Talk 1: Princeton Summer School 2015 Evidence for the chiral anomaly in a Dirac Semi-metal

Jun Xiong, Tian Liang, Max Hirschberger, Wudi Wang, N. P. Ong Department of Physics, Princeton Univ.

Satya Kushwaha, Jason Krizan, R. J. Cava Department of Chemistry, Princeton Univ.

Supported by MURI, ARO, NSF, Moore Foundation

1. Gap-closing route to 3D topological Dirac semimetal 2. Band structure of Na3Bi 3. The Chiral anomaly 4. Charge pumping in Na3Bi 5. An axial current plume

Jun Xiong Kushwaha Tian Liang Jason Krizan Hirschberger Bob Cava NPO

Quantum Spin Hall sys.

Topological Matter Cornucopia

Z2 TRI systems Bi2Se3

Bi2Te2Se

HgTe-CdTe

Topological Dirac semimetals Kagome

Pyrochlore

Strong correlation solids

2D layered systems

Berry phase, Chern

flux

Cd3As2 Na3Bi Eu2Ir2O7

HerbertSmithite

Cu(1,3 carboxylate)

Bi2Te3

Topological Crystalline Insulator

PbSnSe, PbSnTe

MoS2, WTe2

Iridates, SmB6

Weyl metals TaAs, NbAs, NbP

InAs-GaSb

Closing the gap in a 3D semi-conductor Murakami, New J. Phys. (‘07). Murakami. Kuga, PRB (’08) Murakami, Physica E(‘11) Okugawa, Murakami, PRB(’14)

Closing the gap in a semiconductor by pressure, doping etc

Anticrossing?

Discrete point nodes

Line node? Weyl nodes

Courtesy: S. Murakami

m

B.J. Yang and N. Nagaosa, Nat Comm.2014

Dirac node protection by Point-Group Symmetry

Wan, Turner, Vishwanath, Savrasov, PRB 2011 Young, Kane, Mele et al. PRL 2012 Wang, Dai et al., PRB 2012 Fang, Gilbert, Dai, Bernevig, PRL 2012

Nodes stable C 3

C 4

Two Cases 1. Dirac node is protected by TRS and IS if pinned to zone corners (non-symorphic case; has glide planes or screw axes). 2. Inclusion of point-group symmetry Cn extends protection to anywhere on symmetry axis (topological Dirac semimetal).

unstable

Na3Bi Cd3As2

Only 2 bands, derived from Na-3s and Bi-6p, lie near Fermi energy. Large spin-orbit coupling (SOC) leads to band crossing at K± = (0, 0, ±kD)

Crossings protected against gap formation --- |S) and |P) states belong to different irreducible representations of C3.

We end up with 2 Dirac nodes centered at K±

The band structure of Na3Bi (Wang, Dai, Fang et al. PRB 2012)

kz

|S, ±½ )

|P, ±3/2)

60 meV kD

E

Na-3s

Bi-6p

Na3Bi

Protected Dirac nodes in semimetal Na3Bi Identified by Wang, Dai, Fang et al. PRB 2012 Wang, Dai, Fang et al. PRB 2013

C 3

C 4 Cd3AS2

Inst. of Physics

Photoemission on Na3Bi and Cd3As2

Liu, Chen et al, Science 2014 Neupane, Hasan et al., Nat Comm. 2014 Borisenko, Cava et al., PRL 2014

Point group C3

kz

|S, ±½ )

|P, ±3/2)

kD

Dirac cone resolves into two Weyl nodes with opposite chiralities χ = ±1

The low-E Hamiltonian, close to node K+, reduces to

H resolves into two 2x2 Weyl Hamiltonians H1, H2

The chirality is

𝐻𝐻1 = 𝐤𝐤 ∙ 𝐯𝐯1 ∙ 𝛕𝛕 = v(𝑘𝑘𝑥𝑥𝜏𝜏1 − 𝑘𝑘𝑦𝑦𝜏𝜏2 + 𝑘𝑘𝑧𝑧𝜏𝜏3)

𝜒𝜒1 = −1, 𝜒𝜒2 = +1

𝜒𝜒 =det 𝐯𝐯

v

We have a superposition of two Weyl nodes at B = 0

E 𝑆𝑆, ½𝑃𝑃, 3/2𝑆𝑆,−½𝑃𝑃,−3/2

or aij

Initial results on Na3Bi

Deep purple crystals Rapidly oxidizes in ambient air (30 s)

Large linear MR similar to Set B Cd3As2 samples EF 400 mV above node

Jun Xiong et al., submitd

S. Kushwaha et al., APL 2015

Jun Xiong Kushwaha Krizan

H-linear MR and step-profile of Hall angle tan θH

Conventional ρxx(B) ~ [1+ (µB)2] tan θH ~ µB

In Na3Bi ρxx(B) ~ B, linear MR tan θH has a step-function profile

Jun Xiong et al., PRL submitd

Prominent de Haas van Alven oscillations

From dHvA oscillations, we infer kF, m* and Tdingle EF (400 mV) lies above the Lifshitz energy EL (60 mV) Non-trivial inverted band regime

Jun Xiong et al., PRL submitd

Dirac node, Weyl nodes and Chiral Anomaly

Breaking Time Reversal symmetry in magnetic field

Search for the Chiral anomaly and axial current

Chiral anomaly in Quantum Field Theory

𝛻𝛻. 𝐉𝐉𝟓𝟓 + 𝜕𝜕𝜌𝜌5𝜕𝜕𝜕𝜕 = 𝑊𝑊

Anomaly-free condition In a chiral theory, all the anomalies must cancel for it to be renormalizable. This imposes stringent constraints, e.g. number of generations. In Glashow-Weinberg-Salaam theory, exact cancellation involving 10+ diagrams has been called “magical”.

1969 Charged pions decay leptonically, 𝜋𝜋+ → 𝜇𝜇+ + ν But, neutral pion decays electromagnetically

𝜋𝜋0 → 𝛾𝛾 + 𝛾𝛾 Chiral anomaly accelerates decay by a factor 300 million! Missing factor 9x hint of 3 quark colors

u u u

Spoiler role in chiral gauge theories Ruins chiral charge conservation and renormal. Topological in origin

Intro to QFT, Peskin Schroeder QFT in a Nutshell, Tony Zee

Adler-Bell-Jackiw anomaly

axial current

Nielsen and Ninomiya (Phys. Lett. 1983) proposed chiral anomaly may be observable in (1+1)d or (3+1)d crystals ---- spacetime dim. must be even

Nielsen and Ninomiya (Phys. Lett. 1983) Obersvation of anomaly in a crystal?

E

k

v 𝑖𝑖𝛾𝛾𝜇𝜇𝜕𝜕𝜇𝜇Ψ = 0

For massless Dirac fields, the natural repres. is chiral (Weyl) representation. However, coupling to EM fields breaks chiral symmetry.

In cond-mat, left and right movers are linked by states deep in Fermi sea However, 1) chirality in two branches is significant 2) “Steerability” by B field 3) What is the relaxation time?

Berry curvature of Weyl nodes

𝐀𝐀 = −𝑖𝑖 𝑢𝑢𝐤𝐤|𝛻𝛻𝐤𝐤|𝑢𝑢𝐤𝐤

F

Wan, Turner, Vishwanath, PRB 2011 Burkov, Hook, Balents, PRB 2011 Son, Spivak, PRB 2013

qy

qx

E Chern flux F

Dirac TRS

Weyl nodes

Berry curvature

… and 80+ theory uploads on arXiv

𝐅𝐅(𝐤𝐤) = 𝛻𝛻 × 𝐀𝐀(𝐤𝐤)

Chirality

Proposed (2011) existence of Weyl nodes in iridates (Vishwanath et al.) reawakened strong interest in the Nielsen Ninomiya prediction.

Charge pumping and the chiral anomaly

W = − 𝐿𝐿2

2𝜋𝜋ℓ𝐵𝐵2𝐿𝐿𝐿𝐿𝑘2𝜋𝜋

= −𝑉𝑉 𝐿𝐿3

4𝜋𝜋2ℏ2𝐄𝐄.𝐁𝐁

Nielsen, Ninomiya, Phys. Lett. 1983 Wan, Turner, Vishwanath, PRB 2011 Burkov, Hook Balents, PRB 2011 Son, Spivak, PRB 2013 Parameswaran et al. PRX 2014

In large-B regime, with E||B, charge is pumped between Weyl nodes at the rate

Chiral anomaly engenders large, negative longitudinal MR Locked to B field

In weak B, charge pumping gives (Son and Spivak, PRB 2013)

τa is relaxation time for pumped current

E(k)

E B

Landau levels

kz 0 χ=1 χ=-1

τa

Non-metallic resistivity and negative longitudinal MR in Na3Bi

Long-term annealed crystals with EF much closer to node Jun Xiong, S. Kushwaha, Krizan,

Fermi energy lies 30 meV above node Striking negative longitud. MR (LMR)

New set of samples

Cava

A conventional 1-band model does not lead to MR, regardless of band anisotropy.

𝜌𝜌𝑖𝑖 = 𝑚𝑚𝑖𝑖

𝑛𝑛𝑒𝑒2𝜏𝜏0

Standard Boltzmann equation with anisotropic mass tensor (elliptical FS) in arbitrarily large, tilted, field B = (B1, B2, B3)

Longitudinal MR is rare in conventional metals

Diagonal elements are independent of B absence of MR

B 3

2

1

Solution for resistivity tensor

Measure FS caliper by Shubnikov de Haas (SdH) oscillations

FS volume determined by SF in good agreement with desity n inferred from RH

Oscillation amplitude affected by background subtraction, But extrema positions in field only weakly affected.

A test for the chiral anomaly -- B is locked to E Jun Xiong, S. Kushwaha et al., submitted

Negative MR appears only when B is locked to E. Test: if E is rotated by 90o (right panel), neg. MR shifts to new direction of E. For weak B, this locking is novel and unexpected in semiclasscl transport

Some dirty laundry: Raw curves (unsymmetrized)

Evidence for some contamination from large Hall signal, but doesn’t affect intrinsic MR pattern

A narrow plume of chiral current, B in-plane Jun Xiong, S. Kushwaha et al., submitted

Enhanced cond. in a narrowly collimated beam for B in the x-y (horizontal) plane

Width of chiral conductivity “plume”, B normal to plane

Enhanced cond. in a narrowly collimated beam for B rotated in the x-z (vertical) plane

Relaxation time τa of the pumped axial current

Determine intervalley scat. lifetime using Son-Spivak expression

In semiclassical regime, Weyl node conductivity grows as B2 (Son, Spivak, PRB ‘13)

From fit, we find 𝜏𝜏a = 40-80 𝜏𝜏0

τa

cond

ucta

nce

The axial current relaxation lifetime τa

Fit to Son-Spivak expression (τa / τ0 ) = 40-60, where τ0 is Drude lifetime Why is axial current relaxation strongly suppressed? 1) Charged impurities are well screened (Thomas-Fermi factor ) for large valley scattering (∆k >> kF ). 2) Violation of chiral symmetry is weak at low B, so chiral coupling to impurities is weak.

An interesting question: Does large ratio (τa / τ0 ) reflect near-conservation of chiral charge?

Zeeman energy and separation of Weyl nodes with B

At 6 Tesla, what is node displacement ∆kN?

S,-½ S,½ P,-3/2 P, 3/2 E |S, ± ½⟩ |P,±3/2 ⟩

∆kz

Separation may be small if δg < 40

𝛿𝛿𝛿𝛿 = (𝛿𝛿𝑆𝑆−𝛿𝛿𝑃𝑃)/2 Δ𝑘𝑘𝑁𝑁 = 𝛿𝛿𝛿𝛿𝜇𝜇𝐵𝐵𝐵𝐵/ℏ𝑣𝑣

Zeeman term

B

𝜕𝜕𝑛𝑛𝑣𝑣𝜕𝜕𝜕𝜕 = 𝐷𝐷

𝜕𝜕2𝑛𝑛𝑣𝑣𝜕𝜕𝑧𝑧2 + Γ

𝜕𝜕𝑛𝑛𝑎𝑎𝜕𝜕𝑧𝑧

𝜕𝜕𝑛𝑛𝑎𝑎𝜕𝜕𝜕𝜕 = 𝐷𝐷

𝜕𝜕2𝑛𝑛𝑎𝑎𝜕𝜕𝑧𝑧2 −

𝑛𝑛𝑎𝑎𝜏𝜏𝑎𝑎

+ Γ 𝜕𝜕𝑛𝑛𝑣𝑣𝜕𝜕𝑧𝑧

𝜏𝜏0𝜏𝜏𝑎𝑎

= 𝐸𝐸𝐹𝐹2

20𝑊𝑊2 ≪ 1

Burkov arXiv: 1505.01849v2

Near conservation of chiral charge leads to slow diffusion modes. Predicts a long axial current relaxation time, especially for undisplaced Weyl nodes (Dirac case)

The axial relaxation time τa is currently poorly understood, but can now be measured in LMR experiments. Is matrix element for impurity scattering (+|Vimp|-) sensitive to chirality?

Calculation of lifetime τa

Signature of chiral anomaly: B locks direction of axial current

In conventional transport, we cannot “rotate” FS parameters, e.g. scattering rate anisotropy, by rotating direction of a weak B.

Locking of observed axial to B (even in weak B) seems to be a signature characteristic of the chiral anomaly. Axial plume direction fixed by B (and E)

Observation of negative, longitudinal MR is necessary but insufficient.

Axial plume

Summary

Transport experiments on Dirac semimetals In Cd3As2, we observe very high mobility (9 million cm2/Vs) in zero B. Appears to be protected by a zero-B mechanism. Giant MR observed when protection is lifted. In Na3Bi, in samples with EF ~ 30 mV, we see evidence for the chiral anomaly. Signature: The enhanced-conductivity “plume” is locked to the direction of B (and E). YES Estimated inter-valley lifetime is 40-60 x longer than Drude value. Why so large? A surprise: Width of plume is much narrower than anticipated by theory.

Recent reports of Negative Longit MR in semimetals

1. Neg. longit. MR in Bi1-xSbx, Li et al., PRL 2013 --- Accidl band x’ing 2. Neg. longit MR in Cd3As2, Tian Liang et al., Nat Mat. 2015 --- Dirac SM 3. Neg. longit. MR in ZrTe5, Brookhaven Nat. Lab. arXiv --- QSHE? 4. Neg. longit. MR in PdCoO2, Balicas, Maeno, Hussey et al. arXiv --- ?? 5. Neg. longit. MR in TaAs, NbAs, S. Jia and Hasan, arXiv -- Weyl metals Other systems showing large, negative MR 6. PbSnTe, Tian Liang --- Topol xtalline insulator 7. GdPtBi Hirschbirger --- Dirac metal with strong exchange

Thank you

End

Jun Xiong Kushwaha Tian Liang Jason Krizan Hirschberger Bob Cava NPO

Determining the mobility in Cd3As2

Hall conductivity curves collapse to universal curve when plotted in dimensionless variables. Bmax equals 1/µ

Protection of nodes against gap formation

2. Retain the two orbitals |S, 1/2) and |P, 3/2) near node. With spin, we have a 4x4 Hamiltonian

Entries fixed by TRS and P (inversion symm.) At crossing, bands do not mix because they belong to different representations of C3 rotation group

Dai et al., PRB 85, 195320 (2012) Bernevig et al., PRL (2012)

3. Point group symmetry (PGS) e.g. C3, dictates that off-diagonal terms B(k) ~ kzk+

2

Near node, H decomposes to two diagonal 2x2 blocks (Weyl fermions)

1. Gap inversion pulls (Na 3s) S band 0.3 eV below HH (Bi 6p) P band Na 3s

Bi 6p (HH)

If TRI is restored Weyl points of opposite chirality mutually annihilate Left with a gapped spectrum

Point group symmetry prevents annihilation -- we have a 4-comp Dirac metal

β-crystobalite BiO2 A3Bi (A = Na, K, Rb) Cd3As2

Dirac semimetal

E(k)

Gapped spectrum

Young, Kane et al., PRL (2012) Wang, Dai et al., PRB (2012) Bernevig et al., (2013)

kx

kz

kz

Each node is a 4-comp massless Dirac point Has 4-fold degeneracy (lifted by breaking of TRI or Inv.) 3D Analog of graphene

Dirac

0

Giant MR in Cd3As2

Ultrahigh mobility results from unknown protection mechanism against 2kF scattering B field removes protection, giving giant MR

Transport quantities in 7 samples of Cd3As2

Borisenko, Cava, PRL 2013

vF = (6 – 10) x 105 cm/s SdH oscillations provide more accurate meas.

16 x 105 cm/s (Hasan)

Conductivity tensor is anomalous

Conventional σxx(B) = neµ/[1+(µB)2] ~ 1/B2

σxy(B) = neµ2B/[1+ (µB)2] ~ 1/B Differ by one power of B

In Na3Bi Both σxx and σxy ~ 1/B

Dual Dirac nodes in Cd3As2 Wang, Dai, Fang et al. PRB 2012 Wang, Dai, Fang et al. PRB 2013

C 4

Change in Resistivity and Conductivity Tensors with mobility in Cd3As2

Increasing mobility

Hall

cond

uctiv

ity σ

xy

ρ xx a

nd ρ

yx

Field scale changed by ~80

Tian Liang, Quinn Gibson et al. Nat. Mater. 2015

B (T)

B (T)

B (T)

B (T)

Direct measurement of mobility

Peaks in σxy move to lower B as mobility µ increases. µ correlates with residual conductivity over 3 decades

mean mobility

Mobility || axis 1

Tian Liang, Quinn Gibson et al. Nat. Mater. 2015

Weyl Semimetal

• Superficially, 3D analog of graphene, but w important differences:

• No mass term Mσz possible (compare Boron Nitride), so always massless

• Only way to remove nodes is by pairwise annihilation

• Requires broken TRI or broken Inversion • Must come in pairs with χ = ±1

• Regard as monopole source or sink of Berry curvature (Chern) flux

2-spinor 3D space

Chirality χ

F

Berry curvature

Wan, Turner, Vishwanath, PRB 2011

qy

qx

E

Chern flux

Na3Bi

Wang, Dai et al., PRB 85, 195320 (2012) Bernevig et al., PRL (2012)

The Dirac semimetal Na3Bi

Protection of nodes against gap formation

2. Retain the two orbitals |S, 1/2) and |P, 3/2) near node. With spin, we have a 4x4 Hamiltonian

Entries fixed by TRS and P (inversion symm.) At crossing, bands do not mix because they belong to different representations of C3 rotation group

Dai et al., PRB 85, 195320 (2012) Bernevig et al., PRL (2012)

3. Point group symmetry (PGS) e.g. C3, dictates that off-diagonal terms B(k) ~ kzk+

2

Near node, H decomposes to two diagonal 2x2 blocks (Weyl fermions)

1. Gap inversion pulls (Na 3s) S band 0.3 eV below HH (Bi 6p) P band Na 3s

Bi 6p (HH)

IOP, Beijing

|S,½⟩, |P,3/2 ⟩, |S,-½ ⟩, |P,-3/2 ⟩

4x4 basis set

0 kx

kz

4-comp Dirac

Point Group C4

H-linear MR in Na3Bi

Deep purple crystals Rapidly oxidized in ambient air (30 s) Contacts for resistance expts. attached inside argon-flow glove box

Large linear MR similar to Set B Cd3As2 samples

Jun Xiong et al., PRL submitd

Determine the mobility from peaks in σxy

Anomalies: H-linear MR and step-profile of Hall angle tan θH

Conventional ρxx(B) ~ [1+ (µB)2] tan θH ~ µB

In Na3Bi ρxx(B) ~ B, linear MR tan θH has a step-function profile

Jun Xiong et al., PRL submitd

Conductivity tensor is anomalous

Conventional σxx(B) = neµ/[1+(µB)2] ~ 1/B2

σxy(B) = neµ2B/[1+ (µB)2] ~ 1/B Differ by one power of B

In Na3Bi Both σxx and σxy ~ 1/B

Charge pumping and the chiral anomaly

Charge is pumped from left to right-handed Weyl node at the rate

E(k) E B

Landau levels

kz 0

Q

χ=1 χ=-1

Review: Hosur and Qi, arXiv:1309

Chiral (Adler-Bell-Jackiv) anomaly

Decay of neutral pi-meson π0 2 ϒ, non-conservation of axial current Why are all neutrinos left-handed? Goldstone mode of quark condensate Topological origin

W = − 𝐿𝐿2

2𝜋𝜋ℓ𝐵𝐵2𝐿𝐿𝐿𝐿𝑘2𝜋𝜋

= −𝑉𝑉 𝐿𝐿3

4𝜋𝜋2ℏ2𝐄𝐄.𝐁𝐁

Cd3As2 Resistivity vs. Temperature --- RRR shows huge variation

Set A (crystals) show 1000 fold difference in RRR. Some crystals show very low residual resistivity 20 nΩcm Correlated with large resistivity anisotropy (30) Set B samples have higher resistivity

Tian Liang, Quinn Gibson, et al. Nat. Mater. 2015

Shubnikov de Haas oscillations (Set B) Tian Liang, Quinn Gibson et al. Nat. Mater. 2015

Chiral anomaly? Negative, longitudinal MR detected at θ = 0, but swamped by positive MR term (EF is much too high).

IOP, Beijing

|S,½⟩, |P,3/2 ⟩, |S,-½ ⟩, |P,-3/2 ⟩

4x4 basis set

0 kx

kz

4-comp Dirac

Point Group C4

3/2 -3/2

1/2 -1/2

∆kz

Linear MR from Zeeman splitting of FS? Dai et al, PRL (2012)c |S,½⟩, |P,3/2 ⟩, |S,-½ ⟩, |P,-3/2 ⟩ 4x4 basis

B field also breaks TRI Weyl nodes move apart, Lifshitz transition

-3/2

3/2 1/2

-1/2

∆kz

EF

3/2

-3/2

1/2

-1/2

E

Tests need higher B and higher mobility