THE CHIRAL ANOMALY, DIRAC AND WEYL SEMIMETALS, AND FORCE-FREE MAGNETIC FIELDS

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1 THE CHIRAL ANOMALY, DIRAC AND WEYL SEMIMETALS, AND FORCE-FREE MAGNETIC FIELDS Gerald E. Marsh Argoe Natioal Laboratory (Ret) ABSTRACT The chiral aomaly is a purely quatum mechaical pheomeo that has a log history datig back to the late 1960s. Surprisigly, it has recetly made a macroscopic appearace i codesed matter physics. A brief itroductio to the relevat features of this aomaly is give ad it is show that its appearace i codesed matter systems must ivolve force-free magetic fields, which may help explai the log curret relaxatio times i Dirac ad Weyl semimetals. PACS: Gj; j; Pq

2 A 2015 paper by Xiog, et al. 1 reported that the chiral aomaly, usually cosidered a purely quatum mechaical pheomeo, ca be see i the Dirac semimetal Na 3 Bi. The pheomeo appears i this material whe the applied electric field ad magetic fields are parallel. Because ew macroscopic quatum effects are rare it is importat to explore the implicatios of this observatio. Some termiology ad basics: Whe the mass is set equal to zero i the Dirac equatio it decouples ito two equatios kow as the Weyl equatios that have two compoet spiors as solutios; these have charalities of χ = ±1. Now defie the Hamiltoia of the Dirac semimetal H(k) i terms of the spior basis {I, σ 1, σ 2, σ 3 }. If there is a k 0 such that the Hamiltoia satisfies H(k 0 ) = 0, i the viciity of k 0 the cotiuity of Hamiltoia implies that it ca be writte as H ( k) = h( k) : v. If h i are the compoets of h, the bad structure of the Hamiltoia, E ( k) h 2 1 h 2 2! =! h3, is called the Dirac coe, ad if h(k) is a liear fuctio of k, the coe i h-space also forms a coe i k-space. A example of a Dirac coe is show i the Fig. 1(a). (a) Figure 1. (a) A Dirac coe. The origi is said to have a Dirac ode, ad the Fermi level is located where the apexes meet. The upper coe represets the coductio bad ad the lower the valece bad. (b) Whe a magetic field is applied to a Dirac semimetal it breaks the symmetry of the crystal ad causes a Dirac ode to split ito two chiral Weyl odes. (b) The Dirac coe illustrated i Fig. 1(a) correspods to a Dirac semimetal because there is o gap betwee the two coes, which would become hyperbolae whe a gap is preset. A ormal isulator has a gap ad a three dimesioal topological isulator is characterized by the bulk of the material havig a gap while the surface does ot. 2

3 A Dirac semimetal, such as Na 3 Bi, is a three dimesioal system with a Dirac coe havig a double degeeracy at the Fermi eergy; a Weyl semimetal has its valece ad coductio bads touchig each other at isolated poits, aroud which the bad structure forms o-degeerate three dimesioal Dirac coes. The apexes of the Dirac coes are called Weyl odes. Low eergy quasiparticle excitatios i Weyl semimetals give the first example of the appearace of massless Weyl fermios i ature. Figure 1(b) shows the bad structure of a Dirac semimetal whe a strog magetic field is applied. If there is o electric field preset, chirality is preserved at the two odes. If, however, a electric field is applied charge will flow betwee the odes, ad the chiral aomaly will ot vaish. The charge trasfer rate depeds o the chirality χ (see Eq. (3.2) below). The stadard textbooks o topological isulators expad o these defiitios ad o the topological ature of Weyl odes ad their relatio to Berry curvature. 2, 3 The first sectio of this paper explais some aspects of the chiral aomaly ad the secod explais the coectio with force-free magetic fields ad their relevace to the chiral aomaly observed i the semimetal Na 3 Bi. The third sectio looks at the relaxatio of such fields i a medium with a o-zero resistivity. 1. Chiral Aomaly I classical physics there is said to be a symmetry whe the actio S(ψ) is ivariat uder the trasformatio ψ ψ + δψ, while i quatum mechaics the path itegral is( ) # DW e W must be ivariat for a symmetry to be preset. The trasformatio from classical to quatum mechaics does ot always retai a give symmetry. Otherwise said: Symmetries i terms of classical, commutig variables may ot be retaied whe expressed i terms o o-commutig quatum variables. Such a symmetry is said to have a quatum symmetry aomaly. 3

4 The quatum symmetry aomaly of iterest here is the axial aomaly, which violates the coservatio of axial curret. The o-coservatio of chirality was discovered i the late 1960s by Adler 4 ad Bell ad Jackiw, 5 There is a detailed discussio of the origis of the pheomeo i the textbook by Zee, 6 ad a very clear explicatio relevat to this work has bee give by Jackiw. 7 The axial vector curret is defied as J = For massless fermios, J 5 satisfies the cotiuity equatio 2 x J 5 = 0. Now defie 1 ( 5 P I )! = 2! c ad }! = P! } so that 5 c} =!}!!; the if } is a classical or quatum field operator the trasformatio 5 ici! ii } " e, }! " e }! (1.1) is a map betwee differet solutios of ic 2 x }! = 0. If oe ow couples this equatio to a exteral gauge field A, ic ( 2 x + ia( x)) }( x) = 0, (1.2) the for a sigle Fermi field couplig to A the axial vector curret J 5 obeys the aomalous cotiuity equatio 1 o 2 x J 5 = F () x F (), x 8 2 ) o r (1.3) o 1 oab where ) F = 2 e Fab is the dual of the field tesor F () x = 2 x A () x -2 o o o x A() x. For o-abelia fields, A = A a Ta ad the T a are ati-hermitia matrices satisfyig the a c Lie algebra commutators with structure costats f abc ; i.e., [ Ta, Tb] = fab Tc. Note that the structure costats, f abc, are ormalized by trtatb =- dab/ 2. For o-abelia fields, Eq. (1.3) becomes 1 o 2 x J 5 = tr F () x F (). x 8 2 ) o r c (1.4) The chiral aomaly i quatum field theory comes from two triagle Feyma diagrams associated with the decay of the π 0 particle [6]. If A correspods to the electromagetic four potetial, the Eq. (1.3) becomes 4

5 1 2 x J 5 = E B. 4 2 : r (1.5) It is this form of the aomaly that is resposible for the observatios of Xiog, et al. whe the electric ad magetic fields i Na 3 Bi have colliear compoets. Note that the aomaly vaishes whe the electric ad magetic fields are perpedicular; its ovaishig depeds o the compoet of B parallel to E. If the medium caot sustai a Loretz force the fields must be either perpedicular or parallel. It is the parallel case that is of iterest here. The isight that the chiral aomaly should appear i crystals is due to Nielse ad Niomiya. 8 For a topical review of the electromagetic respose of Weyl semimetals see Burkov Chiral Aomaly ad Force-Free Magetic Fields This sectio gives a short itroductio to force-free magetic fields where the curret is parallel to the magetic field, implyig that the Loretz force vaishes. I the experimet by Xiog, et al., the same coditio, that the curret produced by a applied electric field be parallel to the magetic field, is also required for the o-vaishig of the chiral aomaly. The origi of the curret 10 is the E-field parallel to B, which breaks chiral symmetry ad results i a axial curret. I the Dirac semimetal Na 3 Bi the effect of the aomaly was observed whe the applied electric field ad magetic field were aliged. Xiog, et al. suggested that the large egative magetoresistace observed implied a log relaxatio time for the curret. Sice the o-vaishig of the aomaly depeds oly o E < B ot vaishig, the cofiguratio of the field resposible for the aomaly iterior to the Na 3 Bi crystal is likely to be force-free. This is because the curret associated with E is parallel to B, ad this curret is itself a source for a azimuthal magetic field that combies with the logitudial magetic field applied to the Na 3 Bi to twist the flux. It is force-free because the curret associated with E is parallel to the twisted field. It will be see below that force-free fields have a helicity associated with them that is related to the eergy stored i the field. This opes up the possibility that the decay of 5

6 such fields may explai the log axial curret relaxatio time i Dirac ad Weyl semimetals without ivokig quatum mechaical processes. Fields with E < B are closely related to the force-free magetic field equatios d # B = ab with costat α. 11 I the experimet of Xiog, et al., the applied electric field produces a curret so that, because it is oly the compoet of the electric field parallel to the applied magetic field that yields a o-zero chiral aomaly, this curret is parallel to the applied magetic field. This meas that the field is force-free. As a cosequece, sice the electric field correspods to a curret, E < B meas that E = βb, where β is a scalar fuctio. If β is assumed to be a costat, Maxwell s equatios ca be used to show that b =! i, so there are o real solutios. If β is assumed to oly be a fuctio of time, E = βb ad Maxwell s equatios show that d # B = o b 2 B b + 1 (2.1) This equatio tells us is that if E = β(t)b, the B must satisfy the force-free field 2 1 equatio. The fuctio bb+ o ( 1) - i Eq. (2.1) is actually a costat, call it α, as is show i Appedix 1 of [11]; ad this restricts the form of β to If A or B vaishes, b Be i Be - Ae + Ae -iat iat = -iat iat. (2.2) b =! i so that there are o real solutios; If A =! B, the b = ta at or b = cot at respectively. Equatio (2.1) ca the be writte as d # Br () = abr (). (2.3) Thus, ay magetostatic solutio to the force-free field equatios ca be used to costruct a solutio to Maxwell s equatios with E parallel to B. This is true i free space (where the solutios are stadig waves, which have a vaishig Poytig vector) or whe E geerates a electric curret parallel to a exteral magetic field as i the experimet of Xiog, et al. 6

7 3. The Chiral Aomaly ad Curret Relaxatio Lifetime The log axial curret relaxatio time i Dirac ad Weyl semimetals is poorly uderstood ad is thought to be due to ear coservatio of chiral charge. Burkov 12 foud that there is a couplig betwee the chiral ad total charge desity, but this leads to a large egative magetoresistace oly whe the chiral charge desity is a early coserved quatity with a log relaxatio time. Cosider the form of the chiral aomaly give by Eq. (1.5). Usig E =- 2tA ad itegratig over both space ad time gives the helicity H = - # dx 3 ( A: B). (3.1) The itegral o the right had side is the helicity of the field, A: B beig the helicity desity. It plays a importat role i the relaxatio of magetic fields. Because helicity is a topological ivariat there are coditios uder which it is coserved, but here, as will be see below, the chiral aomaly provides a mechaism for the decay of helicity that may help explai the log curret relaxatio time. Kiji, Kharzeev, ad Warriga 13 have show that a chirality imbalace i systems with charged chiral fermios will geerate a electric curret i a exteral magetic field; they call this the Chiral Magetic Effect. Because this curret also acts as a source for a magetic field, the curret flowig alog the magetic field will twist the magetic flux ad iduce helicity ito the field. Xiog, et al. [1] have demostrated the coverse where a applied electric curret causes a charge to flow from oe chiral ode to aother of opposite chirality. That is, applicatio of E betwee the two chiral Weyl odes 3 e W = E: B, r & < B causes a charge pumpig rate W (3.2) where =! 1 idicates the chirality. The chiral imbalace referred to above ca be 1 5 foud by defiig the umber desities LR, = 2V # dx} ( 1! c )}, where V is the volume ad L correspods to the mius sig ad R to the plus. By itegratig the total 7

8 axial vector curret J 5 over space ad time oe ca the obtai the differece i left ad right chiral particles; i.e., # # L- R = dx( 2J5) = 2 dx( E: B). 4r (3.3) The itegrad of the itegral o the right had side of this equatio is the chiral aomaly give by Eq. (1.5). Now differetiatig Eq. (3.3) with respect to time gives d dt 1 3 (L- R) = 2 # dx( E: B). 4r (3.4) Note that the itegratio is ow over a 3-volume. If oe ow assumes the scalar potetial vaishes ad substitutes E =- 2 t A ito Eq. (3.4), ad the itegrates with respect to time, (L- R) may be expressed i terms of the helicity As a result, Eq. (3.4) ca be writte as L- R =- 2 # dx( A: B) =- 4r 4r d dt 1 dh (L- R) =- 2 dt. 4r 2 H. (3.5) (3.6) A similar expressio is readily derivable from the force-free field equatio d # B = ab, where α is agai a costat. The magetic field eergy E due to currets J i a volume V is give by E = 1 2 # V J: AdV. (3.7) By takig the dot product of A with the force-free field equatios ad usig Eq. (3.7) oe obtais 1 E = a A: BdV = 2 V # 1 2 ah. (3.8) 8

9 Now takig the derivative with respect to time ad idetifyig dh/dt with the same quatity i Eq. (3.6) gives d dt 1 de (L- R) =- 2 dt. 2ra (3.9) Thus, for force-free magetic fields, the chage i the differece of the umber of left ad right haded chiral particles ca be related to the chage i eergy. The mechaism by which the chiral aomaly allows the decay of helicity ca be foud by takig the time derivative of the helicity desity ad expadig 2 t ( A: B). Usig the homogeeous Maxwell equatios, oe ca the derive the expressio 2t ( A: B) + d: ( U B+ A# E) = -2E: B. (3.10) This is a cotiuity equatio where A: B is the helicity desity, ( U B+ A# E) is the helicity curret (the flux of helicity), ad see to make sese by writig the itegral form of Eq. (3.10): # # # - 2 E: B is a helicity sik. The latter ca be 2t A: BdV + d: ( UB+ A# E) dv = -2 E: BdV. V V V (3.11) The itegral o the right had side of this equatio represets the resistive decay of helicity ( E = hj v where h is the resistivity ad j v is the curret per uit area). The rate of relaxatio is determied by h. The itegrad is proportioal to the chiral aomaly of Eq. (1.5). Summary After discussio of some aspects of the chiral aomaly ad its form whe F o is the electromagetic field tesor, it was show that i a coductig medium such as Na 3 Bi whe E < B the field must take the form of a force-free magetic field. It was the show that the curret relaxatio time i such media will deped o the decay of helicity, which i tur depeds o the chiral aomaly ad the resistivity of the medium. It is 9

10 likely that this mechaism has some bearig o the log axial curret relaxatio time i Dirac ad Weyl semimetals. 10

11 1 J. Xiog, et al., Evidece for the chiral aomaly i the Dirac semimetal Na 3 Bi, Sciece 350, (2015). 2 Shu-Qig She, Topological Isusators (Spriger-Verlag, Berli 2012). 3 F. Ortma, S. Roche, ad S.O. Valezuela, Eds. Topological Isulators: Fudametals ad Perspectives (Wiley-VCH Verlag GmbH & Co. KGaA 2015). 4 S. L. Adler, Phys. Rev. 177, 2426 (1969). 5 J. S. Bell ad R. W. Jackiw, Nuov. Cim. A60, 4 (1969). 6 A. Zee, Quatum Field Theory i a Nutshell (Priceto Uiversity Press Priceto, 2010), Sectio IV.7. 7 R. W. Jackiw, Axial Aomaly, It. J. Mod. Phys. A 25.4, (2010). 8 H. B. Nielse ad M. Niomiya, The Adler-Bell-Jackiw Aomaly ad Weyl Fermios i a Crystal, Phys. Lett. B 130, (1983). 9 A. A. Burkov, Chiral aomaly ad trasport i Weyl metals, J. Phys: Codes. Matter 27, (2015), [arxiv: ]. 10 J. Xiog, et al., Sigature of the chiral aomaly i a Dirac semimetal-a curret plume steered by a magetic field, [arxiv: ] (2015). 11 G. E. Marsh, Force-Free Magetic Fields: Solutios, Topology ad Applicatios (World Scietific Publishig Co. Pte. Ltd. Sigapore 1996). 12 A. A. Burkov, Negative logitudial magetoresistace i Dirac ad Weyl metals Phys. Rev. B 91, (2015). (arxiv: v2 [cod-mat.mes-hall]). 13 K. Fukushima, D.I. Kharzeev, ad H.H. Warriga, The Chiral Magetic Effect, arxiv: v1 [hep-ph] (2008). 11