1. Berry phase in solid state physics
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

2. Taiwan 2

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

4. 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

5. 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

6. 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?

7. 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

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

9. 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

10. 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

11. 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

12. 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

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

14. 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

15. 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

16. 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

17. 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

18. Note: Back action on the slow variableθ 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

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

21. 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 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

23. 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

24. 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)

25. 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

26. 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)

27. 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?

28. 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

29. 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

30. 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

31. 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

32. 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

33. 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 … …

34. 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

35. 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

36. 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

37. Semiclassical dynamics in solid2π
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

38. 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)

39. 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

40. 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,

41. 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

42. 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

43. 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)

44. 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)

45. 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

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

47. a b c d
Brillouin zone
Proof:

48. 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!

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

50. 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

51. 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

52. 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

53. 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, (2008)
2 (or 3) mechanisms:
In reality, it’s not so clear-cut !

54. 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…”

55. 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. – Princeton
Karplus-Luttinger mechanism:

56. 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

57. 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 !

58. 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

59. 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

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

61. 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)

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

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