Tutorial: Berry phase and Berry curvature in solids

Tutorial: Berry phase and Berry curvature in solids Justin Song Division of Physics, Nanyang Technological University (Singapore) & Institute of High Performance Computing (Singapore) Funding: (Singapore)

references tutorial style Kane - Topological band theory (see website) Ong & Lee - Geometry and the anomalous Hall effect in Ferromagnets (see website) +.. references and reviews Xiao, Chang, Niu - Berry phase effects on electronic properties (Rev Mod Phys) Nagaosa, Sinova, Onoda, MacDonald, Ong, - Anomalous Hall effect (Rev Mod Phys) Bernevig (with Hughes) - Topological Insulators and topological superconductors (Book) +..

Quasiparticles in a crystal Energy bands in a crystal E /a /a k Energy Momentum Mass Energy bands Quasi-momentum Effective mass

Quasiparticles in a crystal: Berry curvature Energy bands in a crystal E Energy Momentum Energy bands Quasi-momentum /a /a k Mass Effective mass Emergent quantum mechanical property: Berry curvature (self-rotation of wavepackets) Electron wavepacket traveling through certain special crystals

(Self-) Rotation enables transverse motion Magnus effect: F"

(Self-) Rotation enables transverse motion Magnus effect: Gyroscopes: F" mg

(Self-) Rotation enables transverse motion Lorentz force:!! B!! Semiclassical equations of motion ṗ = ev B Lorentz force

(Self-) Rotation enables transverse motion Lorentz force: Drifting cyclotron orbits:!! B!! Semiclassical equations of motion ṗ = ev B + ee Lorentz force

Anomalous velocity and Berry curvature (p) Electron wavepacket traveling through certain special crystals Semiclassical equations of motion Group velocity ẋ = d" + ṗ dp ṗ = dv dx + ẋ B Lorentz force

Anomalous velocity and Berry curvature (p) Electron wavepacket traveling through certain special crystals Group velocity Semiclassical equations of motion Anomalous velocity ẋ = d" + ṗ dp ṗ = dv dx + ẋ B Lorentz force Skewed trajectory:

Velocity Energy bands in a crystal; depends on k group velocity: E /a /a k band n v g = hn v ni = 1 ~ @ n (k) @k

Velocity: intra- vs inter-band matrix elements Energy bands in a crystal; depends on k group velocity: E /a /a k band n v g = hn v ni = 1 ~ @ n (k) @k interband matrix element (no real transition) ~hn v n 0 i = hn @H @k n0 i

Berry curvature and equations of motion Energy bands in a crystal; depends on k group velocity: E /a /a k band n v g = hn v ni = 1 ~ @ n (k) @k interband matrix element (no real transition) equations of motion: ẋ k = 1 ~ with Berry curvature @ k @k + ee ~ (k) ~hn v n 0 i = hn @H @k n0 i (k) =i X hn @ ki H n 0 ihn 0 @ kj H ni hn @ kj H n 0 ihn 0 @ ki H ni ( n n 0) 2 n6=n 0

Consequences 1. Intrinsic transverse component of current (anomalous Hall effect) Electron wavepacket traveling through certain special crystals Modern interpretation in terms of Berry curvature (and self-rotation): Chang, Niu (96), Sundaram, Niu (99) 1I. Filled bands can carry a current (e.g. QHE) E /a /a k E F just group velocity: J = e ~ with Berry curvature/anomalous vel. : J = e2 ~ Z Z @ dk =0 @k (k) E dk

Plan Part I. anomalous velocity phenomenology of Berry curvature and anomalous velocity Part II. Berry phase and Berry curvature in Bloch bands Bloch wavefunctions, Berry connection and Berry curvature Part III. filled bands and topology Berry flux in filled bands

Quasiparticles in a crystal Energy bands in a crystal; depends on k E /a /a k Energy Momentum Mass Energy bands Quasi-momentum Effective mass Bloch s theorem: n,k(r) =e ik r u n,k (r)

Quasiparticles in a crystal Energy bands in a crystal; depends on k Bloch wavefunctions also depend on k E /a k /a /a /a k u k Bloch s theorem: n,k(r) =e ik r u n,k (r)

Berry connection and Berry phase distance/ connection between wavefunctions: (p 3 ) (p 2 ) (p 1 ) e i 12 = h (p 1) (p 2 )i h (p 1 ) (p 2 )i h ij =Im i logh (p i ) (p j )i Berry, Proc. Roy. Soc. Lond. A (84)

Berry connection and Berry phase distance/ connection between wavefunctions: (p 3 ) (p 2 ) (p 1 ) e i 12 = h (p 1) (p 2 )i h (p 1 ) (p 2 )i h ij =Im phase around closed path: i logh (p i ) (p j )i = 12 + 23 + 31 h i =Im logh (p 1 ) (p 2 )ih (p 2 ) (p 3 )ih (p 3 ) (p 1 )i ih (p) @ p (p)i p Berry, Proc. Roy. Soc. Lond. A (84)

Berry connection and Berry phase Berry phase (phase across whole loop) Berry connection (phase accumulated over small section): d(p) A = ihu p r p u p i Berry phase: B = I dp A Berry connection Berry, Proc. Roy. Soc. Lond. A (84)

Observing the Berry phase area enclosed Mikitik, Sharlai PRL (99,04) Shubnikov de-haas (SdH) Oscillations in graphene Fan diagram for graphite graphite data: Zhang, Small, Amori, Kim PRL (05) Data: Zhang, Tan, Stormer, Kim Nature (05) Nice review: Young, Zhang, and Kim (14)

Aharonov-Bohm phase (AB phase) Aharonov-Bohm phase B Aharonov-Bohm effect Aharonov, Bohm (59) ' AB = I d` d` A image from Feynman Lectures (62)

Aharonov-Bohm phase (AB phase) and Berry phase Aharonov-Bohm phase Berry phase B d(p) d` ' AB = I d` A B = I dp A Berry connection: A = ihu p r p u p i

Berry curvature acts as a magnetic field in p space Aharonov-Bohm phase Berry phase net Berry curvature B d(p) d` ' AB = I d` A B = I dp A = Z Berry curvature: dpr p A Berry connection: A = ihu p r p u p i

Berry curvature acts as a magnetic field in phase space B d` d(p) Group velocity Semiclassical equations of motion Anomalous velocity ẋ = d" + ṗ dp ṗ = dv dx + ẋ B Lorentz force A = ihu p r p u p i =r p A

Duality In a magnetic field: with Berry curvature: [v i,v j ]=i ijk e~b k m 2 [r i,r j ]=i ijk k minimal coupling ( peierls substitution ) modified minimal coupling (modified peierls substitution ) p! p ea(r)/c r! r A(p) in a single band see for e.g., Xiao, Shi, Niu, PRL (2005), Xiao, Chang, Niu Rev. Mod Phys. (2010)

example: two-band model

Example: two-band model and graphene Hamiltonian: (single valley) Spinor'type+ wavefun1on:+ H = v p all on A site p i adapted from Park and Marzari (2011) all on B site Z C = r p A Berry phase: h k @ k k idk =

Example: two-band model and graphene Hamiltonian: (single valley) Spinor'type+ wavefun1on:+ H = v p all on A site p i Berry curvature vanishes: = r p A adapted from Park and Marzari (2011) all on B site

Example: two-band model and gapped graphene Hamiltonian: H = v p + z (single valley) d(p) =(p x, p y, ) wavefunction: all on A site broken inversion symmetry E 2 p i all on B site p

Example: two-band model and gapped graphene Hamiltonian: H = v p + z (single valley) d(p) =(p x, p y, ) wavefunction: all on A site broken inversion symmetry recall stokes type formula = r p A p i Berry curvature for gapped Dirac system ± (p) = ±~2 v 2 with 2 3 p p = p v 2 p 2 + 2 all on B site

Example: two-band model and gapped graphene Hamiltonian: H = v p + z (single valley) d(p) =(p x, p y, ) wavefunction: all on A site broken inversion symmetry recall stokes type formula = r p A Berry curvature for gapped Dirac system ± (p) = ±~2 v 2 with 2 3 p p = p v 2 p 2 + 2 p i all on B site net Berry curvature

Example: two-band model and gapped graphene Solid angle swept / 2 = net Berry curvature (p) = 1 2 ij ˆd p @ iˆdp @ j ˆdp all on A site net Berry curvature p i all on B site

Example: two-band model and gapped graphene Symmetry constraints Time reversal symmetry: Inversion symmetry: n ( k) = n (k) n ( k) = n (k) all on A site net Berry curvature p i all on B site

Example: two-band model and gapped graphene Symmetry constraints Time reversal symmetry: Inversion symmetry: n ( k) = n (k) n ( k) = n (k) all on A site net Berry curvature valley K valley K p i anded$ all on B site Berry curvature of opposite sign Xiao, Yao, Niu, PRL (07)

Example: two-band model and gapped graphene Graphene% velocity depends on amplitude + phase on A/B sub lattice v p = h (p) (p)i

Example: two-band model and gapped graphene velocity depends on amplitude + phase on A/B sub lattice v p = h (p) (p)i Graphene% e.g. velocity vanishes at band extrema all on A site E 2 (0) (p)i p all on B site

Example: two-band model and gapped graphene velocity depends on amplitude + phase on A/B sub lattice v p = h (p) (p)i Graphene% all on A site Apply electric field, wave function is perturbed (p)i = (0) (p)i + i spinor cants p i velocity: v p = @ p @p + ee ~ (p) all on B site For more details: see Xiao, Chang Niu, Rev Mod Phys (10) For mapping to Bloch spinors, scattering, and, trajectories, see also: Lensky, JS, Samutpraphoot, Levitov PRL (15)

Gapped Dirac material :transition metal dichalcogenides (TMDs) Circularly polarized light absorption in MoS 2 Hall effect at zero magnetic field K K 0 + Le#$Handed$ Right$Handed$ KF Mak, K McGill, JW Park, PL McEuen, Science (2014) Hall effect tunable by photo excitation

Plan Part I. anomalous velocity phenomenology of Berry curvature and anomalous velocity Part II. Berry phase and Berry curvature in Bloch bands Bloch wavefunctions, Berry connection and Berry curvature Part III. Filled bands and topology Berry flux in filled bands

Quantum Hall effect R H = h/ e 2 R H Measurement Standard h/e 2 = 25812.807557(18) R xx Figure adapted from nobelprize.org B

Hall conductivity and wrapping spheres Hall current: J x = e2 ~ Z (k) E y f( k )dk in two-dimensions, and for filled band: xy = e2 h Z d 2 k 2 (k) p i wrapping over sphere total solid angle of sphere is fixed: 4

Quantum Hall effect and sphere wrapping R H = h/ e 2 R H Measurement Standard h/e 2 = 25812.807557(18) R xx Figure adapted from nobelprize.org B Note that the QHE has many faces: see also Halperin (Edge states), and Laughlin (Flux threading)

after some fancy footwork they arrived at: and a quantized Hall conductance Berry curvature (k) =r k A= r k hu k r k u k i David Thouless see also Thouless, Niu, Wu (83)

Important for applications in topological pastries topological cinnamon roll topological bagel

Important for applications in topological matter

Zero-field quantum Hall effect: Chern insulator all on A site net Berry curvature Time reversal symmetry: Symmetry constraints Inversion symmetry: p i n ( k) = n (k) n ( k) = n (k) all on B site

Zero-field quantum Hall effect: Chern insulator tight-binding graphene type model with complex second neighbor hopping

Zero-field quantum Hall effect: Chern insulator

Realizing Haldane model and imaging Berry curvature Jotzu,, Esslinger, Nature (2014)

Electronic chirality without magnetic field (Quantum) Anomalous Hall effect: B =0 Cr 0.15 (Bi 0.1 Sb 0.9 ) 1.85 Te 3 Zhang, et al, Science (2013)

Topological materials Topological Insulators Surfaces of 3D TIs: Bi Se, Bi Te, Bi Sb 2 3 2 3 x 1-x, Topological Crystalline Insulators: Sn Te, Magnetic Topological Insulators: Cr-doped BiSbTe Hg xcd 1-xTe Quantum Wells, InAs/GaSb QWs 3D Dirac/Weyl Experimentally Observed: Cd 3As 2, Na 3Bi, TiBiSe TaAs, 2 Type II Weyl semimetals (candidates): WTe2, MoTe2 Proposed in TI stacks; HgCdTe Stacks Nodal-line semimetals 2D Dirac Materials (materials that host Berry curvature) Graphene heterostructures: G/hBN, dual-gated Bilayer graphene, Transition metal dichalcogenides: MoS 2, WS 2, WSe 2, MoSe 2, MoTe,. 2