arXiv:2105.14941v1 [cond-mat.mes-hall] 31 May 2021 oteaoayo h eta xa etrcret[29] current vector axial neutral the of anomaly the to anomaly. chiral and of breaking symmetry physics reveal helicity the of the may of it presence relevance Furthermore, deep the the fields. in magnetic Dirac signature and massive transport electric of its breaking and symmetry fermions helical investiga- an the between deserves on relation it tion symmetry, close chiral magne- the and helical negative of the Because produce also [28]. may mas- toresistance of Andreev fermions imbalance recently, Dirac helicity Very the sive that mass. proposed finite Spivak bro- and a been by already obviously explicitly has is symmetry ken chiral anomaly the chiral as of challenged magnetore- mechanism negative with exhibit the materials also sistance, [23–27] topological gap more band and finite [15–22]. in more semimetals anomaly as Dirac chiral However, and of semimetals existence Weyl the gapless longi- significant support negative a to as A signature regarded was [8–14]. fermions magnetoresistance physics Dirac tudinal matter massless condensed for in anomaly research recent chiral revived In on semimetals interests Weyl helicity. ques- of of discovery related a breaking the closely raises years, symmetry is This anomaly explicit opposite chiral the an 7]. the to not by [6, or energy differ whether negative tion and the energy for positive iden- sign become In the chirality for field. and helicity electric tical heli- the an mo- The case in massless of explicitly fermions. the broken direction Dirac is for the symmetry conserved at cal also spin is and particle the electric tion the represents of helicity of presence the However, projection the field. in magnetic fluctuation and quantum induced the a breaking as by regarded symmetry is spontaneous theory It of field consequence [1–5]. is quantum physics particle and in elementary effect, subject and mechanical rich quantum extraordinarily purely an a is fermions nti ae,floigteJci-ono approach Jackiw-Johnson the following paper, this In Introduction eia ymtyBekn n unu nml nMsieDi Massive in Anomaly Quantum and Breaking Symmetry Helical edter n iheeg physics. energy th and high breaking and symmetry theory helical field of the between direction ma relation of close the properties a magneto-transport at anomalous as reflects circulating magnetoconductivity only symme current longitudinal positive helical charge dependent condense the a in from means the consequences c from derived physical which anomaly the pertinent be chiral The of the can breaking. bottom understand anomaly the to f route chiral and correction alternative band The anomalous valence the the cancels gap. of fully top breaking the the symmetry breaki of at symmetry helical anomaly states the two the from contribution to the approach diverg discover the Jackiw-Johnson for equations the two present following we Here fields. magnetic eia ymtyo asv ia emosi rknexplic broken is fermions Dirac massive of symmetry Helical eateto hsc,TeUiest fHn og Pokfula Kong, Hong of University The Physics, of Department h hrlaoayo asesDirac massless of anomaly chiral The unWnWn,B u n hnQn Shen Shun-Qing and Fu, Bo Wang, Huan-Wen Dtd ue1 2021) 1, June (Dated: fmto.B aigavnaeo h adroperators ladder the of advantage direction taking the By at spin motion. particle of the of projection the defines odqatmnmes h operator The numbers. quantum good er ln the along metry potential vector eedn eaiemgeoeitnei asv Dirac mass- massive the materials. in to magnetoresistance rise the gives negative termed effect dependent field, The circulat- magnetic effect. the current magnetic of charge helical direction a helical the along the to Physically, ing leads breaking anomaly. breaking symmetry chiral helical provides symmetry of of chirality physics role the and in the helicity into the form insight for identical deep The equations the case. the massless of of in and equivalent equation energy become higher the equations but the two in current, The constant vector current. as axial helical coef- coefficient the the the for revises equation keep strongly term the mass in The ficient in breaking gap. symmetry band explicit the the by field. fluctuation cancelled electric quantum exactly the is the from of correction presence anomalous The the in breaking symmetry efidtedsotniyo eiiya h momentum the at helicity fields. of magnetic Dirac q discontinuity and massive the electric heli- for find of We the presence current the of vector in divergence axial fermions the and for current equations cal the derive we rlt,w assume we erality, ieDrcfrin nafiiemgei field magnetic finite a in fermions Dirac sive z etrpotential vector where = Π n al arcso ria n pndge ffree- of degree spin and orbital of dom. matrices Pauli ing z − eiiyi antcfield magnetic a in Helicity direction, 0 = ~ m sv ia aeil,btas eel the reveals also but materials, Dirac ssive gi trbtdt h cuac fthe of occupancy the to attributed is ng γ atraetehlclmgei effect magnetic helical the are matter d q rnpr intr.Tedsoeynot discovery The signature. transport hsc fcia nml nquantum in anomaly chiral of physics e neo eia n xa etrcurrents vector axial and helical of ence ttezrt adulvl ed otehelical the to leads levels Landau zeroth the at 0 o unu utaini h band the in fluctuation quantum rom steDrcms and mass Dirac the is on fve ftehlclsymmetry helical the of view of point + = tyi h rsneo lcrcand electric of presence the in itly e h antcfil,adtemass the and field, magnetic the eta xa etrcret We current. vector axial neutral A τ od ogKn,China Kong, Hong Road, m 1 σ nuto ad h explicit The band. onduction r raig tpoie an provides It breaking. try steknmtclmmnu ihthe with momentum kinematical the is A A 0 H x 0 and osntbektetasainsym- translation the break not does ( = B> eB and = ∗ γ − γ 0 z By, i 0 γ direction, = ic h rsneo the of presence the Since . i Π 0 − , i v 0) + iτ esatwt h mas- the with start We steeetv velocity. effective the is ihu oso gen- of loss Without . 2 mv a Fermions rac σ i 2 q with
x Σ and · τ = Π B q and z ln the along r still are τ 0 σ σ be- · (1) Π 2

† a = (Πx iΠy)/√2eB~ and a = (Πx + iΠy)/√2eB~ 3 [30], we can− obtain the eigenvalues and eigenstates for the operator [31], 2

~2 2 ~2 2 2 2 1 σ Π n, qx, qz,χn = χn qz + n Ω /v n, qx, qz,χn · | i | (2)i p 0 where n =0, 1, 2, are the indices of the Landau levels,

−1··· E/mv and Ω √2vℓ is the cyclotron frequency with the -1 ≡ B magnetic length ℓB = ~/eB . χn stands for the helicity of massive Dirac fermions: χ = 1 for n> 0 and χ = -2 p n ± 0 sgn(qz) for n =0. The sign change of χ0 around qz =0 −is a peculiar feature of the Landau level of n =0 (see the -3 -3 -2 -1 0 1 2 3 black dots in Fig. 1). In the basis of helicity eigenstates, the helicity operator is expressed as h 2 v qz/mv hˆ = χ τ n, q , q ,χ n, q , q ,χ . (3) n 0 | x z ni h x z n| n,qx,qz,χn Figure 1. The energy dispersion spectrum of the Landau lev- X els with helicity distribution (blue line for right handed and The helicity operator commutes with the Hamiltonian, red for left handed helicity). The two black dots indicate the = 0 [h,Hˆ 0]=0, thus the helical symmetry survives in a fi- discontinuity of helicity in the Landau levels of n . nite magnetic field. In the helicity basis, the Hamiltonian is reduced to an effective one-dimensional system H0 = 2~2 2 ~2 2 2 χn v qz + n Ω τ3 + mv τ1. Thus the energy eigen- values are ε = ζ v2~2q2 + m2v4 + n~2Ω2, where e2 i p nζχn z ∂ ρ + ∂ ji = E B ψγ¯ 0[h,ˆ Vˆ ]ψ (6) ζ = +1 for the conduction band and 1 for the va- t h i h −2π2~2 · − ~ p lence band. The corresponding eigenstate− for each Lan- D E where ρ and ji are the expectation values of helical dau level is [31] h h density and current density at zero temperature. The

φnζχ first term in the right-hand side of Eq. (6) is given by cos n n,ζ,χ ; q , q = 2 n, q , q ,χ , (4) the anomalous correction from the quantum fluctuation n x z φnζχn x z n | i ζ sin 2 ! ⊗ | i ˆ ~−1 ˆ0 ˆ3 S(z)= i lim [jh(z,ǫ)ǫ3 jh(z,ǫ)ǫ0]∂zV (r) (7) ~2 2 2~2 2 ǫα→0 − where cos φnζχn = χn n Ω + v qz /εnζχn . These eigenstates are orthogonal to each other as for small, but nonzero ǫ0 and ǫ3. The divergence of the ′ ′ ′ ′ ′ p ′ ′ ′ ′ n , ζ ,χn; qx, qz n,ζ,χn; qx, qz = δnn δζζ δχnχn δ(qx helical density as 1/ǫ3 is caused by the infinity of the h ′ | i − qx)δ(qz qz). All the Landau levels with different qx are Fermi sea in the valence bands, which was first encoun- − ~ degenerated with the degeneracy nL = eB/2π per unit tered in the anomaly of neutral axial vector current [29]. area in the x-y plane. Besides each Landau level has Besides, the second term in the right-hand side of Eq. additional double degeneracy for helicity when n> 0. (6) comes from the explicit helical symmetry breaking. Continuity equation for helicity The presence of an In the basis of the eigen energy, we find [31] electric field breaks the helical symmetry for the massive Dirac fermions. Consider the electric potential V (r) = [h,ˆ Vˆ ] = i2eE δ(q ) 0,ζ,χ ; q , q 0,ζ,χ ; q , q 0 z z | 0 x zih 0 x z| eE r for a uniform electric field E. Since the helic- qx,qz X ity operator· is a function of momentum, which does not (8) commute the position operator r, [h,ˆ Vˆ ] = 0. To estab- for the Landau levels of n =0. It is noted that there exits lish the equation of the divergence of helical6 currents, we a delta function, which originates from the discontinuity follow the Jackiw-Johnson approach to the anomaly of of helicity around qz = 0, i.e., ∂qz sgn(qz)=2δ(qz) (see the neutral axial vector current [4, 29], and define the Fig.1). The chemical potential determines the occupancy gauge-invariant helical currents of the two states n =0, ζ = ,χ0; qx, qz at qz =0: each 2 | ±e i 2 2 E B ǫ ǫ state may contribute one 2π ~ , which is exactly ˆi ¯ α iˆ α −iφ(t,ǫ0) · jh(z) = lim ψ(rα + )γ hψ(rα )e . (5) the breaking term for chiral anomaly. Thus it will give ǫ →0 2 α 2 − 2 e a nonzero term (sgn(µ)+1) 2 2 E B. Combining with 2π ~ · t+ǫ0/2 ~ ¯ † 0 the term of anomalous correction, we obtain the equation with φ(t,ǫ0) = t−ǫ0/2 V (rα)dt/ . ψ and ψ = ψ γ are the Dirac spinors. The local density and current are ob- for the divergence of the helical density ρh and current R 0 jh in a more compact form as, tained by taking ǫ to be small. ρˆh = limǫ→0 jh(z,ǫ) and ˆi i 2 jh = limǫ→0 jh(z,ǫ). Utilizing the time-dependent Dirac e ∂ ρ + j = C E B. (9) equation, the divergence of helical currents is given by t h ∇ · h h 2π2~2 · 3

1.0 an alternative approach to derive the chiral anomaly from the point of view of the helical symmetry breaking. Un- 0.5 like the helicity, the chirality operator is independent of momentum, and the presence of an electric field does not break the chiral symmetry. h/5 0.0

C Pseudoscalar density and continuity equation of chi- rality for massive Dirac fermions Ch In the presence of a -0.5 finite mass m, γ5 does not commute with the finite mass C5 † 5 2 term, ψ [γ ,H0]ψ = 2imv nP with the pseudoscalar ¯ 5 − -1.0 density nP = ψiγ ψ [34, 35]. After including the con- − -4-2 0 2 4 tribution from the quantum fluctuation, the divergence of the axial vector currents is given by [29, 34] μ/mv 2 2 e 2 2 C 5 ∂ ρ + j = E B mv n . (12) Figure 2. Comparison of the coefficients h/ in the equa- t 5 5 2~2 ~ P tions for the divergence of the helical current and axial vector ∇ · 2π · − h i currents in Eqs. (9) and (14). In the absence of an electric field, all diagonal elements of np vanish in the helicity basis, and the off-diagonal

2 elements connect the conduction and valence bands with Here Ch = sgn(µ) for µ mv , and Ch = 0 for µ < the same Landau index. The expectation value of pseu- mv2. Thus the explicit| symmetry| ≥ breaking term and| | the doscalar density nP is equal to zero for a free gas of anomalous correction are exactly cancelled in the right Dirac fermions. Inh thei presence of electric field, the elec- hand side of Eq. (6) within the gap. The sign change tric potential may couple the two states of the same mo- in the conduction and valence bands is caused by the mentum and Landau index. Due to the double degener- opposite velocities of fermions with identical helicity at acy of the states of the Landau levels of n> 0, it is found the direction of the magnetic field. that only the nondegenerated Landau levels of n =0 con- For the higher Landau levels of n > 0, the diagonal n ˆ tribute to the nonzero value of P . The perturbation elements of [h, zˆ]n always vanish and the off-diagonal el- approach up to the linear electrich fieldi E gives ements may contribute to higher order corrections from the electric field (see details in Ref. [31]). Thus Eq. (9) e2 n = (1 C ) E B. (13) holds for a finite magnetic field and a weak electric field. h P i − 5 4π2~mv2 · It is one of the key results of this paper. Chiral anomaly of the massless fermions The helical m2v4 2 The coefficient C5 = 1 µ2 for µ mv and C5 = symmetry breaking may provide an alternative approach − | |≥ 0 otherwise [36]. Then,q the equation for the axial vector to derive the chiral anomaly for massless Dirac fermions. currents becomes [31] In the basis of the eigen energy levels, the chirality oper- ator γ5 = iγ0γ1γ2γ3 becomes e2 5 ∂ ρ + j = C E B. (14) γ = ζχ n,ζ,χ ; q , q n,ζ,χ ; q , q . t 5 5 5 2~2 n| n x zih n x z| ∇ · 2π · n,ζ,χn,qx,qz X Again the symmetry breaking term caused by the mass (10) 2 e For m = 0, γ commutes with the Hamiltonian and cancels the anomalous correction 2~2 E B from quan- 5 2π · is conserved. In the conduction band of ζ = +1, tum fluctuation out when the valence band is fully filled. 5 γ P+ = hPˆ + but in the valence band of ζ = 1, Clearly, the chirality is not a good quantum number for 5 − γ P− = hPˆ−, where the band projection operator a nonzero mass. We cannot define a chirality-dependent − . Thus potential via the energy levels of free fermions as we do Pζ = n,qx,qz ,χn n,ζ,χn; qx, qz n,ζ,χn; qx, qz the helicity and chirality| becomesih identical in the| conduc- for the helicity, but the chirality density are still closely tion bandP and opposite in the valence band. Substituting related to the helicity density. Consider a tiny helical the relation into Eq. (9), one can obtain the continuity potential µh near the chemical potential µ. It is found m2v4 equation for chiral anomaly, that ρ = 1 2 ρ . This demonstrates that the two 5 − µ h quantitiesq tend to be equal when µ is much larger than 2 e2 the band gap 2mv or the mass approaches zero. When ∂tρ5 + j5 = sgn( µ ) E B (11) µ is located near the band bottom, the chirality density ∇ · | | 2π2~2 · approaches zero. Thus the equations for helical current where ρ5 and j5 are the chirality density and the corre- and axial current density are consistent with each other. sponding axial vector current, respectively. This provides A straightforward comparison of the coefficients on the 4 right hand side in Eq. (9) and (14) is presented in Fig. longitudinal magnetoconductivity is consistent with the 2. It illustrates clearly the difference and connection be- result for massive and massless cases [12, 17, 28] while tween the two equations for a different chemical potential the transverse magnetoresistance gives rise to the pla- Helical magnetic effect For a nearly free gas of mas- nar Hall effect [41]. In the Born approximation, the sive Dirac fermions, the helicity density is equal to zero. inter-helicity scattering time is found to reach a maxi- If the helicity balance is broken, the chemical potentials mal value at m =0, and decays with the increase of the for fermions of different helicity deviate from the equilib- mass, which results in a mass-dependent magnetocon- rium value µ: one increases and another decreases, i.e., ductance in Dirac materials [31]. In the quantum limit µ± = µ µh/2. If there is no other interaction, the helic- regime, the density of state at fermi energy µ is found ± µ ity is still conserved. The helicity-dependent current for as g(µ)= 2 2 2~2 , and the corresponding longitudi- 2π ℓB v qF 3 χ = 1 can be calculated independently. The electric h e v v~qF nal magnetoconductivity becomes σ = 2 2 τ B, current± density is given by the difference of the helical zz 2π ~ µ h which is a linear function of the magnetic field once τ is currents for two distinct helicities [31], h a constant in the massless limit. For a moderate strong magnetic field, the density of states will oscillate with the e2 magnetic field, and there are quantum oscillations in the j = ( µ µ− )B. (15) 4π2~2 | +| − | | magnetoconductivity. Discussion and conclusion The full cancellation of This means that if two chemical potentials are not equal, the explicit symmetry breaking and the anomalous cor- a charge current circulates at the direction of the mag- rection in the band gap reflects the quantum anomaly netic field. We term the field dependent current as the in the massive Dirac fermions. The anomalous correc- helical magnetic effect for massive Dirac fermions. The tion arises by introducing the gauge invariant currents in effect is equivalent to the chiral magnetic effect when the Eq. 5, and is an electromagnetic response from the infi- mass m approaches to zero as the the helicity and chiral- nite Dirac sea. Here it is worth of pointing out that the ity become identical [28, 37–39]. approach is different from the method based on the vari- One remarkable transport consequence of the he- ation of charge density of particles in the zeroth Landau lical magnetic effect is the magnetoconductivity in levels [15, 28], in which all other negative energy levels Dirac/Weyl semimetals. In the presence of impurity scat- are neglected and is actually irrelevant to physics of the tering, the inter-helicity scattering process can maintain quantum anomaly. This point can be further clarified a nonzero helical charge imbalance near the Fermi en- in the following example. Consider the nonrelativistic ergy µ in the background of the electromagnetic field. Pauli Hamiltonian for a free electron gas in a magnetic The scattering potentials are functions of position, and field, H = (σ Π)2 /2m = Π2/2m + e~ Bσ . The helic- do not commute with the helicity operator. Thus we as- P 2m z ity is conserved,· [σ Π,H ]=0. The helical symmetry sume the scattering potentials V are not so strong such P s breaking in an electric· field leads to an identical conti- that the averaged value of [h,ˆ V ] is still negligible. With a s nuity equation for the divergence of helical density and characteristic relaxation time τ between different helical h current as in Eq. (9) for µ > 0 [31], which is consistent electrons, one can introduce a relaxation term in the con- 2 e ρh with the picture of the Landau levels. However, it is un- tinuity equation, ∂ ρ = sgn(µ) 2 2 E B . For the t h 2π ~ · − τh related to the physics of quantum anomaly since there equilibrium state ∂tρh = 0, the solution for ρh is found 2 e are no infinites negative energy states at all. The helical as ρ = sgn(µ) 2 2 E Bτ [40]. When µ µ , the h 2π ~ · h | h| ≪ | | symmetry breaking in this system may also give rise to corresponding helical chemical potential can be found as a negative longitudinal magnetoresistance. Of course it µ 2ρh , where g(µ) is the total density of states at the h ≈ g(µ) should be noted that the effect disappears if the Zeeman Fermi energy. Then, the helical magnetic effect leads to field is absent, i.e., in H = Π2/2m. a nonzero field-dependent current density as In summary, we derived the two equations for the di- e 4 τh vergence of helical current and axial vector current in jHME =( ) E BB. (16) π~ 4g(µ) · electric and magnetic fields. We discovered the disconti- nuity of helicity at the zeroth Landau levels leads to the Accordingly, the helicity-induced magnetoconductivity is 4 helical symmetry breaking in the presence of the elec- h e given by σij = 4π4~4g(µ) τhBiBj . The inter-helicity scat- tric field. The occupancy of the states at the top of the tering time τh is determined by the impurity scattering valence band and the bottom of the conduction band con- 2 potentials. This equation is valid from the weak mag- e tributes one in the unit of 2π2~2 E B in the equation of netic field to the quantum limit regime. In the weak the divergence of the helical currents.· The anomalous magnetic field, the density of state at the Fermi energy is corrections from the quantum fluctuation for both he- µqF 2 2 4 ~ g(µ) = π2v2~2 with qF = µ m v /v . The matrix licity and chirality are cancelled exactly by the explicit element of magnetoconductivity− tensor due to the heli- 2 2 3 symmetry breaking to guarantee the conservation laws ph e e v cal magnetic effect reads σ = 2 τ B B . The ij 4π ~ µv~qF h i j when the chemical potential is located within the energy 5 gap. In the case of higher energy or tiny mass µ mv2, Soc. Jpn. 87, 041001 (2018). the equations for helicity and chirality become| equivalent| ≫ [15] H. Nielsen and M. Ninomiya, The Adler-Bell-Jackiw (only differed by a sign for positive and negative energy). anomaly and Weyl fermions in a crystal, Phys. Lett. B 130 This provides an alternative route to understand the chi- , 389 (1983). [16] D. T. Son and B. Z. Spivak, Chiral anomaly and classical ral anomaly from the point of view of the helical sym- negative magnetoresistance of Weyl metals, Phys. Rev. B metry breaking. 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