Topological response in Weyl metals. Anton Burkov

Transcription

1 Topological response in Weyl metals Anton Burkov NanoPiter, Saint-Petersburg, Russia, June 26, 2014

2 Outline Introduction: Weyl semimetal as a 3D generalization of IQHE. Anomalous Hall Effect in metallic ferromagnets and its connection to Weyl semimetals. Universality of AHE in Weyl metals.

3 Quantum Anomalous Hall Insulator Thin film of 3D TI: gapless Dirac surface states protected by TRS

4 Quantum Anomalous Hall Insulator Break TRS by doping with magnetic impurities

5 Quantum Anomalous Hall Insulator T xy = e2 2h B xy = e2 2h xy = T xy + B xy = e2 h

6 Quantum Anomalous Hall Insulator b< b> xy =0 xy = e2 h Haldane, 1988 Yu et al., 2010 Chang et al., 2013

7 From 2D QAH insulator to 3D Weyl semimetal Stack of 2D QAH insulators, separated by ordinary insulator spacers. d D S

8 From 2D QAH insulator to 3D Weyl semimetal ehk z LêD S k zd C c (k z )= 1 xy = C v (k z )=1 Z G/2 G/2 b> S + D dk z 2 e 2 h = e2 h G 2 Trivial generalization of IQHE to 3D. Kohmoto, Halperin, Wu, 1992

9 From 2D QAH insulator to 3D Weyl semimetal ehk z LêD S S D <b< S + D k zd -0.5 xy = e2 h K Exchange of Chern numbers at gap closing points Nontrivial generalization of IQHE to 3D. Wan et al., 2011 AAB & Balents, 2011 Xu et al., 2011

10 Weyl nodes Weyl nodes are magnetic monopoles in momentum space. Band dispersion near the nodes must be linear (Atiyah-Bott- Shapiro construction): H = ± k Monopole sources of Berry curvature: = ± k 2k 3

11 Weyl nodes Momentum space monopoles are indestructible. Can only be created and annihilated in pairs of opposite charge at k=0 or at the BZ boundary.

12 Fermi arcs Fermi arcs are chiral surface states that connect projections of the monopole locations on the surface Brillouin zone. Hall conductivity of Weyl semimetal may be associated with the Fermi arc surface states.

13 Weyl metal Weyl metal: Fermi surface breaks up into sheets, each enclosing a single Weyl node. Each Fermi surface sheet has a nonzero Chern number, equal to the topological charge of the enclosed node. Are there measurable consequences, i.e. can Weyl metal be distinguished from ordinary FM metal? ehk z LHarb. unitsl k zd C = 1 Z ds 2

14 Chiral anomaly ehk z LHarb. unitsl Nielsen & Ninomiya, 1983 L R k zd + r j L,R ±E B Numbers of left- and right-handed electrons not separately conserved in the presence of electric and magnetic fields.

15 Chiral anomaly k z Nielsen & Ninomiya, 1983 k z Chiral Anomaly: the zero mode LL is chiral and has to cross the Fermi energy at the Weyl node locations. Every momentum point between the nodes contributes a quantum of Hall conductance e 2 /h

16 This work Is there a qualitative difference between AHE in Weyl metal and a regular FM metal?

17 Anomalous Hall Effect AHE is a contribution to the Hall resistivity of a ferromagnetic metal, that does not vanish even when the external magnetic field is absent. xy = R 0 B +4 R s M FIG. 1. The Hall effect in Ni data from Smith, From Pugh and Rostoker, 1953.

18 Intrinsic Anomalous Hall Effect Broken TR and SO interactions are needed, otherwise many distinct sources of AHE are possible. Recent work has shown that geometrical electronic structure properties likely very important in many materials: intrinsic AHE. Sundaram & Niu, 1999 Jungwirth, Niu, MacDonald, 2002 Nagaosa et al., 2002

19 Intrinsic Anomalous Hall Effect Bloch wave functions form a fiber bundle over BZ. Fiber bundle geometry can be characterized by Berry connection and Berry curvature. A nk = ihnk r k nki nk = r k A nk Gauge invariance then requires modification of the position operator: r nk = ir k + A nk And corresponding correction to band velocity: v nk = e ~ E nk

20 Intrinsic Anomalous Hall Effect Intrinsic anomalous Hall conductivity is given by the integral of the anomalous velocity over all occupied states: xy = e2 ~ X n Z d 3 k (2 ) 3 n F ( nk ) z nk In an ordinary metallic FM this is significantly modified by impurity scattering: extrinsic contribution to AHE. What about Weyl metal?

21 Weyl metal: model 2D Dirac fermions with kz-dependent mass. H ± (k) =v F (ẑ ) k + m ± (k z ) z m ± (k z )=b ± q 2 S + 2 D +2 S D cos(k z d) One of the masses changes sign at the locations of the Weyl nodes.

22 Weyl metal: model b =0 ehk z LêD S k zd <b< S D ehk z LêD S k zd S D <b< S + D b> S + D ehk z LêD S

23 Topological response in Weyl metal Can evaluate anomalous Hall conductivity as long-wavelength, low-frequency limit of the coefficient of a 3D Chern-Simons term (path integral version of the Streda formula): S = X q,i z0 (q,i ) A 0 ( q, i ) ˆq A (q,i )

24 Order of limits Total anomalous Hall conductivity is the DC limit of the interband optical Hall conductivity: xy = lim i!0 lim q!0 1 q (q,i ) Opposite order of limits gives a part of the total Hall conductivity, which is a thermodynamic equilibrium property: II xy =lim q!0 lim i!0 1 q (q,i ) µ

25 Fermi surface part of AHE The difference is a purely transport property, which vanishes in equilibrium and which can be associated with states on the Fermi surface: I xy = II xy xy This is analogous to the difference between the Drude weight and the superfluid weight. Scalapino, White, Zhang, 1993

26 AHE in Weyl metal (q, ) = = + Density response function is given by the sum of ladder diagrams (noncrossing approximation).

27 AHE in Weyl metal (q, ) = I (q, )+ II (q, ) Fermi surface contribution, containing the diffusion pole Z I Z 1 (q, ) =2e 2 v F 1 P ( Z d 2 i dn F ( ) d P 0x (q, i, + +i ), i, + +i ) hg R ( + )G A ( )i (8) Contribution of all states below the Fermi energy, including completely filled bands II (q, ) =4ie 2 v F Z 1 1 d 2 i n F ( )ImP 0x (q, +i, + +i ) P ( + i, + +i ) hg R ( + )G R ( )i Details in arxiv:

28 FIG. 2. (Color online). Plot of the anomalous Hall conductivity versus the Fermi energy for the same system as in Fig. 1. The plateau-like feature in xy correlates with the range of the Fermi energies, for which the Fermi surface consists of two separate sheets, each enclosing a single Weyl node. and AHE in Weyl metal In a Weyl metal both the Fermi surface contribution, and the equilibrium contribution from conduction band vanish. xy = e2 h K 2 ehk z LHarb. unitsl k zd s xy h dêe e FHarb. unitsl W duc as b we t = the and the

29 We can now finally evaluate the anomalous Hall We can now evaluate thedi usive anomalous Hall ductivity. We finally willmetal focus on the limit re AHE in Weyl ductivity. on the similar. di usive At limit res as ballistic We limitwill is focus qualitatively this as ballistic is qualitatively At thisof we will alsolimit explicitly include thesimilar. contribution In a Weyl metaltwe=both theofexplicitly Fermi contribution, will also include the contribution of ± pairs bands, surface which simply amounts to rest t=± pairs which simply amounts to rest the index t of in bands, mt, and summing over t. Using Eq and the equilibrium contribution from conduction the index t in mt,that andasumming over t.wick Using Eq and remembering! ia upon rotati 0 0 band vanish. and remembering that A! ia upon Wick rotati 0 the real time, we obtain 0 efharb. unitsl the real3 time, we2 obtain 2 X 2.0 efharb. unitsl e vf mt +t I ehkz LHarb. unitsl F [mt ], xy = e2 v 2 X gt ( F ) mt +t 2.0 I F 2 F F t = g ( ) F [mt ], t t F 1.0 xy malous Hall conductiv2 F t F t malous Hallasconductivme system in0.5fig. 1. and Z /d me as in Fig. of 1. and atessystem with the range kz d 2 X e II Z ates with the range of surface consists of two /d dkz sign[mt (kz )] 2 = X -0.5 xy e2 II 8 surface consists of two Weyl node. /d dkz sign[mt (kz )] = t xy Weyl node. /d t {1 [ F mt (kz ) ]}. (a) {1 [ F mt (kz ) ]}. nctional of m, which Since m+ (kz ) is positive throughout the first BZ, nctional of m,towhich Since (kz ) is positive throughout the first is important note m (kzm ) +changes sign at the Weyl nodes, the BZ, first is important to note m Eq. (kz )(20), changes sign at the Weyl nodes, the firstb mathematical consein which comes from completely filled in Eq.a (20), which comes quantized from completely filled b nmathematical pole in I (q,conse ), is gives universal (almost) contribution

30 Simple explanation A pair of Weyl nodes is a dipole of two topological charges. (b)

31 Simple explanation A pair of Weyl nodes is a dipole of two topological charges. Anomalous Hall conductivity is proportional to the average of the z-component of the Berry curvature field of this dipole. xy = e2 ~ X n Z d 3 k (2 ) 3 n F ( nk ) z nk

32 (b) Simple explanation Close to the nodes the average vanishes (exactly, as long as dispersion may be taken to be linear), since the field forms a hedgehog around each node.

33 Simple explanation Close to the nodes the average vanishes (exactly, as long as dispersion may be taken to be linear), since the field forms a hedgehog around each node. Since this property is related to topology of the nodes, it survives when impurity scattering is introduced. This may be thought of as nonrenormalization of chiral anomaly by finite fermion density and disorder.

34 Conclusions Weyl semimetal provides a nontrivial generalization of IQHE to 3 dimensions. Weyl metal (doped Weyl semimetal) has a universal disorderindependent anomalous Hall conductivity, proportional to the separation between the Weyl nodes in momentum space (dipole moment of the topological dipole).