Anomalous Transport: from Anti de Sitter Space

to Weyl semi-metals

Karl LandsteinerInstituto de Física Teórica UAM-CSIC

Strings 2017, Tel Aviv, 28-06-2017

arXiv:1610.04413 [hep-th]

arXiv:1703.10682 [cond-mat.mtrl-sci]

Outline

• Anomalies and Quantum Currents

• WEYL semi-metals

• Anomalous transport in WSMs

• Summary

Anomalies

DµJµa = ✏µ⌫⇢�

✓dabc96⇡2

F bµ⌫F

c⇢� +

ba768⇡2

R↵�µ⌫R

�↵⇢�

◆

• Good old fashioned garden variety triangle anomalies• Anomaly coefficients detect presence of anomalies• In 3+1 no pure gravitational anomaly• consistent Anomalies can always be shifted into currents

dabc =X

r

qraqrbq

rc �

X

l

qlaqlbq

lc

ba =X

r

qra �X

l

qla

CME CVE

Anomalous Transport

~J✏ =

✓dabc6⇡2

µaµbµc +ba6µaT

2

◆~⌦~J✏ =

✓dabc8⇡2

µbµc +ba24

T 2

◆~Ba

~Ja =

✓dabc4⇡2

µbµc +ba12

T 2

◆~⌦~Ja =

dabc4⇡2

µb~Bc

• Dissipationless currents via magnetic field and rotation• Proportional to anomaly coefficients• Anomaly effects at finite density and temperature• Formulas are correct but need proper interpretation • Clearest in Holography!!

CME CVE

Anomalous Transport

~J✏ =

✓dabc6⇡2

µaµbµc +ba6µaT

2

◆~⌦~J✏ =

✓dabc8⇡2

µbµc +ba24

T 2

◆~Ba

~Ja =

✓dabc4⇡2

µbµc +ba12

T 2

◆~⌦~Ja =

dabc4⇡2

µb~Bc

[Vilenkin] ’79, ‘80[Alekseev, Chaianov, Fröhlich] ‘98[Giovannini, Shaposhnikov] ’97[Newman, Son][Kharzeev, Zhitnisky]

[Kharzeev, Fukushima, Warringa][Son, Surowka]

[Neiman, Oz]

[Jensen, Loganayagam, Yarom]

[Banerjee, Bhattacharya, Bhattacharyya, Dutta, Loganayagam, Surowka][Ermenger, Haack, Kaminski, Yarom][Yee] [Rebhan, Schmitt, Stricker]

[Gynther, K.L.,Pena-Benitez,Rebhan][Megias, Melgar, K.L.,Pena-Benitez]

[Megias, K.L.,Pena-Benitez]

[Newman]

[Chowdhury, David]

[Golkar, Son]

[Hou,Liu,Ren]

Anomalous Transport

✓~J✏~J

◆= L ·

~r�1T

�~ET � ~r

� µT

�!

+

✓�B

�B

◆~B

~Js =1

T~J✏ �

µ

T~J + ⌘B ~B

✏+ ~r · ~J✏ = ~E · ~J

⇢+ ~r · ~J = c ~E · ~BConservation laws

Constitutive relations

Entropy currents+ ~rJs � 0

�B = cµ �B =

✓cµ2

2+ cgT

2

◆⌘B =

✓cµ2

2T+ cgT

◆

[Son, Surowka] [Neiman, Oz]Integration constant

Anomalous TransportInterpretation? Why?

Example: proper CME for Dirac fermion

d5V V = dV 5V = dV V 5 = 2

~J =µ5

2⇡2~B

~J5 =µ

2⇡2~B

CME

CSE

Too good to be true

OK, but useful?

AME Edge currents(Fermi arcs)

~J =µ

2⇡2~B5

Theorem due to Bloch ~J = 0 in ground state [N. Yamamoto]

Quantum Currents from 5D

Black Hole

ds2 = r2(�f(r)dt2 + d⌥x2) +dr2

f(r)r2

Strongly coupled QFT = gravity in 5D

UV

IR

Quantum Currents from 5DAnomaly = Chern Simons term

Zd

5xAL ^ (FL ^ FL) +AR ^ (FR ^ FR)

Zd

5xA5 ^ (3F ^ F + F5 ^ F5)

Difference is Bardeen counter termZ

@d

4x cA5 ^A ^ F

Left-right basis:

Axial-Vector basis:

Quantum Currents from 5DHolographic dictionary:

Jµ =p�gFµr +

1

4⇡2✏µ⌫⇢�A5

⌫F⇢�

~J =µ5

2⇡2~B � A5

0

2⇡2~B

[Gynther, K.L.,Pena-Benitez,Rebhan]

Strict thermodynamic equilibrium: µ5 = A50

(Proper) CME vanishes in strict equilibrium for physical current

Covariant anomaly vs. Consistent anomaly[Bardeen, Zumino]

Quantum Currents from 5D• Anomaly = dynamics of Chern-Simons terms

•Grav. Anomaly = dynamics of grav. Chern-Simons terms

•“Extrinsic curvature” K appears• QFT anomaly knows only 4D curvature• K-terms does not contribute at r=∞ (UV)

�

AdSd5xA ⇥ F ⇥ F �

�

�(AdS)d4x�(F ⇥ F )

Z

AdSd

5xA ^R

(5) ^R

(5) !Z

@(AdS)d

4x ✓

⇣R

(4) ^R

(4) +D(K ^DK)⌘.

Quantum Currents from 5D• BUT: at the black hole horizon

⌅K ⇤⌅K ⇥ r4Hf �(rH)2

� ⌥A

�t(⌅� ⌥A)

ds2 = r2f(r)(�dt2 + 2 �Ad�xdt) + r2d�x2 +dr2

r2f(r)

• Suggestive form

⇥K �⇥K = 64�2T 2 ⌥Eg. ⌥Bg

• Following Hawking

•Holography suggests in deep IR ⇥µJµ =T 2

12⌥Eg. ⌥Bg

•A new anomaly ??? ⇤µJµ =e2

4�2�E. �B

•NO! not anomaly but quantum current (Luttinger) T 2

12⌃Eg. ⌃Bg = ⌃� ⌃Jq

•The Chiral Vortical Effect (CVE) !!

Quantum Currents from 5D

⇧Jq =b

12T 2⇧�

�⇥⇥T

T⇥⇥� ⇥v

[Megias, Melgar, K.L.,Pena-Benitez][Azayanagi, Loganayagam, Ng, Rodriguez],[Chapman, Neiman, Oz],[Grozdanov, Poovuttikul]hydro+geometry: [Jensen,Loganayagam,Yarom], global anomaly: [Golkar, Sethi]

•Current Jµ =p�gFµr � �

2⇡G✏µ⌫⇢�K�

⌫D⇢K��

[Copetti, Fernandez-Pendas, K.L.]

Applications: WeylSemiMetal

TaAs[Huang, Xu, Belopolski,Hasan] Nature Comm.

Wikipedia

Hiroyuki Inoue, András Gyenis, Zhijun Wang, Jian Li, Seong Woo Oh, Shan Jiang,Ni Ni, B. Andrei Bernevig,and Ali Yazdani, was published in the March 11, 2016 issue of the journal Science

Figure 3 | Longitudinal magnetotransport – Chiral anomaly-induced negative

magnetoresistance. (a) Energy spectrum of left- and right-handed chirality fermions (red and

blue, respectively) in parallel applied electric and magnetic fields. In the zeroth Landau level,

left-handed particles and right-handed antiparticles have been produced, leading to an

additional topological current. (b) Temperature dependence of the NMR in parallel magnetic

fields. (c) Angle-dependent MR at 300 K. (d) Positive magneto-conductance at 300 K reveals

a parabolic low-field regime that evolves into a linear regime under high magnetic fields.

Anna Corinna Niemann, Johannes Gooth, Shu-Chun Wu, Svenja Bäßler, Philip Sergelius, Ruben Hühne, Bernd Rellinghaus, Chandra Shekhar, Vicky Süß, Marcus Schmidt, Claudia Felser, Binghai Yan, Kornelius Nielsch

Qiang Li (Brookhaven Natl. Lab.), Dmitri E. Kharzeev (Brookhaven Natl. Lab. & SUNY, Stony Brook), Cheng Zhang, Yuan Huang (Brookhaven Natl. Lab.), I. Pletikosic (Brookhaven Natl. Lab. & Princeton U.), A.V. Fedorov (LBNL, ALS), R.D. Zhong, J.A. Schneeloch, G.D. Gu, T. Valla

Zr5Te

NbP

Applications: WSM

�µ�iDµ + �5A

5µ

� = 0

CME in WSM

Conclusions

NMR and NTMR in WSMNMR = Negative Magnetoresistivity

In equilibrium CME vanishes, Induce non-equilibrium steady state

⇥5 =1

2�2⌥E. ⌥B � 1

⇤5⇥5

⇥5 = ⇤5µ5 ⌥J = ⇤ ⌥E +µ5

2⇥2⌥B

�J =�

⇥ +⇤5B2

(2�2)⌅5

⇥�E

NMR and NTMR in WSM

[J. Zaanen, “Electrons go with the flow in exotic materials”, Science Vol. 351, 6277]

In our holographic calculations, we have not included dissipation e↵ects. It would be

very interesting to test the dissipation e↵ects holographically. Recently there has been a lot of

work in including momentum dissipation in holography. These include the lattice construction

which breaks the translational symmetry explicitly (e.g. [46, 47, 48, 49, 50]) and massive

gravity which breaks the di↵eomorphism symmetry in the bulk (e.g. [51, 52, 53]). Besides

momentum dissipations, we also need to include energy and charge dissipations.

In [30], a bulk massive gauge theory was studied in the chiral anomalous fluid (see also

[54]). The massive gauge theory breaks the U(1) gauge symmetry in the bulk and leads

to charge dissipation for the boundary theory. In a follow up paper, we plan to study the

fluid/gravity analysis of this theory (similar to [8, 9]) to get the charge relaxation time from

the hydrodynamic modes [55].

The holographic energy dissipation e↵ects have not been considered so far. As we argued

in the paper, the holographic zero density system is automatically a system with energy not

conserved for the charge carriers. To encode energy dissipations at finite density, we can as

well mimic the way that momentum dissipations are introduced, such as the Q lattice [49]

or massive gravity constructions [51]. It is possible to combine all the momentum, energy

and charge dissipations holographically to test the formula in this work and we would like to

consider this in future work.

Figure 5: Schematic depiction of an inter valley scattering event. Such an event will lead to axial

charge relaxation. But if the two Weyl cones are at di↵erent chemical potentials (as they are in

parallel external electric and magnetic fields) inter valley scattering will also lead to energy relaxation

since �✏ ⇡ µ5�⇢5.

Finally we would like to point out that in the context of Weyl metals inter-valley scattering

does indeed lead to energy relaxation. A schematic picture of an intervalley scattering event

33

If WSM is not strongly coupled, hierarchy of scattering times

⌧inner < ⌧inter < ⌧ee

Kills Kills Is irrelevant ~P ⇢5, ✏5

NTMR via CMECoupled charge and energy transport of chiral currents

~J = GE~E +GT

~rT

GE = ⌧5a2�

det(⌅)

✓@✏

@T� µ

@⇢

@T

◆B2

GT = ⌧52aga�det(⌅)

@⇢

@TB2

Large B (ultraquantum limit):•GE linear in B•GT vanishes

⇢ =|B|4⇡2

µ

[Spivak, Andreev], [Lundgren, Laurell, Fiete] kinetic theory[Lucas, Davison, Sachdev] chiral fluids

arXiv:1703.10682

~J✏ =⇣a�

2µ2 + agT

2⌘~B

~J = a�µ ~B

NMR and NTMR in NbPJohannes Gooth, Anna Corinna Niemann, Tobias Meng, Adolfo G. Grushin, Karl Landsteiner, Bernd Gotsmann, Fabian Menges, Marcus Schmidt, Chandra Shekhar, Vicky Sueß, Ruben Huehne, Bernd Rellinghaus, Claudia Felser, Binghai Yan, Kornelius Nielsch

Experimental signatures of the mixed axial-gravitational anomaly in the Weyl semimetal NbP

arXiv:1703.10682 [cond-mat.mtrl-sci]

Angle dependence NMR and NTMR show B2 at small B NMR ~ linear for large B field NTMR vanishes for large B field

Summary• Anomalies have moved from Hep-Th to Cond-Mat

• Rich anomaly induced transport phenomenology (CME, CVE, CSE, AME, AHE,NMR, and now NTMR)

• WSMs allow experimental observation of effect of gravitational anomaly

• Holography is ideal tool to investigate anomalous transport

• Holographic model of WSM (Poster of J. Fernandez-Pendas) with Band-bending predicts Hall viscosity

Thank You!

[Y. Liu, K.L.], [Y. Liu, K.L., Y.W. Sun]^2, [Copetti,Fernandez-Pendas,K.L.] [Grignani, Marini, Pena-Benitez, Speziali],[Ammon, Heidrich, Jimenez-Alba, Moeckel]