitianoperators the smooth modifications of the system cannotlead to the change of the values of both N and N (3)3 .Different values of the topological invariants N and N (3)3 determine the topological classification of Weyl fermions.Namely, we define those with N = N (3)3 = +1 as theleft - handed particles. Notice, that our definition of N differs by sign from that of [2]. The fermions with N = − N (3)3 = 1 may be defined as the left - handed anti -particles. Those with N = N (3)3 = − N = − N (3)3 = − R : Ψ → − Ψ) isidentical to unity, which reflects invariance of fermionicsystems under the CPT.
TABLE I: Weyl fermions and values of topological invariants.fermion type N N (3)3 left - handed particles +1 +1right - handed particles -1 -1left - handed anti - particles +1 -1right - handed anti - particles -1 +1 From the definition of the topological invariants N and N (3)3 it follows that the sum of the values of N or N (3)3 over all Fermi points is equal to zero provided thatmomentum space is compact. The proof of this state-ment is as follows. The sum of N ( N (3)3 ) over the Fermipoints is given by Eq. (13) (Eq. (15)) with hypersurfaceΣ surrounding all Fermi points. If total momentum spaceis compact, Σ may be deformed smoothly to a point. Theresult is, obviously, zero. We deal with compact momen-tum space if lattice systems are considered with compactBrillouin zone, and, in addition, the axis of imaginarytime is discretized. The latter condition is not alwaysfulfilled, and we deal in condensed matter systems with-out interactions with the Green functions of the form ofEq. (14). Then at ω → ±∞ the expression standing inthe integrals of Eq. (15) tends to zero. As a result thementioned topological theorem is back, and the sum of N (3)3 over all Weyl points is zero. However, in the pres-ence of interactions the dependence of the Green functionon ω may be more complicated. Then the sum over theWeyl points of N may appear to be nonzero while thesum of N (3)3 remains vanishing. In the latter case theweakened topological theorem allows breakdown of theconventional state with equal number of left and right- handed fermions in the lattice models. Namely, thesituation is possible, when there are two left - handedfermions, but one of them is of the ”particle” type, whileanother one is of the type of ”anti - particles”. Thissituation is considered in the next Section. V. TOY MODEL
Let us consider the toy model, which illustrates uncon-ventional properties of the systems with the Weyl pointsof the type of ”anti - particles”. In this model the parti-tion function is given by Z = Z D Ψ D ¯Ψexp (cid:16) i Z dt X x ¯Ψ x ( t ) ˆ Q ( i∂ t , ˆ p )Ψ x ( t ) (cid:17) (21)with ˆ Q ( p , p ) = p (cid:16) α cos p + ( α − p (cid:17) (22) − (cid:16) α sin p − ( α − − cos p + cos p ) (cid:17) σ e − iǫ − sin p σ e − iǫ − (sin p + 2 − cos p − cos p ) σ e − iǫ Here α is a real - valued parameter. In this system thereare two Fermi points K ± : K + : p = 2 arctg (1 /α −
1) = 2 φ α , p = p = 0 K − : p = π, p = p = 0Operator ˆ Q may be written in the form:ˆ Q ( p , p ) = p α − α + 1 (cid:16) p cos ( p + φ α ) − (sin ( p − φ α ) + sin φ α cos p ) σ e − iǫ − sin p σ e − iǫ + (sin p + 2 − cos p − cos p ) σ e − iǫ √ α − α + 1 (cid:17) Close to the Fermi points we have: K + : ˆ Q ( p , p ) √ α − α + 1= (cid:16) p cos φ α − q σ e − iǫ − q σ e − iǫ + q σ e − iǫ √ α − α + 1 (cid:17) and K − : ˆ Q ( p , p ) √ α − α + 1= (cid:16) − p cos φ α + q σ e − iǫ − q σ e − iǫ + q σ e − iǫ √ α − α + 1 (cid:17) Here q = p − K ± . One can see that for 0 < α ≤ K + : N = N (3)3 = 1and K − : N = − N (3)3 = 1Both Weyl points are left - handed. But one of them isof the type of ”particles” while the other is of the typeof the ”anti - particles”. At α = 1 we deal with theFermi points separated in momentum space by π . When α approaches to zero the two Weyl points approach eachother and merge for α = 0 at the position of K − . Thisis the marginal Weyl point with N = 2 and N (3)3 = 0.In its small vicinity we have: K − : ˆ Q ( p , p )= p q −
12 ( q + p ) σ e − iǫ − q σ e − iǫ + q σ e − iǫ One can calculate directly the values of N and N (3)3 forthis Weyl point using machinery developed in AppendixC of [41]. VI. NON - HOMOGENEOUS ANDINTERACTING SYSTEMS
In the systems with weak inhomogeneity we come topartition function (for N , N (3)3 = ± Z = Z D Ψ D ¯Ψ De ik DB k e iS [ e ja ,B j , ¯Ψ , Ψ] (23)with S = S [ e, B ]+ 12 (cid:16) i Z dt X x ¯Ψ x ( t ) e e ja ˆ σ a −→ ˆ D j e − iǫ Ψ x ( t )+ ¯Ψ x ( t ) ←− ˆ D + j e e ja ˆ σ a e − iǫ Ψ x ( t ) (cid:17) , (24) In case of lattice models we require that all fields dependingon coordinates almost do not vary at the distances of the orderof lattice spacing. Under these conditions in the majority ofexpressions the sum over the lattice points may be replaced byan integral. where the sum is over a, j = 0 , , , σ ≡
1, and ˆ D is the covariant derivative that includes the U (1) gaugefield B . S [ e, B ] is the part of the effective action that de-pends on e and B only. Now both fields e and B dependon coordinates. We denote hereˆ D +0 = − ∂ t e iǫ + iµ, ˆ D + k = − ∂ k + iB k One can see, that this is the Hermitian conjugation ex-cept for the factor e iǫ . In principle, one may include intocovariant derivative D the spin connection. However, itsappearance, may be taken into account effectively by arenormalization of e and B . Namely, the spin connectionmay be represented as C a,k σ a , where C a,k is complex -valued, and a, k = 0 , , ,
3. Then we come to¯Ψ e kb C a,k σ b σ a Ψ + ( h.c. )= ¯Ψ e kb ( C b,k + iC a,k ǫ bad σ d )Ψ + ( h.c. )= ¯Ψ( e kb Re C b,k − e kb Im C a,k ǫ bad σ d )Ψ (25)Here ( h.c. ) means the hermitian conjugation comple-mented by transformation Ψ → ¯Ψ. One can check thatcombinations e kb Re C b,k and e kb Im C a,k ǫ bad may be ab-sorbed by redefinition of emergent gauge field.As above one may define the two topological invariants.Let us introduce the Green functionˆ G = 2 e e ja ˆ σ a i ˆ D j + i ˆ D + j e e ja ˆ σ a Next, we define its Wigner transformation G ( M ) W ( p, x ) = R d re ipr h x + r/ | ˆ G | x − r/ i .After Wick rotation we introduce p = iω = iP , and P j = p j for j = 1 , , x = iX , X j = x j , and denotethe Euclidean Wigner transformation of Green function: G W ( P, X ) = G ( M ) W ( p, x ). We also define Q W that obeys Q W ∗ G W = 1. Here by ∗ we denote the Moyal product ∗ = e i ( ←−− ∂ Xi −−→ ∂ Pi −←−− ∂ Pi −−→ ∂ Xi ) The first topological invariant is given by N = 13! 4 π | V | Z Σ Z d X Tr h G W ( P, X ) ∗ dQ W ( P, X ) ∗ ∧ dG W ( P, X ) ∗ ∧ dQ W ( ω, P, X ) i (26) | V | is the three - volume of the system. Here Σ is thethree - dimensional hypersurface surrounding the givensingularity M ( i ) of expression standing inside the inte-gral. This expression resembles the one of [38]. The gen-eral procedure for the construction of such invariants hasbeen proposed in [36]. In [37] such topological invarianthas been considered for the interacting QHE systems.In addition, we define Q H,W = iω − Q W ( P, X ) (cid:12)(cid:12)(cid:12) ω =0 and G H,W obeys Q H,W ∗ G H,W = 1Then the second topological invariant can be defined as N (3)3 = 13! 4 π | V | Z Σ Z d X Tr h G H,W ( P, X ) ∗ (27) dQ H,W ( P, X ) ∗ ∧ dG H,W ( P, X ) ∗ ∧ dQ H,W ( ω, P, X ) i Again, N (3)3 and N may differ from each other due to thenontrivial components of the emergent vierbein e a . Forthe weakly dependent on coordinates e ka ( X ) and B (suchthat sign det × e ka and sign det × e ka do not depend on X ) we still have N = sign det × e ja and N (3)3 = sign det × e ja Those two invariants provide topological classificationof vierbeins for the non - homogeneous case. In the pres-ence of interactions we are to use the complete interactingGreen function ˆ G : h x | ˆ G | y i = − iZ Z D Ψ D ¯Ψ De ik DB k e iS [ e ja ,B j , ¯Ψ , Ψ] Ψ( x ) ¯Ψ( y ) (28)instead of the non - interacting one in Eqs. (26) and (27).Moreover, we may use equivalently the Green function ofthe original multi - fermion system h x , t | ˆ G | x , t i = − iZ Z DψD ¯ ψD Φexp (cid:16) iR [Φ] (29)+ i Z dt X x ¯ ψ x ( t )( i∂ t − ˆ H (Φ) e − iǫ ) ψ x ( t ) (cid:17) ψ x ( t ) ¯ ψ x ( t )and its Wigner transform.This will give us the modified expressions for the abovetopological invariants. Σ of these expressions is the hy-persurface in phase space ( P, X ) that surrounds a singu-larity of expression standing inside the integral. Positions M ( i ) of distinct singularities play the role of the Fermisurfaces/Fermi points in case of the non - homogeneoussystems. Each M i is reduced to a Fermi point for thecase of homogeneous system. N and N (3)3 defined viaintegrals of Eq. (26) and Eq. (27) with Σ surround-ing M ( i ) are responsible for the topological stability ofthe given singularities. Only the singular transformationthat leads to a change of the values of N and N (3)3 maybring the singularity of one type to the singularity of an-other type. Correspondingly, in the cases N = ± N (3)3 = ± P M ( i ) N = 0 and P M ( i ) N (3)3 = 0 if the corre-sponding Euclidean momentum space is compact. Thisis always true in lattice regularized models, when the axisof imaginary time is discretized as well as the coordinatespace. In this case in the non - homogeneous case thenumber of the left - handed fermions coincides with thenumber of the right - handed fermions. In condensedmatter systems with noncompact axis of ω this topologi-cal theorem may be broken partially as it was explainedabove, and P M ( i ) N = 0 while P M ( i ) N (3)3 = 0. VII. CONCLUSIONS
In this paper we discuss topological classification ofemergent Weyl fermions in multi - fermion systems. InEuclidean space - time there would be only one topo-logical invariant responsible for stability of Fermi points.This is N of Eq. (13) or Eq. (26). Correspondingly,there exist two types of low energy emergent Weyl spinorfields with minimal values of N - those that describe left- handed and right - handed particles/antiparticles. Thelow energy subsystem (of the given multi - fermion sys-tem) incident in a vicinity of the given right - handedWeyl point cannot be transformed continuously to thatof the left - handed Weyl point.In the real - time dynamics the operator ˆ Q (inverse ofthe Green function) is Hermitian up to an infinitely smallcorrection that points out the way the poles are treatedin Feynmann diagrams . Now we are able to define thetwo distinct topological invariants composed of ˆ Q and ˆ G .The second invariant is that of Eq. (15) or Eq. (27).In the class of Hermitian operators we cannot transformsmoothly those ˆ Q with different values of N (3)3 to eachother.We come to an unexpected conclusion: in generalcase in multi - fermion systems with minimal values of N , N (3)3 = ± e component may become negative, and the Weyl fermion More precisely, operator ˆ Q is Hermitian, but its inverse ˆ G is con-sidered in space of generalized (rather than ordinary) operator- valued functions, and the mentioned would be infinitely smallcorrection to ˆ Q actually has the meaning of the proper definitionof ˆ Q − . In space of ordinary operators the inverse to ˆ Q does notexist. of a marginal type with N = − N (3)3 appears. We feelthis instructive to call the field describing the correspond-ing Weyl fermion the field of ”the anti - particle” type,and the Fermi point itself may be referred to as the anti- Weyl point. This is natural because charge conjugationbeing applied to the ordinary (left or right - handed) Weylfermion (with N = N (3)3 ) results in the Weyl fermionwith N = − N (3)3 , i.e. in the Weyl fermion of the typeof ”anti - particle” existing in a vicinity of anti - Weylpoint.The appearance of anti - Weyl points with N = − N (3)3 have sense only in the presence of the conventional Weylfermions with N = N (3)3 . Without the latter R trans-formation Ψ → − Ψ , ¯Ψ → ¯Ψ brings Weyl fermions to thetype of ”particles” with N = N (3)3 . At the same timeif both types of Weyl fermions co - exist, the interestingphenomena may occur. For example, a couple of Weylfermions ( N , N (3)3 ) = (+1 , −
1) and (+1 , +1) may mergegiving marginal Weyl point with ( N , N (3)3 ) = (+2 , α of the system it is possible to bring it tothe state, in which the Weyl point and the anti - Weylpoint merge giving the marginal Weyl point with N = 2and N (3)3 = 0. (The way to calculate the values of N and N (3)3 for this Fermi point may be read off from Ap-pendix C of [41].) Notice, that if imaginary time axis isdiscretized as well as coordinate space, the sum over theWeyl points of both N and N (3)3 is equal to zero. As aresult, in such systems the total number or Weyl pointsis equal to the total number of anti - Weyl points, whilethe total number of left - handed Weyl/anti - Weyl pointsis equal to the total number of right - handed Weyl/anti- Weyl points. This theorem is broken partially in thementioned toy model, in which the imaginary time axisis not discretized, and the axis of ω remains open.In practise the anti - Weyl points may appear in solidsin the presence of sufficiently strong inter - electron inter-actions. Therefore, prediction of such materials cannotbe given on the basis of DFT calculations only. We needfor the purpose of engineering of such materials (in addi-tion to the DFT calculation of energy bands) the meth-ods that take into account interactions between Blochelectrons. One may also suppose that existence of anti -Weyl points may be found in fermionic superfluids, pos-sibly, under specific external conditions. Notice, that inthe presence of the anti - Weyl points the equality be-tween the numbers of left - handed and right - handedfermions is broken. Now the lattice systems are allowedwith the left - handed fermions only, but then the num-ber of Weyl points and the number of anti - Weyl pointsshould be equal.It is worth mentioning that in addition to the topo-logical classification of Fermi points presented here thereshould exist the complimentary topological classificationof the systems, in which the energy bands cross eachother at the points in momentum space, but when thoseWeyl points form various forms of Fermi surfaces ratherthan the Fermi points. In the case when the values of N (and N (3)3 ) are minimal = ± N and N (3)3 is also possible, but it is out of the scope of the present paper.Finally, we would like to notice that the classifica-tion presented here may be relevant for the high energyphysics and applications of quantum field theory to cos-mology (see [34], [35] and references therein). Then theappearance of the four (rather than two) topologicallydistinct types of Weyl fermions may be assumed fromthe very beginning. 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