Berry phase in solid state physics

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03/10/09 _at_ 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
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Taiwan
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Paper/year with the title Berry phase or
geometric phase
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• Introduction (30-40 mins)
• Quantum adiabatic evolution and Berry phase
• Electromagnetic analogy
• Geometric analogy
• Berry phase in solid state physics

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Fast variable and slow variable
H2 molecule
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
  ?(t)
• (Kinetic energies from Pi are neglected)
• solve time-independent Schroedinger eq.

snapshot solution
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Adiabatic evolution of a quantum system

• After a cyclic evolution

• Energy spectrum

E(?(t))
n1
x
n
x
n-1
Dynamical phase
0
?(t)

• Phases of the snapshot states at different ?s
  are independent and can be arbitrarily assigned

• Do we need to worry about this phase?

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• Fock, Z. Phys 1928
• Schiff, Quantum Mechanics (3rd ed.) p.290

No!
Pf
Consider the n-th level,
Stationary, snapshot state
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One problem
does not always have a well-defined
(global) solution
Vector flow
Vector flow
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M. Berry, 1984 Parameter-dependent phase NOT
always removable!
Index n neglected

• Berry phase (path dependent)

Berrys face

• Interference due to the Berry phase

Phase difference
1
1
a
b
a
C
interference
2
-2
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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
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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
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Example spin-1/2 particle in slowly changing B
field

• Real space

• Parameter space

(a monopole at the origin)
Berry curvature
Level crossing at B0
E(B)
Berry phase
B
spin solid angle
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Experimental realizations
Tomita and Chiao, PRL 1986
Bitter and Dubbers , PRL 1987
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Why Berry phase is often called geometric phase?

• Anholonomy angle (or defect angle)

Parallel transport
R
Eg., for a spherical triangle, aA/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
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Geometry behind the Berry phase
Why Berry phase is often called geometric
phase?
Examples

• Nontrivial fiber bundle Simplest example
  Möbius band

• Trivial fiber bundle ( a product space)

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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

For fiber bundle
? 0
Euler characteristic ?
For 2-dim closed surface
? -2
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Geometric phase vs topological phase

• Aharonov-Bohm (AB) phase (1959)

• Berry phase (1984)

Independent of the speed (as long as its slow)
Independent of precise path
Some people call both phases Geometric
phase, some others call both phases Berry phase

• Berry phase Geometric phase
• Aharonov-Bohm phase Topological phase

In this talk
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Note Back action on the slow variable
State of the whole system
slow
fast
Fast S Slow f
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

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• Introduction
• Berry phase in solid state physics

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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)

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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)

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• 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

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• 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)

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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-
L2?R
A periodic lattice with lattice constant L ?The
electron would feel no resistance
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Persistent charge current in a small metal ring
(II)
After one circle, the electron gets a phase
AB phase
free
Elastic/inelastic scatterings
Persistent current from an electron
confirmed in 1990 (Levy et al, PRL)
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Persistent spin current in a metal ring (Loss et
al, PRL 1990)
textured B field

• After circling once, an electron gets
• an AB phase 2pF/F0
• a Berry phase (1/2)O(C) (from the
  texture)

IS
free
Elastic/inelastic scatterings
O/2
-p
p
? persistent charge and spin current
Confirmation?
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• 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

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Magnetic cluster (particle, molecule) with a
single domain
reversal
particle

• S102 - 106

E
Thermal activation
M
Quantum tunneling
molecule

• S10

Fe8
E
S
Figs from Wernsdorfers talk
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Berry phase and quantum tunneling (I)
For example,
Spin space
Phase difference between 2 paths Berry phase
JO2pJ
if Jhalf integer ? Jp,3p,5p ? No tunneling
due to destructive interference
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Berry phase and quantum tunneling (II)
S-10 ? 8
S-10 ? 9
S-10 ? 10
Wernsdorfer and Sessoli, Science 1999
Shrinking solid angle as B is increasing
?2pJ ? 0
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• 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

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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

• However, the change of P is well-defined

Experimentally, its ?P thats measured


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Modern theory of polarization
One-dimensional lattice (?atomic displacement in
a unit cell)
Resta, Ferroelectrics 1992
Ill-defined
However, dP/d? is well-defined, even for an
infinite system !
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Berry phase and electric polarization
Dirac comb model
g15
g24


Rave and Kerr, EPJ B 2005
0
b
a
Lowest energy band
?1
? g20
?1p
r b/a
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• 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

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Semiclassical dynamics in solid
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
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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 Seffrc(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)
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Semiclassical dynamics with Berry curvature
Anomalous velocity
Cell-periodic Bloch state
Berry curvature
Wavepacket energy
Zeeman energy due to spinning wavepacket
Bloch energy
If B0, then dk/dt // electric field ? Anomalous
velocity ? electric field
Simple and Unified

• (integer) Quantum Hall effect
• (intrinsic) Anomalous Hall effect
• (intrinsic) Spin Hall effect

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• Why the anomalous velocity is not found earlier?

• In fact, it had been found by
• Adams, Blount, in the 50s

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• 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

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Quantum Hall effect (von Klitzing, PRL 1980)
classical
3
sH (in e2/h)
2
1
1/B
Increasing B
z
2 DEG
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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)
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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)
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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 B0
LLs
Magnetic flux (in F0) / plaquette
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Bloch energy E(k)
Berry curvature O(k)
C1 1
C2 ?2
C3 1
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Proof
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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!
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cyclotron orbits (LLs) in graphene
? QHE in graphene
Dirac cone
B
E
Cyclotron orbits
k
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• 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

Mokrousovs talks this Friday
Buhmanns next Thu (on QSHE)
Poor mens, and womens, version of QHE, AHE, and
SHE
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Anomalous Hall effect (Edwin Hall, 1881) Hall
effect in ferromagnetic (FM) materials
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
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Intrinsic mechanism (ideal lattice without
impurity)

• Linear response
• Spin-orbit coupling
• magnetization

gives correct order of magnitude of ?H for Fe,
also explains thats observed in some data
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Smit, 1955 KL mechanism should be annihilated by
(an extra effect from) impurities
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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
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However,
Science 2001
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Old wine in new bottle
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• classical Hall effect

• Lorentz force

• anomalous Hall effect

charge
spin
EF

• Berry curvature
• Skew scattering

?
?
y
0
L

• spin Hall effect

• Berry curvature
• Skew scattering

No magnetic field required !
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Murakami, Nagaosa, and Zhang, Science 2003
Intrinsic spin Hall effect in semiconductor

• Spin-degenerate Bloch state due to Kramers
  degeneracy
• ? Berry curvature becomes a 2x2 matrix
  (non-Abelian)

Band structure
The crystal has both space inversion symmetry
and time reversal symmetry !
Spin-dependent transverse velocity ? SHE for holes
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Only the HH/LH can have SHE?
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Observations of SHE (extrinsic)
Observation of Intrinsic SHE?
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• 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
L(k)
O(k)
Orbital moment
Berry curvature
(Not in this talk)
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References
Reviews

• Chang and Niu, J Phys Cond Matt 20, 193202
  (2008)
• Xiao, Chang, and Niu, to be published (RMP?)

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Thank you!
Slides http//phy.ntnu.edu.tw/changmc/Paper
Reviews

• Chang and Niu, J Phys Cond Matt 20, 193202
  (2008)
• Xiao, Chang, and Niu, to be published (RMP?)