Helical Symmetry Breaking and Quantum Anomaly in Massive Dirac Fermions Huan-Wen Wang, Bo Fu, and Shun-Qing Shen∗ Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China (Dated: June 1, 2021) Helical symmetry of massive Dirac fermions is broken explicitly in the presence of electric and magnetic fields. Here we present two equations for the divergence of helical and axial vector currents following the Jackiw-Johnson approach to the anomaly of the neutral axial vector current. We discover the contribution from the helical symmetry breaking is attributed to the occupancy of the two states at the top of the valence band and the bottom of the conduction band. The explicit symmetry breaking fully cancels the anomalous correction from quantum fluctuation in the band gap. The chiral anomaly can be derived from the helical symmetry breaking. It provides an alternative route to understand the chiral anomaly from the point of view of the helical symmetry breaking. The pertinent physical consequences in condensed matter are the helical magnetic effect which means a charge current circulating at the direction of the magnetic field, and the mass dependent positive longitudinal magnetoconductivity as a transport signature. The discovery not only reflects anomalous magneto-transport properties of massive Dirac materials, but also reveals the close relation between the helical symmetry breaking and the physics of chiral anomaly in quantum field theory and high energy physics. Introduction The chiral anomaly of massless Dirac we derive the equations for the divergence of the heli- fermions is a purely quantum mechanical effect, and is cal current and axial vector current for massive Dirac an extraordinarily rich subject in quantum field theory fermions in the presence of electric and magnetic fields. and elementary particle physics [1–5]. It is regarded as a We find the discontinuity of helicity at the momentum consequence of spontaneous symmetry breaking induced qz = 0 at the zeroth Landau levels leads to the helical by the quantum fluctuation in the presence of electric symmetry breaking in the presence of the electric field. and magnetic field. However, the helicity represents the The anomalous correction from the quantum fluctuation projection of the particle spin at the direction of mo- is exactly cancelled by the explicit symmetry breaking in tion and is also conserved for Dirac fermions. The heli- the band gap. The mass term strongly revises the coef- cal symmetry is broken explicitly in an electric field. In ficient in the equation for the axial vector current, but the massless case the helicity and chirality become iden- keep the coefficient as constant in the equation of the tical for the positive energy and differ by an opposite helical current. The two equations become equivalent in sign for the negative energy [6, 7]. This raises a ques- the higher energy and massless case. The identical form tion whether or not the chiral anomaly is closely related of the equations for the helicity and chirality provides to the explicit symmetry breaking of helicity. In recent deep insight into the role of helical symmetry breaking years, the discovery of Weyl semimetals revived research in the physics of chiral anomaly. Physically, the helical interests on chiral anomaly for massless Dirac fermions symmetry breaking leads to a charge current circulat- in condensed matter physics [8–14]. A negative longi- ing along the direction of the magnetic field, termed the tudinal magnetoresistance was regarded as a significant helical magnetic effect. The effect gives rise to the mass- signature to support the existence of chiral anomaly in dependent negative magnetoresistance in massive Dirac gapless Weyl semimetals and Dirac semimetals [15–22]. materials. However, as more and more topological materials with Helicity in a magnetic field We start with the mas- finite band gap [23–27] also exhibit negative magnetore- sive Dirac fermions in a finite magnetic field B along the sistance, the mechanism of chiral anomaly is obviously z direction, challenged as the chiral symmetry has already been bro- − H = γ0 γiΠ + mv2 (1) ken explicitly by a finite mass. Very recently, Andreev 0 i arXiv:2105.14941v1 [cond-mat.mes-hall] 31 May 2021 0 i and Spivak proposed that the helicity imbalance of mas- where γ = τ1σ0 and γ = iτ2σi with τ and σ be- sive Dirac fermions may also produce negative magne- ing Pauli matrices of orbital− and spin degree of free- toresistance [28]. Because of the close relation between dom. m is the Dirac mass and v is the effective velocity. the helical and chiral symmetry, it deserves an investiga- Π = ~q + eA is the kinematical momentum with the tion on the helical symmetry breaking of massive Dirac vector potential A = ( By, 0, 0). Without loss of gen- fermions and its transport signature in the presence of erality, we assume eB− > 0. Since the presence of the electric and magnetic fields. Furthermore, it may reveal vector potential A does not break the translation sym- the deep relevance of the helicity symmetry breaking and metry along the x and z direction, qx and qz are still the physics of chiral anomaly. good quantum numbers. The operator Σ Π = τ0σ Π In this paper, following the Jackiw-Johnson approach defines the projection of the particle spin at· the direction· to the anomaly of the neutral axial vector current [29] of motion. By taking advantage of the ladder operators 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).