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Quantum Critical Behavior
of
Disordered Itinerant Ferromagnets
D. Belitz – University of Oregon, USA
T.R. Kirkpatrick – University of Maryland, USA
M.T. Mercaldo – Università di Salerno, Italy
S.L. Sessions – University of Oregon, USA
• Introduction: the relevance of quantum fluctuations
• phenomenology of QPT
• Quantum critical behavior of disordered itinerant FM
• Theoretical backgrounds:
• Hertz Theory
• Disordered Fermi Liquid
• The new approach
• Microscopic theory
• Derivation of the effective action
• Behavior of observables
• Experiments
• Conclusion
• QPT = Phase transitions
at T=0, driven by a nonthermal control parameter
• quantum mechanical
fluctuation are relevant
Experiments have revealed that rich new
physics occurs near quantum critical points
Fig. by T. Vojta
from cond-mat/0010285
Quantum criticality 2
In the yellow region (quantum critical region) the
dymical properties fo the system are influenced by the
presence of the QCP
Dashed lines are crossover lines
 Quantum Critical Behavior of Itinerant Ferromagnets
The FM transition in itinerant electron systems at T=0 was the
first QPT to be studied in detail [Hertz PRB 14, 1165 (1976)]
Hertz theory: due to the mapping dd+z, the transition in the
physically interesting dimension is mean field like.
This conclusion is now known to be incorrect
The reason for the breakdown of the mean-field
theory is the existence, in itinerant electron
systems, of soft or massless modes other than
the order parameter fluctuations.
 Disordered Fermi Liquid
Disordered e-e correlations lead to non analyticities in electron systems.
3D
 The conductivity has a T -temperature dependence [Altshluler, Aronov]

(T )  0 1 const. T

 The density of states has a  -energy dependence [Altshuler, Aronov]

N ()  NF 1  const. 

 The phase relaxation time has a 3/2 –energy dependence [Schmid]
These effects are known as weak localization effects
• Starting Model


0
0
S   d dx a x,  a x,   dH ()
H ()   dx


1
2m

 a x,   x     a x,  
2
2
1
dx dy u x  y  a x, b y ,  b y ,  a x,  

2
We keep explicitly all soft modes
F Critical modes (magnetization)
F Diffusive modes (particle-hole excitations which exist
in itinerant electron systems at T=0)
Belitz, Kirkpatrick, Mercaldo, Sessions, PRB 63, 174427 (2001); ibidem 174428 (2001)
• Interaction term
Sint=Sint(s) + Sint(t)
S
(s)
int
(t )
Sint

 s
2

 t
2
1
dx u (x)

0 d dx nc x, nc x, 
2
nc x,    a x,  a x, 



d dx ns x,   ns x,  

0
ns x,    a x,   ab  b x, 

t 
 
Using the Hubbard-Stratonovich transformation we
decouple the Sint(t)
term and introduce explicitly the
magnetization in the problem
S
(t )
int


  d dx M x,   M x, 
0



 2t  d dx M x,  ns x, 

0
• Rewrite the fermionic degrees of freedom in terms of
bosonic matrix fields
This formulation is particularly well suited for a separation of
soft and massive modes
• Write explicitly soft and massive modes
• Integrate out massive modes
Aeff = AGLW[M] + ANLM[q] + Ac[M,q]
details
power counting analysis
of the effective action
Ma's method to
identify simple fixed
points
We use physical arguments to
determine which coupling
constants should be marginal,
and then check whether this
choice leads self consistently
to a stable fixed point
Hertz fixed point is unstable
M  0,
1
v ,
2
zc  4
New fixed point marginally stable
M  4  d ,
v
1
, zc  d
d 2
N.B. there are 2 time scales in the problem:
the diffusive time-scale (zd=2)and the critical one (zc=4 or d)
pert.
theory
Asymptotic critical behavior is
not given by simple power laws
ln g (ln b)
ln b
ln g (ln b)
zd  2 
ln b
zc  d 
ln g (ln b)
  4d 
ln b
1
ln g (ln b)
 d 2

ln b
Power laws with scale
dependent critical exponents
b RG length rescaling
factor
[ln(c(d ) x)]2 
g ( x)  exp

ln
(
d
/
2
)


The values of critical exponents are reflected in the
behavior of the single particle density of states and
the electrical conductivity across the transition
At T=0
of states
N(F +)=NF[1+Density
m<qnm(x)qnm(x)>i   +i0 ]
n
To obtain the dependence at T0 one can use scaling
theory:
• The leading correction to N, N, can be related to a correlation function that has
scale dimension – ( d – 2 )
Expect scaling law:
N(t,,T) = b-(d-2) FN(tb1/,  bzc,Tbzc)
( d 2) / d


  1 F 


N ()  N F 1  cN  g d ln  




 F





d=3

lnln(1/ )2
N ()  NF 1 c  e
1/ 3

At T=0 =8/G
conductivity
To obtain the dependence at T0 one can use scaling theory:
 consist of a backgorund part that does not scale (since this quantity is
unrelated to magnetism  [] = 0) and a singular part that does [] = – (d – 2)
• unrelated
to magnetism
[] = 0 = b-(d-2) F(tb1/, Tbzd, hbzc)
Expect scaling
law: 
(t,T,u)
• perturbation theory   depends on critical dynamics (z) and on
( d 2 ) / d
leading irrelevant
operator u


 T 1 TF 



(T )   0 1  c  g d ln T 
• u is related to diffusive
electrons  [u] = d – 2
T


 F






d=3
(T )  0 1  c T e
1/ 3
ln ln(1/ T ) 2

Exp 1
Disordered Itinerant FM
Fe1-xCoxS2
DiTusa et al. 2003
cond-mat/0306
This compound
shows a PM-FM QPT
for x~0.032
Measure of conductivity for different applied fields
For H=0,  ~ T0.3
The unusual T dependence of  is a reflection of the critical behavior of the
ordering spins
Exp 1
Scaling plot of the conductivity
Fe1-xCoxS2
DiTusa et al. 2003
Thermodynamics quantities
The thermodynamic properties near the phase transition
can all be obtained by a scaling ansatz for the free energy.
Natural scaling ansatz for f (free energy density):
f(t,T,h) = b-(d+zc) f1(tb1/, Tbzc, hbzc)
+ b-(d+zd) f2(tb1/, Tbzd, hbzc)
m(t )  t g  d 12 lnm1t  f / mh  t 
2 /(d  2 )
  =2 
m(h)  h g  ln   f / T
1+   = –d/z
1
1


T
2
2
C
c

 C (T )  g  d ln 
T   f / h
1
d
 s (t )  t
1
1 2/ d
h
C
s
2
m  h21/   = zc/2
s  t 
 =1
Feedback of critical behavior on weak-localization corrections
of relevance for
• (indirect) measurements of ferromagnetic quantum critical behavior
• understanding breakdown of Fermi liquid behavior in the vicinity of QCPs
Conductivity and DOS acquire stronger corrections to Fermi-liquid behavior
• Quantum critical behavior for disordered itinerant
ferromagnets has been determined exactly
• Measurements of conductivity and density of states in
the vicinity of the quantum critical point are the easiest
way to experimentally probe the critical behavior