can be moved in energy/momentum but not individually destroyed; in general: topological Fermi surfaces

Transcription

1

2

3

4 nodes protected against gapping can be moved in energy/momentum but not individually destroyed; in general: topological Fermi surfaces physical realization: stacked 2d topological insulators

5 C=1 3d top ins. class AII 2d top ins. class A C=-1 Burkov and Balents, 11

6 C=1 3d top ins. class AII 2d top ins. class A C=-1 Burkov and Balents, 11

7 nodes protected against gapping can be moved in energy/momentum but not individually destroyed; in general: topological Fermi surfaces physical realization: stacked 2d topological insulators surface states ( Fermi arcs )

8 nodes protected against gapping can be moved in energy/momentum but not individually destroyed; in general: topological Fermi surfaces physical realization: stacked 2d topological insulators surface states ( Fermi arcs ) non-conservation of charge at individual nodes (axial anomaly) generates unconventional transverse response: chiral magnetic effect, CME and anomalous Hall effect, AHE

9 transverse response from anomalies acts in nodal space is external vector potential is (3+1)-dimensional constant internal vector potential. Coupled to axial current. Can be gauged out by anomalous gauge transformation

10 transverse response from anomalies cont d effective gauge field action (Burkov & Zyusin)

11 transverse response from anomalies cont d effective gauge field action (Burkov & Zyusin) variation yields axial current µ j µ,a = µ F µ F

12 transverse response from anomalies cont d effective gauge field action (Burkov & Zyusin) variation yields current response j = µb j = b E CME (?) AHE

13 Effective theory of the disordered Weyl metal Minneapolis, May 1st, 2015 Alexander Altland, Dmitry Bagrets (:( not here ) disorder qualitative field theory construction discussion arxiv: , PRL to be published

14 disorder qualitative

15 diffusive transport Q: do anomalies survive internode scattering? (Burkov et al. 14, Son & Spivak 13, )

16 diffusive transport Q: do anomalies survive internode scattering? (Burkov et al. 14, Son & Spivak 13, ) = 2 + charge/axial charge coupling axial charge = n n a a. internode coupling substitution of first into second eq. DoS

17 disorder at the Weyl nodes Q: how does disorder affect the nodes? (Fradkin 86, Syzranov et al.14)

18 disorder at the Weyl nodes Q: how does disorder affect the nodes? (Fradkin 86, Syzranov et al.14) impurity line dimensional analysis: disorder irrelevant in d=3

19 disorder at the Weyl nodes Q: how does disorder affect the nodes? (Fradkin 86, Syzranov et al.14) impurity line dimensional analysis: disorder irrelevant in d=3 but only at the clean fixed point?

20 this talk describe physics at large distance scales in the supercritically disordered system suspect: 3d Anderson metal with added topological signatures

21 field theory

22 field theory (I) replica functional and disorder averaging (II) stationary phase, symmetry breaking, Goldstone modes (III) anomaly and gauge structure (IV) gradient expansion

23 field theory (I) replica functional and disorder averaging (II) stationary phase, symmetry breaking, Goldstone modes (III) anomaly and gauge structure (IV) gradient expansion

24 replica functional & disorder averaging replica index discriminates between retarded and advanced replica rotation symmetry: action invariant under

25 field theory (I) replica functional and disorder averaging (II) stationary phase, symmetry breaking, Goldstone modes (III) anomaly and gauge structure (IV) gradient expansion

26 field theory (I) replica functional and disorder averaging (II) stationary phase, symmetry breaking, Goldstone modes (III) anomaly and gauge structure (IV) gradient expansion

27 stationary phase analysis disorder averaged functional apple =(2/ )v (1 Fradkin 86 / ) replica rotation symmetry spontaneously broken if system is supercritically disordered

28 Goldstone modes action invariant under change of magnetization axis rotation around magnetization axis

29 field theory (I) replica functional and disorder averaging (II) stationary phase, symmetry breaking, Goldstone modes (III) anomaly and gauge structure (IV) gradient expansion

30 field theory (I) replica functional and disorder averaging (II) stationary phase, symmetry breaking, Goldstone modes (III) anomaly and gauge structure (IV) gradient expansion

31 anomaly and gauge structure

32 anomaly and gauge structure -> anomaly

33 anomaly and gauge structure -> anomaly generator of Goldstonemode fluctuations Goldstone mode generators T 1 [Ĥ,T]! T i T A i Transformation implies (cf. Volovik & Yakovenko 89) non-abelian gauge theory with gauge group

34 field theory (I) replica functional and disorder averaging (II) stationary phase, symmetry breaking, Goldstone modes (III) anomaly and gauge structure (IV) gradient expansion

35 field theory (I) replica functional and disorder averaging (II) stationary phase, symmetry breaking, Goldstone modes (III) anomaly and gauge structure (IV) gradient expansion

36 gauge field expansion note: regulator action essential (cf. Redlich 84)

37 gauge field expansion note: regulator action essential (cf. Redlich 84) lowest order expansion

38 discussion I

39 layered quantum Hall action

40 layered quantum Hall action

41 layered quantum Hall action coupling constants longitudinal conductivity Hall conductivity disorder averaged Chern number of n-th layer

42 layered quantum Hall action action describes layered quantum Hall systems

43 layered quantum Hall action dg xx d ln L = g xx g xx, dg xy d ln L = g xy, Ohmic scaling (Wang, 97)

44 layered quantum Hall action dg xx d ln L = g xx g xx, dg xy d ln L = g xy, Ohmic scaling (Wang, 97)

45 discussion II

46 CS action higher order expansion in A/derivatives projector onto retarded/advanced sector a strange animal: not a Hopf term, but also no genuine CS-term

47 building faith in CS action gauge (non-)invariance: under action changes as CS compensated by regulator action

48 building faith in CS action gauge (non-)invariance: under action changes as CS compensated by regulator action quadratic expansion reproduces axial diffusion propagator

49 quadratic expansion con t quadratic action from CS action S (2) [B,B ]= 2 DoS Z d 3 x tr B Pauli in nodal space D@ 2 + i! + 1 s (1 + n 1 ) i B n 3 B produces axial diffusion mode diffusion constant internode = 2 a = n n a a.

50 Bulk CME substitute into action, where a 3 = a(x) 3, a i = 1 2 3ij Bx j,i=1, 2 source field static magnetic field B in 3-direction lim R!0 i 4 R a(x) Z = hj 3 (x) ( H)i = B! Bµ 2 2 two nodes &

51 summary strong disorder phase is 3d Anderson metal with stable Hall response coefficient (AHE), and topological term supporting bulk CME and axial modification of diffusion surface theory not understood work to be done

52 summary strong disorder phase is 3d Anderson metal with stable Hall response coefficient (AHE), and topological term supporting bulk CME and axial modification of diffusion surface theory not understood work to be done