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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]
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