Adler-Bell-Jackiw anomaly in Weyl semi-metals: Application to Pyrochlore Iridates
aa r X i v : . [ c ond - m a t . s t r- e l ] A ug Adler-Bell-Jackiw anomaly in Weyl semi-metals: Application to Pyrochlore Iridates
Vivek Aji
Department of Physics and Astronomy, University of California, Riverside, CA 92521
Weyl semimetals are three dimensional analogs of graphene where the energy of the excitations area linear function of their momentum. Pyrochlore Iridates ( A Ir O ) with A =yttrium or lanthanideelement) are conjectured to be examples of such a system, with the low energy physics describedby twenty four Weyl nodes. An intriguing possibility is that these materials provide a physical real-ization of the Adler-Bell-Jackiw anomaly. In this letter we investigate the properties of pyrochloreiridates in an applied magnetic field. We find that the dispersion of the lowest landau level dependson the direction of the applied magnetic field. Consequently the velocity at low energies can bemanipulated by changing the direction of the applied field. The resulting anisotropy in longitudinalconductivity is investigated. Graphene[1] and topological insulators[2–4] have pro-vided the venue for condensed matter realizations, out-side of liquid Helium[5], of nontrivial phenomena origi-nally in the realm of high energy physics. Massless rel-ativisitic fermions[6], Klein tunneling [6], Theta vacuum(i.e. axion electrodynamics) [7], and Majorana modes[7, 8] are a few examples. A common feature of thesesystems is that the low energy physics is described bya two component Dirac Hamiltonian with the fermionicmomentum is confined to two dimensions. Recently py-rochlore iridates have been conjectured to realize theAdler-Bell-Jackiw (ABJ)[9, 10] chiral anomaly, addinganother example to the growing list.Wan et al. [11] explored the possibility of the threedimensional analog of graphene being realized in the py-rochlore iridates. These materials have a large spin or-bit couplings and are in a regime of intermediate cor-relations, making them promising candidates to realizetopological insulators[12, 13]. They have a magneticground state[14, 15] and, within a LSDA+U+SO calcu-lations, conjectured to be semi-metals. Most strikinglythe low energy physics is described by the Weyl equa-tion, which is the two component version of the Diracequation. There are 24 Weyl nodes, three around each L point ([111] and equivalent directions) in the Brilloiunzone (see fig.1). Nodes related either by inversion or re-flection about the { xy, yz, zx } planes have opposite chi-rality. Consequently the material is expected to havean anomalous hall response to applied uniaxial pressureand is susceptible to charge ordering in large magneticfields[16].In a quantizing magnetic field, the Lowest LanduaLevel (LLL) is a linear function of the magnitude of amomentum, with the sign determined by the band struc-ture. If an electric field is applied parallel to the magneticfield, the Adler-Bell-Jackiw (ABJ)[9, 10] axial anomalyleads to an anomalous magneto-conductance. The ori-gin of this effect is in the production of Weyl fermionsof a given chirality and an equivalent annihilation of theopposite chirality[17]. This translates to a transfer ofparticles from one Weyl node to another of opposite chi-rality at a constant rate. To reach a steady state, this is xx xx xx L KX Z Y
FIG. 1: The low energy physics of pyrochlore iridates is de-scribed by linearly dispersing fermionic modes near nodes inthe band structure. The position of Weyl nodes in the Bril-louin zone is shown in the figure. There are three nodes, withthe same chirality, located in the vicinity of the L points, andnodes related by inversion have opposite chirality. balanced by inter-node scattering due to impurities.In the absence of the magnetic field, the intra-nodescattering is quite effective in relaxing the momentum.In the presence of a large field, such that only the LLLis occupied, intra-node scattering is suppressed due to alack of phase space. The only scattering mechanism avail-able are processes that involve different nodes of oppositechirality. Since these nodes are located at different pointin the Brillouin zone, Nielsen and Ninomeya [17] arguedthat the corresponding scattering rate is much smallerimplying a large magneto-conductivity.In this letter we focus on the anomalous magneto-conductivity expected in Weyl semi-metals [11, 17]. Ingeneral the Hamiltonian takes the form ± ~q · V · ~σ where ~q isthe momentum, V is a real matrix, ± label right handed(RH) and left handed (LH) chirality, and ~σ = { σ x , σ y ,σ z } . We first address the question of the dispersion ofthe LLL for arbitrary V . We find that the energy is de-termined by the component of the momentum parallelto the applied field. Rather surprisingly the velocity ofthe mode depends on the direction of the magnetic field.This implies that the low energy dispersion and densityof states can be manipulated by varying the direction ofthe magnetic field.The low energy Hamiltonian in the vicinity the Weylnodes of the proposed topological semimetal state is H ( ~q ) = (cid:18) ∆ + q z m − q ⊥ m (cid:19) σ z (1)+ (cid:0) βq z + c q ⊥ cos (3 θ ) (cid:1) σ y + c q ⊥ sin (3 θ ) σ x where the local z − axis is taken along the Γ − L directionand the local x − axis along the L − K direction in theBrillouin zone. Notice that these direction rotate withrespect to the global coordinates from one L point to an-other. At the Weyl nodes, there is a degeneracy amongtwo states with opposite symmetry under inversion. Thephysics of the two state system is captured by the σ ma-trices. Define q z and ~q ⊥ as the displacements of thenode away from the L point parallel and perpendicularto the Γ − L line respectively. θ is defined as the angle ~q ⊥ makes with the x − axis. The nodes are located at θ = pπ/ q z = ± c q ⊥ /β (positive(negative) for odd(even) values of p ), and q ⊥ satisfying the equation ∆+ q z / m − q ⊥ / m = 0.In the vicinity of these nodes the Hamiltonian can beexpanded and written as H i ( δ~q ) ≈ δ~q · V i · ~σ (2)where δ~q = ~q − ~q , and V i is a real matrix whose en-tries depend on the node index i . For the Hamiltonian ineq.1 the matrix V is generically not symmetric and doesnot have all real eigenvalues. This is quite unlike thecase considered in the context of the ABJ anomaly[17]or the anomalous hall effect [16]. For a diagonal V ma-trix, the direction of the momentum, whose magnitudedetermines the energy if the lowest Landau level, is par-allel to the applied magnetic field. Since momentum doesnot commute with the electromagnetic vector potential,and the fact that the latter is always transverse to the ~B field, suggest that this property is generally valid. Tounderstand the nature of the ABJ anomaly in iridates,we first verify that this conjecture holds for arbitrary V matrices.The procedure for computing the energy is sketchedout here (details provided elsewhere [18]). Considera single Weyl node in an applied magnetic field, ~B . We can always rotate the coordinate system sothat the ~B lies in the xz − plane. In this referenceframe ~B = B { sin ( θ ) , , cos ( θ ) } . Using a Landaugauge we write the corresponding vector potential as B {− cos ( θ ) y, , sin ( θ ) y } . Given this choice, the sys-tem is translationally invariant in the x and z direc-tions. The wave-function of the LLL has the form { u, v } φ ( y ) e − iδk x x − iδk z z , where u and v are constants.Rather remarkable, the energy ǫ , calculated with this choice of gauge, can be written in a gauge invariant formas ǫ = − Det [ V ] k adj [ V ] · ~B k δ~q · ~B (3)= − sgn ( Det [ V ]) δ~q · ~B k V − · ~B k where Det [ V ] and adj [ V ] are the determinant, and ad-jugate of the matrix [ V ]. The sign of the dispersion, andhence the chirality, is determined by the determinant.The energy is inversely proportional to the projection ofthe deviation of the momentum in the direction of theapplied field.The results in eqn.3 is the key result and we will ex-plore the consequences in the rest of the letter. The mostimportant feature of the dispersion is the dependence ofthe velocity of the low energy excitations on the direc-tion of the applied field. This anisotropy is inheritedfrom the underlying band structure. While most systemshave inherent anisotropies, what makes Weyl semi-metalsunique is that they have linearly dispersing modes, theleft handed and right handed branches of which are lo-cated at distinct point in the Brillouin zone. All smallmomentum scattering are suppressed leading to a largemagneto-conductivity. Furthermore the ability to changethe velocity implies that the low energy density of statescan also be manipulated. As such all thermodynamic andtransport properties are sensitive to the direction of theapplied field. Here we focus on the ABJ anomaly.Let us now consider the case of the pyrochlore iridates.Given the Hamiltonian (eqn.1) we construct the relevantmatrices. To make further progress we use parametersthat best fit the LSDA+U+SO calculations [11, 16]: m = m = 0 . − , c = c = 1 . β = 0 . . ~q is dimensionless). These parameters give q ⊥ =0 .
48 and q z = ± . θ = { , π/ , − π/ } are V θ =0 = q ⊥ − q ⊥ q ⊥ q z (4) V θ = ± π/ = ∓ √ q ⊥ − q ⊥ q ⊥ − q ⊥ ± √ q ⊥ ∓√ q ⊥ q z The system possesses three fold rotation symmetry aboutthe Γ − L axis and the three matrices are related by120 degree rotation about the z − axis. If U is a rota-tion matrix for 2 π/ z − axis,than UV θ =0 = V θ =2 π/ , UV θ =2 π/ = V θ = − π/ and UV θ = − π/ = V θ =0 . In other words, knowing one issufficient to generate the others. The determinant ofthese three matrices are all equal and given by Det [ V ] i = − q ⊥ − q ⊥ q z .Since V − = adj [ V ] /Det [ V ], the adjugate matriceshave the same transformation properties as the inversematrix. Given the three matrices near the [111] point, allother can be constructed by symmetry. The matrices forthe Weyl points related by inversion to those near [111]are obtained by changing the sign of the third columnor equivalently the sign of σ z . This remains true for allnodes related by inversion as the local z − axis changessign. The nodes near [¯1¯11] have the same structure asthose near [111] while those near [¯111] have the sign ofthe first column changed. This ensures the geometry andhelicity obtained within LSDA+U+SO calculations [11].Pyrochlore iridates have cubic symmetry. Thus theanomalous Hall is zero unless an uniaxial pressure is ap-plied to break the symmetry[16]. The same propertyleads to an isotropic density of states as a function ofthe direction of the magnetic field. To get anomalousresponse in thermodynamic properties from lowest Lan-dau level, one needs to have Weyl nodes in systems thatare inherently anisotropic. The layered heterostructureof normal and topological insulators[19] is one example ofsuch systems. Remarkably, even for an isotropic systemthe transport properties can be anisotropic as we showbelow.Since the dispersion of the LLL is in the direction of theapplied magnetic field, only the response to an electricfield, ~E = E ˆ B applied parallel to the magnetic fieldwill be considered. We will assume that the magneticfield is strong enough so that only the LLL is occupied.The dispersion being linear, elastic scattering within asingle node is suppressed due to the lack of phase space.Stated differently no momentum relaxation is possiblewithin a node as states with opposite velocities do notexist. However the scattering between two nodes cannotbe ignored. In the presence of the electric field, thereis a generation of RH particles and annihilation of LHparticles. The rate of production is given by the rate ofchange of energy, vδ ˙ q , times the density of states, eB/v ~ .Since δ ˙ q = eE , we get ˙ N || = δ ˙ qeB/ ~ = e E B/ ~ . Thisrate has to be balanced by the scattering between nodeswith the opposite chirality to maintain a steady state.Weyl nodes always come in pairs with opposite chiral-ity. Before considering all 24 Weyl nodes, we first lookat a single pair whose energies are given by ǫ = ± vδq || .For impurity scatterers, where the transition probabilitybetween states with different moment is independent oftheir momenta, the scattering rate, τ − is proportionalto the density of states and is given by1 τ = C eBv ~ (5)where C is a constant determined by the strength of scat- tering potential. We will comment on more general formsof scattering later. The ABJ anomaly requires an imbal-ance in particle number at the RH and LH Weyl nodes.If the difference in chemical potential between the RHand LH nodes is ∆ µ , than energy balance requires∆ µ = eE vτ (6)This is because over the scattering time τ , the momentumchanges by eE τ . Since the change in energy is v timeschange in momentum, the net energy transferred fromone cone to the other is eE τ v . A steady state is achievedif this transfer is balanced by the difference in chemicalpotential.The difference in chemical potential leads to a currentgiven by J A = nev (7) J A = eBv ~ ∆ µev = e ( eB/ ~ ) eE vτ = C − e v E The subscript A refers to the anomalous response. So farwe have only considered momentum independent scatter-ing rate. Let us look at the response in the presence ofscreened charged impurities characterized by matrix ele-ments of the form 1 / ( q + κ ). The internode scatteringprobes intermediate to large momenta, as the nodes arephysically separated in the Brillouin zone. The transi-tion probabilities, which are proportional to the squareof the matrix elements, fall of as 1 / | ~q | giving a large τ . In contrast, the conductivity in zero magnetic field ismuch smaller as it is dominated by intra-node scatter-ing ( ∼ /κ ). Thus the conductivity in the presence ofthe magnetic field can be much larger than that in zerofield. The results in eq.6 and eq.7, and the argument oflarge magneto-conductance, are the main conclusions ofNielsen and Ninomeya [17].Having reviewed the expected nature of the magneto-conductance, we return to the discussion of the iridates.The key feature of eq.7 is its dependence on the veloc-ity v . As we have seen in our discussion of the disper-sion of the LLL, the velocity can be tuned by changingthe direction of the applied field. Moreover the differentWeyl nodes have different Fermi wave-vector and differ-ent density of states. Since the longitudinal conductivitydepends on the scattering rate, we study two cases. Wefocus on momentum independent processes which eitherscatter only between nodes with opposite velocities oronly between nodes of opposite chirality. Detailed stud-ies of how various scattering processes will reveal the un-derlying anisotropy will be part of future efforts, but allof them are sensitive to the underlying dispersion whichis the source of the phenomena. Θ Ú v i FIG. 2: Top: The variation of the conductivity, for scatteringpotential that only couple nodes that are related by inver-sion, as a function of the direction of magnetic field is shownas a three dimensional spherical plot. The radial distance ofa point on the figure from the origin is a measure of the con-ductivity for the magnetic field along that direction. Bottom:A cut along the Z − K direction is shown. Weyl nodes related by time reversal have opposite ve-locities and the bands are related by inversion. For scat-tering processes that conserve the corresponding quan-tum numbers, scattering between these nodes domi-nate. The scattering rate is proportional to the den-sity of states of the nodes and the total conductivity is σ = P i C − e v i . In fig.2 the conductivity is plottedas a function of the magnetic field and a cut along the Z − K direction, where both the maximum and minimumvalues are obtained, is shown. The anisotropy, define as( σ max − σ min ) / ( σ max + σ min ) is ∼ ∼ Θ Ú È v i È FIG. 3: Top: The variation of the conductivity, for scatteringpotentials that couple all nodes, as a function of the directionof magnetic field is shown as a three dimensional sphericalplot. The radial distance of a point on the figure from theorigin is a measure of the conductivity for the magnetic fieldalong that direction. Bottom: A cut along the Z − K directionis shown. is obtained even in a system that has cubic symmetry.For topological-normal insulator heterostructure where[ V ] is diagonal, but has different velocities in-plane asopposed to perpendicular to the plane, the density ofstates is also anisotropic. In such systems even the lowenergy thermodynamics will depend on the direction ofthe applied field. A more general study of the low energybehavior of specific heat, susceptibility and the effect ofmomentum dependent scattering will be part of a futureeffort in characterizing the novel behavior of Weyl semi-metals due to their unique zero Landau level dispersion.In this letter we have considered the nature ofmagneto-conducatnce in Weyl semi-metals. In the pres-ence of an applied field the energy of the LLL is obtained.We find that the the level always disperses linearly withrespect to the momenta parallel to the applied field. Thisresult is general and holds even for systems with arbitraryWeyl hamiltonians. The velocity of the low energy modesdepends on the direction of the applied field. This fea-ture is exploited in the context of the pyrochlore iridates.Using the symmetries of the crystal, we derive the Weylequation at the 24 nodes. Applying magnetic fields in dif-ferent directions allows us to manipulate the low energyphysics, leading to anisotropic transport properties.VA acknowledges the support of University of Califor-nia at Riverside through the initial complement. V.A.thanks A.Vishwanath for helpful suggestions and com-ments. [1] K. S. Novoselov1, A. K. Geim1, S. V. Morozov, D. Jiang,Y. Zhang, S. V. Dubonos, I. V. Grigorieva and A. A.Firsov, Science , 666 (2004).[2] C. L. Kane and E. J. Mele, Phys. Rev. Lett. , 146802(2005).[3] L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. ,106803 (2007).[4] R. Roy, Phys. Rev. B , 195322 (2009).[5] G. Volovik, Universe in a Helium droplet, Oxford Uni-versity Press (2003) .[6] A. H. Castro Neto, F. Guinea, N.M.R. Perez, K.S.Novoselov and A.K. Geim, Rev. Mod. Phys. , 109(2009).[7] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. , 3045 (2010).[8] L. Fu and C. L. Kane, Phys. Rev. Lett. , 096407(2008).[9] S. L. Adler, Phys. Rev. , 2426 (1969).[10] J. S. Bell and R. Jackiw, Nuovo Cim. A60 , 47 (1969).[11] X. Wan, A. M. Turner, A. Vishwanath, and S. Y.Savrasov, Phys. Rev. B , 205101 (2011).[12] A. Shitade, H. Katsura, J. Kunes, X.-L. Qi, S.-C. Zhangand N. Nagaosa , Phys. Rev. Lett. , 256403 (2009).[13] D. Pesin and L. Balents, Nature Physics , 376 (2010).[14] N. Taira, M. Wakeshima, and Y. Hinatsu, Journal ofPhysics: Condensed Matter , 5527 (2001).[15] K. Matsuhira, M. Wakeshima, R. Nakanishi, T. Yamada,A. Nakamura, W. Kawano, S. Takagi and Y. Hinatus,Journal of the Physical Society of Japan , 043706(2007).[16] K.-Y. Yang, Y.-M. Lu, and Y. Ran, Phys. Rev. B ,075129 (2011).[17] H. B. Nielsen and M. Ninomiya, Physics Letters B ,389 (1983).[18] V. Aji, unpublished, manuscript in preparation (unpub-lished).[19] A. A. Burkov and L. Balents, Phys. Rev. Lett.107