Characteristics of Chiral Anomaly in View of Various Applications

PHYSICAL REVIEW D 97, 016018 (2018)

Characteristics of chiral anomaly in view of various applications

Kazuo Fujikawa Quantum Hadron Physics Laboratory, RIKEN Nishina Center, Wako 351-0198, Japan

(Received 17 May 2017; revised manuscript received 3 December 2017; published 29 January 2018)

In view of the recent applications of chiral anomaly to various fields beyond particle physics, we discuss some basic aspects of chiral anomaly which may help deepen our understanding of chiral anomaly in particle physics also. It is first shown that Berry’s phase (and its generalization) for the Weyl model H ¼ vFσ⃗· p⃗ðtÞ assumes a monopole form at the exact adiabatic limit but deviates from it off the adiabatic limit and vanishes in the high frequency limit of the Fourier transform of p⃗ðtÞ for bounded jp⃗ðtÞj.An effective action, which is consistent with the nonadiabatic limit of Berry’s phase, combined with the Bjorken-Johnson-Low prescription, gives normal equal-time space-time commutators and no chiral anomaly. In contrast, an effective action with a monopole at the origin of the momentum space, which describes Berry’s phase in the precise adiabatic limit but fails off the adiabatic limit, gives anomalous space-time commutators and a covariant anomaly to the gauge current. We regard this anomaly as an artifact of the postulated monopole and not a consequence of Berry’s phase. As for the recent application of the chiral anomaly to the description of effective Weyl fermions in condensed matter and nuclear physics, which is closely related to the formulation of lattice chiral fermions, we point out that the chiral anomaly for each species doubler separately vanishes for a finite lattice spacing, contrary to the common assumption. Instead, a general form of pair creation associated with the spectral flow for the Dirac sea with finite depth takes place. This view is supported by the Ginsparg-Wilson fermion, which defines a single Weyl fermion without doublers on the lattice and gives a well-defined index (anomaly) even for a finite lattice spacing. A different use of anomaly in analogy to the partially conserved axial-vector current is also mentioned and could lead to an effect without fermion number nonconservation.

DOI: 10.1103/PhysRevD.97.016018

I. INTRODUCTION e2 ∂ ψγ¯ μγ ψ 2imψγ¯ ψ E⃗ B⃗ μð 5 Þ¼ 5 þ 2 · ð1Þ In the treatment of topological properties in condensed 2π matter physics, one often uses the adiabatic Berry’s phase holds for the fundamental electron in condensed matter in [1] induced at the crossing point of two levels in the band an arbitrary small domain of space-time (such as the structure. See Ref. [2] for a review of topological effects in tangent space of a curved space) independently of frequen- condensed matter physics. Chiral anomaly, which was cies carried by the gauge field, which may include the established in particle theory [3–6], has also been used Coulomb potential provided by surrounding charged par- to elucidate the properties of “Weyl fermions” in condensed ticles in addition to external field Aμ. matter and nuclear physics. See, for example, Refs. [7–10] It has been recently argued, relying on some preceding and references therein. analyses [14,15], that a “kinematic” derivation of chiral It is clear from its original derivation [1] that the anomaly from Berry’s phase is possible in an effective monopole-type behavior of Berry’s phase is valid only theoretical model [16]. If this derivation, which is based on in the precise adiabatic limit [11]. On the other hand, chiral anomalous commutation relations of space-time variables anomaly is believed to be a short distance effect [12], and in induced by Berry’s phase, is valid, it would open up a fact only the high frequency components of fermion completely new perspective for the subject of quantum variables are essential in the (Euclidean) evaluation of anomalies. The purpose of the present paper is first to chiral anomaly [13]. Thus the relation [3,4] examine the foundation of this kinematic derivation and then, second, to discuss some basic issues related to the applications of the chiral anomaly in condensed matter and Published by the American Physical Society under the terms of nuclear physics. the Creative Commons Attribution 4.0 International license. ’ Further distribution of this work must maintain attribution to We first show that Berry s original model is exactly the author(s) and the published article’s title, journal citation, solved if the time dependence of parameters is suitably and DOI. Funded by SCOAP3. chosen. This solution enables us to study Berry’s phase in

2470-0010=2018=97(1)=016018(15) 016018-1 Published by the American Physical Society KAZUO FUJIKAWA PHYS. REV. D 97, 016018 (2018) the nonadiabatic and nontopological domain and shows To analyze the behavior of Berry’s phase away from the that, in particular, Berry’s phase (its nonadiabatic gener- precise adiabatic limit quantitatively, we discuss an exactly alization [17]) vanishes in the nonadiabatic limit when solvable model which is defined by measured in a natural manner. Using this knowledge of the nonadiabatic behavior, it is shown that the mere presence of H ¼ −ℏμB⃗ðtÞσ⃗ ð2Þ a Weyl fermion does not induce anomalous equal-time commutation relations of space-time variables in the with Bjorken-Johnson-Low prescription. In contrast, an effective action with a postulated pointlike monopole at the B⃗ðtÞ¼Bðsin θ cos φðtÞ; sin θ sin φðtÞ; cos θÞ; ð3Þ origin of the momentum space, which describes Berry’s phase in the precise adiabatic limit but fails off the adiabatic where φðtÞ¼ωt with constant ω, B, and θ are constants limit, gives rise to anomalous space-time commutators and a and σ⃗stand for Pauli matrices. This model is identical to the covariant form of anomaly to the gauge current [16].Itiswell original model analyzed by Berry [1], except for the choice known that the magnetic field modifies commutators such as of specific time dependence (or independence) of param- in the Landau level, and thus the appearance of anomalous eters so that we can solve the model exactly. The exact space-time commutators for an assumed monopole in the solution of the Schrödinger equation, iℏ∂tψðtÞ¼HψðtÞ,is momentum space is not surprising. We regard the anomaly given by thus obtained as an artifact of the postulated monopole and Z not a consequence of Berry’sphase. i t ψ t w t − dt0w† t0 H − iℏ∂ w t0 ; In the application of the chiral anomaly in condensed ð Þ¼ ð Þ exp ð Þð t0 Þ ð Þ ℏ 0 matter physics and related fields, the species doublers are treated as physical objects unlike in conventional lattice ð4Þ gauge theory where they are unphysical nuisances. We point out that the chiral anomaly generated by γ5 for species where doublers vanishes for each doubler separately in the chiral ! 1 θ − α e−iφðtÞ symmetric lattice gauge theory with a finite lattice spacing, cos 2 ð Þ w ðtÞ¼ ; contrary to the common assumption. Instead, the general þ 1 sin 2 ðθ − αÞ form of the pair creation associated with the spectral flow ! 1 −iφðtÞ appears; it is emphasized that a picture of spectral flow is sin 2 ðθ − αÞe w−ðtÞ¼ : ð5Þ drastically changed if the Dirac sea has a finite depth. This − cos 1 ðθ − αÞ view is supported by the Ginsparg-Wilson fermion that 2 describes a Weyl fermion on the lattice without species This solution is confirmed by evaluating doublers and gives a well-defined index related to the chiral anomaly even for a finite lattice spacing. † iℏ∂tψðtÞ¼fiℏ∂tw ðtÞþw ðtÞ½w ðtÞðH − iℏ∂tÞw ðtÞg The present study is motivated by the recent excitement Z i t in the subject of effective Weyl fermions in condensed − dt0w† t0 H − iℏ∂ w t0 × exp ð Þð t0 Þ ð Þ matter and nuclear physics [2,7–10,14–16], but we believe ℏ 0 that it contains physical and technical aspects which will † ¼fiℏ∂tw ðtÞþw ðtÞ½w ðtÞðH − iℏ∂tÞw ðtÞ interest the wider audience. † þ w∓ðtÞ½w∓ðtÞðH − iℏ∂tÞw ðtÞg Z II. BERRY’S PHASE i t − dt0w† t0 H − iℏ∂ w t0 × exp ð Þð t0 Þ ð Þ We discuss here some general properties of Berry’s ℏ 0 phase, which are relevant to the analyses in the present ¼ HψðtÞ; ð6Þ paper, using a simple soluble model. The precise adiabatic Berry’s phase is defined only in an ideal mathematical † where we used w∓½H − iℏ∂tw ¼ 0 and the completeness limit. We thus adopt the exact nonadiabatic phase (hol- w w† w w† 1 onomy) in the manner of Aharonov and Anandan for the relation þ þ þ − − ¼ . α general Schroedinger problem [17], which contains enough The parameter is given by freedom to describe a possible genuine monopole, as our η α θ − α definition of Berry’s phase. The adiabatic Berry’s phase is sin ¼ sinð Þð7Þ naturally defined from the nonadiabatic phase in the α θ= η θ adiabatic limit. The generic term “geometric phase” is or, equivalently, tan ¼ sin ð þ cos Þ, with the param- η thus more appropriate, but we follow the common practice eter defined by and use the term “Berry’s phase” except for the case where η ≡ 2ℏμB=ℏω μBT=π it is appropriate to make a distinction. ¼ ð8Þ

016018-2 CHARACTERISTICS OF CHIRAL ANOMALY IN VIEW OF … PHYS. REV. D 97, 016018 (2018) and the period T ¼ 2π=ω. The parameter η → ∞ implies the such as B → 0 with fixed T. It is emphasized that we do not adiabatic limit, and η → 0 implies the nonadiabatic limit. assign physical reality to our “monopole”; we define The exact extra phase factor in (4) for one period of the monopole operationally, and we say that there is a motion monopole if one finds a well-defined phase corresponding Z to a monopole. i T † To analyze the vanishing of the monopole more quanti- exp − dtw ðtÞð−iℏ∂tÞw ðtÞ ℏ Ω =Ω 1 − θ − α = 0 tatively, we define the ratio þ mono ¼ð cosð ÞÞ 1 − θ Ω =Ω η2 2 1 θ ¼ exp ½−iπð1 ∓ cosðθ − αÞÞ ð9Þ ð cos Þ which approaches þ mono ¼ cos 2 for η → 0 sincewe have θ − α ≃ η sin θ from (7) in this limit. We defines Berry’s phase thus have for the upper half-sphere with θ ¼ π=2, I dB⃗ Ω =Ω 1=2 η2; Ω ≡ 2 A⃗ B⃗ dt þ mono ¼ð Þ ð13Þ ð Þ dt which is an indicator how the monopole flux vanishes in the ¼ 2πð1 ∓ cosðθ − αÞÞ; ð10Þ nonadiabatic limit η ¼ 2μB=ω → 0. This in particular shows where Berry’s connection is defined by that

A⃗ B⃗ w† t −i∂ w t : Ω =Ω ∼ 1=ω2 ð14Þ ð Þ¼ ð Þð B⃗Þ ð Þ ð11Þ þ mono We tentatively normalize Ω in this section to agree with the for ω → ∞. [If one starts with the south pole, one may use Ω0 2π 1 θ Ω Ω solid angle subtended by the moving spin. It is known [18] mono ¼ ð þ cos Þ and − and the fact that is defined that our phase agrees with the nonadiabatic phase in the mod 4π; then, one obtains the same ratio.] Physically, this manner of Aharonov and Anandan. The exact expression of shows that the movement of the spin does not follow the rapid Ω in (10) as it stands is not topological because of the movement of B⃗or very weak B⃗;itiswellknownthatBerry’s α “ ” Ω presence of , which modifies the magnetic flux þ, for phase is understood as the solid angle subtended by the example, from the monopole value moving spin (see, for example, Ref. [18]) Ω 2π 1 − θ : mono ¼ ð cos Þ ð12Þ ψ † t σψ⃗ t ð Þ ð Þ ’ w† t σ⃗w t The topology of Berry s phase is thus only approximate for ¼ ð Þ ð Þ the generic situation η < ∞ for which α ≠ 0. For the precise ¼ðsinðθ −αÞcosφ;sinðθ −αÞsinφ;cosðθ −αÞÞ: ð15Þ adiabatic limit T → ∞ [11] (and thus η → ∞) for which α → 0 Ω → 2π 1 ∓ θ in (7),wehave ð cos Þ, namely, approaches the phase generated by a spurious monopole In passing, we mention that, using the hidden-local gauge symmetry inherent in the second quantized formulation, one located at the origin of the parameter space B⃗¼ 0. The limit T → ∞ can parametrize any solution of the Schroedinger equation implies that one cannot define the winding number iℏ∂ ψ t; x⃗ Hˆ t ψ t; x⃗ Hˆ t since it takes an infinite amount of time to make one turn; this t ð Þ¼ ð Þ ð Þ for a general Hamiltonian ð Þ, trivial topology is also seen by the vanishing of Berry’sphase which is cyclic (i.e., periodic up to a phase factor), in the by deforming it smoothly to the nonadiabatic limit using the form Z Z exact solution, as shown below. i t In the nonadiabatic limit η ¼ μBT=π → 0, which ψðt;x⃗Þ¼wðt;x⃗Þexp − dt d3xw†ðt;x⃗ÞHˆ ðtÞwðt;x⃗Þ ℏ 0 includes the cases B → 0 with fixed T or T → 0 with Z Z B α → θ ’ Ω t fixed , we have in (7), and thus Berry s phase − dt d3xw† t;x⃗iℏ∂ w t;x⃗ ; Ω → 0 4π ð Þ t ð Þ ð16Þ become trivial constants, or . Namely, the 0 “magnetic monopole” disappears for η → 0 which includes the approach to the spurious monopole position B → 0 with with a suitable function wðt; x⃗Þ with wð0; x⃗Þ¼wðT;x⃗Þ [18]. fixed T or T → 0 with fixed B in the exact solution; Our exact solution in (4) has this general structure. The physically, this means that the nonadiabatic phase cannot nonadiabatic phase (holonomy) in the manner of Aharonov describe a genuine monopole in the nonadiabatic domain and Anandan [17] is then written as Z Z i T ψ 0; y⃗† dt d3xψ † t; x⃗iℏ∂ ψ t; x⃗ ψ T;y⃗ arg ð Þ exp ℏ ð Þ t ð Þ ð Þ Z Z 0 T 1 3 † ¼ − dt d xw ðt; x⃗Þð−iℏ∂tÞwðt; x⃗Þ: ð17Þ ℏ 0

016018-3 KAZUO FUJIKAWA PHYS. REV. D 97, 016018 (2018)

This shows that Berry’s phase used in the present paper in exist, is topological with respect to both ðT;jBjÞ for (11) for the spatially independent problem with wðtÞ is the each θ. This fact shows that the derivation of a genuine nonadiabatic phase [18]. It is known that the nonadiabatic monopole from the adiabatic Berry’s phase is a very crude phase for the spin system agrees with the solid angle approximation. subtended by the moving spin as in (15) (see, for example, The parameter domain is too strongly constrained in the Ref. [18]). It is also possible to consider that our formula (9) above mathematical adiabatic limit [11] to analyze equal- is measuring the generally defined geometric phase (17) time commutators in our application. We thus work with using the specific magnetic field (3). See also an old paper of the exact nonadiabatic phase (holonomy) [17] that is E. Majorana [19]. geometric, giving a solid angle drawn by the moving spin In application of Berry’s phase to condensed matter as in (15). This geometric phase has enough parameters to physics [2,9,10], one analyzes a 2 × 2 energy matrix describe a possible genuine monopole if it should be hðp⃗ðtÞÞ which is a truncation of band structure to the contained in the geometric phase as a subclass. It should two levels crossing at p⃗ðtÞ¼0. The variable p⃗ðtÞ specifies be emphasized that we do not assume the physical reality of the movement of the electron along the levels in the band our monopole. We measure the geometric phase (flux from d ⃗ ⃗ ’ structure; in a semiclassical picture, dt p⃗¼ e½E þ v⃗F × B. a monopole) using Berry s connection (11) operationally. This two-level truncation of the multiband structure is When the geometric phase agrees with the solid angle given expected to be valid at sufficiently close to the level- by a monopole, we say that there is a monopole. If the crossing point. It is generally expanded as geometric phase vanishes, we say that the monopole disappeared. The same is true in the next section, where ’ y0ðtÞ 0 we measure the geometric phase using Berry s connection hðp⃗ðtÞÞ ¼ þ σ⃗· y⃗ðtÞ; ð18Þ defined in terms of momentum. 0 y0ðtÞ We have shown that the geometric phase thus defined approaches the monopole value at the adiabatic limit, for where σ⃗stand for Pauli matrices; ðy0; y⃗Þ are functions of example, ω → 0 with fixed nonvanishing jBj, but as soon as p⃗ðtÞ. Ignoring the common term y0ðtÞ, one may consider a one is away from the exact adiabatic limit, the geometric special (Weyl) case y⃗¼ vFp⃗with vF a suitable constant, and one obtains the Hamiltonian with pseudospin, phase deviates from the monopole value. The solid angle which determines the geometric phase shrinks to 0 in the nonadiabatic limit η → 0, for example, for jBj → 0 with H ¼ vFσ⃗· p⃗ðtÞ; ð19Þ fixed finite ω or ω → ∞ with jBj ≤ B0 with a constant B0 which is the case we discuss below and mathematically as (15) shows. The geometric phase vanishes in the non- identical to the above motion of spin in a rotating magnetic adiabatic limit, while the flux from a genuine monopole, if field. A more general case is analyzed in Ref. [9]. The it should exist, does not vanish. This limiting behavior is property of Berry’s phase explained above in (14) shows crucial for our analyses in the next section. that Berry’s phase vanishes in the nonadiabatic limit ω ¼ 2π=T → ∞ with fixed B. This property is crucial in the III. COMMUTATORS IN BJL PRESCRIPTION analysis of possible anomalous commutation relations To analyze the possible anomalous commutation rela- ’ ω → ∞ induced by Berry s phase since the limit deter- tions induced by anomalies which are not recognized by mines the equal-time commutation relations, which imply naive canonical manipulations [3–6], one needs to use high frequencies by the uncertainty principle, of the machinery which does not rely on the canonical argument. ⃗ variable involved in Berry’s phase such as BðtÞ or p⃗ðtÞ The BJL prescription provides such a scheme and has been by the Bjorken-Johnson-Low (BJL) prescription [20]. used extensively to analyze commutators related to anoma- Before proceeding further, we emphasize that the notion lies [21–24]. We emphasize that the BJL prescription has of topology of an adiabatic phase is “self-contradictory” in worked for all the known cases of anomalous commutators the sense that topology implies its robustness against the associated with anomalies. general variation of dynamical variables and parameters, To be explicit, we start with the time ordered correlation while the adiabatic phase is well defined only in the precise function of dynamical variables such as adiabatic limit. Mathematically, one may first take the Z ∞ precise adiabatic limit jBj → ∞ with fixed T ¼ 2π=ω or iωt ⋆ T → ∞ B dte hT xkðtÞxlð0Þi; ð20Þ with fixed j j, as was done by B. Simon [11], and −∞ examine the remaining topological structure, namely, a Ω 2π 1 − θ T⋆ T monopole with mono ¼ ð cos Þ in our model; to be where stands for the so-called covariant product, precise, this monopole is topological (invariant) with which does not specify the precise equal-time limit t ¼ 0 of respect to T for each θ when jBj → ∞ is first taken and the correlation function. We assume that an explicit form of topological with respect to jBj for each θ when T → ∞ is the correlation function is known by the path integral first taken. In comparison, a genuine monopole, if it should evaluation, for example. The basic observation of BJL is

016018-4 CHARACTERISTICS OF CHIRAL ANOMALY IN VIEW OF … PHYS. REV. D 97, 016018 (2018) that, if the above correlation vanishes for ω → ∞, one can quadratic expansion of the Lagrangian around the origin replace T⋆ by the conventional T product which is assumed of the space of dynamical variables to analyze the com- to specify the equal-time limit precisely. This criterion is mutation relations. We thus consider the quadratic form of regarded as an analog [13] of Riemann-Lebesgue lemma in (21) around ðx;⃗p⃗Þ¼ð0; p⃗FÞ by redefining the momentum the Fourier transform; if the function fðtÞ is smooth and ðp⃗− p⃗FÞ → p⃗, wellR defined around t ¼ 0, the large frequency limit of ∞ dteiωtf t Z −∞ ð Þ vanishes. We can thus identify the canonical 1 1 p⃗2 T T⋆ S dt p x_ F x x_ − Ω p p_ − ; product from the quantity defined by the product. ¼ k k þ 2 lk l k 2 lk l k 2m ð22Þ We now examine an explicit effective action for the electron F ∂ A 0 − ∂ A 0 Ω ∂ A p⃗ − where lk ¼ l kð Þ k lð Þ and lk ¼ l kð FÞ Z 2 ðp⃗− p⃗FÞ ∂kAlðp⃗FÞ. One can then confirm the basic correlation S ¼ dt pkx_k þ Akðx;⃗ tÞx_k − Akðp⃗Þp_ k − ; 2m Rfunction by first integrating over p⃗ in the path integral Dx⃗Dpx⃗ kðtÞxlð0Þ exp½ði=ℏÞS as ð21Þ Z which played a central role in the analyses of anomalous iωt ⋆ dte hT xkðtÞxlð0Þi commutation relations induced by Berry’s phase in – −1 2 −1 Refs. [14 16]. This action is constructed to reproduce ¼ iℏf½1 þ imωΩ mω þ iωFg the equations of motion in the precise adiabatic limit. iℏ A x;⃗ t 1 imωΩ ; kð Þ is the electromagnetic potential of which the ¼ 2 2 ð þ Þ ð23Þ explicit time dependence is assumed to be very slow in mω þ iωF − mω ΩF our analysis. The Berry connection Akðp⃗Þ stands for an analog of (11) defined by the Hamiltonian (19), the where it is understood that the kl matrix element of the expression of which has a well-defined monopole form right-hand side is taken in the following, together with the only in the precise adiabatic limit η ¼ 2vFjp⃗j=ℏω → ∞, equation of motion pk ¼ m½δkl − ðΩFÞklx_l. namely, sufficiently slow time dependence of p⃗ðtÞ and The correlation (23) vanishes for ω → ∞, and thus we ⋆ sizable jp⃗ðtÞj. Customarily, an exact monopole form is can replace T by the canonical T. We then multiply the assumed for the general movement of dynamical variables both sides by −iω and consider the large ω limit. We then on the premise that one applies the action only to the obtain approximately adiabatic movement of variables [14–16]. Z We follow this procedure for a moment but come back to a iωt lim dte fhTx_kðtÞxlð0Þi þ δðtÞ½xkð0Þ;xlð0Þg more precise reexamination of (21) later. We choose the ω→∞ p⃗− p⃗ 2= 2m form of the free Hamiltonian ð FÞ ð Þ in (21) by 1 redefining the momentum to avoid the (obvious) non- iℏ Ω; ¼ 1 − ΩF ð24Þ adiabatic limit caused by the vanishing momentum. The parameter p⃗F is an arbitrary parameter, which does not d exist in the action of the original literature [14–16], and it where we used dt hTxkðtÞxlð0Þi ¼ hTx_kðtÞxlð0Þiþ does not necessarily imply the Fermi momentum p⃗F.To δðtÞ½xkð0Þ;xlð0Þ. We thus conclude ensure the true adiabatic limit, one needs to constrain the time dependence of dynamical variables also (“adiabatic” 1 x 0 ;x 0 iℏ Ω implies time dependence), but it is not done in a simple ½ kð Þ lð Þ ¼ 1 − ΩF ð25Þ manner. The applicability of this action (21) beyond the precise and adiabatic movement of dynamical variables is not established, but we tentatively assume that this action is used for Z the general movement of the electron to evaluate equal-time iωt dte hTx_kðtÞxlð0Þi commutation relations. It is also shown later that this action is nonlocal in time (contains infinitely higher derivative ℏω Ω ¼ ð1 þ imωΩÞ − iℏ : terms) and thus no canonical quantization is applicable. mω2 þ iωF − mω2ΩF 1 − ΩF We next observe that canonical commutation relations are not modified by any static potential in quantum ð26Þ mechanics. Only the terms with an explicit time derivative such as the kinetic energy term are essential, and the We then multiply both sides of this relation by −iω and absolute size of the potential such as the harmonic examine the behavior for ω → ∞. Repeating this pro- oscillator potential does not matter. We can use the cedure, we obtain

016018-5 KAZUO FUJIKAWA PHYS. REV. D 97, 016018 (2018) I Z 1=m 1=m 1 x_ 0 ;x 0 −iℏ F Ω ; Akðp⃗Þp_ kdt ¼ ΩdS ¼ π; ð29Þ ½ kð Þ lð Þ ¼ 1 − ΩF þ 1 − ΩF 1 − ΩF 1=m 1=m x_ 0 ; x_ 0 −iℏ F and this value of Ω appearing in the commutator (25) ½ kð Þ lð Þ ¼ 1 − ΩF 1 − ΩF determines the coefficient of “chiral anomaly” in the 1=m 1=m 1 F F Ω ; derivation of Ref. [16]; any deviation from the monopole þ 1 − ΩF 1 − ΩF 1 − ΩF ð27Þ value would invalidate the derivation of anomaly. Since the Hamiltonian (19) is valid only in the vicinity of the crossing point of two levels in the band structure, jp⃗j is much smaller ẍ 0 ;x 0 − x_ 0 ; x_ 0 and ½ kð Þ lð Þ ¼ ½ kð Þ lð Þ. We thus reproduce the than jp⃗Fj, and thus ω contained in p⃗ðtÞ is required to be quantized version of Poisson brackets in Refs. [15,16] very small to satisfy adiabaticity. Note that for any value x_ 0 ; x_ 0 except for the last term in ½ kð Þ lð Þ. This agreement jp⃗j ≤ jp⃗Fj implies that the Poisson bracket implicitly treats the variables with time derivatives preferentially, irrespective 2vFjp⃗j=ℏω ≤ η ¼ 2vFjp⃗Fj=ℏω → 0 ð30Þ of the size of the potential term. More importantly, the agreement of the two schemes (to the order relevant to the in the limit ω → ∞, and this momentum value belongs to analysis in the present paper) shows that both schemes use the nonadiabatic domain in the analysis of commutators. the behavior of dynamical variables in the extremely The point-type monopole structureR of Ω we assumed in ω → ∞ nonadiabatic region with in an essential way, for (22), which always satisfies ΩdS ¼ π, is not valid for which the derivation of the effective action (21) completely large ω, and it is strongly suppressed in the nonadiabatic fails. Here, we briefly comment on what we mean by the limit with small η typically in the form (if one uses the high frequency region. When one defines correct value of Berry’s phase) I Z Z 2 iωt Akðp⃗Þp_ kdt ¼ ΩdS ≃ πη ð31Þ FðωÞ¼ dte h0jTxkðtÞxlð0Þj0i; ð28Þ

as in [18] which goes as 1=ω2 for large ω. We tentatively Pwe confirm Fð∞Þ¼0, but limω→∞ωFðωÞ∼h0j½xk;xlj0i¼ assumed that (21) with a monopole at the origin of the nðh0jxkjnihnjxlj0i−h0jxljnihnjxkj0iÞ; namely, all the momentum space is valid for the general class of dynamical intermediate states contribute with equal weight. If one variables and derived (25) and (27). But the derivation of 2 2 cuts off the frequency such as FðωÞ ∼ exp½−ω =ωc, for the effective action (21) itself completely fails for the example, no nontrivial equal-time commutators appear. frequency regions used in the BJL analysis. This is an ingenious insight of BJL. To summarize the above analysis, one recognizes that the The denominator of the basic correlation function (23) direct use of Berry’s phase and the indirect use of Berry’s is written as ð1 − imωΩ þðimωΩÞ2 − :::::Þmω2 þ iωF; phase through an effective action and equal-time commu- namely, it contains an infinite series in ω which shows tators are very different; for the direct use of the action in that the theory is nonlocal in time and thus no canonical (21), one obtains an equation of motion with a monopole at quantization is possible [25], although the action (21) is the origin of the momentum space. One can choose the time formally written in terms of x⃗ and p⃗ and thus looks dependence of dynamical variables and the value of jpj superficially canonical; it is no more canonical if one adds such that the approximateH adiabatic conditionR is satisfied. a pointlike monopole at the origin of the momentum space. One can thus maintain Akðp⃗Þp_ kdt ¼ ΩdS ≃ π as in The path integral defines a correlation function even for a (29). Namely, one can choose the systems which satisfy the theory nonlocal in time, but one cannot convert it to adiabaticity conditions approximately [14]. To my knowl- canonical formulation. A consequence of this nonlocality edge, all the successful applications of Berry’s phase in the is that we have an infinite tower of commutation relations in past are the direct use of the action. But the model with a (27) containing arbitrary higher time derivative terms. The pure monopole (21) totally fails to describe Berry’s phase assumption of a pointlike monopole in momentum space, off the adiabatic limit implied by the exact solution of which gives rise to the correlation (23), leads to anomalous Berry’s model analyzed in Sec. II. space-time commutation relations similar to the noncom- For the indirect use of the action (22) to derive the mutative space-time [25]. This clear recognition of non- anomalous commutators, one finds that the identical form locality is an advantage of the BJL method. Rof anomalous commutators with the monopole value We recall that the monopole-type structure of Berry’s ΩdS ¼ π appears irrespective of the value of pF even phase is valid only in the precise adiabatic limit η ¼ for pF ∼ 0. Mathematically, the extremely nonadiabatic 2vFjp⃗j=ℏω → ∞ [11]. Only in this limit, we can have limit ω → ∞ is important to determine the commutators, (for the half-sphere) and in this limit, the assumed pointlike monopole in the

016018-6 CHARACTERISTICS OF CHIRAL ANOMALY IN VIEW OF … PHYS. REV. D 97, 016018 (2018) momentum space controls the commutators independently In conclusion, we choose the action (32) which is of the adiabatic or nonadiabatic domain. Once an explicit consistent with the exact solution of Berry’s model and form of an effective action is written, either the Poisson the BJL analysis as a prediction of Berry’s phase to evaluate bracket or BJL prescription automatically determines the equal-time commutators, and we obtain no anomalous form of commutators. space-time commutators. Physicists tend to assume the We want to incorporate the applicability condition of monopole form of Berry’s phase everywhere, which is Berry’s phase faithfully in the evaluation of commutators. actually valid only in the precise adiabatic limit [11,26]. The only way to do so is to choose the action which is valid The analysis of Berry’s phase in the nonadiabatic and in the extreme nonadiabatic domain to be consistent with nontopological domain is very common in chemistry [27]. the BJL prescription. This action may not be accurate in the One may recall that the treatment of molecular systems adiabatic limit. Since Berry’s phase, when measured in a using the Born-Oppenheimer approximation led to the natural manner described in Sec. II, vanishes for ω → ∞ as original discovery of the geometric phase in the form of in (14) irrespective of any finite jpj or jBj, we define the the Longuet-Higgins phase change rule [27] in chemistry. effective action without Berry’s phase, which gives accurate equations of motion in the nonadiabatic domain where IV. CHIRAL ANOMALY FROM A MONOPOLE the BJL method is applicable, The mere presence of a Weyl fermion does not induce Z p⃗2 noncommutative space-time, and thus no anomaly is S dt p x_ A x;⃗ t x_ − : induced by Berry’s phase. However, a genuine pointlike ¼ k k þ kð Þ k 2m ð32Þ monopole placed at the origin of momentum space induces noncommutative space-time, as we have analyzed above This gives the standard free nonrelativistic motion of the in (25) and (27), in agreement with the analysis in electron. One can confirm that the BJL method and Poisson Refs. [15,16]. Based on this latter assumption, the authors brackets applied to (32) reproduce the ordinary commuta- of Refs. [16] discussed the anomalous equal-time commu- 0 tion relations for x⃗and p⃗which are obtained by setting tation relations of the charge density operator j ¼ nðxÞ Ω ¼ 0 in (25) and (27). This analysis confirms that no induced by the anomalous space-time commutators. They anomalous equal-time space-time commutation relations then derived a kinematic relation which formally agrees 0 are induced by the mere presence of a Weyl fermion (such with the covariant chiral anomaly by an analysis of ½j ;H. as a massless Weyl neutrino.) Here, we discuss the mathematical consistency of their Ideally, one may want to have an effective action which formulation, since this problem is analogous to the well- incorporates the exact information of Berry’s phase for known interesting but controversial problem of an interplay general movement of dynamical variables, but it is not of monopole and anomaly in the fermion-monopole system feasible. We thus chose two explicit effective actions, the in the presence of a genuine pointlike monopole (Callan- action (21) which gives a correct equation in the precise Rubakov effect) [28]. It is interesting to see if a monopole at adiabatic limit with a monopole value of Ω but fails in the the origin of the momentum space gives a flux correspond- nonadiabatic domain, and the action (32) which gives a ing to the correct chiral anomaly. correct equation in the precise nonadiabatic limit with Since they derived the possible anomaly using equal- vanishing Ω ¼ 0 but fails off the exact nonadiabatic time commutation relations of currents [16], we first briefly domain. From the consistency with the BJL prescription, summarize the known basic properties of equal-time we choose the commutators given by the action (32) as commutators associated with the conventional formulation physical ones. of chiral anomaly. To avoid the Schwinger term, we usually As for the action (21), we suggest that it should be use the Gauss-law operator defined by regarded as an action for a charged particle with a 0 _ k monopole located at the origin of the momentum space. G ¼ j − ∂kA ð33Þ [If the action (21) or (22) should be shown to those unfamiliar with Berry’s phase, they would recognize it of chiral Abelian gauge theory in the gauge A0 ¼ 0. First of as an action for a particle with a monopole placed at the all, no anomalous equal-time commutation relation of the origin of the momentum space without any constraint on Gauss operator for Abelian chiral gauge theory is known in the movement of variables.] The appearance of anomalous conventional formulation [21–24,29], namely, space-time commutators in Refs. [15,16] is understood as an artifact that they used an effective action (21) or (22) 0 _ k 0 _ k which describes a genuine pointlike monopole placed at the ½j ðt; x⃗Þ − ∂kA ðt; x⃗Þ;j ðt; y⃗Þ − ∂kA ðt; y⃗Þ ¼ 0: ð34Þ origin of the momentum space. It is well known that the magnetic field modifies equal-time commutators such as in The anomalous commutation relation of the Gauss operator, the Landau level. which is closely related to the commutator in Ref. [16],

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i 1 1 ⃗ ⃗ using the anomalous space-time commutators; the first term ∂tGðt; x⃗Þ¼ ½H; Gðt; x⃗Þ ¼ − E · B; ð35Þ ℏ 3 4π2 with σ⃗ arises form (25), and the second term with B⃗ x_ 0 ;x 0 arises from (27), namely, ½ kð Þ lð Þ combinedR with has been discussed in Refs. [13,29]. This relation shows that ⃗ pk ¼ m½δkl − ðΩFÞ x_l. The parameter k ¼ð1=2πÞ dS · the Gauss operator, which is the generator of the time kl Ω⃗ independent gauge transformation, is time dependent for corresponds to a monopole located at the origin of the k 1 chiral Abelian gauge theory and that the Hamiltonian H is momentum space [ ¼ for the full sphere integral in gauge noninvariant, and thus theory is inconsistent,asiswell (29)]. They then assume the form of the Hamiltonian known. To have a consistent theory in continuum, one needs Z to have a vectorlike theory such as the conventional QED if H0 ¼ H þ d3xϕðx⃗Þnðx⃗Þð39Þ one does not increase the number of fermion species. The relation (35) is reduced to ⃗ with E⃗¼ −∇ϕðx⃗Þ. H has a rather involved expression but 1 1 the detailed expression of H is not relevant for the ∂ jμ − E⃗ B⃗ μ ¼ 3 4π2 · ð36Þ evaluation of anomaly. They then evaluate

0 ∂tnðx⃗Þ¼i½nðx⃗Þ;H if one uses the equation of motion for Aμ. The extra factor 1=3 −1=2 ⃗ ⃗ k (and due to chiral projection) compared to the standard ¼ −∇ · j⃗þ ∇ × σ⃗þ B⃗ · E⃗ form of anomaly (1) with m ¼ 0 is a characteristic of 4π2 consistent anomaly. This difference arises from the fact that k μ −∇⃗ j⃗0 B⃗ E⃗ j in the standard anomaly (1) does not couple to gauge field ¼ · þ 4π2 · ð40Þ Aμ; one can thus impose gauge invariance for currents appearing inside the Hamiltonian (or Lagrangian) and collect with j⃗0 ¼ j⃗þ E⃗× σ⃗. The relation the nonconservation (anomaly) only to the external non- gauge current. In contrast, the current appearing in the Gauss- k ∂ n x⃗ ∇⃗ j⃗0 B⃗ E⃗ law operator in (35) is generated by a variational derivative of t ð Þþ · ¼ 4π2 · ð41Þ the action with respect to Aμ, and thus a preferential treatment of one of currents is not allowed. is then identified with the anomalous identity with a More precisely, when one evaluates the anomaly by covariant anomaly. The choice k ¼ −1 corresponds to equal-time commutators, one generally encounters the the left-handed convention we have used in this paper commutator of the form so far, 1 ½jμðt; x⃗Þ;jνðt; y⃗Þ: ð37Þ ∂ jμ − E⃗ B;⃗ μ ¼ 4π2 · ð42Þ

From this commutator which treats the two currents on μ μ for the chiral current j ¼ ψγ¯ ½ð1 − γ5Þ=2ψ, which is equal footing, one immediately recognizes that it is μ obtained from (1) combined with ∂μðψγ¯ ψÞ¼0 if one impossible to impose gauge invariance on one of the sets m ¼ 0. currents and collect the nonconservation (anomaly) to The essential commutator in the analysis in Ref. [16] is the other. This is the reason why only the consistent form 0 thus given by [with j ðxÞ¼nðxÞ] of anomalies appeared in the analysis of equal-time commutation relations in the past [23,24,29]. This is the 0 0 ½j ðt; x⃗Þ;j ðt; y⃗Þ: ð43Þ mathematical consistency condition, which has nothing to do with physics. From a point of view of Feynman 1=3 As we have already shown in (37), it is mathematically diagrams, this factor in the present Abelian gauge impossible to treat two currents on unequal footing in the theory is also understood as a Bose symmetrization factor evaluation of the equal-time commutator. The appearance of a triangle diagram, which means all three currents are of the covariant anomaly in the evaluation of the commu- treated on equal footing. tator is a mathematical inconsistency. One might still argue Coming back to the analysis in Ref. [16], the authors that the derivation of anomaly in Ref. [16] is different from argued for the derivation of a kinematic relation by an the conventional one, and all the currents involved can have analysis of j0;H with j0 x n x . To be more explicit, ½ ð Þ¼ ð Þ covariant anomalies. Even in this case, all the currents they first evaluate 0 including j ðxÞ¼nðxÞ inside the Hamiltonian (39) are k anomalous (not conserved), and thus their Hamiltonian is n x⃗;n y⃗ −i ∇⃗ σ⃗ B⃗ ∇⃗δ x⃗− y⃗ gauge noninvariant, and the theory is physically incon- ½ ð Þ ð Þ ¼ × þ 2 · ð Þð38Þ 4π sistent just as (35).

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One may conclude that a covariant form of anomaly inevitable appearance of species doublers seems to be appears in the gauge current in the derivation of chiral assumed in condensed matter physics [10], presumably anomaly from a pointlike monopole located at the origin of because the simplest nearest neighbor discretization momentum space. of a massless fermion in the tight-binding approximation In passing, we comment on the evaluation of the (latticed model) suggests such behavior. conventional gauge theoretical anomalies by emphasizing The notion of species doublers means that a single the adiabatic treatment and thus compared with the idea of “local” fermion and thus a single local current defined on adiabatic phases [30–32]. Our understanding of this the lattice actually describe the multiple species of fer- approach, although very interesting by giving a novel mions, i.e., doublers, in momentum space in the limit physical picture, is that it illustrates that chiral anomaly a → 0. For example, if one attempts to define a left-handed in gauge field theory is evaluated using various formula- massless fermion tions of field theory, both in the Lagrangian and Hamiltonian formalisms. The actual evaluation of chiral ψ LðxÞ¼½ð1 − γ5Þ=2ψðxÞð44Þ anomaly itself in Ref. [31], for example, is identical to the conventional field theoretical evaluation. Also, it appears to on the lattice, the species doubling implies that one be not easy to understand the fact that the local form of the inevitably obtains both a left-handed fermion and a basic identity (1) is valid even in a curved space-time from a right-handed fermion in momentum space in the limit purely adiabatic picture. a → 0. Similarly, ψ RðxÞ¼½ð1 þ γ5Þ=2ψðxÞ implemented Similarity of topological properties appearing in chiral on the lattice inevitably induces both a left-handed fermion ’ anomaly and Berry s phase have been discussed in and a right-handed fermion in momentum space in the limit Refs. [33,34]. The Hamiltonian formalism initiated by a → 0. The chiral fermion such as ψ LðxÞ, which is inflicted Nelson and Alvarez-Gaume [33] was very influential. with chiral gauge anomaly (35) in continuum theory and is The analysis of topology in Ref. [33] [in particular, of thus inconsistent by itself, contains no chiral gauge SU 2 ð Þ anomaly] has been examined in great detail in anomaly when placed on the lattice which cuts off the Ref. [35]. The conclusion of Ref. [35] is summarized in the short distances. Both ψ L and ψ R can in principle appear form: In the fundamental level, the difference between the simultaneously; for a → 0, we then have multiple species ’ two notions, anomaly and Berry s phase, is stated as of charged fermions (such as the massless “electron” and follows; the topology of given gauge fields leads to level “muon,” although we originally intended to define only the crossing in the fermionic sector in the case of chiral anomaly, electron, and thus the construction is inconsistent). In any and the inevitable failure of the adiabatic approximation is case, the notion of species doubling implies that we obtain essential to establish the existence of anomaly, whereas the at least twice as many fermions in momentum space in the level crossing in the matter sector leads to the topology of a → 0 ’ limit than the original intention. Berry s phase only when the precise adiabatic approximation On the other hand, the common knowledge in lattice holds. These two adiabatic conditions are not compatible. gauge theory nowadays is that a recent progress of the The analysis of anomaly by Nelson and Alvarez-Gaume [33] ’ Ginsparg-Wilson fermion allows a definition of a single is perfectly valid without referring to Berry sphase. Weyl fermion without doublers on the lattice [38]. See Ref. [13] for further references. The chiral current in such a V. CHIRAL SYMMETRY AND theory is exponentially local [39] (stated intuitively, it SPECIES DOUBLING contains nearest neighbor couplings, next nearest neighbor We would like to analyze the species doublers, which are couplings, and so on ...; namely, the construction is not commonly associated with an assertion of a pairwise manifestly local but actually becomes local in the limit appearance of level crossings in the Brillouin zone. In a → 0) without any species doublers. In this formulation, the recent discussions of Weyl fermions [8–10] and also in exact chiral symmetry is realized by an operator that is the earlier work [7], the chiral anomaly corresponding to deformed from γ5 for finite a; one obtains a nontrivial (42) is assumed for a Weyl fermion at each level-crossing Jacobian as a symmetry breaking factor under chiral point such as (19) in the band structure by identifying it as a transformation in the path integral formulation of chiral species doubler. identity [40], which gives a correct anomaly in the The fermion theory defined on a lattice with a finite continuum limit [13]. For small momenta close to the spacing a removes the entire short distances [12] and thus origin of the momentum space, the ordinary Weyl spectrum high frequencies [13] that are responsible for the anomaly. is realized; namely, one can identify a low-energy chiral This fact is reflected in the appearance of species doublers excitation mode on the (hypercubic) lattice without dou- in momentum space which ensure the absence of anomaly blers. Thus we no longer have the “no-go theorem” against even in the limit a → 0 for a manifestly chiral invariant a single chiral fermion on the lattice. formulation [36,37]. This is symbolically stated as the In condensed matter and related fields, the species absence of a “neutrino” on the lattice [37], and the related doublers are treated as physical objects, in contrast to

016018-9 KAZUO FUJIKAWA PHYS. REV. D 97, 016018 (2018) Z Z π=2a 3π=2a lattice gauge theory where the species doublers are dp −ipx dp −ipx ψ LðxÞ¼ e ψ LðpÞþ e ψ LðpÞ unphysical nuisances. It is then interesting to ask if the −π=2a ð2πÞ π=2a ð2πÞ Z absence of anomaly in latticized theory with species π=2a dp ð0Þ 1 −iϵ− ðpÞtþipx doublers is caused by the cancellation of well-defined ¼ e ψ LðpÞ −π=2a ð2πÞ anomalies produced by each doubler or none of doublers Z π=2a dp 0 has anomalies for finite a. We would like to show that none iπx1=a −iϵð ÞðpÞtþipx1 þ e e þ ψ Lðp þ π=aÞ of doublers has well-defined anomalies, as is expected from −π=2a ð2πÞ the fact that each doubler is defined only in part of the iπx1=a ≡ eL x e μR x 48 Brillouin zone of momentum space and thus not a well- ð Þþ ð ÞðÞ defined local field. Instead, a pair production associated −π=2a ≤ p<3π=a with spectral flow in the Dirac sea with finite depth takes by choosing the Brillouin zone .We e x μ x place. We emphasize that our analysis below is based on defined formally two fields Lð Þ and Rð Þ, although they ψ x lattice gauge theory, although we hope that it may have are actually part of a single field Lð Þ; when one discusses some implications on condensed matter and related fields. short distance properties such as the chiral anomaly of ψ LðxÞ, which imply the maximum extension in momentum space by the uncertainty principle, one cannot separate eLðxÞ A. Species doubling and spectral flow and μRðxÞ. We emphasize that eLðxÞ and μRðxÞ separately To understand the following discussions intuitively, it is cannot define well-defined fields for a ≠ 0 since half of the instructive to consider a d ¼ 1 þ 1-dimensional lattice Brillouin zone is missing in them. Nevertheless, this notation fermion. We thus consider a model Hamiltonian is useful to understand the following discussions. For example, in the picture of the spectral flow of ð0Þ ap r ϵ− ðpÞ¼− sin ap=a with all the negative energy states H σ sin σ 1 − ap ¼ 3 a þ 1 a ð cos Þð45Þ 0 ≤ p ≤ π=a being filled initially, a particle creation at the momentum close to p ¼ 0 implies a hole creation close to p ¼ π=a; namely, we have an inevitable pair production, with a constant r which is usually called the Wilson parameter. Note that HðpÞ has a period 2π=a in p.If ψ L þ ψ¯ L or eL þ μ¯ R: ð49Þ one recalls that the chiral matrix is given by γ5 ¼ σ3 in this notation, the first term is chiral invariant, but the second r ≠ 0 term does not commute with γ5 and thus breaks chiral For in (45), we have no more chiral symmetry symmetry. The chiral symmetric Hamiltonian is thus given generated by γ5. We have only a single massless solution, H p 0 by setting r ¼ 0, since ð Þ¼ means two linearly independent terms of HðpÞ in (45) vanish simultaneously. To be explicit, we have two eigenvalues, sin ap H0 ¼ σ3 : ð46Þ a sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ap 2 r 2 ϵ p sin 1 − ap ; ð Þ¼ a þ a ð cos Þ ð50Þ We have the energy spectrum of H0 as

ϵ p 2π=a ϵ p ϵ p p 0 sin ap that show ð þ Þ¼ ð Þ.Wehave ð Þ¼j j ϵð ÞðpÞ¼ ; ð47Þ p 0 ϵ p 2r p π=a a near ¼ and ð Þ¼a near ¼ , namely, no more species doubling for r=a ≫ 1. We thus have a single massless Dirac fermion, but the drawback of this con- ϵð0Þ p γ 1 where ð Þ correspond to chirality 5 ¼ , respec- struction is that the chiral symmetry of the Hamiltonian is ϵð0Þ p 2π=a ϵð0Þ p tively. Note that ð þ Þ¼ ð Þ. This exhibits completely lost. The recent progress in the Ginsparg- ϵð0Þ p Wilson fermion is that one can now construct a fermion the spectrum similar to the Weyl fermion for small j ð Þj, ð0Þ model that defines a single massless Dirac fermion with an and ϵ ðpÞ¼0 at p ¼ 0 and p ¼ π=a in the Brillouin exact chiral symmetry without doublers in the limit a → 0. ϵð0Þ p p π=a zone. Moreover, þ ð Þ near ¼ has the same In the sequel, we discuss the chiral anomaly using two ð0Þ structure as ϵ− ðpÞ near p ¼ 0, and similarly starting with pictures of Weyl fermions, namely, species doublers and ð0Þ ϵ− ðpÞ. Thus, we have effectively two Weyl fermions with the Ginsparg-Wilson fermion. We first discuss the chiral γ5 ¼ 1 and two Weyl fermions with γ5 ¼ −1; namely, we anomaly using species doublers and then the Ginsparg- have species doubling of the Weyl fermion for chiral Wilson fermion to support our view. Our analysis is invariant theory in the limit a → 0. performed in the framework of lattice gauge theory without ð0Þ We may write the solution corresponding to ϵ− ðpÞ in referring to the details of specific fields such as condensed the form matter and nuclear physics.

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B. Chiral anomaly and spectral flow hypercubic symmetry), but the essential aspect we want to We generalize the Hamiltonian (19) to a full-fledged discuss is correctly captured by this simplified model (53). Weyl-type wave equation By multiplying the chiral projectors, we find from (53) ∂ 1 ∂ 1 ∓ γ5 i − v σ⃗ u t; x⃗ 0; ψ L;RðxÞ¼ ψðxÞ ∂t F · i ∂x⃗ ð Þ¼ ð51Þ 2 iπx =a ¼ eL;RðxÞþγ3γ5e 3 μR;LðxÞ: ð54Þ where vF > 0 or vF < 0 corresponds to right-handed or ℏ 1 left-handed fermions, respectively. We set ¼ for ψ x simplicity. Since this equation is defined as an effective This shows that the left-handed fermion Lð Þ, for exam- a ≠ 0 ple, when placed on the lattice becomes a superposition of equation in the Brillouin zone with , we attempt to e x μ x simulate two fermions Lð Þ and Rð Þ. Two fermions are actually this equation by an effective lattice gauge theory ψ x treating the pseudospin σ⃗as a real spin. The purpose of this part of the single field Lð Þ, although they are displaced in momentum space. In the limit 1=a → large, the coherence simulation is to obtain possible new insights into con- e x μ x densed matter and related fields with regards to quantum between Lð Þ and Rð Þ is lost (and quantum tunneling anomaly. between these two states in momentum space is sup- We thus start with a massless Dirac fermion in continuum, pressed), and thus two fermions behave approximately as two Weyl fermions. We have approximately μ iγ ð∂μ − iAμÞψ ¼ 0; ð52Þ S ¼ ψ¯ LDψ L with the electromagnetic gauge field Aμ. We then put this ≃ e¯LDeL þ μ¯ RDμR ð55Þ equation on the hypercubic lattice, for example, with γ manifest chiral symmetry generated by 5. There are two ψ x ways to latticize the continuum gauge theory. The first one and similarly for Rð Þ. Note that the local chiral current in coordinate space carries no anomaly for a ≠ 0 [36,37], is the Hamiltonian formalism that exhibits the energy- ∂ ψ¯ γμψ x 0 momentum dispersion in the band diagram nicely. In this μð L LÞð Þ¼ . The semiclassical spectral flow on the ψ scheme, it is known that the minimum number of species basis of L in (54) shows that only the pair production from doublers appears [7], namely, one species doubler for each the vacuum such as chiral component that is sufficient to ensure the absence of chiral anomaly even in the limit a → 0. The other formu- ψ L þ ψ¯ L or eL þ μ¯ R ð56Þ lation is the Euclidean Lagrangian formalism, which generally contains more species doublers than the Hamiltonian is allowed. See the detailed discussion in the context of ΔN 0 formulation, but the analysis of chiral anomaly is more (49). Thus, the selection rule ψ L ¼ or transparent. Also, it is this Euclidean formulation that led to a formulation of the Ginsparg-Wilson fermion containing ΔðNe þ Nμ Þ¼0 ð57Þ no species doublers by deforming the generator of chiral L R symmetry from γ5. d 4 is satisfied, although two (would-be) Weyl fermions eLðxÞ We use here the ¼ Euclidean formulation, although μ x the physical contents in the band diagram of condensed and Rð Þ are far apart by matter theory are not directly seen in this formulation [to see the band structure, we need to consider the domain of Δp ≃ π=a ð58Þ small energy momentum combined with Wick rotation, and then (52) is realized]. In this notation, we have [see (48)] in momentum space for 1=a → large. The analysis of possible anomaly produced by each of the Weyl fermions iπx =a ψðxÞ¼eðxÞþγ3γ5e 3 μðxÞ; ð53Þ eLðxÞ and μRðxÞ separately is not needed in this pair creation other than the anomaly-free conservation law μ where eðxÞ and μðxÞ are “Dirac fermions” defined in ∂μðψ¯ Lγ ψ LÞðxÞ¼0. the momentum domain −π=2a ≤ kμ < π=2a when the In some applications, however, one may want to know an Brillouin zone is chosen as −π=2a ≤ kμ < 3π=2a.We explicit form of the (possible) anomaly for each Weyl emphasize that we have actually only a single physical fermion separately [7,9,10], but the chiral anomaly for each field ψðxÞ defined in the entire Brillouin zone; see (48) for a doubler with γ5 as the generator of chiral symmetry related discussion. Two massless Dirac fermions eðxÞ and generally vanishes for a ≠ 0. This is understood by recall- μðxÞ are displaced on the lattice in the z-direction by π=a. ing the fact that the anomaly as a Jacobian (a local version We have actually more species doublers in d ¼ 4 Euclidean of the index) is expressed using a general form of the chiral formulation (due to the time discretization and also the invariant lattice Dirac operator D [13],

016018-11 KAZUO FUJIKAWA PHYS. REV. D 97, 016018 (2018) Z d4p D†D the Ginsparg-Wilson fermion discussed in the next section γ e−ipx − eipx 0; lim 4 tr 5 exp 2 ¼ ð59Þ M→∞ B ð2πÞ M that contains only a single species, one can intuitively understand that the pair production corresponds to Trγ5 ¼ for a fixed finite lattice spacing a. Here, B is the Brillouin 0 in (81), while the anomaly TrΓ5 ≠ 0 in (82) in the next zone (or any subdomain), the volume of which is finite for section projects out the hole in the deep Dirac sea. To have finite a. This vanishing of anomaly is a local version of the anomaly on the lattice, it is essential to eliminate the holes vanishing index at the bottom of the energy spectrum, thus effectively realizing the infinitely deep Dirac sea. Trγ5 ¼ 0; ð60Þ The difference of the presence or absence of anomaly is that we would have well-defined (63) in the presence of which holds very generally on the lattice. See Eq. (81) for anomaly, while in the absence of separate anomaly, we the Ginsparg-Wilson fermion in the next section. Only after have taking the precise limit a → 0 in (59) first, for which each μ species doubler is treated as a full-fledged field, one can ∂μðe¯Lγ eLÞðxÞ¼0; define the ordinary chiral anomaly for each species doubler μ ∂μðμ¯ Rγ μRÞðxÞ¼0: ð64Þ separately by suitably choosing the subdomains of B. The absence of anomaly for each doubler separately with In the picture of spectral flow, the first relation in (64) a lattice cutoff and thus with a finite number of degrees of shows the cancellation of a particle near the fermi surface freedom is also understood by an argument of spectral flow by a hole deep inside the Dirac sea, and the second relation by considering the simple example (48). Let us concentrate shows the cancellation of a hole near the surface by a on the field eLðxÞ that has the spectrum ϵ−ðpÞ¼ “particle” deep inside the Dirac sea. But in both cases (63) − sin ap=a defined only in half of the Brillouin zone and (64), we always satisfy −π=2a

016018-12 CHARACTERISTICS OF CHIRAL ANOMALY IN VIEW OF … PHYS. REV. D 97, 016018 (2018) 1 VI. GINSPARG-WILSON FERMION Γ ≡ γ 1 − aD : 5 5 2 ð74Þ We recapitulate some representative properties of the Ginsparg-Wilson fermion. Using the mathematically pre- D cise formulation of the Ginsparg-Wilson fermion, we It is known that one can construct the operator which illustrate that the states at the ultraviolet region such as satisfies the Ginsparg-Wilson relation and Hermiticity con- γ D † γ D the states at the bottom of the Dirac sea are crucial to dition ð 5 Þ ¼ð 5 Þ and contains a single species in the analyze the phenomena of anomaly and spectral flow on the Brillouin zone, which is mainly concentrated in the sub- −π=2a ≤ p < π=2a lattice which has the energy bound at ∼j1=aj, thus domain μ without any doublers [38]. supporting our view presented in Sec, V. We can also One can show [42] that all the normalizable eigenstates H ≡ γ D understand that the apparently low-energy property such as of 5 , the Atiyah-Singer index theorem is controlled by the Hϕ λ ϕ ; ultraviolet cutoff in the lattice formulation. n ¼ n n ð75Þ The Ginsparg-Wilson fermion is defined by X on the lattice with a finite spacing are categorized into the following three classes using the basic relation derived S ¼ ψ¯ ðxÞDðx; yÞψðyÞð66Þ x;y from (67), Γ H HΓ 0; on the d ¼ 4 hypercubic lattice, for example. This fermion 5 þ 5 ¼ ð76Þ operator satisfies the Ginsparg-Wilson relation [41] 1 with Γ5 ¼ γ5 − 2 aH: (i) Zero modes (n states), γ5D þ Dγ5 ¼ aDγ5D: ð67Þ Hϕ 0; γ ϕ ϕ ; We then define two projection operators, n ¼ 5 n ¼ n ð77Þ

(ii) Highest states (N states) with Γ5ϕn ¼ 0, 1 ˆ 1 P ¼ ð1 γ5Þ; P ¼ ð1 γˆ5Þ; ð68Þ 2 2 2 Hϕn ¼ ϕn; γ5ϕn ¼ϕn; ð78Þ 2 a with γˆ5 ¼ γ5ð1 − aDÞ that satisfies γˆ5 ¼ 1 using the Ginsparg-Wilson relation (67). We define the chiral com- respectively; ponents by (iii) Remaining paired states with 0 < jλnj < 2=a, ψ Pˆ ψ; ψ¯ ψ¯ P ; R;L ¼ L;R ¼ ð69Þ Hϕn ¼ λnϕn;HðΓ5ϕnÞ¼−λnðΓ5ϕnÞ; ð79Þ which satisfy and the sum rule

n N n N S ¼ ψ¯ LDψ L þ ψ¯ RDψ R ð70Þ þ þ þ ¼ − þ − ð80Þ

D P DPˆ P DPˆ holds. This sum rule is a result of using ¼ þ − þ − þ. We define the chiral transformation X γ ϕ ;γ ϕ n N − n N 0 Tr 5 ¼ ð n 5 nÞ¼ þ þ þ ð − þ −Þ¼ n ψ → eiαγˆ5 ψ; ψ¯ → ψ¯ eiαγ5 ð71Þ ð81Þ under which the action is invariant since that holds even for non-Abelian Yang-Mills fields. The chiral symmetry breaking (nontrivial Jacobian) in γ5D þ Dγˆ5 ¼ 0: ð72Þ (73) is described by the Atiyah-Singer index theorem [42] X Under the transformation (71), we have a Jacobian of the Γ ϕ ; Γ ϕ n − n : Tr 5 ¼ ð n 5 nÞ¼ þ − ð82Þ form n 1 −2i α γ γˆ −2i αΓ ; The difference of Γ5 and γ5 is very important; Γ5 projects exp Tr 2 ð 5 þ 5Þ ¼ exp½ Tr 5 ð73Þ N out the highest states in (78). The Hilbert space of continuum theory in the present scheme is defined by N a → 0 where projecting out those states in the precise limit .

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The difference of Γ5 and γ5 has been already used to discuss implied by the basic identity (1) as a low-energy matrix the difference between the spectral flow and chiral anomaly element between two states jni and jn0i by preserving the μ in Sec. V, where the role of the states at the bottom of the fermion number ∂μðψγ¯ ψÞ¼0. The basic task in this N Dirac sea is identified with the role of the states ; approach is to find an operator that may dominate 0 spectral flow without anomaly corresponds to (81), and the hn j2imψγ¯ 5ψjni at low energies such as the helicity operator anomaly (82) is realized only when one projects out N .It ↔ n0 iψ † x⃗ 1 σ⃗∇ ψ x⃗ n is also interesting that the apparently low-energy statement h j ð Þ 2 ð Þj i with the Pauli two-component ψ x⃗ of the index theorem is controlled by the ultraviolet cutoff if spinor ð Þ, for example. The dominating operator is n − n N − N presumably related to the Coulomb interaction with sur- one uses (81), þ − ¼ − þ. n − n rounding particles, which is analogous to the strong inter- The index þ − is given by the Chern-Pontryagin number in the continuum limit for non-Abelian gauge action (QCD) in the PCAC. This scheme does not induce 1 ⃗ ⃗ “fermion number nonconservation.” It would be very inter- theory, which is an analog of the integral of 2 E · B. The 4π esting if one can formulate this approach in condensed matter index vanishes for Abelian theory in the absence of a or nuclear physics. Historically, the PCAC motivated the idea magnetic monopole, but a local version of TrΓ5 gives the of “anomaly-matching” at the quark level and nucleon level correct anomaly in the continuum limit [13], calculations. Z 4 2 In conclusion, we have shown that the mere presence of a d p −ipx ðγ5DÞ ipx lim tr e Γ5 exp − e Weyl fermion does not induce anomalous equal-time M→∞;a→0 2π 4 M2 B ð Þ commutation relations of space-time variables if the non- 1 adiabatic behavior of Berry’s phase is carefully taken into ¼ E⃗· B:⃗ ð83Þ 4π2 account. The effective theory with a monopole placed at the origin of the momentum space, which fails to describe The Ginsparg-Wilson fermion is tightly constrained by the Berry’s phase in the nonadiabatic domain, gives rise to a ≠ 0 consideration of the (local) index even for . Thus, a anomalous space-time commutators and thus chiral single Weyl fermion obtained by projecting (66) to a chiral anomaly to the gauge current [16]. We regard this appear- component without doublers ance of the anomaly as an artifact of the postulated monopole and not a consequence of Berry’s phase [45]. S ψ¯ Dψ ¼ L L ð84Þ We have also shown that the chiral anomaly vanishes for each species doubler separately in the scheme which treats is not defined quantum mechanically in the path integral even species doublers in lattice theory as physical objects for a for a ≠ 0 for the gauge group that contains gauge anomaly a ≠ 0 U 1 finite lattice spacing . Instead, a general form of such as the ð Þ electromagnetism [43,44]. This corre- spectral flow in the Dirac sea with finite depth takes place. sponds to the inconsistency of continuum chiral gauge theory The Ginsparg-Wilson fermion which is free of species in (35). In many respects, the Ginsparg-Wilson fermion is doubling supports this view. close to the continuum fermion in Minkowski space. We expect that the ideas developed in particle theory such as chiral anomaly and Ginsparg-Wilson fermions will VII. DISCUSSION AND CONCLUSION find more interesting applications in the fields such as We finally mention an alternative use of the anomaly in condensed matter and nuclear physics. analogy to the partially conserved axial-vector current hypothesis (PCAC), which played an important role in the history of chiral anomaly [3,4]. This approach starts with ACKNOWLEDGMENTS I thank K. Fukushima for calling the work [16] to my e2 0 0 ⃗ ⃗ attention and for a useful comment. I also thank H. Suzuki hn j2imψγ¯ 5ψjni ≃ −hn j E · Bjnið85Þ 2π2 for a discussion on chiral anomaly.

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