1. Non-Hermitian quantum mechanics and localization
Naomichi Hatano
Aoyama Gakuin University, Japan
Collaborators
D.R. Nelson (Harvard)
A. Zee (UC Santa Barbara)
N.H. and D.R. Nelson:
PRL 77, 570 (1996)
PRB 56, 8651 (1997)
PRB 58, 8384 (1998)
N.H.:
Physica A 254, 317 (1998)

2. Why non-Hermitian? - provides a new viewpoint of Hermitian physics
- expresses physical
phenomena effectively
(1) Anderson localization
Localization length
(2) Resonant states
Resonance lifetime
(3) Flux-line pinning in
type-II superconductors
Vortex depinning transition

3. Outline 1. Non-Hermitian Anderson model and delocalization
2. Motivation I ____
Localization length
3. How does the
delocalization happen?
4. Numerical examples
5. Motivation II ____
Resonant states
6. Motivation III ____
Vortex depinning in
type-II superconductors

4. H  ( p  i g ) m V x Non-Hermitian Anderson model Continuum model2 m V x
g: “imaginary vector
potential” (constant)
V(x): random scalar
potential
g=0: (conventional)
1-electron Anderson model

5.   H    e e  Lattice model t  x  e x  x  e x  V x x 2g  e x  e x   2 x   1   e  g  e x  e x     V x x x x
asymmetric hopping vs. randomness

6. (iii) inverse localization length: = gc
Non-Hermitian delocalization
g=0: localized states
(i) g: delocalized
at g = gc
(ii) delocalization ~
complex eigenvalue
(iii) inverse localization
length: = gc

7. Imaginary gauge transformation
How does it happen?
Imaginary gauge transformation g H  ( p  i ) 2 m V x
: real, fixed H    g

8. integrable for g only

9. Motivation (1) Localization length of Hermitian Anderson model
Hermitian case ( g = 0 ):

10. Delocalization of eigenfunction Note: Particle Density H     
1D: L x = (Diagonalized 1000  1000 matrix)
Note: Particle Density H  R   L      ÷

11. Complex spectrum and delocalization g g real eigenvalue
1D: L x = (Diagonalized 1000  1000 matrix)
real eigenvalue
localized state
g
complex eigenv.
delocalized state
g

12. Localization length 1D: L x = 1000 (Diagonalized 1000  1000 matrix)
Lloyd model (Lorentzian random distribution):
Hirota & Ishii (1971), Brezin & Zee (1998)

13. Motivation
(3) Vortex pinning in
high-Tc superconductors

14. Path-integral mapping
Nelson and Vinokur, PRB 48, (1993) TM : T (   ) 
exp(  H Z   D x  cl E      e  f L H i

15. Pinning and localization
localized fn.
depinning
delocalized fn.

16. Motivation Resonance lifetime Unintegrable eigenfn.  ( r ) ~ e
(2) Resonant states
Resonance lifetime
Unintegrable eigenfn. 
res ( r ) ~ e i k   as  

17. H  V r ;   x Aguilar, Balslev & Combes, Comm. Math. Phys. 22 (1971) i g ) 2 m V r ;   x
Aguilar, Balslev & Combes,
Comm. Math. Phys. 22 (1971)
complex rotation:

18. Other applications random gauge field CDW pinning
Chen et al.,PRB 54, (1996)
Fokker-Planck equation
Nelson and Shnerb, PRE 58, 1383 (1998)
Dirac Fermion in
random gauge field
Mudry et al. PRL 80, 4257 (1998)
PT symmetry
Bender and Boettcher, PRL 80, 5243 (1998)
complex spectrum  PT symmetry breaking
Interactions
Lehrer and Nelson, cond-mat/
Boson crystal and delocalization