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Chap 17 Transport phenomena and Fermi liquid theory
• Boltzmann equation
• Onsager reciprocal relation
• Thermal electric phenomena
• Seebeck, Peltier, Thomson…
• Classical Hall effect, anomalous Hall effect
• Theory of Fermi liquid
• e-e interaction and Pauli exclusion principle
• specific heat, effective mass
• 1st sound and zero sound
Dept of Phys
M.C. Chang
Boltzmann equation
• Distribution function: f(r,k,t)
(“g” in Marder’s)
d 3k
Number of electrons within d3r and d3k
f (r , k , t )d r  2

around (r,k) at time t
(2 )3
3
For example,
d 3k
j ( r , t )  e  2
rf (r , k , t )
(2 )3
• Evolution of the distribution function
t  t  dt
f (r , k , t )  f (r  dr , k  dk , t  dt )
=f ( r , k , t ) 
f
f
f
 dr 
 k  dt
r
t
k
without collision
the phase-space density does not
change in the comoving frame
k
df f
f
f
 r  k 
0
dt t
r
k
Phase space is incompressible (Liouville’s Theorem)
Larger △r  smaller △k
r
supplementary
With collision
(due to disorder… etc):
f
f
f  f 
r  k 
 
t
r
k  t coll
Boltzmann eq.
～ Source/drain
• Transition rate for an electron at k → k’:
Wk, k’ (calculated by Fermi golden rule)
In a crowded space, one needs to
consider occupancy and summation
Wk ,k '   f k (1  f k ' )Wk ,k '
k'
• On the other hand, the transition rate for
an electron to be scattered into k
 (1  f
k
) f k 'Wk ',k
Grosso, SSP, p.404
k'
 f 
     (1  f k ) f k 'Wk ',k  f k (1  f k ' )Wk ,k ' 
 t coll k '
=  ( f k '  f k )Wk ',k
if Wk ,k '  Wk ',k
(for elastic scattering)
k'
see Marder, Sec 18.2 for more
supplementary
 f 
    ( f k '  f k )Wk ',k
 t coll k '
(valid for uniform E, B, T fields)
if f k  f k0  C  k , then
 f 
   C   (k  k ')Wk ',k
 t coll
k'
=  C  k (1  kˆ  kˆ ')W

(k  k ')
k ', k
k'
=
where
1

f k  f k0
Only the component of k’ // k
contribute to the integral
(if W depends only on θ)

  (1  kˆ  kˆ ')Wk ',k
Transport relaxation time
k'
Note: for inelastic scatterings, detailed balance requires
(1  f k0 ) f k0'Wk ',k  f k0 (1  f k0' )Wk ,k '
f 0 (k ) 
1
 ( k )    / k BT

e
 ( k ) 
e
k BT
1
 ( k ')  
Wk ',k  e
k BT
Wk ,k '
k
θ
k’
• Relaxation time approximation
f (r , k , t )  f 0 (r , k )
 f 


 
 t coll
  (k )

f 0 (r , k ) 

1
  (k )   (r ) 
exp 
 1
k
T
(
r
)
B


Unperturbed
equilibrium dist
Relaxation
(allows energy dependence)
Density and temperature
gradients are allowed
• Boltzmann eq.
f
f
f df
f  f0
r k 


t
r

k dt

f rk (t ) 
1


t

dt ' f rk0 (t ')e  (t t ')/
0
0
 ( t t ')/ df rk (t ')
or = f rk   dt 'e

dt '
t
       T
 f
f rk (t )  f rk0   dt 'e  (t t ')/ vk   
 k 0

T
r
 r
 
= f0   f
Let’s call this
t

“Chamber’s formulation”
df 0
f
f
r 0 k  0
dt
r
k
f 0 f 0  f 0


r
 T r T

T
r
       T  f 0
= 



r
T
r  

f 0
f
 r 0

k
1st, consider a system with electric field and temperature gradient,
but no magnetic field
d 3k
• Electric current j (r , t )  e  [dk ] rf (r , k , t ), [dk ]  2
(2 )3
density
=  e  [dk ]vk f 0  e  [dk ]vk  f

r
1 
k
k  eE


      T  f 0
 ( t t ')/
dt
'
e
v

eE






r
T

r

 

    T  f0 , G  E  1  Electrochemical
= v  eG 

“force”
T
r  
e r

For steady
f 
perturbations
t

     T    f 0 
 j (r , t )  e  [dk ] v v  eG 


T
r    

 T 
σ is conductivity tensor
=L11G  L12  
L11 is σ
,
T


• Thermal current j Q (r , t )  [dk ](   )r  f

density

     T    f0 
   [dk ] (   )v v  eG 


T

r

   
 T 
 L 21G  L 22  
L 22 is κT κ is thermal
,
 T 
conductivity tensor
Coefficients of transport (matrices)
 f 
L11  e 2  [dk ] v v   0 
  
 f 
L 21  e  [dk ] ( k   )v v   0 
  
 f 
L12  e  [ dk ] ( k   )v v   0 
  
 f 
L 22   [dk ] ( k   ) 2 v v   0 
  
They are of the form
Define energy-resolved conductivity
 f 
 ij( )  e 2  [dk ] ( k   ) vi v j   0 
  
L11  Λ (0)
1
L12  L 21 
Λ (1)
(  e)
1
L 22 
Λ (2)
2
(  e)
 ij ( )  e 2  [dk ]  vi v j ( k   )
then
 f 
 ij( )   d  (   )   0   ij ( )
  
One example of
Onsager relation
T=0,
Λ (0)  σ ( F )
Λ
(1)

Λ (2) 
2
3
2
3
(k BT ) 2 σ '( F )
H.W.
(k BT ) 2 σ ( F )
More on the conductivity
consider ▽T=0.
Ohm’s law:
j  σG
Fick’s law for “diffusion current”:
j  D
2
3

n

*
2m
 F 2  F n 2  F 


r
3 n r 3  r
3  F
 j  D
2 F r
  F 
2/3
1 
e r
f
• At T=0,   0    (   F )
  
• For electron gas at low T,
2
GE
   en 
σ  F
 j
e r
1 2 F
1
 D
σ 2
σ
(  e) 3 
e g ( F )
 ij  e2  [dk ]  vi v j (   F )
=e 2  F g ( F ) vi v j
vi v j
FS
FS
is an integral over the FS
Dij    F vi v j
FS
For isotropic diffusion,
Dxx    F
Einstein relation
(for degenerate electron gas)
1 2
v
3
FS
1
 vF2  F
3
• Alternative form of conductivity
 f 0 

  
 ij  e2  [dk ] vi v j  

 f 
vi v j   0   vi
k j
  
 f 0 
 f 0 


v


 i  

k
  
j 

  ( f 0 vi )
vi 
 ij  e   [dk ]  
 f0


k

k


j
j 
If τ a
2
const.
=e   [dk ] f 0  m
2
ne 2
 op  ij
m
*1
ij
～ free electron gas
f0
 f 0 
v
v



i j 
  *
   k mij (k )
k

N
mijop
Optical effective mass
= m* if the carriers are
near the band bottom
• Fourier’s law on thermal conduction
 T 
Note: this would induce electric current.
j Q  κT  L 22  

To remedy this, see Marder, p.496.
 T 
L 22  2 k B2T
κ

σ
(Wiedemann-Franz law for metals)
2
T
3 e
supplementary
More on the Onsager reciprocal relation (1931)
互易關係
• Transport processes near equilibrium
(linear transport regime)
 
ji   ij  
 x
j




(Ohm’s Law)
 
ji  Dij  
 x
j




(Fick’s Law)
 T
j   ij  
 x
j

...



(Fourier’s Law)
Q
i
• They are of the form
J i  Lij X j
Thermodynamic
“flow”

Thermodynamic
“force”
Thermodynamic
conjugate variables
• “kinetic coefficient” L is symmetric:
For example, if Ex drives a current Jy,
then Ey will drive a current Jx.
• The Onsager relation is a result of
fluctuation-dissipation theorem, plus
time reversal symmetry.
• A specific example:
the conductivity tensor of a crystal is
symmetric, whatever the crystal
symmetry is.
supplementary
Simultaneous irreversible processes
Note:
For example,
For many transport processes near
equilibrium, the entropy production is
a product of flow and force
 j   L11 L12   G 

 Q

L
L
j

T
/
T

21
22

 


also of the form:
 Y
J i  Lij  
 x
j

TSirr  J i X i



Sirr 
T
Xi X j  0
∴Entropy production ～
a thermodynamic potential
Same symmetry relation applies to this
larger α matrix
→ LT12=L21
• if force 1 (e.g., a temperature gradient) drives a
flow 2 (diffusion current), then force 2 (density
gradient) will drive a flow 1 (heat current) !
Precursors of the symmetry relation
Lij
Nature likes to stay at the lowest potential
→ minimum entropy production
(only in the linear regime)
1968
(D.G. Miller, J Stat Phys 1995)
• Stokes (1851), anisotropic heat conduction
• Kelvin (1854), thermoelectric effect
•…
http://www.ntnu.no/ub/spesialsamlingene/tekhist/tek5/eindex.htm
Thermoelectric coupling
(1) Seebeck effect (1821)
Seebeck found that a compass needle would be deflected by a closed loop
formed by two metals joined in two places, with a temperature difference
between the junctions.
or
• In the absence of electric current
Longitudinal T gradient →
electrochemical potential
(in metals or semiconductors)
 0   L11 L12   G 


 Q   L
L
j

T
/
T
   21
22  

 G  T
溫差導致電位差
• Seebeck coefficient
(aka thermoelectric power) 熱電功率
   L11 
1
L12
 2 kB2T  '

T
3 e 
typical values observed: a few μV / K
(Bi: ~ 100 μV / K)
(2) Peltier effect (1834)
Peltier found that the junctions of dissimilar
metals were heated or cooled, depending
upon the direction of electrical current.
• In a bi-metallic circuit without T gradient,
a current flow would induce a heat flow
(in metals or semiconductors)
 j   L11 L12   G 
 Q
 
L
L
j

21
22
 0 
 
 jQ   j
  L 21  L11    T
1
(If L11 and L21 commute)
• In practical applications of Seebeck/Peltier,
the figure of merit (dimensionless) is (Prob 7)
 2
ZT 
T

High electric conductivity and low
thermal conductivity is good.
(>< Wiedemann-Franz law)
• Bi2Te3 ZT ~0.6 at room temperature
If ZT~4, then thermoelectric refrigerators will
be competitive with traditional refrigerators.
Thermoelectric cooling
Peltier element:
Bismuth Telluride
(p/n type connected in series)
advantages
• Solid state heating and cooling –
no liquids. CFC free.
• Compact instrument
• Fast response time for good
temperature control
Seebeck vs Peltier
Hall effect (1879): classical approach
dv
v
v
 eE  e  B  m*
dt
c

B  Bzˆ; dv / dt  0 at steady state
m*
 Ex 
 m* /  eB / c   vx 

  v   e  E 
*
 eB / c m /   y 
 y
j  env
 m*
 Ex   ne 2
 
 Ey    B

 nec
B
nec
m*
ne 2

 j 
 1
  x   0 
  jy 
 c


 1
0
σρ 
2 
1  c   c
1
c 

1 
 1 c 
c 1

 0 

1 
 c
nec / B 
 0
c 1

 

0
 nec / B

c   jx 
 
1   jy 
ne2
0  *
m
m*
eB
0  2 , c  *
ne 
mc
ρxy
B
dk
e
 eE  vk  B
dt
c
Hall effect: semiclassical approach
Recall “Chamber’s
formulation” (without
density and T gradients)
f
 f   dt 'e  (t t ')/ vk    k  0


 
t
1

  f 
 e  dt 'e  (t t ')/ vk   E  vk  B    0 

c

   
t
We can now only count on vk for magnetic effect
e
k  eE  v  B
c

B
v
Bk E B


c
e B2
B2
ExB drift
Assume E⊥B
v
Bk
EB
E   E 2  
k
c
e
B
e B2
e  (t t ')/ k (t ') 
EB
 f  c 2 
B



d  (t t ')/
1
e
k (t ')  e  (t t ')/ k (t ')
dt '


1 t
 f0 
 ( t t ')/
k (t )  k (t )  
k (t )
 , where k (t )   dt 'e





Q1:How do we get the
next order terms?
j  e  [dk ]vk  f 0   f 


f 0
EB
=ce 2  [dk ] k  k
B
k
EB

=ce 2  [dk ]   k  k
B
 k 



f0   f0 



 0

nec
σ 
 B

 0


nec
B
0
0
1
c
1
c 
0
2

0


0


1


If the open orbit is along the
x-direction (in real space), then
EB
EB
[
dk
]
f


nec
0
B2 
B2


0
 0


 1
0  0 

 c

1
 0





Q2: What if the orbit is
not closed?
• For ωcτ>>1, the first term is zero
 j  ec
 1
2





 c
 1
σ  0 
 c

 0


1
c
0
0

0


0

1



 

 1
σ  0 
 c
 0



1
c
1
c 
0
2

0


0

1


See Kitel, QTS, p.244
Q3: What about ωcτ<<1?
supplementary
Anomalous Hall effect (Edwin Hall, 1881):
Hall effect in ferromagnetic (FM) materials
FM material
ρH
saturation
slope=RN
RAHMS
H
The usual Lorentz
force term
 H  RN H   AH ( H ),
Anomalous term
 AH ( H )  RAH M ( H )
Ingredients required for a successful
theory:
• magnetization (majority spin)
• spin-orbit coupling
(to couple the majority-spin direction
to transverse orbital direction)
supplementary
“Intrinsic” AHE due to the Berry curvature
Mn5Ge3
Zeng et al PRL 2006
dk
 eE
dt
dr 1 

 k  ( k )
dt
k
1  e
=
 E  ( k )
k
A transverse current
To leading order,
j  e  [dk ] vk f 0

e2
=   [dk ]
f 0   [dk ] (k ) f 0  E
k
e
  AH 
e2

filled
After averaging over long-wavelength spin
fluctuations, the calculated anomalous Hall
conductivity is roughly linear in M. The
S.S. refers to skew scattering.
• Skew scattering from an impurity
[dk ]  z (k )
transition rate
Wks k 's'
 
  SO s's  k ' k

• classical Hall effect
charge
B
EF
 Lorentz force
+++++++
↑↓
y
0
L
• anomalous Hall effect
↑
↑↑↑↑↑↑↑
↑↑↑↑
EF
 Berry curvature (int)
↓
B
charge
spin
 Skew scattering (ext)
y
↑↑
↑↑↑↑
0
L
• spin Hall effect
spin
EF
↑
 Berry curvature (int)
↓
↑↑↑↑↑↑↑
↑↑↑↑
↑↑↑↑
 Skew scattering (ext)
↑↑↑↑↑↑↑
No magnetic field required !
y
0
L
Thermo-galvano-magnetic phenomena
j e  σ ( B) E  α ( B)  T 
-yT
j Q  β( B) E  κ ( B)  T 
Bz
σT (  B )  σ ( B)
Onsager
relations
Jx
β ( B)  α ( B)
T
κ T ( B)  κ ( B)
• Expand to first order in B,
Ohm
Hall
Nernst
1826
1879
1886
The effect of B on
thermo-induced
electric current
j e   E  RE  B    T   N  T   B
Onsager relations
j Q   E  NE  B    T   L  T   B
The effect of B on
electric-induced
thermo current
Ettingshausen Fourier Leduc-Righi
1886
1807
(Thermal Hall effect)
1887
Landau, ED of continuous media, p.101
Beyond thermo-galvano-magnetic phenomena
Optical
(O)
Thermal
(T)
• E-T: Thomson effect, Peltier/Seebeck effect
Mechanical
(M)
• E-B: Hall effect, magneto-electric material
• E-B-T: Nernst/Ettingshausen effect,
Leduc-Righi effect
• E-O, B-O: Kerr effect, Faraday effect,
photovoltaic effect, photoelectric effect
Electric
(E,P)
Magnetic
(B,M)
• E-M, B-M: piezoelectric effect/electrostriction,
piezomagnetic effect/magnetostriction
• M-O: photoelasticity
N
E
B
• ...
solid state refrigerator
solid state sensor
solid state motor, artificial muscle
...
Landau and Lifshitz, Electrodynamics of continuous media
Scheibner, 4 review articles in IRE Transations on component parts, 1961, 1962
TABLE 1-3 Physical and Chemical Transduction Principles. (from “Expanding the vision of sensor materials” 1995)
Input (Primary)
Signal
Output (Secondary) Signals
Mechanical
Thermal
Electrical
Magnetic
Radiant
Mechanical
(Fluid) Mechanical
effects; e.g.,
diaphragm,
gravity balance.
Acoustic effects;
e.g., echo sounder.
Friction effects;
e.g., friction
calorimeter.
Cooling effects;
e.g., thermal flow
meter.
Piezoelectricity.
Piezoresistivity.
Resistive.
Capacitive.
Induced effect.
Magnetomechanical
effects; e.g.,
piezomagnetic
effect.
Photoelastic
systems (stressinduced
birefringence).
Interferometer.
Sagnac effect.
Doppler effect.
Thermal
Thermal
expansion; e.g.,
bimetallic strip,
liquid-in-glass and
gas thermometers.
Resonant
frequency.
Radiometer effect;
e.g., light mill.
Electrical
Electrokinetic and
electromechanical
effects; e.g.,
piezoelectricity,
electrometer, and
Ampere's Law.
Joule (resistive)
heating. Peltier
effect.
Charge collectors.
Langmuir probe.
Biot-Savart's Law.
Magnetic
Magnetomechanical
effects; e.g.,
magnetostriction,
and
magnetometer.
Thermo-magnetic
effects; e.g.,
Righi-Leduc effect.
Galvano-magnetic
effects; e.g.,
Ettingshausen
effect.
Thermo-magnetic
effects; e.g.,
EttingshausenNernst effect.
Galvano-magnetic
effects; e.g., Hall
effect, and
magnetoresistance.
Magneto-optical
effects; e.g.,
Faraday effect,
and CottonMouton effect.
Seebeck effect.
Thermoresistance.
Pyroelectricity.
Thermal (Johnson)
noise.
Chemical
Thermo-optical
effects; e.g., liquid
crystals. Radiant
emission.
Reaction
activation; e.g.,
thermal
dissociation.
Electro-optical
effects; e.g., Kerr
effect, Pockels
effect. Electroluminescence.
Electrolysis.
Electro-migration.
Landau’s theory of Fermi liquid
Why e-e interaction can usually be ignored in metals?
2
•
K
U
K
1
e2
, U
m r2
r
me 2
r
r

2
aB
Typically, 2 < U/K < 5
• Average e-e separation in a metal is about 2 A
Experiments find e mean free path about 10000 A (at 300K)
At 1 K, it can move 10 cm without being scattered! Why?
• A collision event:
k1
k3
k2
k4
• Calculate the e-e scattering rate using Fermi’s golden rule:
1


2

f | Vee | i
2
 ( Ei  E f )
2
 k3 , k4 | Vee | k1 , k2
i, f
Scattering
amplitude
f | Vee | i
Ei  E1  E2 ; E f  E3  E4
2
The summation is over all possible
initial and final states that obey
energy and momentum conservation
Pauli principle reduces available states for the following reasons:
If the scattering amplitude |Vee|2 is roughly of the same order for all k’s,
then
 1
Vee
2
 1
k1 , k2 k3 , k4
E1+E2=E3+E4;
k1+k2=k3+k4
• 2 e’s inside the FS cannot scatter with each other
(energy conservation + Pauli principle), at least one of
them must be outside of the FS.
Let electron 1 be outside the FS:
• One e is “shallow” outside, the other is “deep” inside
also cannot scatter with each other, since the “deep” e
has nowhere to go.
• If |E2| < E1, then E3+E4 > 0 (let EF=0)
But since E1+E2 = E3+E4, 3 and 4 cannot be very far from
the FS if 1 is close to the FS.
Let’s fix E1, and study possible initial and final states.
1
2
3
(let the state of electron 1 be fixed)
• number of initial states = (volume of E2 shell)/Δ3k
number of final states = (volume of E3 shell)/Δ3k
(E4 is uniquely determined)
• τ-1 ～ V(E2)/Δ3k x V(E3)/Δ3k
← number of states for scatterings
V ( E2 )  4 kF2 | k2  kF |
V ( E3 )  4 kF2 | k3  kF |
∴τ-1 ～ (4π/Δ3k)2 kF2|k2-kF|×kF2|k3-kF|
Total number of states for particle 2 and 3 = [(4/3)πkF3/ Δ3k]2
• The fraction of states that “can” participate in the scatterings
= (9/kF2) |k2-kF|× |k3-kF|
～(E1/EF)2
(1951, V. Wessikopf)
In general
 1  2   2  k BT 
Finite temperature:
～ (kT/EF)2 ～ 10-4 at room temperature
→ e-e scattering rate

T2
• need very low T (a few K) and very pure sample to eliminate thermal and
impurity scatterings before the effect of e-e scattering can be observed.
2
Landau’s theory of the Fermi liquid (1956)
assumptions
• Strongly interacting fermion system
→ weakly interacting quasi-particle (QP) system
• 1-1 correspondence between fermions and QPs
(fermion, spin-1/2, charge -e).
～ a particle plus
its surrounding,
finite life-time
Q: Is this trivial?
• adiabatic continuity: As we turn off the interaction, the QPs smoothly
change back to noninteracting fermions.
1962
• The following analysis applies to a neutral, isotropic FL, such as He-3.
Another application:
He-3
TF=7 K
Similarity and difference with free electron gas
• QP distribution (at eq.)
at T  0,
For a justification,
see Marder
1
f k 
e( k   )/ kBT  1
  ( F   k ) (  =  F at T  0)
f k  f k0
  ( F   k0 )
← if no other ext perturbations
kF is not changed by interaction!
• Due to external perturbations the distribution will deviate from
the manybody ground state (no perturbation) at T=0
• Thermal
 f k  f k ( k )   ( F   k0 )
• Non-thermal (T=0)
 f k   (    k )   ( F   k0 )
(density perturbation,
magnetic field… etc)
• In general
• QP energy
 f k  f k ( k )   ( F   k0 )
 k     ukk '  f k ' '
0
k
 '
k ' '
interaction between QPs
near FS
In absence of
other QPs
• Total number
ukk' 'is an effective
N   f k
k
N kF3
  2
V 3
11
u
 '
kk '
 uk' 'k 
E[ f ]  E[0]    k  f k
• Total energy
k
 E[0]    k0  f k 
k
1
ukk' ' f k  f k ' '  O( f 3 )

2 k
k ' '
This form is not good for
charged FL (with longrange interaction)
• If there is no magnetic field, nor magnetic order, then  k is
independent of σ, and
'
ukk' depends
only on the relative spin directions.
uk1k2'  k1 , k2 | Vee | k1 , k2
For example,
ukk' '
(forward scattering amplitude)
1
4 e2

2
 V  (k  k ', 0) k  k '

0

if    '
if    '
Quinn, p.384
(recall the Fock
interaction in ch 9)
• Fermi velocity
 k0
vF 
k
• Effective mass
m* 
k  kF
kF
vF
Note: The use of  k follows
Coleman’s note, Baym and
Pethick etc, but not Marder’s.
We don’t want these quantities to
depend on perturbation
0
• DOS
D* ( F ) 
See ch 7

1
V
  (   
0
k
F
)
k
1
4 3

dS
m* k F
 2 2
 k  k0 
• Specific heat
dE    k  f k     f k ,  f k
k
k
0
k
f ( k0 )

T
T
(to lowest order)
0
E
0 f ( k )
 CV 
   k
T V k
T
   d ( k0   )
k
f ( )
T
f ( k0 )
=V  d D ( )
See ch 6
T
same as non-interacting
f ( k0 )  2 2
*
*
 VD ( F )  d
= k BTVD ( F ) result except for the
T
3
effective mass.
*
Heavy fermion material (CeAl3)
C is proportional to T, but the
slope gives an effective
mass 103 times larger!
He-3
ρ is proportional to T2,
also a FL behavior
Specific is linear in T below 20 mK
Giamarchi’s note, p.88
(30 bar)
Effective mass of a QP (I)
(total) “Particle” current
JN  
k
k
k
f k    f k
m
k m
(1)
Z.Qian et al, PRL 93, 106601 (2004)
Effective mass of a QP (II)
JN 
On the other hand,
give particles an
active boost
(with p-h excitations)
dE
dk
(see next page)
k  k  dk
f k  f k dk ,  f k  dk 
f k
k
f 

f k0'
dE    k  dk  k 

k 

k
 k '



 J N =  k f k    vk   ukk' 'vk ' ' ( k0'   F )  f k
k
k
k 
k ' '

(1)=(2)
→
(δf is arbitrary)
(2)
k
k
 ' k '
 *   ukk ' *  ( k0'   F )
m m k ' '
m '
If m* is spin-indep,
m*
 ' k F  k F '
0


1

u

(

kk
'
k
' F )
2
(nonmagnetic FL)
m
kF
k ' '
then

(an integral
over the FS)
(Only for QPs near the Fermi surface)
see Fradkin’s note,
Pathria p.296
(a passive
“boost”)
Nozieres and Pines, p.37
Introducing Fermi liquid parameters
 '
• Moments of ukk ' over the FS provide the most important information
about interactions (e.g., see the previous m* formula )
ukk'  ukk'  ukks '  ukka '
let
ukk'  ukk'  ukks '  ukka '
and decompose
u
k
θ
k’
s
kk '
ukka '
For spherical FS,
ukk’ depends only
on θ



 s 2 1 1
u

u
kk
'
kk
'
  u P (cos  ) u 
d cos  P (cos  )

1

0
2
2





1
u

u
2

1
a

a
kk
'
kk
'
  u P (cos  )
u

d
cos

P
(cos

)


2 1
2
0
s
• Dimensionless parameters
F  VD ( F )u , F  VD ( F )u
s
For
example,
*
s
a
*

a
A small set of parameters for
various phenomena

m*
k '2 dk '


0
 1V 
d

'
u

u
cos

'

(

kk '
kk '
k' F )
3
m
 2 
1
=1+ F1s
3
H.W.
determined from specific heat.
m*/m～3 for He-3
More on the effective mass
• recall
m*
1
 1+ F1s
m
3
pF pF  F1s  pF
 * 


m
m  3  F1s  m
Backflow correction
(to ensure current conservation)
•
m* k F
m*
1 *
s
*
 1+ VD ( F )u1 , D ( F )  2 2
m
3

m
 m* 
V
1  D ( F )u1s
3
diverges when
V
D( F )u1s  1 (～ Mott transition)
3
Compressibility of Fermi liquid
1 V
 
V P
1 n
 T  2
n  T
Note:
dP  SdT  nd 
At fixed S or T
(little difference near T=0)
T  n
  

At T=0,  f k       k     F   k0
=



1

P T n


Before
compression

 1/ n 
P N ,T
1 n
n 
T

1 n
 2
P T n 
T
   k 
F
 f k

 f k ' ' 
    F   k0  1   ukk ' '


 
 k ' '
Note: Slightly different
from Marder’s (see
Baym and Pethick, p.11)
Both k and k’ lie on FS,
and Ukk’ depends only on
cosθ, ∴ Ak is indep. of k.
≡ Ak (indep. of σ if not magnetized)
 A   ukk' '
k ' '
F0s
 A
1  F0s
 f k ' '
  ukk' '   F   k0'  1  A 

k ' '
= F0s
Dependence of various quantities on δμ
 f k     F  
0
k
 1  A 
=   F   k0 
1

1  F0s
 k   ukk' ' f k ' '
k ' '
Note:
• For attractive interaction,
Fs  0
If F s  1 , then κT diverges,
0
and FS will become unstable to
deformation (spontaneous breaking
of rotational symmetry).
F0s
=

1  F0s
also,
N
V

1
1
*

f

D
(

)


k
F
s
V k
1  F0
n
1
 D* ( F )
 V
1  F0s
D* ( F ) 1
1 n
 T  2

n  V ,T
n 2 1  F0s
• This is called Pomeranchuk instability
(1958). For example, nematic FL.
向列型
 T m* / m
 0 
 T 1  F0s
0910.4166
Deformation of Fermi sphere
and the FL parameters
 '
0
  ukk
'   k '   F 
k ' '

1
 ' k F  k F '
u
 ( k0'   F )

kk '
2
3 k ' '
kF
From Coleman’s note
summary
For He-3,
F1s  6
(larger
effective mass)
F0a  0.5
(more spin
polarizable)
compressibility
κ
F0s  10
(less
compressible)
From Coleman’s note
Travelling wave: firstly, 1st sound (i.e., the usual pressure wave)
Velocity of the 1st sound
c1 
so
P

P
,
 s

s,N
  mn
V P
1

n V s  s
1/2

n
s 
=
1

F
 0 
*
mD
(

)

F

m*k F
k F3
*
D ( F )  2 2 , n  2

3
1/2
 m*

 c1  vF  1  F0s  
 3m


F0S=10.8 for He-3,
determined by measuring C1
vF
3
(w/o interaction)
Zero sound (predicted by Landau, verified by Wheatley et al 1966)
• usual sound requires ωτ<<1 (mean free path ℓ<<λ)
when ωτ→1 , sound is strongly absorbed
• when ωτ>>1, sound propagation is again possible
• zero sound is a collisionless sound ～ plasma wave in charged FL
Can exist
at 0 K
no thermal equilibrium in each volume element
• to get the zero sound, one can increase ω or decrease T (to increase τ)
Oscillation of Fermi sphere
1st sound
t
zero sound
 fk 
cos 
s  cos 
γ
t
(egg-like shape)
Giamarchi’s note, p.102
Boltzmann-like approach (requires
f
f
f
r  k 
0
t
r
k
consider
(collisionless)
Instead of the semiclassical
equations, one uses
rk 
No r-dependence
hidden in ε.
f k  f k0   f k (r , t ),
   F , q  kF )
 k
;
k
f k0     k0   F 
 f k   k ei ( qr t ) (indep of spin)

f k
f
 r  k  i (  vk  q ) f k
t
r
To order δf
f k
 k
f k0  k
k

 vF

 vF    k0   F 
r
 k
r
k
 k
 f k ' '
  ukk' '
 iq  ukk' ' f k ' '
r k ' '
r
k ' '
 (  v  q ) k  vk  q   k0   F   ukk' ' k '  0
k ' '
  k     k0   F 
vk  q
ukk' ' k '

  vk  q k ' '
k 
 k
r
let  k     k0   F  k
then k 
 u
vk  q
ukk' ' k ' (1)

  vk  q k ' '
 k '   ukk '      F  k '
 '
kk '
 '
k ' '
k' '
=
0
k'
d '
a( ')k '
4

V * 
D ukk '  ukk'
2
d

a ( )  F0s
4
a ( ) 
Assume a(θ)
～const.
let   (q , k )
F0s
cos 

1 
d
cos

,
s

2 
s  cos 
vq
F0s 
s s 1 
=

1

ln


2 
2 s 1 
s s 1
1
or ln
1  s
2 s 1
F0

1
F0s
decompose

k ( )   P (cos  )
0
d '
 4 k ( ')  0

d  vk  q 
(1)  0   F0s
 0

4



v

q
k



• only F0s > 0 (repulsion) can have a solution
• for
F0s  0, S  1
  svF q  c0  svF  3c1
• when QP velocity > C0 (s<1), the integral has a pole
at cos  0  s , a QP would emit “supersonic” zero sound
Analogies:
• Supersonic shock wave
• Cherenkov EM radiation from
“superluminal” charged particles
Transition from the 1st
sound to the zero sound
Dispersion of zero sound in He-3
(from neutron scattering exp’t)
vF
Superfluid
transition
Vollhardt and Woelfle, p.45
Aldrich et al, PRL 1976
coherent
In addition to collective excitations (zero sound, plasma),
there are also particle-hole excitations
incoherent
particle-hole excitation:
 k q
k
For charged
FL only
(more in ch 23)
 k q   k 
2
 2k
2m
F
qq
2

2
 2k
2m
F
2k  q  q 

2m
2
2
q  q2 
Q: what is the particle-hole band for 1-dim electron liquid?
H. Godfrin et al, Nature 2012
From Altshuler’s slide
Beyond the Fermi liquid
Quasiparticle decay rate at
T = 0 in a Fermi Liquid:
•
•

 2 F
 2
    F  log   F  
 e e    


d 3
d 2
d 1
For d =3,2, from    it follows that   /  , i.e., the QPs are well
ee
F
determined and the Fermi-liquid approach is applicable.
For d =1,  ee is of the order of /  , i.e., the QP is not well defined and
the Fermi-liquid approach is not valid.
→ Tomonaga-Luttinger liquid in 1D (1950,1963)
For more, see Marder, Sec 18.6
Giamarchi, p.6
Features of a Luttinger liquid (cited from wiki)
• Even at T=0, the particles' momentum distribution
For electron,
not for QP
function has no sharp jump (Z=0). (in contrast to QP dist)
• Charge and spin waves are the elementary excitations of
Z
the Luttinger liquid, unlike the QPs of the FL (which carry
both spin and charge).
• spin density waves propagate independently from the
charge density waves (spin-charge separation).
Luttinger liquids reported in literatures
• electrons moving along edge states in the fractional
Quantum Hall Effect (Th: Kane and Fisher 1994; Ep: 1996)
朝永 振一郎
• electrons in carbon nanotubes (McEuen group, 1998)
• 'quantum wires' defined by applying gate voltages to a
two-dimensional electron gas. (Auslaender et al, 2000)
Non-Fermi-liquid in 2D?
Luttinger