Localization, interaction and the qHe
Adrianus M.M. Pruisken
Mikhail A. Baranov
Igor S. Burmistrov
Mikhail Voropaev
Physical objectives:
 quantum theory of metals and insulators
 integer and fractional quantum Hall effect
2D spinless electrons

+ quenched disorder Vdis (r )
 
+ e-e interaction U 0 (r  r )
+ ( magnetic field B)
The fermionic path integral




S   d r  d  (r , )[     H 0  Vdis (r )] (r , ) 

 


 


1



, )
d
r
d
r
d


(
r
,

)

(
r
,

)
U
(
r

r
)

(
r
,

)

(
r
0


2
1
The effective action:

Matsubara frequency space
   n   T (2n  1)
1
1

Electronic scattering processes   el   n   el

Replica trick N r  0

Q field method


 ( r , )    ( r , )
 
  (r ,  n )  (r ,  m )  Qnm
(r )
replica indices

space variable
r
Matsubara frequency indices
Nonlinear constraint
nm

Q 2 (r )  1
2
The effective action

Seff [Q, A ]  S [Q, A]  S F [Q]  SU [Q, A ]

free electron part

 2  xy
 xx
S [Q, A]  
tr [ D, Q] 
 ab  tr Q [ Da , Q] [ Db , Q]

8
8

singlet interaction term

S F [Q]   z T   tr I n Q tr I nQ  4tr 
  ,n


Q

the (Coulomb) term
  

i nB 

2
1
SU [Q, A ]   T    tr I n Q(r ) 
A0 (r , n )  2
B(r , n ) 
2

T
T

 ,n 


  

 U r -r   tr I


n
i nB 



1


Q(r ) 
A0 (r ,- n )  2
B(r ,- n )
T
T 

A.M. Finkelstein, JETP Lett. 37, 517 (1983), Sov. Phys. JETP 59, 212 (1984)
A.M.M. Pruisken, M.A. Baranov, and B. Škorić, Phys. Rev. B 60, 16807 (1999)
3
The physical parameters of the effective theory:
 xx ,  xy
mean field conductances
z
singlet interaction amplitude
T
temperature
U (q) 

1  U 0 (q )
…
bare interaction
U 0 (q)
 n

 
n
nB 
B
thermodynamic DoS
…
4
Cutoff N m in frequency space
Q  T 1 T
I 
 
n mk
    k m,n

 mk
 m    mk
mk  sign m    k ,m
F algebra
5
U(1) gauge invariance

covariant derivative

D i
 
 A (r , n ) I n
 ,n

gauge transformations
A  A    
Q  W 1Q W
where

W  exp i


 ( n ) I 

 ,n



n
6
Overview
1.Perturbative RG
 one loop
A.M. Finkelstein, JETP Lett. 37, 517 (1983), Sov. Phys. JETP 59, 2121 (1984)

two loop
M.A. Baranov, A.M.M. Pruisken, and B. Škorić, Phys. Rev B 60, 16821 (1999)
M.A. Baranov, I.S. Burmistrov, and A.M.M. Pruisken, Phys. Rev B 66, 075317 (2002)
2.Nonperturbative RG functions (instanton gas)
A.M.M. Pruisken and M.A. Baranov, Europhys. Lett. 31, 543 (1995)
3.Complete microscopic theory of Luttinger liquid of
quantum Hall edge
A.M.M. Pruisken, M.A. Baranov, and B. Škorić, Phys. Rev B 60, 16838 (1999)
B. Škorić and A.M.M. Pruisken, Nucl. Phys. B 559, 637 (1999)
4.Linear response and continuity equation at quantum
level
A.M.M. Pruisken, M.A. Baranov, and I.S. Burmistrov, to be published
5.Crossover RG functions (singlet and triplet cases)
A.M.M. Pruisken, M.A. Baranov, and I.S. Burmistrov, to be published
7
F algebra
N m finite
N m infinite
finite size Q
infinite size Q
free electrons
interacting electrons
one particle problem
many-body problem
ordinary -model
Finkelstein theory
Very different physics…
How can it be undestood?
The effective action ( g  1 /  xx )

1
2


Seff [Q]  
tr
[

Q
]


z
T
c
tr
I
Q
tr
I
n
nQ  4tr 
  
8 g
 ,n
c0
free electrons:
short ranged e-e interaction:
the Coulomb interaction:

Q

0  c 1
c 1
8
Background field methodology
[analogous to A.M. Polyakov, Phys. Lett. B 59, 79 (1975) for vector model]
Q  t 1Q t
,
Q are “fast” modes
“large” matrices
t  SU (2 nm N r )
t are “slow” modes
“small” matrices
By integrating over Q we obtain the action in terms of
q  t 1 t
the local quantity
exp Seff [q]   DQ exp Seff [t 1Q t ]
9
bare theory
background field
action
Seff [Q ]

g  g ( L0 )

g   g (L )
c  c( L0 )

c  c(L)
z  z ( L0 )

z   z (L)
S eff [q]
RG functions
dg
  ( g , c)
d ln L
dc
  c ( g , c)
d ln L
d ln z
  z ( g , c)
d ln L
10
Crossover RG functions
(in 2+ dimensions, T=0)
dg
1 c


 g  2 g 2  f 
ln( 1  cf )
d ln L
c


dc
  gc(1  cf )
d ln L
d ln z
  gcf
d ln L
where
NOTE:
f 
L2
L L
2
N m is finite
f 0:
short distances:
2
m

Lm2  8 2 zTg N m
and
f=0 at
T 0
Free particle theory ( N m finite)
L  Lm
dg
 g  O( g 3 )
d ln L
dc
  gc
d ln L
d ln z
0
d ln L
11
f  1:
Many-body theory ( N m infinite )
L  Lm
Long distances:
dg
 1 c

 g  2 g 2 1 
ln( 1  c)
d ln L
c


dc
  gc(1  c)
d ln L
d ln z
  gc
d ln L
Free particles
Short-ranged
Long-ranged(Coulomb)
interaction
interaction
N m finite
N m infinite
N m infinite
f=0
f=1
f=1
c<1
c=1
No F invariance
F invariance
Fermi liquid theory *
Non-Fermi liquid theory
No F invariance
*
NOTE: also for f<1
12
Conclusions

Frequency cutoff N m = crossover between free
electron approximation and many body theory

RG, F invariance, F algebra are fundamental tools to
understand this crossover

General knowledge of theory at finite N m
(renormalizability, background field methodology)
can be extended to include problem of interacting
electrons

Finkelstein theory is fundamental theory of
localization and interaction effects
13