1. Recent progress in the theory of Anderson localization
Akira Furusaki (RIKEN)
理化学研究所
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)

2. 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

3. Introduction
New universality classes
Scaling approach in 1D
2D (symplectic class)

4. 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

5. Scaling theory (gang of four, 1979)
Conductance changes when system size is changed.
Metal:
Insulator:
All wave functions are localized below two dimensions!
A metal-insulator transition at g=gc is continuous (d>2).

6. 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

7. Anderson metal-insulator transition is a 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

8. Conductance is a random variable.
average, variance, and higher moments
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, …..)

9. Introduction
New universality classes
Scaling approach in 1D
2D (mostly symplectic class)

10. New universality classes (1) BdG
(Altland & Zirnbauer 1997)
Bogoliubov-de Gennes quasiparticles in a superconductor
random Hamiltonian
no self-consistency
particle-hole symmetry
energy eigenvalues:
is a special point.
New universality classes near E=0
time reversal symmetry spin rotation symmetry CI C
DIII D

11. Class CI: Disordered d-wave superconductors
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
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)

12. 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)
energy eigenvalues:
is a special point.
time reversal symmetry spin rotation symmetry
chiral orthogonal (BDI)
chiral unitary (AIII)
chiral symplectic (CII)

13. 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)

14. Introduction
New universality classes
Scaling approach in 1D (functional RG)
2D (mostly symplectic class)

15. Random-matrix approach to transport in quasi-1D wires
(Dorokhov, 1982; Mello, Pereyra, & Kumar, 1988; Beenakker, 1997)
: # of channels
mean free path:
Localization length:
Transfer matrix:
Eigenvalues of are
Lie group
Landauer conductance:
“radial coordinates”
symmetric space (E. Cartan)
coset

16. Distribution function of
Let’s imagine “time” and “coordinate variable of n-th particle”.
“time” evolution of motion of “particles”
Brownian motion of in symmetric space

17. Diffusion equation for “particles” = Fokker-Planck equation (DMPK equation)
standard & BdG classes
chiral classes

18. functional RG equation
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)

19. Chiral universality classes
(diffusive regime) no weak-localization correction
even-odd effect
odd N:
Dyson singularity
even N:

20. 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)

21. 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)

22. symplectic o u
Symplectic universality class
(1) Zirnbauer (1992)
(2) Brouwer & Frahm (1996) corrected Zirnbauer’s result:
(3) Ando & Suzuura (2002) found in nanotubes
Odd number of Kramers pairs
(4) Takane (2004): Fokker-Planck equation with N=2m+1

23. Symplectic universality classo u
Symplectic universality class
(1) Zirnbauer (1992)
(2) Brouwer & Frahm (1996) corrected Zirnbauer’s result:
(3) Ando & Suzuura (2002) found in nanotubes
Odd number of Kramers pairs
(4) Takane (2004) considered Fokker-Planck equation with N=2m+1
(5) Kane-Mele model for graphene
(2005)
(no disorder)
with disorder ???
(Onoda, Avishai & Nagaosa, 2006)

24. Introduction
New universality classes
Scaling approach in 1D
2D (attempt to understand symplectic class)

25. Anderson transitions are continuous phase transitions driven by disorder.
At ordinary continuous phase transition points without disorder,
correlation length
scale invariance
conformal invariance (Polyakov, 1970)
2D conformal field theory (BPZ, 1984)
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?

26. 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?

27. Multifractality: scaling behavior of moments of (critical) wave functions
Critical wave function at a metal-insulator transition point
multifractal exponents
In a metal
fractal dimension
Continuous set of independent and universal critical exponents
: anomalous scaling dimensions
singularity spectrum
: measure of r where

28. 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

29. Surface multifractality (Subramaniam et al., 2006)
bulk q
corner
disordered sample
with open boundaries
In a metal

30. 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)

31. Multifractality & field theory (Duplantier & Ludwig, 1991)
local random events at position
scaling operator in a field theory
Conformal mapping

34. SU(2) model for the symplectic class
(Asada, Slevin, Ohtsuki, 2002)
Tight-binding model on a 2D square lattice

35. Numerical simulations
SU(2) model
system size
# of samples
For each sample we keep only one eigenstate with E closest to 1. h

36. bulk, surface, and corner multifractal spectra
(Obuse et al., cond-mat/ )
Bulk, Surface, Corner（　　 ）, Whole Cylinder
Bulk, surface, and corner are all different.
Surface contributions dominate at large |q| in the whole cylinder

37. surface
Colored thin curves: Conformal Invariance !!
rounding of cusps at
finite-size effect

38. Summary Open questions:
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)

39. Acknowledgments: Piet Brouwer (Cornell) Ilya Gruzberg (Chicago)
Christopher Mudry (Paul Scherrer Institut)
Andreas Ludwig (UC Santa Barbara)
Shinsei Ryu (UC Santa Barbara)
Hideaki Obuse (RIKEN)