Weyl semimetals from chiral anomaly to fractional chiral metal

Transcription

1 Weyl semimetals from chiral anomaly to fractional chiral metal Jens Hjörleifur Bárðarson Max Planck Institute for the Physics of Complex Systems, Dresden KTH Royal Institute of Technology, Stockholm J. Behrends, A. G. Grushin, T. Ojanen, JHB Phys. Rev. B 93, (2016) F. Arnold, A. G. Grushin, JHB, C. Felser, E. Hassinger, B. Yan Nat. Comm. 7, (2016) T. Meng, A. G. Grushin, K. Shtengel JHB arxiv: (to appear in PRB)

2 Weyl Fermions

3 Weyl fermions My work always tried to unite the truth with the beautiful, but when I had to choose one or the other, I usually chose the beautiful Dirac equation Massless Weyl fermions: Hermann Weyl ( ) PhD Göttingen 1908 with Hilbert Göttingen ETH Zürich Göttingen Princeton H. Weyl Z. Phys. 1929

4 Accidental crossing of two bands in 3D band structure is a Weyl point Note! Type II Weyl semimetals see, e.g., Soluyanov et al Nature 2015 C. Herring, Phys. Rev E. Witten arxiv:

5 At low energy Weyl semimetals are described by Weyl fermions E k H = p = p x x + p y y + p z z Reviews: Turner and Vishwanath arxiv: ; Hosur and Xi C. R. Phys. 2013

6 Scalar disorder is irrelevant for 3D Weyl nodes 2D 3D ( ) /(e 2 /h ) SCBA K JHB, Tworzydlo, Brouwer, Beenakker, PRL 2007 Nomura, Koshino, Ryu PRL 2007 Fradkin PRB 1986 (x2) Sbierski, Pohl, Bergholtz, Brouwer PRL 2014

7 Weyl node is a topologically stable monopole of Berry flux k x,y k z Berry phase Berry curvature topological invariant Volovik JETP Lett. 2014; The Universe in a Helium Droplet 2003

8 In momentum space the monopoles always come in pairs k x,y k z Nielsen & Ninomiya Nucl. Phys. B 1981 Volovik JETP Lett. 2014; The Universe in a Helium Droplet 2003

9 The minimal model of a Weyl semimetal has two Weyl nodes E k z Reviews: Turner and Vishwanath arxiv: ; Hosur and Xi C. R. Phys. 2013

10 Weyl semimetals necessarily break time reversal and/or inversion symmetry Inversion symmetry Time reversal symmetry TRS + IS Dirac semimetal Wan, Turner, Vishwanath, Savrasov PRB (2011)

11 Weyl (and Dirac) semimetals exist! Dirac semimetals 2014 Na 3 Bi Xu et al. Science 2015 Liu et al. Science 2014 Kushwaha et al. APL Mat Cd 3 As 2 Neupane et al. Nature Comm Borisenko et al. PRL 2014 Yi et al. Sci. Rep Liu et al. Nature Mater Liang et al. Nature Mater He et al. PRL 2014 Weyl semimetals 2015 TaAs NbAs Weng et al. PRX 2015 (th) Xu et al. Science 2015 Yang et al. Nature Physics 2015, Lv et al. Nature Physics 2015, PRX 2015 Xu et al. Nature Physics 2015 See Claudia Felser s talk! Images from: Weng et al. PRX 2015

12 Fermi Arcs

13 In momentum space the monopoles always come in pairs k x,y k z Nielsen & Ninomiya Nucl. Phys. B 1981 Volovik JETP Lett. 2014; The Universe in a Helium Droplet 2003

14 In some parts of momentum space, Dirac string cuts a plane odd times k x,y k z Volovik JETP Lett. 2014; The Universe in a Helium Droplet 2003

15 In some parts of momentum space, Dirac string cuts a plane odd times k x,y k z Volovik JETP Lett. 2014; The Universe in a Helium Droplet 2003

16 Topological surface states appear for certain momentum values E k z E k x,y y C =1 C =0 Reviews: Turner and Vishwanath arxiv: ; Hosur and Xi C. R. Phys. 2013

17 The Fermi arc separates occupied and unoccupied surface states E k z k y k z k y k z Wan, Turner, Vishwanath, and Savrasov PRB 2011

18 Quantum oscillations from Fermi arcs y B v k x k z + - L B ) E + v - y x Potter, Kimchi, Vishwanath, Nature Comm Baum, Berg, Parameswaran, Stern PRX 2015 Moll, Nair, Helm, Potter, Kimchi, Vishwanath, Analytis, Nature 2016

19 Chiral anomaly

20 Chiral anomaly in a (n L + n (n L n R )= e2 2 2 ~ 2 E B n L v n R Nielsen & Ninomiya Phys. Lett. B 1983; Volovik JETP Lett 1999; Aji PRB 2012 see also Burkov and references within

21 Zero Landau level is chiral Nielsen & Ninomiya Phys. Lett. B 1983

22 Zero Landau level is = Nielsen & Ninomiya Phys. Lett. B 1983

23 Semiclassical Boltzmann theory gives the same answer D. T. Son and B. Z. Spivak PRB (2013)

24 The chiral anomaly induced chiral chemical potential tilts the Fermi arc k y k z k y k z Behrends, Grushin, Ojanen, JHB PRB 2016

25 ARPES Weyl semimetal Z (a) I(k Equilibrium k,!) = dx X n n,kk (x) 2 ( n,kk!)w(x)f(! µ kk ) (b) Chiral anomaly (b) Chiral anomaly Cuts in momentum space use tight binding model from Vazifeh and Franz PRL 2013 Behrends, Grushin, Ojanen, JHB PRB 2016

26 ARPES Weyl semimetal (c)cuts in momentum space I = I ca I eq k y =0 use tight binding model from Vazifeh and Franz PRL 2013 Behrends, Grushin, Ojanen, JHB PRB 2016

27 Dirac semimetal as two copies of a Weyl semimetal Young et al. PRL 2012, Behrends, Grushin, Ojanen, JHB PRB 2016

28 ARPES Dirac semimetal (a) Equilibrium ( (b) Chiral anomaly use tight binding models from Wang et al. PRB 2012; PRB 2013 Behrends, Grushin, Ojanen, JHB PRB 2016

29 Experimental estimates Na 3 Bi E = 10 4 Vm 1 B =1mT v / c = 10 4 µ = µ L µ R 10 mev Weyl semimetal But, ARPES doesn t like B! Magnetic substrate Pump-probe Dirac semimetal discussion of scattering times: Parameswaran et al. PRX 2014 Behrends, Grushin, Ojanen, JHB PRB 2016

30 Transport: negative magnetoresistance

31 Semiclassical Boltzmann theory and conductance Chiral magnetic effect: Negative magnetoresistance D. T. Son and B. Z. Spivak PRB (2013)

32 Negative magnetoresistance in TaP Nielsen & Ninomiya Phys. Lett. B 1983, Aji PRB 2012, Son and Spivak PRB 2013, Burkov PRL 2014, PRB 2015 Arnold, Grushin, JHB, Felser, Hassinger, Yan Nat. Comm. 7, (2016)

33 TaP has several electron and hole pockets and no well defined chirality Arnold, Grushin, JHB, Felser, Hassinger, Yan Nat. Comm. 7, (2016)

34 Negative magnetoresistance also for electron pockets in quantum limit

35 Gyrotropic magnetic effect Zhong, Moore, Souza PRL 2016 Ma, Pesin PRB 2015

36 Fractional chiral metal Meng, Grushin, Shtengel, JHB arxiv: (to appear in PRB)

37 A 4D construction of a Weyl semimetal Meng, Grushin, Shtengel, JHB arxiv:

38 A 4D construction of a Weyl semimetal Meng, Grushin, Shtengel, JHB arxiv:

39 In presence of interactions first go to quantum limit and bosonize Meng, Grushin, Shtengel, JHB arxiv:

40 Coupled wires construction Meng, Grushin, Shtengel, JHB arxiv:

41 Fractional quantum Hall state obtained for correlated hopping C. L. Kane, R. Mukhopadhyay, and T. C. Lubensky, PRL 2002 Meng, Grushin, Shtengel, JHB arxiv:

42 The bulk is gapped out by the correlated hopping Electromagnetic response: Meng, Grushin, Shtengel, JHB arxiv:

43 Field theory from minimal coupling and gauge invariance Meng, Grushin, Shtengel, JHB arxiv:

44 Strong coupling and electromagnetic response: fractional chiral anomaly Meng, Grushin, Shtengel, JHB arxiv:

45 Action is an anisotropic part of the isotropic 4+1D Chern-Simons action compare with Meng, Grushin, Shtengel, JHB arxiv:

46 Surface field theory fractional chiral metal 3+1D surface state constructed from copies of the edge theory of 2+1D Chern-Simons theory, labeled by Consistent with current algebra analysis of the 4+1D Chern-Simons field theory K. Gupta and A. Stern, Nucl. Phys. B 1995 Meng, Grushin, Shtengel, JHB arxiv:

47 Collaborators Jan Behrends Adolfo G. Grushin UC Berkeley Kirill Shtengel UC Riverside Teemu Ojanen Aalto Helsinki Tobias Meng Frank Arnold Binghai Yan Elena Hassinger ( ) and more

48 Summary: chiral anomaly visualized through the blue note and fractional chiral metals Open PhD and Postdoc positions at KTH J. Behrends, A. G. Grushin, T. Ojanen, JHB Phys. Rev. B 93, (2016) F. Arnold, A. G. Grushin, JHB, C. Felser, E. Hassinger, B. Yan Nat. Comm. 7, (2016) T. Meng, A. G. Grushin, K. Shtengel JHB arxiv: (to appear in PRB)