Quantum Oscillations in the Chiral Magnetic Conductivity
QQuantum Oscillations in the Chiral Magnetic Conductivity
Sahal Kaushik ∗ and Dmitri E. Kharzeev
1, 2, 3, † Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794, USA Department of Physics, Brookhaven National Laboratory, Upton, New York 11973-5000, USA RIKEN-BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973-5000, USA
In strong magnetic field the longitudinal magnetoconductivity in 3D chiral materials is shown toexhibit a new type of quantum oscillations arising from the chiral magnetic effect (CME). Thesequantum CME oscillations are predicted to dominate over the Shubnikov-de Haas (SdH) ones inchiral materials with an approximately conserved chirality of quasiparticles at strong magnetic fields.The phase of quantum CME oscillations differs from the phase of the conventional SdH oscillationsby π/ The chiral magnetic effect (CME) [1] (see [2, 3]for reviews and additional references) is a macroscopicquantum transport phenomenon induced by the chiralanomaly. In 3D systems possessing chiral fermions,an imbalance between the densities of left- and right-handed fermions generates a non-dissipative electric cur-rent along the direction of an external magnetic field.The CME has been predicted to occur in Dirac and Weylsemimetals (DSMs/WSMs) [1, 4–9], and has been re-cently experimentally observed through the measurementof negative longitudinal magnetoresistance in DSMs [10–13] as well as in WSMs [14–19].The CME electric current flowing along the externalmagnetic field B in the presence of a chiral chemical po-tential µ is given by j CME = e π µ B . (1)The chiral chemical potential µ = ( µ R − µ L ) / ρ = e π E · B − ρ τ V , (2)where ρ = ρ R − ρ L is the difference between the densitiesof the right-handed and left-handed fermions, and thesecond term is introduced to take account of the chiralitychanging transitions with a characteristic time τ V . Ifthe chirality flipping time τ V is much greater than thescattering time τ , the left-handed fermions and the right-handed fermions can exist in a steady state with differentchemical potentials µ L and µ R . ∗ Electronic address: [email protected] † Electronic address: [email protected]
In a uniform and constant magnetic field, the energiesof the lowest Landau levels are (cid:15) = − vp z for left-handedand (cid:15) = + vp z for right-handed fermions, see Fig. 1 ( v isthe Fermi velocity; we assume that magnetic field B isalong the z -axis). The energies of excited Landau levelsare E = ± v (cid:112) p z + 2 eBn for n ≥
1, for both chiralities.The density of Landau levels in the xy plane is eB/ π ,whereas the density of states in the z -direction is p z / π .Because the lowest Landau level is not degenerate in spin,it has right-handed fermions of positive charge travelingalong B and left-handed ones of negative charge travelingin the opposite direction. This induces the CME current(1).The density of the chiral charge ρ is related to thechiral chemical potential µ through the chiral suscepti-bility χ ≡ ∂ρ /∂µ – at small µ , ρ = χµ + ... so that µ (cid:39) χ − ρ . Note that in the absence of chirality losscorresponding to τ V → ∞ the CME current would growlinearly in time – in other words, it would behave as asuperconducting current, see [27] for a discussion.At finite τ V , the density of the chiral charge saturatesat the value ρ = e / π E · B τ V , and the longitudinalCME conductivity for parallel E and B is given by σ CME = e B π χ ( iω + 1 /τ V ) , (3)where ω is the frequency of an external field.In the presence of a Fermi surface with a chemical po-tential µ = ( µ R + µ L ) /
2, in weak magnetic fields with2 eBv (cid:28) µ the Landau quantization can be ignored,and the chiral susceptibility is given by χ = µ π v + T v . (4)The DC CME conductivity [1, 5, 6, 10] is then σ CME = e v τ V B π ( µ + π T / . (5)In strong magnetic fields with 2 eBv (cid:29) µ , only thelowest Landau level contributes, and µ = eB/ vπ , sothe DC CME conductivity has a linear dependence on B : σ CME = e v τ V B π . (6) a r X i v : . [ c ond - m a t . m e s - h a ll ] M a r p z Ε p z Ε FIG. 1: The energy structure of left-handed and right-handedfermions in a magnetic field. The red line indicates µ . In a real material, the conductivity is the sum of theCME conductivity and the Ohmic conductivity, σ zz = σ CME + σ Ohm .As is well known, the Shubnikov - de Haas (SdH) oscil-lations appear due to the quantized Landau levels and thepresence of Fermi surface; they have been observed in thetransverse magnetoconductivity of some Dirac and Weyl(semi)metals [14, 28, 29]. The phase of the SdH oscilla-tions depends on the Berry curvature; this fact can beused to distinguish materials with massive carriers fromDirac and Weyl materials [14, 29–31]. The oscillatingpart of the transverse conductivity has the form σ xx = A ( B ) cos (cid:20) π (cid:18) B B − γ + δ (cid:19)(cid:21) , (7)where γ is 0 for Dirac and Weyl carriers and 1 / δ varies between − / / µ and the density of chiral charge ρ .Our treatment will apply to Dirac materials and Weylmaterials in which the Weyl points have the same energyand Fermi velocity. The assumptions used this work are τ V (cid:29) τ (i.e. the chirality flips are relatively rare), T (cid:28) µ , µ (cid:28) T, µ and 2 eBv (cid:28) µ ; we set ¯ h = 1. The density of states in energy E for each chirality is g ( E ) = E L π v (cid:34) ∞ (cid:88) n =1 Θ( E − nE L ) (cid:115) E E − nE L (cid:35) (8)where E L = 2 eBv is the difference in the squares of theLandau level energies. The factor of 2 is because we haveparticles traveling in both directions for higher levels.The total number density of particles, for each chiralityis given by ρ R,L ( µ, T ) = (cid:90) E + E − g ( E ) f ( µ − E, T ) dE, (9)where f ( V, T ) = e V/T e V/T +1 is the Fermi distribution func-tion, and E − , E + are the cutoff energies. Therefore, χ = ∂ρ ∂µ = 2 ∂ρ L,R ∂µ R,L = 2 (cid:90) E + E − g ( E ) f (cid:48) ( µ − E, T ) dE (10)yielding χ = E L π v (cid:90) ∞−∞ (cid:34)
12 + ∞ (cid:88) n =1 Θ( E − nE L ) (cid:115) E E − nE L (cid:35) × f (cid:48) ( µ − E, T ) dE (11)For small fields (when many Landau levels contribute),the sum can be approximated by an integral, and werecover (4): χ ( B = 0) = µ π v + T v . (12)Let us now evaluate the quantum corrections to this ex-pression that will be responsible for the quantum oscilla-tions in CME conductivity. We are concerned only withenergies close to µ , so (cid:115) E E − nE L ≈ (cid:115) µ µ + 2 µV − nE L (13)= (cid:114) µ V − ( n − x )( E L / µ )) , (14)where x = µ /E L and V = E − µ .We can define the contribution of the n th level to thesusceptibility χ n as χ n = E L π v (cid:90) ∞−∞ Θ( E − nE L ) (cid:115) E E − nE L f (cid:48) ( µ − E, T ) dE (15) ≈ E L π v (cid:90) ∞−∞ Θ( V − ( n − x )( E L / µ )) × (cid:114) µ V − ( n − x )( E L / µ )) f (cid:48) ( V, T ) dV. (16)Now, in the expression χ = ∞ (cid:88) n =1 χ n + 12 χ (17)we can extend the sum to −∞ and approximate the con-tribution of the fictitious negative levels by an integral: χ ≈ ∞ (cid:88) n = −∞ χ n − (cid:90) −∞ χ n dn (18)= ∞ (cid:88) n = −∞ χ n − (cid:90) ∞−∞ χ n dn + (cid:90) ∞ χ n dn (19)= ∞ (cid:88) n = −∞ χ n − (cid:90) ∞−∞ χ n dn + µ π v + T v (20)We can then use the Poisson summation in χ = µ π v + T v + ∞ (cid:88) l =1 (cid:60) ( χ l ) (21)to evaluate the Fourier transform of χ n , χ n = α (cid:20) ( x − n ) E L µ (cid:21) (22)where α ( z ) = E L π v (cid:90) ∞−∞ Θ( V + z ) (cid:114) µ V + z )) f (cid:48) ( V, T ) dV (23) ≡ E L π v ( β (cid:63) γ )( z ) , (24)with β ( z ) = Θ( z ) (cid:112) µ z and γ ( z ) = f (cid:48) ( z, T ). Here we haveused the fact that f (cid:48) ( V, T ) is even in V . So according tothe Poisson summation, χ l = 2 µE L √ π exp( − πilx )˜ α (cid:18) πl µE L (cid:19) . (25)From the convolution theorem,˜ α ( k ) = E L π v √ π ˜ β ( k )˜ γ ( k ) . (26)The Fourier transforms are˜ β ( k ) = (cid:114) µ | k | + ik | k | , (27)˜ γ ( k ) = (cid:114) π kT sinh( πkT ) , (28)and χ l = µE L π v (1 + i ) √ l (cid:16) l π µTE L (cid:17) sinh (cid:16) l π µTE L (cid:17) exp( − πilµ /E L ) . (29) In a real material, the scattering caused by impuritiessmears the Landau levels. The density of states is thusthe convolution of (8) with a Lorentzian distribution π Γ Γ +( E − E ) , where Γ is the Dingle factor. Therefore,we must multiply each harmonic in the oscillating termby factor of exp( − πl Γ µ/E L ) (the Fourier transform ofthe Lorentzian): χ ≈ µ π v + T v + µE L π v ∞ (cid:88) l =1 √ l (cid:16) l π µTE L (cid:17) sinh (cid:16) l π µTE L (cid:17) × exp( − πl Γ µ/E L )[cos(2 πlµ /E L ) + sin(2 πlµ /E L )](30) B ( T ) ρρ FIG. 2: ρ zz /ρ vs B for µ = 150 meV, v = c/ T = 1 .
74 K,Γ = 0 . τ V /τ = 20. The solid line represents thefull prediction taking account of the quantum CME oscilla-tions, see (32); the dashed line represents only the SdH oscil-lations given by (31). The quantum CME oscillations becomelarger than the SdH oscillations at B (cid:39) Since χ oscillates as a function of magnetic field, theCME conductivity also acquires these quantum oscilla-tions. The Ohmic conductivity also oscillates with B ;these oscillations for 3D chiral materials have been eval-uated using the chiral kinetic theory in [32]. In our no-tations, σ zz ( B ) σ ≈ τ V τ E L µ − E L µ − π E L µ ∞ (cid:88) l =1 l / × (cid:16) l π µTE L (cid:17) sinh (cid:16) l π µTE L (cid:17) exp( − πl Γ µ/E L ) × [cos(2 πlµ /E L ) − sin(2 πlµ /E L )] , (31)where τ is the (chirality-preserving) scattering time and σ ≡ σ ( B = 0) = µ e τ π v . The τ V τ E L µ term, which isquadratic in B , comes from the CME conductivity; toaccount for the variation of χ with B , we should nowinclude a factor of µ π v χ in this term. Note that in weak / B ( / T ) Δσσ FIG. 3: The residue of σ zz after subtracting the constant andquadratic in B contributions, plotted as a function of 1 /B for µ = 150 meV, v = c/ T = 34 . . τ V /τ = 20. The solid line represents the prediction of (32)while the dashed line represents the predictions of (31) thatignore the quantum CME oscillations. magnetic fields, according to (4), this factor is equal tounity, µ π v χ = 1, but quantum corrections to χ givenby (30) will now induce additional oscillations in lon-gitudinal magnetoconductivity. All other terms in (31)represent the Ohmic conductivity. Therefore, the totallongitudinal conductivity as a function of E L = √ eBv is σ zz ( B ) σ ≈ µ π v χ τ V τ E L µ − E L µ − π E L µ ∞ (cid:88) l =1 l / (cid:16) l π µTE L (cid:17) sinh (cid:16) l π µTE L (cid:17) exp( − πl Γ µ/E L ) × [cos(2 πlµ /E L ) − sin(2 πlµ /E L )] , (32)where the chiral susceptibility χ that enters the secondterm oscillates with B according to (30). When the tem- perature or the Dingle factor are large enough so thatthe first term in the Fourier series dominates, the longi-tudinal conductivity is given by σ zz ( B ) σ ≈ (cid:18) τ V τ − (cid:19) (cid:18) BB (cid:19) −− A ( B ) (cid:20) cos (cid:18) B B + π (cid:19) + π τ V τ BB cos (cid:18) B B − π (cid:19)(cid:21) , (33)where B = µ / ev and A ( B ) is a positive non-oscillating factor which represents the effects of the tem-perature and the Dingle factor. For a material with µ = 150 meV and Fermi velocity v = c/ B is B ≈
48 T.When chirality flipping time is much longer than thescattering time τ V /τ (cid:29)
1, and in strong magnetic field,the quantum CME oscillations dominate over the SdHones; these CME oscillations have a phase of − π/
4. Onthe other hand, in weak fields the SdH oscillations aredominant, with the phase of π/ π/ [1] K. Fukushima, D. E. Kharzeev, and H. J. Warringa,Phys. Rev. D78 , 074033 (2008), 0808.3382.[2] D. E. Kharzeev, Prog. Part. Nucl. Phys. , 133 (2014),1312.3348.[3] D. E. Kharzeev, K. Landsteiner, A. Schmitt, and H.-U.Yee, Lect. Notes Phys. , 1 (2013), 1211.6245.[4] D. T. Son and N. Yamamoto, Phys. Rev. Lett. ,181602 (2012), 1203.2697.[5] D. T. Son and B. Z. Spivak, Phys. Rev. B88 , 104412(2013), 1206.1627.[6] A. A. Zyuzin and A. A. Burkov, Phys. Rev.
B86 , 115133(2012), 1206.1868.[7] G. Basar, D. E. Kharzeev, and H.-U. Yee, Phys. Rev.
B89 , 035142 (2014), 1305.6338.[8] M. Vazifeh and M. Franz, Physical Review Letters , 027201 (2013).[9] P. Goswami and S. Tewari, Physical Review B , 245107(2013).[10] Q. Li, D. E. Kharzeev, C. Zhang, Y. Huang, I. Pletikosic,A. V. Fedorov, R. D. Zhong, J. A. Schneeloch, G. D. Gu,and T. Valla (2014), 1412.6543.[11] H.-J. Kim, K.-S. Kim, J.-F. Wang, M. Sasaki, N. Satoh,A. Ohnishi, M. Kitaura, M. Yang, and L. Li, Physicalreview letters , 246603 (2013).[12] J. Xiong, S. K. Kushwaha, T. Liang, J. W. Krizan,M. Hirschberger, W. Wang, R. Cava, and N. Ong, Science , 413 (2015).[13] C.-Z. Li, L.-X. Wang, H. Liu, J. Wang, Z.-M. Liao, andD.-P. Yu, Nature communications (2015).[14] X. Huang, L. Zhao, Y. Long, P. Wang, D. Chen, Z. Yang, H. Liang, M. Xue, H. Weng, Z. Fang, et al., PhysicalReview X , 031023 (2015).[15] Z. Wang, Y. Zheng, Z. Shen, Y. Zhou, X. Yang, Y. Li,C. Feng, and Z.-A. Xu, arXiv preprint arXiv:1506.00924(2015).[16] C. Zhang, S.-Y. Xu, I. Belopolski, Z. Yuan, Z. Lin,B. Tong, N. Alidoust, C.-C. Lee, S.-M. Huang, H. Lin,et al., arXiv preprint arXiv:1503.02630 (2015).[17] X. Yang, Y. Li, Z. Wang, Y. Zhen, and Z.-a. Xu, arXivpreprint arXiv:1506.02283 (2015).[18] C. Shekhar, F. Arnold, S.-C. Wu, Y. Sun, M. Schmidt,N. Kumar, A. G. Grushin, J. H. Bardarson, R. D. d. Reis,M. Naumann, et al., arXiv preprint arXiv:1506.06577(2015).[19] X. Yang, Y. Liu, Z. Wang, Y. Zheng, and Z.-a. Xu, arXivpreprint arXiv:1506.03190 (2015).[20] S. Adler and J. Bell, Nuovo Cimento A , 47 (1969).[21] R. Jackiw, Phys. Rev. (1969).[22] H. B. Nielsen and M. Ninomiya, Physics Letters B ,389 (1983).[23] S. Zhong, J. E. Moore, and I. Souza, Physical review letters , 077201 (2016).[24] A. Cortijo, D. Kharzeev, K. Landsteiner, and M. A. Voz-mediano, Physical Review B , 241405 (2016).[25] Z. Song, J. Zhao, Z. Fang, and X. Dai, arXiv preprintarXiv:1609.05442 (2016).[26] D. Kharzeev, Y. Kikuchi, and R. Meyer, arXiv preprintarXiv:1610.08986 (2016).[27] D. E. Kharzeev, arXiv preprint arXiv:1612.05677 (2016).[28] T. Liang, Q. Gibson, M. N. Ali, M. Liu, R. Cava, andN. Ong, Nature materials , 280 (2015).[29] J. Hu, J. Liu, D. Graf, S. Radmanesh, D. Adams,A. Chuang, Y. Wang, I. Chiorescu, J. Wei, L. Spinu,et al., Scientific reports , 18674 (2016).[30] H. Murakawa, M. Bahramy, M. Tokunaga, Y. Ko-hama, C. Bell, Y. Kaneko, N. Nagaosa, H. Hwang, andY. Tokura, Science , 1490 (2013).[31] I. A. Luk’yanchuk and Y. Kopelevich, Physical reviewletters , 256801 (2006).[32] G. M. Monteiro, A. G. Abanov, and D. E. Kharzeev,Physical Review B92