Recent%20progress%20in%20the%20theory%20of%20Anderson%20localization

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Recent progress in the theory of Anderson
localization

• Akira Furusaki (RIKEN)

??????
Collaborators
Piet Brouwer (Cornell)
Ilya Gruzberg (Chicago)
Christopher Mudry (PSI)
Andreas Ludwig (UC Santa Barbara)
Shinsei Ryu (UC Santa Barbara)
Hideaki Obuse (RIKEN)
Arvind Subramaniam (Chicago)
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Anderson localization
Anderson (1957)
A non-interacting electron in a random potential
may be localized.
Gang of four (1979) scaling theory
Weak localization
P.A. Lee, H. Fukuyama, A. Larkin, S. Hikami, .
well-understood area in condensed-matter physics
Unsolved problems
Theoretical description of critical points
Scaling theory for critical phenomena in
disordered systems
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• Introduction
• New universality classes
• Scaling approach in 1D
• 2D (symplectic class)

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A non-interacting electron moving in random
potential
Quantum interference of scattering waves
Anderson localization of electrons
extended
localized
localized
localized
E
Ec
extended
critical
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Scaling theory (gang of four, 1979)
All wave functions are localized below two
dimensions!
A metal-insulator transition at ggc is
continuous (dgt2).
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3 symmetry classes (orthogonal, unitary,
symplectic)
symplectic class ? time-reversal,
spin-rotation
spin-orbit interaction
anti-localization
critical point in 2D
Metal-insulator transition in 2D
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Anderson metal-insulator transition isa
continuous quantum phase transition driven by
disorder

• Dimensionality d
• Symmetry of Hamiltonian
• time-reversal symmetry
• SU(2) rotation symmetry in spin space
• Wigner-Dyson ensemble of random matrices
• time reversal
  symmetry spin rotation symmetry
• orthogonal
• unitary
• symplectic

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average, variance, and higher moments
Conductance is a random variable.
In diffusive regime, fluctuations are universal
Universal Conductance Fluctuations (Lee Stone,
Altshuler 1985)
Beyond diffusive regime and near a critical
point, moments become large.
RG flows of high-gradient operators in NLsigma
model
(Altshuler, Kravtsov,
Lerner 1986, )
We need RG of the whole distribution function
Functional RG
Successful example Fokker-Planck eq. for
Lyapunov exponents for 1D wires
cf elastic manifolds in random potential
(Le Doussal, Wiese, ..)
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• Introduction
• New universality classes
• Scaling approach in 1D
• 2D (mostly symplectic class)

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New universality classes (1) BdG
(Altland Zirnbauer 1997)
Bogoliubov-de Gennes quasiparticles in a
superconductor
random Hamiltonian
no self-consistency
particle-hole symmetry
New universality classes near E0
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Gorkov, Kalugin (1985) Schmitt-Rink, Miyake,
Varma (1986) P.A. Lee (1993) Senthil, Fisher
(1999)
Class CI Disordered d-wave superconductors
SR ? TR ?
localization length
density of states
weak-localization
2D
Class C in magnetic field
disorder
SR ? TR
spin insulator
Spin (thermal) quantum Hall fluid
spin QHF
2D
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Class D
SR TR
Majorana ferimons in random potential
random-bond Ising model, Moore-Read pfaffian
state, etc.
vortex in p-wave DIII-odd, B (D.A. Ivanov, 2001)
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New universality classes (2) chiral
Random-hopping models (electrons hop between A
and B sublattices only)
Dirac fermion coupled to random vector potential
(Ludwig et al., 1994 Mudry, Chamon Wen, 1996)
chiral universality classes
(chiral RMT in QCD)
time reversal symmetry spin rotation
symmetry
chiral orthogonal (BDI)
chiral unitary (AIII)
chiral symplectic (CII)
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1D random-hopping model
random XY chain (via Jordan-Wigner tr.)
real-space RG integrating out a bonding
(singlet) state on the strongest bond
random-singlet phase
(Dasgupta S.-k. Ma, 1980 Bhatt P.A. Lee,
1982 D.S. Fisher, 1994)
(Westerberg, AF, Sigrist, P.A. Lee, 1995)
abundance of low-energy excitations
1D Dyson singularity
(Dyson, 53)
2D Gade singularity
(Gade, 1993 Motrunich, Damle Huse, 2002
Mudry, Ryu AF, 2003)
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• Introduction
• New universality classes
• Scaling approach in 1D (functional RG)
• 2D (mostly symplectic class)

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Random-matrix approach to transport in quasi-1D
wires
(Dorokhov, 1982 Mello, Pereyra, Kumar, 1988
Beenakker, 1997)
Transfer matrix
Eigenvalues of are
Lie group
Landauer conductance
radial coordinates
symmetric space (E. Cartan)
coset
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Distribution function of
Lets imagine time and
coordinate variable of n-th particle.
time evolution of motion of particles
Brownian motion of in symmetric space
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Diffusion equation for particles
Fokker-Planck equation (DMPK equation)
standard BdG classes
chiral classes
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Diffusion equation for particles
Fokker-Planck equation (DMPK equation)
fixed point
metal
functional RG equation
Describes RG flows from weak (diffusive) to
strong-coupling (localized) regime.
universal scaling behavior
(average) density of states
(Titov, Brouwer, AF, Mudry, 2001)
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Chiral universality classes
(diffusive regime) no weak-localization
correction
even-odd effect
odd N
Dyson singularity
even N
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BdG universality classes
(diffusive regime)
weak-localization corrections
SR ? (CI, C)
SR (DIII, D)
Fokker-Planck equations can be solved exactly for
U, chU, CI, DIII classes
(by mapping to free fermions)
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Alternative approach 1D (supersymmetric)
non-linear sigma model
equivalent to Fokker-Planck approach at
exact results SUSY method
standard classes (Zirnbauer 1992)
CI, DIII, chU (Lamacraft, Simons, Zirnbauer
2004)
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Symplectic universality class
(1) Zirnbauer (1992)
(2) Brouwer Frahm (1996) corrected Zirnbauers
result
(3) Ando Suzuura (2002) found in nanotubes
Odd number of Kramers pairs
(4) Takane (2004) Fokker-Planck equation with
N2m1
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Symplectic universality class
(1) Zirnbauer (1992)
(2) Brouwer Frahm (1996) corrected Zirnbauers
result
(3) Ando Suzuura (2002) found in nanotubes
Odd number of Kramers pairs
(4) Takane (2004) considered Fokker-Planck
equation with N2m1
(5) Kane-Mele model for graphene
(2005)
(no disorder)
with disorder ???
(Onoda, Avishai Nagaosa, 2006)
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• Introduction
• New universality classes
• Scaling approach in 1D
• 2D (attempt to understand symplectic class)

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Anderson transitions are continuous phase
transitionsdriven by disorder.
Examples of critical points in 2D Anderson
localization
symplectic class, QHE(unitary class, class C,
class D)
What kind of field theory describes a 2D critical
point driven by disorder?
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Field theory for 2D critical points driven by
disorder?
(1) Quantum Hall plateau transitions (unitary
class)
Nonlinear sigma model (Pruisken, ..)
Super spin chain (D.H. Lee, .)
WZNW model (Zirnbauer, Tsvelik et al.,)
(2) Quantum Hall plateau transitions (class C)
equivalent to classical percolation (Gruzberg,
Ludwig, Read 1999)
(3) Symplectic class (spin-orbit scattering)
Q Conformal invariance at these critical points?
Q What kind of CFTs describe the disordered
critical points in 2D?
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Multifractality scaling behavior of moments of
(critical) wave functions
Critical wave function at a metal-insulator
transition point
multifractal exponents
fractal dimension
Continuous set of independent and universal
critical exponents
singularity spectrum
measure of r where
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To examine the presence of conformal invariance,
Consider disordered samples with open boundaries
(surface),
Change the shape of samples and see how wave
functions change.
Surface multifractality
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Surface multifractality (Subramaniam et al.,
2006)
disordered sample with open boundaries
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Surface critical phenomena in conventional phase
transitions
At conventional critical points
Surface critical exponents are different from
bulk exponents
conformal mapping
Boundary CFT (Cardy 1984)
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Multifractality field theory (Duplantier
Ludwig, 1991)
local random events at position
scaling operator in a field theory
Conformal mapping
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SU(2) model for the symplectic class
(Asada, Slevin, Ohtsuki, 2002)
Tight-binding model on a 2D square lattice
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Numerical simulations
SU(2) model
For each sample we keep only one eigenstate with
E closest to 1.
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bulk, surface, and corner multifractal spectra
(Obuse et al., cond-mat/0609161)
Bulk, surface, and corner are all
different. Surface contributions dominate at
large q in the whole cylinder .
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surface
Colored thin curves Conformal Invariance !!
rounding of cusps at finite-size
effect
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Summary
Anderson metal-insulator transition as a
disorder-driven quantum phase transition
Functional RG (infinite number of coupling
constants)
Open questions
Field theories for random critical fixed points
in 2D?
Non-unitary CFT
(String/gauge theory duality, AdS/CFT)
SUSY nonlinear sigma model
Interactions
Finkelstein, Altshuler, Aronov, Lee, Fukuyama, .
Weak-coupling (weak-localization) regime is well
understood.
Strong-coupling regime?
Finite-temperature phase transition? (Basko,
Aleiner, Altshuler, 2005)
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Acknowledgments
Ilya Gruzberg (Chicago)
Piet Brouwer (Cornell)
Christopher Mudry (Paul Scherrer Institut)
Andreas Ludwig (UC Santa Barbara)
Hideaki Obuse (RIKEN)
Shinsei Ryu (UC Santa Barbara)