Evidence for the Chiral Anomaly in a Dirac Semi-Metal

BigIdeas Conference, LaJolla, Dec 2015 Evidence for the chiral anomaly in a Dirac Semi-metal

Jun Xiong Kushwaha Tian Liang Jason Krizan Hirschberger Zhijun Wang Tong Gao Wudi Wang Minhao Liu Carina Belvin

Bob Cava Bernevig NPO

1. Intro to Dirac/Weyl semimetal 2. Intro to the chiral anomaly 3. The chiral anomaly in Na3Bi 4. Future Directions in Topological Matter

Supported by MURI, ARO, Moore Foundation, NSF and National High Magnetic Field Lab. 3D Dirac semimetal

Topological Insulator 2D Spin-locked states on surface Bulk insulating

Bi2Se3, Bi2Te 3, Bi2Te 2Se …

kz Topological Dirac semimetal Bulk Dirac conducting states 3D protected nodes on symmetry axis Spin textured

Na3Bi, Cd3As2

Ω Weyl nodes In applied B, each Dirac node B splits into 2 Weyl modes, monopole source and sink of Berry curvature Ω Dirac, Weyl and Chiral Symmetry

Dirac equation (i + ) = 0 Ψ is a 4-spinor 𝜇𝜇 mass Herman Weyl 𝛾𝛾 𝜕𝜕𝜇𝜇 𝑚𝑚 Ψ

(Weyl, 1940s) If m = 0, the Dirac Hamiltonian describes two massless populations.

0 = 0 𝐻𝐻+ 𝐻𝐻 − i) Each has permanent handedness𝐻𝐻 (chirality = ±1) ii) Opposites do not mix (chiral symmetry holds) From Topological Insulators to Dirac/Weyl semimetals

2011 Proposed existence of Weyl nodes in iridate pyrochlores (Vishwanath et al.) 2012 3D Dirac nodes are protected by TRS and IS only if pinned to Γ or BZ corners (Kane, Mele) 2012 Point-group symmetry renders protection anywhere on symmetry axis (Bernevig, Xi Dai, Zhong Fang, …) Na Bi and Cd As (Wang . ) Zhijun Wang 3 3 2 𝑒𝑒𝑒𝑒 𝑎𝑎𝑎𝑎 Na3Bi

C 3 Zhijun Wang, Xi Dai et al, PRB 2012

Wan, Turner, Vishwanath, PRB 2011 Burkov, Hook, Balents, PRB 2011 Son, Spivak, PRB 2013 … and 80+ theory uploads in 2013 on arXiv

Resurgent interest in the Nielsen Ninomiya prediction. Sparked a race to observe the chiral anomaly in solids The band structure of Na3Bi (Wang, Dai, Fang et al. PRB 2012)

Na-3s Only 2 bands, derived from Na-3s and E |S, ±½ ) Bi-6p, lie near Fermi energy.

60 meV Spin orbit interaction leads to kD crossings at K± = (0, 0, ±kD) kz

|P, ±3/2) Crossings protected against gap formation Bi-6p --- |S) and |P) states belong to different irreducible representations of C3.

We end up with 2 Dirac nodes centered at K± Dirac cone resolves into two Weyl nodes with opposite chiralities χ = ±1 Protected The low-E Hamiltonian, close to node K+, reduces to E |S, ±½ ) , ½ k , 3/2 D 𝑆𝑆, ½ kz 𝑃𝑃, 3/2 𝑆𝑆 − | P, ±3/2) 𝑃𝑃 − H resolves into two 2x2 Weyl Hamiltonians H1, H2

Calculate chirality from velocity matrix

= = v( 𝒗𝒗� + )

𝐻𝐻1 𝐤𝐤 ∙ 𝐯𝐯�1 ∙ 𝛕𝛕 𝑘𝑘𝑥𝑥𝜏𝜏1 − 𝑘𝑘𝑦𝑦𝜏𝜏2 𝑘𝑘𝑧𝑧𝜏𝜏3 det = The chirality is v 𝐯𝐯� 𝜒𝜒 = 1, = +1

1 2 We𝜒𝜒 have− a superposition𝜒𝜒 of two Weyl nodes at B = 0 The Adler Bell Jackiw (or chiral, triangle) anomaly A discrepancy of 300 million in pion decay

pion photon vector current γ γ axial current 0 π Adler Bell Jackiw anomaly

Pions, the lightest hadrons, are long-lived. Charged pions can decay only into leptons. + ν + + However,𝜋𝜋 → 𝜇𝜇 neutral pions break chiral symmetry by decaying into 2 photons (3x108 faster!) + (Adler, Bell, Jackiw, 1969) 0 𝜋𝜋 → 𝛾𝛾 𝛾𝛾 Coupling massless fermions to electromagnetic fields Aμ ruins chiral symmetry -i Provides EM channel for very rapid decay (by 300 million). 𝜕𝜕𝜇𝜇 −→ −𝑖𝑖𝜕𝜕𝜇𝜇 − 𝑒𝑒𝑒𝑒𝜇𝜇 An anomaly is the breaking of a classically allowed symmetry by quantum effects. Chiral anomaly in Quantum Field Theory (QFT)

( ) = axial current 162 𝑒𝑒 𝜇𝜇𝜇𝜇𝜇𝜇𝜇𝜇 𝐴𝐴 𝑥𝑥 2 𝜀𝜀 𝐹𝐹𝜇𝜇𝜇𝜇𝐹𝐹𝛼𝛼𝛼𝛼 Spoiler role and Anomaly-free condition 𝜋𝜋 All the anomalies must cancel for a theory to be renormalizable. Imposes stringent constraints, e.g. number of generations. In Glashow-Weinberg-Salam theory, exact cancellation of all anomalies in each generation of quarks and leptons has been called “magical” (Peskin1).

Topological in origin Closely related to winding number on 3-sphere S3 and the Atiyah Singer index theorem2,3 “Gauss Law” Winding number on S3, 1 Number of zero modes topological charge, = = 2 Second Chern number, + − Instanton number,… 𝑞𝑞 �𝑑𝑑𝑑𝑑 𝐴𝐴 𝑥𝑥 𝑛𝑛 − 𝑛𝑛 Cross section of 3-sphere 1. Intro to QFT, Peskin Schroeder 2. Anomalies in QFT, Bertlmann 3. Geometry, Topology … , Nakahara Nielsen and Ninomiya (1983) Observation of anomaly in a crystal?

Nielsen and Ninomiya (Phys. Lett. 1983) proposed the chiral anomaly should appear in crystals ---- spacetime dim. must be even

E, B v

= k 162 𝑒𝑒 𝜇𝜇𝜇𝜇𝜇𝜇𝜇𝜇 Axial current, 𝐴𝐴 2 𝜀𝜀 𝐹𝐹𝜇𝜇𝜇𝜇𝐹𝐹𝛼𝛼𝛼𝛼 Spectral flow 𝜋𝜋 Observable as a large, negative longitudinal magnetoresistance

Rate at which charge is pumped

= = . 2 2 2 4 3 𝐿𝐿 𝐿𝐿𝐿𝐿𝑘𝑘𝑧𝑧̇ 𝑒𝑒 𝐴𝐴 − 2 −𝑉𝑉 2 2 𝐄𝐄 𝐁𝐁 DOS of one 𝜋𝜋ℓ𝐵𝐵 𝜋𝜋 Rate of increase𝜋𝜋 ℏ of states Landau level along kz in E field Charge pumping and the chiral anomaly

Landau levels E(k) Nielsen, Ninomiya, Phys. Lett. 1983 Wan, Turner, Vishwanath, PRB 2011 Burkov, Hook Balents, PRB 2011 B, E Son, Spivak, PRB 2013 Parameswaran et al. PRX 2014

χ=1 χ=-1 Last Landau Level is chiral in Weyl states 0 kz Chiral anomaly engenders large, negative longitudinal MR

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

= = . 2 2 2 4 3 𝐿𝐿 𝐿𝐿𝐿𝐿𝑘𝑘̇ 𝑒𝑒 𝐴𝐴 − 2 −𝑉𝑉 2 2 𝐄𝐄 𝐁𝐁 𝜋𝜋ℓ𝐵𝐵 𝜋𝜋 𝜋𝜋 ℏ In weak B, charge pumping gives (Son and Spivak, PRB 2013)

τ is relaxation time τ a a for pumped current Steady progress (2 years) in improving crystal growth of Na3Bi

Deep purple crystals, that rapidly oxidize in ambient air (30 s) Jun Xiong, S. Kushwaha, Krizan, Cava

Long-term annealed crystals with EF much closer to node

Fermi energy lies 30 meV above node Striking negative longitud. MR (LMR) Measure FS caliper by Shubnikov de Haas (SdH) oscillations

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

Carrier mobility μ = 3,000 cm2/Vs at 4 K 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 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 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.

Axial plume 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)

Mere observation of negative, longitudinal MR is insufficient. Relaxation time τa of the pumped axial current Determine intervalley scat. lifetime using Son-Spivak expression

conductance

τa

In semiclassical regime, Weyl node conductivity grows as B2 (Son, Spivak, PRB ‘13) From fit, we find = 40-80

𝜏𝜏a 𝜏𝜏0 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.

Question raised by expt. Does large ratio (τa / τ0 ) reflect near-conservation of chiral charge? Vector current is conserved . + = 0 𝜕𝜕𝜌𝜌 𝛻𝛻 𝐉𝐉 But not axial current, because of the anomaly𝜕𝜕𝑡𝑡 A polar . + = 𝜕𝜕𝜌𝜌5 𝛻𝛻 𝐉𝐉𝟓𝟓 𝐴𝐴 with A defined by 𝜕𝜕𝜕𝜕 axial A = = . 2 3 𝐿𝐿 𝐿𝐿𝐿𝐿𝑘𝑘̇ 𝑒𝑒 2 2 2 − 2𝜋𝜋ℓ𝐵𝐵 2𝜋𝜋 −𝑉𝑉 4𝜋𝜋 ℏ 𝐄𝐄 𝐁𝐁 mirror σxx

The conductivity increases by ~ 2 when B = 1 T. Are impurities blind to the axial current? 0 1 B(T) See Burkov, PRB 2015 Future directions in Topological Matter: Phenomena and Materials

1. Chiral anomaly axial current relaxation, Berry curvature electronics, nonlocal current

Materials: Na3Bi, Cd3As2, TaAs, NbP, Sn, HgCdTe, ZrTe5, MoTe 2

2. Majorana fermions = ( + )/ 2 = topological quantum computing,† triplet† p-wave pairing Materials: InSb𝛾𝛾 nanowires,𝑐𝑐 𝑐𝑐 Fe√ nanowires,𝛾𝛾 Fe films, TI surface

3. Spin liquids and quantum magnets Berry curvature induced thermal Hall conductivity Kxy Materials: Herbert smithite, Tb and Yb pyrochlores

4. Interactions: Induced pairing by proximity effect Andreev reflection in Weyl semimetals current-phase relation of supercurrents in topological matter

5. Topological aspects of IQHE and FQHE Dirac fermions at ν = ½ (Vishwanath) Parafermions by proximitizing Central charge c and Kxy in FQHE End

Thank you The Anomaly term

Vector current is conserved . + = 0 polar 𝜕𝜕𝜌𝜌 𝛻𝛻 𝐉𝐉 Axial current is not because of A 𝜕𝜕𝑡𝑡 . + = axial 𝜕𝜕𝜌𝜌5 Expresses conversion between chiral𝛻𝛻 𝐉𝐉populations𝟓𝟓 𝐴𝐴at rate A. 𝜕𝜕𝜕𝜕

The ABJ anomaly term A is given by

= , = 2 𝑒𝑒 𝜇𝜇𝜇𝜇𝜇𝜇𝜇𝜇 2 𝐴𝐴 16𝜋𝜋 𝜀𝜀 𝐹𝐹𝜇𝜇𝜇𝜇𝐹𝐹𝛼𝛼𝛼𝛼 𝐹𝐹𝜇𝜇𝜇𝜇 𝜕𝜕𝜇𝜇𝐴𝐴𝜈𝜈 − 𝜕𝜕𝜈𝜈𝐴𝐴𝜇𝜇

A = . In terms of E, B 3 (see below) 𝑒𝑒 2 2 −𝑉𝑉 4𝜋𝜋 ℏ 𝐄𝐄 𝐁𝐁 Photoemission on Na3Bi and Cd3As2

Point group C3

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

Calculation of lifetime τa

Burkov arXiv: 1505.01849v2, Phys Rev B (2015) Near conservation of chiral charge leads to slow diffusion modes. Predicts a long axial current relaxation time, especially for undisplaced Weyl nodes (Dirac case)

= + 2 𝜕𝜕𝑛𝑛𝑣𝑣 𝜕𝜕 𝑛𝑛𝑣𝑣 𝜕𝜕𝑛𝑛𝑎𝑎 𝐷𝐷 2 Γ 𝜕𝜕=𝑡𝑡 𝜕𝜕𝑧𝑧 + 𝜕𝜕 𝑧𝑧 2 𝜕𝜕𝑛𝑛𝑎𝑎 𝜕𝜕 𝑛𝑛𝑎𝑎 𝑛𝑛𝑎𝑎 𝜕𝜕𝑛𝑛𝑣𝑣 2 𝐷𝐷 − 𝑎𝑎 Γ 𝜕𝜕𝑡𝑡 𝜕𝜕=𝑧𝑧 𝜏𝜏 1 𝜕𝜕𝑧𝑧 20 2 𝜏𝜏0 𝐸𝐸𝐹𝐹 2 ≪ 𝜏𝜏𝑎𝑎 𝑊𝑊

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? Longitudinal MR is rare in conventional metals

A conventional 1-band model does not lead to MR, regardless of band anisotropy. Standard Boltzmann equation with anisotropic mass tensor (elliptical FS) in arbitrarily large, tilted, field B = (B , B , B ) 1 2 3 3 B

2 1

Solution for resistivity tensor

= 𝑚𝑚𝑖𝑖 𝜌𝜌𝑖𝑖 2 𝑛𝑛𝑒𝑒 𝜏𝜏0 Diagonal elements are independent of B  absence of MR Closing the gap in a 3D semi-conductor

Murakami, New J. Phys. (‘07). Closing the gap in a semiconductor by pressure, doping etc Murakami. Kuga, PRB (’08) m Murakami, Physica E(‘11) Okugawa, Murakami, PRB(’14)

Anticrossing? Discrete point nodes

Courtesy: S. Murakami

Line node? Weyl nodes Topological Matter Cornucopia

Berry HgTe-CdTe phase, Quantum Spin Hall sys. Chern InAs-GaSb Strong correlation flux solids

Iridates, SmB6

2D layered Topological Z2 TRI systems systems Crystalline Insulator MoS2, WTe2 Bi2Se3 Bi2Te3

PbSnSe, PbSnTe Bi2Te2Se

HerbertSmithite Topological Dirac Kagome semimetals Eu2Ir2O7 Pyrochlore Cd3As2 Na3Bi Cu(1,3 carboxylate)

Weyl metals TaAs, NbAs, NbP Raw curves (unsymmetrized)

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