Berry phase in solid state physics 03/10/09 @ Juelich Berry phase in solid state physics － a selected overview Ming-Che Chang Department of Physics National Taiwan Normal University Qian Niu Department of Physics The University of Texas at Austin

Taiwan 2

Paper/year with the title “Berry phase” or “geometric phase”

Berry phase in solid state physics Introduction (30-40 mins) Quantum adiabatic evolution and Berry phase Electromagnetic analogy Geometric analogy Berry phase in solid state physics

Fast variable and slow variable H+2 molecule e－ electron; {nuclei} nuclei move thousands of times slower than the electron Instead of solving time-dependent Schroedinger eq., one uses Born-Oppenheimer approximation “Slow variables Ri” are treated as parameters λ(t) (Kinetic energies from Pi are neglected) solve time-independent Schroedinger eq. “snapshot” solution 5

Adiabatic evolution of a quantum system Energy spectrum: After a cyclic evolution E(λ(t)) n+1 x n x n-1 Dynamical phase λ(t) Phases of the snapshot states at different λ’s are independent and can be arbitrarily assigned Do we need to worry about this phase?

Stationary, snapshot state No! Fock, Z. Phys 1928 Schiff, Quantum Mechanics (3rd ed.) p.290 Pf : Consider the n-th level, Stationary, snapshot state ≡An(λ) Choose a  (λ) such that, Redefine the phase, Thus removing the extra phase An’(λ) An(λ) An’(λ)=0

One problem: does not always have a well-defined (global) solution  is not defined here Contour of  Vector flow Vector flow Contour of  C

Berry phase (path dependent) M. Berry, 1984 : Parameter-dependent phase NOT always removable! Index n neglected Berry phase (path dependent) Berry’s face Interference due to the Berry phase Phase difference 1 1 a b a C interference 2 －2

Some terminology Berry connection (or Berry potential) Stokes theorem (3-dim here, can be higher) Berry curvature (or Berry field) Gauge transformation (Nonsingular gauge, of course) Redefine the phases of the snapshot states Berry curvature nd Berry phase not changed

Analogy with magnetic monopole Berry potential (in parameter space) Vector potential (in real space) Berry field (in 3D) Magnetic field Berry phase Magnetic flux Chern number Dirac monopole

Example: spin-1/2 particle in slowly changing B field Real space Parameter space C C (a monopole at the origin) Level crossing at B=0 Berry curvature E(B) Berry phase B spin × solid angle

Experimental realizations : Bitter and Dubbers , PRL 1987 Tomita and Chiao, PRL 1986

Why Berry phase is often called geometric phase? Anholonomy angle (or defect angle) A α Parallel transport R Eg., for a spherical triangle, α=A/R2 Gaussian curvature G ≡ Berry phase ≒ anholonomy angle in differential geometry Berry curvature ≒Gaussian curvature The analogy becomes exact in the language of fiber bundle

Geometry behind the Berry phase Why Berry phase is often called geometric phase? base space fiber space Fiber bundle Examples: Trivial fiber bundle (= a product space) Nontrivial fiber bundle Simplest example: Möbius band R1 x R1 R1 base fiber

Fiber bundle and quantum state evolution (Wu and Yang, PRD 1975) Fiber space: inner DOF, eg., U(1) phase  Base space: parameter space Berry phase = Vertical shift along fiber (U(1) anholonomy) Chern number n χ = 2 For fiber bundle χ = 0 ～ Euler characteristic χ For 2-dim closed surface χ =－2

Geometric phase vs topological phase Berry phase (1984) Aharonov-Bohm (AB) phase (1959) Φ magnetic flux solenoid C Independent of the speed (as long as it’s slow) Independent of precise path Some people call both phases Geometric phase, some others call both phases Berry phase In this talk Berry phase = Geometric phase Aharonov-Bohm phase = Topological phase

Note: Back action on the slow variable θ φ S x y z Illustrative example: A rotating solenoid State of the whole system: slow fast Fast: S; Slow: φ Effective H for slow variable Berry potential Spectrum of slow variable is shifted by An The slow variable may feel a force due to ▽× An

Berry phase in solid state physics Introduction Berry phase in solid state physics 19

Berry phase in condensed matter physics, a partial list: 1982 Quantized Hall conductance (Thouless et al) 1983 Quantized charge transport (Thouless) 1984 Anyon in fractional quantum Hall effect (Arovas et al) 1989 Berry phase in one-dimensional lattice (Zak) 1990 Persistent spin current in one-dimensional ring (Loss et al) 1992 Quantum tunneling in magnetic cluster (Loss et al) 1993 Modern theory of electric polarization (King-Smith et al) 1996 Semiclassical dynamics in Bloch band (Chang et al) 1998 Spin wave dynamics (Niu et al) 2001 Anomalous Hall effect (Taguchi et al) 2003 Spin Hall effect (Murakami et al) 2004 Optical Hall effect (Onoda et al) 2006 Orbital magnetization in solid (Xiao et al) …

Berry phase in condensed matter physics, a partial list: 1982 Quantized Hall conductance (Thouless et al) 1983 Quantized charge transport (Thouless) 1984 Anyon in fractional quantum Hall effect (Arovas et al) 1989 Berry phase in one-dimensional lattice (Zak) 1990 Persistent spin current in one-dimensional ring (Loss et al) 1992 Quantum tunneling in magnetic cluster (Loss et al) 1993 Modern theory of electric polarization (King-Smith et al) 1996 Semiclassical dynamics in Bloch band (Chang et al) 1998 Spin wave dynamics (Niu et al) 2001 Anomalous Hall effect (Taguchi et al) 2003 Spin Hall effect (Murakami et al) 2004 Optical Hall effect (Onoda et al) 2006 Orbital magnetization in solid (Xiao et al) … 22

Berry phase in solid state physics Persistent spin current Quantum tunneling in a magnetic cluster Modern theory of electric polarization Semiclassical electron dynamics Quantum Hall effect (QHE) Anomalous Hall effect (AHE) Spin Hall effect (SHE) Spin Bloch state Persistent spin current Quantum tunneling Electric polarization QHE AHE SHE

Berry phase in solid state physics Persistent spin current Quantum tunneling in a magnetic cluster Modern theory of electric polarization Semiclassical electron dynamics Quantum Hall effect (QHE) Anomalous Hall effect (AHE) Spin Hall effect (SHE)

Persistent charge current in a small metal ring (I) (F Persistent charge current in a small metal ring (I) (F. Hund, Ann Physik 1936) e－ Static disorder R unwrap Diffusive regime Phase coherence length e－ … L=2R = A periodic lattice with lattice constant L ∴The electron would feel no resistance

Persistent charge current in a small metal ring (II) Φ After one circle, the electron gets a phase AB phase I /0 1/2 -1/2 free Elastic/inelastic scatterings Persistent current from an electron: confirmed in 1990 (Levy et al, PRL)

Persistent spin current in a metal ring (Loss et al, PRL 1990) B S e- S C Ω(C) textured B field After circling once, an electron gets an AB phase 2πΦ/Φ0 a Berry phase ± (1/2)Ω(C) (from the “texture”) IS free Elastic/inelastic scatterings Ω/2 -π π → persistent charge and spin current Confirmation?

Berry phase in solid state physics Persistent spin current Quantum tunneling in a magnetic cluster Modern theory of electric polarization Semiclassical electron dynamics Quantum Hall effect Anomalous Hall effect Spin Hall effect

Magnetic cluster (particle, molecule) with a single domain reversal E Thermal activation M Quantum tunneling S～10 molecule Fe8 E S Figs from Wernsdorfer’s talk

Berry phase and quantum tunneling (I) For example, Spin space Phase difference between 2 paths = Berry phase =JΩ=2πJ medium x y z easy hard Tunneling path Degenerate GS if J=half integer → J=π,3π,5π… → No tunneling due to destructive interference 30

Berry phase and quantum tunneling (II) x y z B easy hard S=-10 → 8 S=-10 → 9 S=-10 → 10 Wernsdorfer and Sessoli, Science 1999 Ω (side view) Shrinking solid angle as B is increasing: γ=2πJ → 0

Berry phase in solid state physics Persistent spin current Quantum tunneling in a magnetic cluster Modern theory of electric polarization Semiclassical electron dynamics Quantum Hall effect Anomalous Hall effect Spin Hall effect

Electric polarization of a periodic solid well defined only for finite system (sensitive to boundary) or, for crystal with well-localized dipoles (Claussius-Mossotti theory) P is not well defined in, e.g., covalent crystal: P Unit cell + － Choice 1 … P + － Choice 2 … However, the change of P is well-defined Experimentally, it’s ΔP that’s measured ΔP … …

Modern theory of polarization One-dimensional lattice (λ=atomic displacement in a unit cell) Resta, Ferroelectrics 1992 Iℓℓ-defined However, dP/dλ is well-defined, even for an infinite system ! Berry potential King-Smith and Vanderbilt, PRB 1993 For a one-dimensional lattice with inversion symmetry (if the origin is a symmetric point) (Zak, PRL 1989) Other values are possible without inversion symmetry

Berry phase and electric polarization Dirac comb model g1=5 g2=4 … … Rave and Kerr, EPJ B 2005 b a Lowest energy band: γ1 ← g2=0 γ1=π similar formulation in 3-dim using Kohn-Sham orbitals Review: Resta, J. Phys.: Condens. Matter 12, R107 (2000) r =b/a

Berry phase in solid state physics Persistent spin current Quantum tunneling in a magnetic cluster Modern theory of electric polarization Semiclassical electron dynamics Quantum Hall effect Anomalous Hall effect Spin Hall effect

Semiclassical dynamics in solid 2π E(k) x n n+1 n-1 Limits of validity: one band approximation Negligible inter-band transition. “never close to being violated in a metal” Lattice effect hidden in En (k) Derivation is harder than expected Explains (Ashcroft and Mermin, Chap 12) Bloch oscillation in a DC electric field, cyclotron motion in a magnetic field, … quantization → Wannier-Stark ladders quantization → LLs, de Haas - van Alphen effect

Semiclassical dynamics - wavepacket approach 1. Construct a wavepacket that is localized in both r-space and k-space (parameterized by its c.m.) 2. Using the time-dependent variational principle to get the effective Lagrangian for the c.m. variables 3. Minimize the action Seff[rc(t),kc(t)] and determine the trajectory (rc(t), kc(t)) → Euler-Lagrange equations Wavepacket in Bloch band: Berry potential (Chang and Niu, PRL 1995, PRB 1996)

Semiclassical dynamics with Berry curvature “Anomalous” velocity Cell-periodic Bloch state Berry curvature Wavepacket energy Zeeman energy due to spinning wavepacket Bloch energy If B=0, then dk/dt // electric field → Anomalous velocity ⊥ electric field Simple and Unified (integer) Quantum Hall effect (intrinsic) Anomalous Hall effect (intrinsic) Spin Hall effect 39

Why the anomalous velocity is not found earlier? In fact, it had been found by Adams, Blount, in the 50’s Why it seems OK not to be aware of it? Space inversion symmetry Time reversal symmetry both symmetries For scalar Bloch state (non-degenerate band): When do we expect to see it? SI symmetry is broken TR symmetry is broken spinor Bloch state (degenerate band) band crossing ← electric polarization ← QHE ← SHE ← monopole Also,

Berry phase in solid state physics Persistent spin current Quantum tunneling in a magnetic cluster Modern theory of electric polarization Semiclassical electron dynamics Quantum Hall effect Anomalous Hall effect Spin Hall effect

Quantum Hall effect (von Klitzing, PRL 1980) 2 DEG classical 3 σH (in e2/h) quantum 2 1 Increasing B 1/B LLs Increasing B Density of states energy EF B=0 Each LL contributes one e2/h z EF AlGaAs GaAs CB VB 2 DEG

Semiclassical formulation Equations of motion (In one Landau subband) Magnetic field effect is hidden here Hall conductance Quantization of Hall conductance (Thouless et al 1982) Remains quantized even with disorder, e-e interaction (Niu, Thouless, Wu, PRB, 1985)

Quantization of Hall conductance (II) For a filled Landau subband Brillouin zone Counts the amount of vorticity in the BZ due to zeros of Bloch state (Kohmoto, Ann. Phys, 1985) In the language of differential geometry, this n is the (first) Chern number that characterizes the topology of a fiber bundle (base space: BZ; fiber space: U(1) phase)

Berry curvature and Hofstadter spectrum 2DEG in a square lattice + a perpendicular B field tight-binding model: (Hofstadter, PRB 1976) Landau subband energy Width of a Bloch band when B=0 LLs Magnetic flux (in Φ0) / plaquette

Bloch energy E(k) Berry curvature Ω(k) C1 = 1 C2 = 2 C3 = 1

a b c d Brillouin zone Proof:

Re-quantization of semiclassical theory Bohr-Sommerfeld quantization Would shift quantized cyclotron energies (LLs) Bloch oscillation in a DC electric field, re-quantization → Wannier-Stark ladders cyclotron motion in a magnetic field, re-quantization → LLs, dHvA effect … Now with Berry phase effect!

cyclotron orbits (LLs) in graphene ↔ QHE in graphene Dirac cone B E Novoselov et al, Nature 2005 σH ρL … Cyclotron orbits k

Berry phase in solid state physics Persistent spin current Quantum tunneling in a magnetic cluster Modern theory of electric polarization Semiclassical electron dynamics Quantum Hall effect Anomalous Hall effect Spin Hall effect Mokrousov’s talks this Friday Buhmann’s next Thu (on QSHE) Poor men’s, and women’s, version of QHE, AHE, and SHE

Anomalous Hall effect (Edwin Hall, 1881): Hall effect in ferromagnetic (FM) materials saturation slope=RN H ρH RAHMS FM material The usual Lorentz force term Ingredients required for a successful theory: magnetization (majority spin) spin-orbit coupling (to couple the majority-spin direction to transverse orbital direction) Anomalous term

Intrinsic mechanism (ideal lattice without impurity) Linear response Spin-orbit coupling magnetization gives correct order of magnitude of ρH for Fe, also explains that’s observed in some data

Smit, 1955: KL mechanism should be annihilated by (an extra effect from) impurities Alternative scenario: Extrinsic mechanisms (with impurities) Side jump (Berger, PRB 1970) Skew scattering (Smit, Physica 1955) ～ Mott scattering Spinless impurity e－ anomalous velocity due to electric field of impurity ～ anomalous velocity in KL (Crépieux and Bruno, PRB 2001) Review: Sinitsyn, J. Phys: Condens. Matter 20, 023201 (2008) 2 (or 3) mechanisms: In reality, it’s not so clear-cut !

CM Hurd, The Hall Effect in Metals and Alloys (1972) “The difference of opinion between Luttinger and Smit seems never to have been entirely resolved.” 30 years later: Crepieux and Bruno, PRB 2001 “It is now accepted that two mechanisms are responsible for the AHE: the skew scattering… and the side-jump…”

However, And many more … Karplus-Luttinger mechanism: Science 2001 Science 2003 And many more … Mired in controversy from the start, it simmered for a long time as an unsolved problem, but has now re-emerged as a topic with modern appeal. – Ong @ Princeton Karplus-Luttinger mechanism:

Old wine in new bottle Ideal lattice without impurity Berry curvature of fcc Fe (Yao et al, PRL 2004) Karplus-Luttinger theory (1954) = Berry curvature theory (2001) same as Kubo-formula result ab initio calculation → intrinsic AHE

classical Hall effect ↑↓ +++++++ anomalous Hall effect ↑ ↓ y L ↑↓ charge B +++++++ Lorentz force anomalous Hall effect charge spin B ↑ ↑ ↑ ↑ ↑↑↑↑↑↑↑ ↑↑↑↑ ↑ ↑ EF Berry curvature Skew scattering ↑ ↓ y L spin Hall effect EF y L ↑ ↓ spin ↑↑↑↑↑↑↑ ↑↑↑↑ Berry curvature Skew scattering No magnetic field required !

Spin-degenerate Bloch state due to Kramer’s degeneracy Murakami, Nagaosa, and Zhang, Science 2003: Intrinsic spin Hall effect in semiconductor Spin-degenerate Bloch state due to Kramer’s degeneracy → Berry curvature becomes a 2x2 matrix (non-Abelian) Band structure (from Luttinger model) Berry curvature for HH/LH 4-band Luttinger model The crystal has both space inversion symmetry and time reversal symmetry ! Spin-dependent transverse velocity → SHE for holes

Only the HH/LH can have SHE? : Not really Berry curvature for conduction electron: 8-band Kane model spin-orbit coupling strength : Not really Chang and Niu, J Phys, Cond Mat 2008 Berry curvature for free electron (!): mc2 electron positron Dirac’s theory

Observations of SHE (extrinsic) Science 2004 Nature 2006 Nature Material 2008 Observation of Intrinsic SHE?

Three fundamental quantities in any crystalline solid Summary Spin Bloch state Persistent spin current Quantum tunneling Electric polarization QHE AHE SHE Three fundamental quantities in any crystalline solid E(k) Bloch energy Ω(k) L(k) Berry curvature Orbital moment (Not in this talk)

References 1989 2003 Reviews: Chang and Niu, J Phys Cond Matt 20, 193202 (2008) Xiao, Chang, and Niu, to be published (RMP?)

Thank you! Reviews: Slides : http://phy.ntnu.edu.tw/~changmc/Paper Chang and Niu, J Phys Cond Matt 20, 193202 (2008) Xiao, Chang, and Niu, to be published (RMP?)