Transcript Quasiparticle anomalies near a ferromagnetic instability

Quasiparticle anomalies near
ferromagnetic instability
A. A. Katanin
A. P. Kampf
V. Yu. Irkhin
Stuttgart-Augsburg-Ekaterinburg
2004
Motivation
Study of low-dimensional itinerant ferromagnetism:
• Layered systems
• Thin magnetic films
• Surface electronic states in 3D materials (ARPES)
The properties of layered systems are expected to be
completely different from cubic systems
The T-m phase diagram: 2D vs 3D
T
TC   0
PM
Ordered phase
(Fermi liquid,
well-defined QP)
m
QPT
T
T*0
PM
 = C exp(T*/T)
(Fermi liquid,
well-defined QP)
RC
QPT

3D
ordered phase m
TC0/ln(t/t')
2D
+ small
interlayer
coupling
How do the physical properties evolve in the vicinity of a
ferromagnetic instability ?
Theoretical predictions for NMR
Q() = ( I  , I  ) = i 
d '
T
Im I  | I   '
  '(   ')
AFM:
MF theory:
Q ( ) =
FM: Q()
Q()
2i I ( I  1)
3   A Siz 
 A SQ 
z
PM state, 2D system:
A S 
A SQ 
z
z
Q()
2 I ( I  1)
Q( ) =
, m ()
3 i  m ( )
iA2 S 2

 AS
AS
V. Yu. Irkhin and M. I. Katsnelson,
Z. Phys. B 62, 201 (1986); Eur. Phys. J. B 19, 401 (2001)
Theoretical predictions for ARPES
A(kF,)
A(kF,)
T
T  T*
T*


• similar to an AFM, where the suppression of the spectral weight
is due to opening of a gap
Simple RPA-like calculation
( k , i n ) = U 2T
 ( q, ) =
G0 (k  q, i n  in )  (q, in )
q,in
 0 ( q,  )
0
 2
1  U 0 ( q,  )   q 2  i / q
The source of potential
divergencies
[0 (q,)  A  Bq
2
 iC / q]
3D: Im (kF,0) is only weakly (logarithmically) divergent at the
magnetic phase transition temperature T = Tc

2D

Im (kF ,0)  T   tO((T / t)2/3),
T 0

*
Bare U:  (T  T ) = 
Renorm.
Uef:
 1 (T  T * ) = C exp( T * / T )
Im (kF,0) is divergent at T0.
This type of divergences was discussed earliar in the AFM context
(A. M. Tremblay) and for gauge field theories (P. Lee et al.)
Interpretation of the results
 (T / t ) 2 (  e k )   t 1
Re ( k,  )  
1
Tt
(ln

)
/(


e
)


t


k

The spectral function depends on -ek only
A(k,)
ek = 0
ek  

 Pre-formation of the two split Fermi surfaces already in the
PM phase`at low T<<T*
Qualitative physical picture

What is the nature of the anomalies
found in self-energy and spectral functions ?
Formation of the two pre-split Fermi
surfaces already in PM phase
Formation of dynamic “domains” of
electrons with certain spin projection

Approximations
 We have neglected:
o momentum- and frequency dependence of the interaction;
contributions of the channels of the electronic scattering other
than the ph channel
o self-energy and vertex corrections beyond the RPA-like
diagrams
Functional renormalization-group approach
.
= VT  (GT ST  ST GT)  VT
VT =
.
T
T0
=
=
ST *  dT ' [VT ' * ST ' * GT ' *VT ' ]
T
a1
 ( z) =
a ( z  i1 )
1 2
a ( z  i2 )
1 3
1  ...
T 1/ 2
GT =
in  e k
ST =
in  e k
1
2T 1 / 2 (in  e k ) 2
Self-energy in the fRG

Results (Hubbard model, U = 4t, t'/t = 0.45,
vH band filling n = 0.47, T = 0.1t)
Self-energy and vertex corrections
Two types of corrections to the results of non-selfconsistent approach:

Self-energy corrections in the internal Green
functions

Vertex corrections
(q,in)
(k,in)
Self-energy and vertex corrections
Dyson equations
 ( q, in ) = T
( k , i n ) = T
 (k , k  q; i n , i n  in )G(k , i n )G(k  q, i n  in )
k ,i n
 (k , k  q; i n , i n  in )G(k  q, i n  in )  (q, in )
q,in
(k , k  q; i n , in  i n ) =  (k , k  q; i n , in  i n )[1  U ( q, in )]

Similar to QED: no equation for  !
Ward identity:
 (k, k; i n , i n ) = 1 
(k, i n )
h
 Similar approach was applied by Edwards and Hertz to
the problem of strong FM
Approximations which are used
 (k, k  q; i n , i n )  (k, k; i n , i n )
Justification:  (q,0) 
0
 2  q 2
is strongly enhanced at q=0
+1/N expansion where N is the number of spin components
The self-consistent solution
with vertex and self-energy
corrections:
3
( 20   2   2  120  2  220 )
10
3
 ( k , k , ,  ) =
[2 4  920 2  340 
5020 2
( k ,  ) =
(320  2 2 )  2  120  2  220 ]
Im   0 (  0)
Self-consistent without
vertex corrections
(analogue of FLEX):
1
( k ,  ) = (    61 / 2  0
2
   61 / 2  0 )
 (k , k , ,  ) = 1
Results of the solution
Other observable quantities
The density of states

 (e ) =  A(k, e ) =  de  0 (e ) A(e  e ' )  [  0 (e  )   0 (e  )]
1
2
k
The density of states is split already in PM phase

Static magnetic susceptibility
0 =
0
1  U ef 0
0 = 

 
i
 (k , k ; i n , i n )G (k , i n )2 =  de
k ,i n
f ( z ) =  ( z )G( z )2
n

f (i n  e ) 0 (e )  de

e 
2
2
0 (e )
Triplet pairing
 triplet =
 pptriplet (k ,k ;i n ,i n )G(k , i n )G(k ,i n ) =  de  g (i n ,e ) 0 (e )
k ,i n
i n
triplet
g ( z, z ' ) =  pp
( z, z ' )G ( z  z ' )G (  z  z ' ); g ( z,0) =  f ( z )
g(i,ek)
ek
ek



Enhancement of the triplet pairing amplitude at small , ek  
Quantum critical regime
What about QC regime ?


RC: 
1
exp( T / T )  T
*

T
QC :  1 ~ T 
T*0
RC
QPT
m

There are no solid theoretical
results for the value of exponent 

11/3  the quantum spin fluctuations are less important
than classical, the “inverse qp lifetime” | Im  | T  but there
are no well-defined qps
1/3  the quantum spin fluctuations are more important
than classical, the guess  “scattering rate” | Im  | T 2 / 3
requires verification vs. vertex corrections; coincides with
the result by W. Metzner et. al. near Pomeranchuk
instability
Summary




Ferromagnetic fluctuations invalidate quasiparticle
picture at the paramagnetic FS at low T
New quasiparticles emerge at the points of the Brillouin
zone with ek    ( is the ground-state spin splitting)
The density of states is pre-split at T « T*
Triplet pairing amplitude is greatly enhanced at the
ferromagnetic FSs already in the paramagnetic state
Future perspectives
Non-perturbative semi-analytical tool of investigation of
self-energy and vertex corrections in spin systems with
strong forward scattering –
Ward identity approach + 1/N expansion
Possible extensions and applications:

Inclusion of quantum fluctuations

Long-range ferromagnetic order
More accurate description
of QC regime
Comparison with
experimental ARPES data
Description of criteria of ferromagnetism and spectral
properties of 2D and 3D ferromagnetic systems between
limits of weak (Moriya theory) and strong (saturated)
(Edwards-Hertz approach) ferromagnetism

Extension to vH singularity problem
Possible experimental implication

Layered manganite compound La1+xSr2-xMn2O7
TC=126K
Phys. Rev. Lett. 81, 192 (1998)
Phys. Rev. B 62, 1039 (2000).
Spectral properties in mean-field theory
 Direction of the magnetization along z-axis
A(kF,)
G (k , e ) =
1
  e k    i 0


 Direction of the magnetization perpendicular to the spin
quantization axis:
 = U  ci ci   ci ci   / 2
1
1
1
G (k, e ) = (

)
2   e k    i0   e k    i0
A(k,)


