1. Transport signatures of 3D topological matter:
SPICE workshop, Mainz June 2017
Transport signatures of 3D topological matter:
Glide Hall in nonsymmorphic metal and chiral anomaly
Z.J. Wang
S. Kushwaha
M. Hirschberger
Bob Cava
NP Ong
S.H. Liang
Bernevig
T. Gao
S.H. Liang, S. Kushwaha, T. Gao, M. Hirschberger, J. Li, Z.J. Wang, K. Stolze, B.A. Bernevig, R.J. Cava, N. P. Ong
Departments of Physics and Chemistry, Princeton Univ.
Glide symmetry protection
The nonsymmorphic metal KHgSb
A novel zero Hall state at lowest Landau level
The chiral anomaly in Na3Bi and GdPtBi
Tests for chiral anomaly
Support from
Moore Foundation, ARO, NSF

2. 2D Dirac node protected by time-reversal symmetry (TRS)
Kramer’s theorem k
When Hamiltonian is time reversal symmetric (TRS)
𝐻𝜑=𝐸𝜑, 𝐻𝜓=𝐸𝜓
Θ𝜑= 𝜓, Θ𝜓= −φ
Θ𝜑, Θ𝜓 = 𝜑,𝜓 ∗ =(𝜓, 𝜑
− 𝜓,𝜑 = 𝜓, 𝜑 =0
protected
antiunitarity
,  are orthogonal (hence 2-fold degenerate)
To apply to k space, need the little group.

3. Little group 𝒢 𝐤 and symmetry protection
For space group 𝒢, the subset of operations g that leave k invariant define the little group 𝒢 𝐤 . We have the equation
𝑔 𝐤≐ m k mod G ( 𝑔 ∈ 𝒢 𝐤 )
To apply Kramer’s theorem at k, we need its little group 𝒢 𝐤 (to return to k)
Example: In 2D topological insulators, the TR operator Θ returns k  -k = k+G ≐ k
Θ 𝐤,𝛼 = ±𝑖 |−𝐤, 𝛼 )
provided k is a time-reversal invariant momentum TRIM (Γ and corners of BZ). The surface Dirac cone is protected only if pinned at a TRIM. π -π

4. Nonsymmorphic space groups 𝓖 have glide and screw operations
translate
glide plane
c/2
Glide operation in space group 𝒢
Liu, Zhang, Vanleuven, PRB (2014)
Shiozaki, Sato, PRB (2014)
Young, Kane, PRL (2015)
Fang, Fu, PRB (2015)
mirror plane z z
Mirror
reflection
in xz plane
c/2 O x
Glide combines mirror reflection and translation by half a lattice parameter
𝛕= 𝑐 2 𝐳
𝑔 𝑥 = 𝑀 𝑥 |𝝉
Mirror reflection = Rotation by π and Inversion P
𝑀 𝑥 =−𝑃𝑖 𝜎 𝑥

5. 1. “Sticking” of bands on BZ surface when Θ 𝑔 𝑥 is in little group
The power of glide 1
1. “Sticking” of bands on BZ surface when Θ 𝑔 𝑥 is in little group
C. Herring (1937)
Glide protection of degenerate bands all along UZU
mirror plane
Under TR operation Θ, k  -k
2. Under glide operation gx, -k  k’ ≐ k
𝑘 𝑥 Γ 𝑍 𝑈 𝑋
𝑘 𝑧 𝐤 −𝐤 𝐤′
Θ 𝑔 𝑥 | 𝐤 =𝑝 | 𝐤
Θ 𝑔 𝑥 returns k to itself, but what is the “eigenvalue” p?
𝑔 𝑥 2 = 𝑀 𝑥 2 e 𝑖 𝑘 𝑧 𝑐 = +1
Mirror reflection
𝑀 𝑥 =𝑃𝑖 𝜎 𝑥
Θ 𝑔 𝑥 2 = −1 𝑔 𝑥 2 =−1
Eigenvalue 𝑝 2 is -1 for all k on UZU --- Kramer’s doublet at each k!
Θ 𝑔 𝑥 in little group of k on UZU states k are doubly degenerate (“stick” together).

6. The power of glide 2
2. Möbius band topology (in gx is part of little group)
All states k on UXU are eigenstates of 𝑔 𝑥 with eigenvalues 𝑚 ± ( 𝑘 𝑧 )
𝑘 𝑥 Γ 𝑍 𝑈 𝑋
𝑘 𝑧
𝑔 𝑥 | 𝐤 = 𝑚 ± | 𝐤
𝑚 ± = ±𝑖 e 𝑖 𝑘 𝑧 𝑐 2
𝑚 ± ( 𝑘 𝑧 ) has 2 distinct branches both with period 4π.
After translation of 2π, 𝑚 + 𝑘 𝑧 becomes 𝑚 − 𝑘 𝑧
(topology of edges of a Möbius band) 𝑖 −𝑖 1 −1
Leads to exchange of states and hourglass fermions
(edges exchange in Möbius band after 2π)
E(k) 𝑋 𝑈 𝑍

7. Nature (2016)
KHgSb, KHgAs
Nonsymmorphic
Two pairs of helical states wrap around sides
Hourglass fermions exist on Z-G face

8. Resistivity and Shubnikov de Haas oscillations
S. Kushwaha
S.H. Liang
Special handling needed:
Hg exudes to surface, extreme air sensitivity
Residual density of n-type carriers (loss of Hg)
occupy a ellips. FS at Γ
Large SdH oscillations. Enters lowest Landau level at ~10 Tesla

9. First hint: A sharp anomaly in Hall angleHp
Hp is close to where EF enters lowest Landau level (LLL).
In conventional metal, tan H is generally linear in B even in quantum limit.

10. Clearer picture emerges in pulsed fields
Above Hp, tan 𝜃 𝐻 is strongly T dependent Hp
Further unusual features:
Hall response reaches zero in large B and is locked to zero.
Exponential sensitivity of Hall response to temp.

11. Anomalous Hall conductivity in pulsed field to 63 T
At lowest T (1.53 K), resistivity 𝜚 𝑥𝑥 tends towards saturation.
Hall resistivity 𝜚 𝑦𝑥 approaches zero exponentially.

12. tan 𝜃 𝐻 and 𝜎 𝑥𝑦 decrease exponentially to zero as B incr.
Both quantities “stick” to zero over a large field interval.

13. Dope with bismuth to lower Hp= to 3 -- 4 Tesla
The zero-Hall state is intrinsic to lowest Landau level (LLL).
A large gap  separates LLL from excited state.
 is roughly linear in B

14. Fit 𝜎 𝑥𝑦 vs. B and T to obtain gap  vs B
Fits are solid curves;
Data are open circles
𝜎 𝑥𝑦 = 𝜎 𝑥𝑦 (1) exp − Δ 𝑘𝑇 + 𝜎 𝑥𝑦 (0)
Gives good fits over broad range of T and B. Yields  at each B.
Existence of a gap is incompatible with gapless spectrum in 3D Landau states.

15. Longitudinal conductivity 𝜎 𝑥𝑥 remains finite
In high-field limit, 𝜎 𝑥𝑥 at 1.5 K is finite but small (~ 0.1 e2/h per QSH mode)
Gapped behavior is restricted to Hall response.

16. Ab initio calculation of Landau level spectrum in KHgSb

17. The chiral anomaly in Dirac and Weyl Semimetals
SPICE workshop, Mainz Jun 2017
The chiral anomaly in Dirac and Weyl Semimetals
Jun Xiong
Kushwaha
Tian Liang
Jason Krizan
Hirschberger
Bob Cava
NP Ong
S.H. Liang
Bernevig
Jun Xiong, Tian Liang, Max Hirschberger, Sihang Liang, N. P. Ong
Department of Physics, Princeton Univ.
Satya Kushwaha, Jason Krizan, R. J. Cava
Department of Chemistry, Princeton Univ.
Z.J. Wang
1. Review of the chiral anomaly in Na3Bi and in GdPtBi
2. Current jetting? The squeeze test
3. Giant planar Hall effect and angular MR
Support from
Moore Foundation, ARO, NSF

18. Creation of Weyl states in applied magnetic field
Berry curv 
Weyl FS
Dirac
B field
E(k) E B,
Landau levels kz
χ=1
χ=-1
Lowest Landau level
Schwinger model
Detect axial
current
𝐴=− 𝐿 2 2𝜋 𝓁 𝐵 𝐿𝑒 𝑘𝑧 2𝜋 =−𝑉 𝑒 3 4 𝜋 2 ℏ 2 𝐄.𝐁

19. Chiral anomaly
E(k) E B,
Landau levels kz
χ=1
χ=-1
In strong B, the chiral lowest Landau levels (LLL) in a Weyl metal
realizes the Schwinger model (sea of right-moving and left-moving massless fermions).
With E || B, we produce a large axial current, observed as negative LMR.
Similar to the Adler-Bell-Jackiw (triangle) anomaly in decay of neutral pions to photons 𝜋 0 ⟶2𝛾
Presence of axial current alters
Longitudinal MR
Thermoelectric response
Hall effect
Observed in
Na3Bi, Xiong et al., Science (2015)
GdPtBi, Hirschberger et al., Nat. Mat. (2016)
ZrTe5, T. Valla et al. (BNL), Nat. Phys. (2016)

20. Resemblance between long. MR in Na3Bi and GdPtBi

21. Check for uniformity of current density
Repeat measmt. on Sample G with 10 voltage contacts
Longitud. MR profiles plotted as relative change are closely similar across
all 8 nearest neighbor pairs of contacts
Conclusion: Negative longitude. MR is an intrinsic electronic effect, not a spurious
result of inhomogeneity.

22. The “squeeze” test for current jetting
Chiral anomaly effect: Voltage decreases with B both along spine and side
Pure current jetting: Voltage decreases along side, but increases along spine
Extreme current jetting
Voltage contacts
“spine” y x y
J(y)
mB>>1
mB<<1
cut

23. Apply squeeze test to high-purity elemental bismuth
Pure bismuth
Apply squeeze test to high-purity elemental bismuth
Voltage contacts along the spine
Voltage contacts along the edge
Negative LMR caused by extreme current jetting

24. The “squeeze” test applied to GdPtBi
Compare MR measured with spine and edge contacts in GdPtBi
Voltage contacts along the edge
Voltage contacts along the spine
In GdPtBi, negative LMR is intrinsic and uniform
However, tests on TaAs and NbP fail (to date)

25. A new signature of chiral anomaly (A. Burkov)
A. Burkov, arXiv:
In presence of chiral anomaly, the chemical potential µ for electrons and the “chiral” chemical potential 𝜇 𝑐 obey coupled diffusion equations
(the chiral charge nc decays with lifetime 𝜏). 𝜃 B x y
CME current I
𝐷 𝛻 2 𝜇+ Γ𝐁∙𝛻 𝜇 𝑐 = 0
𝐷 𝛻 2 𝜇 𝑐 + Γ𝐁∙𝛻𝜇= 𝜇 𝑐 𝜏
In a rectangular sample, this leads to an angular magnetoresistance and a giant planar Hall effect.
E(k) E B,
Landau levels kz
χ=1
χ=-1
𝜌 𝑥𝑥 = 𝜌 ⊥ − Δ𝜌 cos 2 𝜃
𝜌 𝑦𝑥 =−Δ𝜌 sin𝜃.cos𝜃,
Δ𝜌= 𝜌 ⊥ − 𝜌 ∥

26. Angular Magnetoresistance in Na3Bi
Sihang Liang, unpubl.
Longitudinal voltage has a cos 2 𝜃 dependence
𝜌 𝑥𝑥 = 𝜌 ⊥ − Δ𝜌 cos 2 𝜃
Also seen in GdPtBi

27. Giant planar Hall effect in Na3Bi
Transverse voltage displays a component that is sin 2𝜃 and symmetric in B
𝜌 𝑦𝑥 =−Δ𝜌 sin𝜃.cos𝜃,

28. Summary Zero-Hall state in system with glide protection
In lowest Landau level (LLL), Hall response approaches zero exponentially in B and T
Fits yield a B-dependent gap  protecting zero-Hall ground state
Zero-Hall state is intrinsic to LLL (from doping study)
Longitudinal conductance remains finite (~ 0.1 of surface mode conductance)
Chiral anomaly
Observed in Na3Bi and GdPtBi
Squeeze test rules out current jetting
Giant planar Hall effect -- a new signature of the anomaly

29. Thank you S.H. Liang T. Gao Bernevig S. Kushwaha Bob Cava NP Ong
M. Hirschberger
Bob Cava
NP Ong
S.H. Liang
T. Gao
Z.J. Wang
Bernevig
Jun Xiong
Thank you