Chap 3
Linear response theory
Ming-Che Chang
Department of Physics, National Taiwan Normal University, Taipei, Taiwan
(Dated: April 6, 2013)
I. GENERAL FORMULATION
A material would polarize, or carry a current under an
external electric field,
P = χe E
j = σE.
and |nI (t0 )i = |n0 i is the states before perturbation.
Substitute Eqs. (7) and (8) into Eq. (3), and keep only
the terms to linear order in H 0 , we have
hA(t)i
(1)
(9)
Z
t
= hAi0 − i
If the electric field is not too strong, then the electric susceptibility and the conductivity are independent of the
electric field. They only depend on the material properties in the absence of the electric field (in equilibrium).
This type of response is called the linear response.
The average of an observable (such as the electric polarization) in equilibrium is
1 X −βE{n}0
e
h{n}0 |A|{n}0 i.
hAi0 =
Z0
0
Z
dt0
t0
= hAi0 − i
t0
hn0 |[AI (t), HI0 (t0 )]|n0 i
e
n
t
0
−βEn
Z0
dt0 h[AI (t), HI0 (t0 )]i0 .
For example, if
Z
H 0 (t) =
dv B(r) · f (r, t) ,
| {z }
| {z }
(10)
operator C−number
(2)
{n}
X
then
0
Under an external field, the states {n} are perturbed to
become {n}, and the average becomes
hAi =
1 X −βE{n}
h{n}|A|{n}i = hAi0 + δhAi.
e
Z
(3)
{n}
Our job is to find out δhAi.
In the following, as long as there is no ambiguity, we
will write the labels of a manybody state {n} simply as
n. Before perturbation,
H0 |n0 i = En0 |n0 i,
(4)
where En0 and |n0 i are eigne-energies and eigenstates of
the manybody Hamiltonian H0 . The external perturbation is assumed to be
0
H = H0 + H (t)θ(t − t0 ).
(5)
That is, the perturbation is turned on at time t0 . After
the perturbation,
H(t)|n(t)i = i
∂
|n(t)i.
∂t
t
UI (t, t0 ) = 1 − i
t0
dt0 HI0 (t0 ) + · · · ,
This can be written as
Z
X
δhA(x)i = dx0
χABα (x, x0 )fα (x0 ),
(12)
α
where x = (r, t), dx0 ≡ dv 0 dt0 , and
χABα (x, x0 ) = −iθ(t − t0 )h[AI (x), BIα (x0 )]i0 .
(13)
Eq (12) is called the Kubo formula, and χABα is called
the response function. Notice that the operators are
written in the interaction picture.
II. DENSITY RESPONSE AND DIELECTRIC FUNCTION
(7)
A. Density response
In this section, we consider the perturbation of electron
density caused by an external electric potential. Before
perturbation,
where
Z
−∞
(6)
In the interaction picture, the perturbed states
|n(t)i = e−iH0 t |nI (t)i
= e−iH0 t UI (t, t0 )|nI (t0 )i,
δhA(r, t)i
(11)
Z
Z t
= −i dv 0
dt0 h[AI (r, t), BI (r0 , t0 )]i0 · f (r0 , t0 )
t0
Z
Z ∞
= −i dv 0
dt0 θ(t − t0 )h[AI (r, t), BI (r0 , t0 )]i0 · f (r0 , t0 ),
(8)
H0 = T + VL + Vee ,
(14)
2
where VL is a one-body interaction, such as the electronion interaction, and Vee is the electron-electron interaction. The perturbation can be written in the following
form,
Z
0
H = dvρe (r)φext (r, t),
(15)
P
where ρe = q s ψs† (r)ψs (r) (q = −e) is the electron
density, and φext is an external potential.
Because of the external potential φext , electron density
hρe i0 → hρe i = hρe i0 + δhρe i.
(16)
Comparing with the Kubo formula, we find the following
replacement necessary,
A
B
f
→
→
→
ρe ,
ρe ,
φext .
(17)
The Kubo formula gives
Z
δhρe (x)i = dx0 χρe (x, x0 )φ(x0 ),
(18)
and the response function is
0
0
0
χρe (x, x ) = −iθ(t − t )h[ρe (x), ρe (x )]i0 .
(19)
Remember that the operators are in the interaction picture, but the subscript I is neglected from now on.
If the unperturbed system H0 is uniform in both space
and time, then
χρe (x, x0 ) = χρe (x − x0 ).
(20)
In this case, the convolution theorem in Fourier analysis
tells us that
δhρe (κ)i = χρe (κ)φext (κ)
where κ ≡ (q, ω), κx ≡ q · r − ωt, and
X
δhρe (x)i =
eiκx δhρe (κ)i,
(21)
Z
0
d(x − x0 )e−iκ(x−x ) χρe (x − x0 )
(25)
Z
0
= −i d(t − t0 )θ(t − t0 )eiω(t−t )
Z
0
×
d(r − r0 )e−iq·(r−r ) h[ρe (r, t), ρe (r0 , t0 )]i0 .
χρe (κ) =
Since the system is uniform
in space, one can perform an
R
extra space integral V10 dr0 to the space integral above,
and use
Z
Z
Z
Z
1
1
dr0 d(r − r0 ) =
dr dr0 .
(26)
V0
V0
Then it is not difficult to see that
Z ∞
i
χρe (κ) = −
dteiωt h[ρe (q, t), ρe (−q, 0)]i0 .
V0 0
(22)
(27)
Notice that ρe (−q, 0) can also be written as ρ†e (q, 0). In
the following, we may sometimes use the particle density
ρ and its response function χρ , which are related to the
electron density and its response function as
ρe = −eρ, χρe = e2 χρ .
(28)
Also, notice that these response functions are related to,
but not exactly the same as, the electric susceptibility χe
introduced at the beginning of this chapter.
B. Dielectric function
The response function connects δρe with φext . However, the dielectric function connects φext with the total
potential φ, which is the sum of φext and the potential
due to material response,
²(κ) =
φext (κ)
.
φ(κ)
(29)
The total particle density is
κ
Z
−iκx
δhρe (κ)i =
dxe
δhρe (x)i;
X
φext (x) =
eiκx φext (κ),
hρi = hρiext + δhρi,
dxe−iκx φext (x).
The summation over k should be understood as
Z
X
1 X dω
=
.
V0 q
2π
κ
q 2 φext (κ) = −4πehρ(κ)iext ,
q 2 φ(κ) = −4πehρ(κ)i.
(23)
The Fourier expansion of the response function is
X
0
eiκ(x−x ) χρe (κ),
(24)
χρe (x − x0 ) =
κ
(30)
which are related to the potentials via the Poisson equations (CGS),
Zκ
φext (κ) =
and
(31)
Notice that quantities such as φext (κ) = φext (q, ω) is
allowed to be frequency-dependent. Also, if one prefers
the MKS system, then just replaces 4π with ²10 .
Combine the equations above, we get
φ(κ) = φext + 4πe2 χρ
φext
.
q2
(32)
3
This leads to
2
1
4πe
=1+
χρ ,
²(κ)
q2
| {z }
(33)
The second term in φ(r) is the induced potential, or the
local field correction. That is, if one calculates the response to the total perturbing potential φ(r), then the
unperturbed system is H̃0 , which is non-interacting.
V (2) (q)
in which V (2) (q) is the Fourier transform of V (2) (r) =
e2 /r.
Instead of using δhρi = χρ φext , an alternative relation
is
δhρi = χ0ρ φ, φ = φext + δφ.
(34)
It’s not difficult to see that
χρ =
χ0ρ
1−
4πe2 0
q 2 χρ
,
(35)
and
²(κ) = 1 −
4πe2 0
χ .
q2 ρ
(36)
The calculation of χρ is based on Eq. (27), in which
one averages over unperturbed manybody states (including electron interactions). A great advantage of using
the alternative response function χ0ρ is that, since the local field correction has been included in φ, one may use
non-interacting manybody states in the calculation of the
response function. This is justified as follows:
The interaction term is, apart from a one-body correction (see Sec. IV.B.1 of Chap 1),
Z
1
Vee =
dvdv 0 V (2) (r − r0 )ρe (r)ρe (r0 ).
(37)
2
Using the mean field approximation, and expand the
charge density with respect to a mean value hρ(r)ie ,
ρe (r) = hρe (r)i + ρe (r) − hρe (r)i .
|
{z
}
(38)
2
Neglecting the (δρe ) term, we have
Z
Vee '
dvdv 0 V (2) (r − r0 )ρe (r)hρe (r0 )i
(39)
Z
1
−
dvdv 0 V (2) (r − r0 )hρe (r)ihρe (r0 )i.
2
The mean-field Hamiltonian under perturbation is
(dropping the second term in Eq. (39))
Z
Z
dvdv 0 V (2) (r − r0 )ρe (r)hρe (r0 )i +
= H̃0 +
We now drop the superscript and subscript 0 that refer
to equilibrium states. Recall that
Z ∞
i
0
χρ (κ) = −
dteiωt h[ρ(q, t), ρ(−q, 0)]i.
(42)
V0 0
In the interaction picture, ρ(q, t) = eiH0 t ρ(q)e−iH0 t . The
summation
I(q; t, 0)
X e−βEn
≡
hn|ρ(q, t)ρ(−q, 0)|ni
Z
n
X e−βEn
=
Z
n,m
ei(En −Em )t hn|ρ(q, 0)|mihm|ρ(−q, 0)|ni,
P
where we have inserted a complete set m |mihm|, and
used e−iH0 t |mi = e−iEm t |mi.
Since both |ni and |mi are manybody states of noninteracting particles, hm|a†ks ak−q,s |ni can be non-zero
only if, when comparing to |ni, the |mi state has one
more electron at state (k, s), but one less electron at
(k − q, s). Therefore, En − Em = −εk + εk−q , a difference of two single-particle energies. This energy factor
can now be moved outside of the m-summation, and
I(q; t, 0)
(44)
X e−βEn X
=
ei(εk−q −εk )t hn|a†k−q,s aks a†ks ak−q,s |ni
Z
n
=
X
ei(εk−q −εk )t
X e−βEn
n
k,s
= 2
X
Z
e
i(εk−q −εk )t
f (εk−q )[1 − f (εk )],
where
f (εk ) =
X e−βEn
n
Z
hn|a†ks aks |ni
(45)
(40)
is the Fermi distribution function (spin-independent
dvρe (r)φext here). It is left as an exercise to show that
f (εk ) =
dvρe (r)φ(r),
where H̃0 = T + VL , and
Z
φ(r) = φext (r) + dv 0 V (2) (r − r0 )hρe (r0 )i.
hn|a†k−q,s ak−q,s (1 − a†ks aks )|ni
k
Z
= H̃0 +
(43)
k,s
δρe (r)
HM F
C. Calculation of χ0ρ
(41)
1
.
1 + eβεk
(46)
Similarly, one can show that
X
ei(εk−q −εk )t f (εk )[1 − f (εk−q )]. (47)
I(−q; 0, t) = 2
k
4
From Eq. (42), we have
Z ∞
i
0
χρ (κ) = −
dteiωt [I(q; t, 0) − I(−q; 0, t)]
(48)
V0 0
Z
2i X ∞
= −
dteiωt ei(εk−q −εk )t [f (εk−q ) − f (εk )].
V0
0
2. High frequency limit
The response function in Eq. (50) can be re-written as
χ0ρ (q, ω) = −
2(εk−q − εk )
2 X
f (εk ) 2
.
V0
ω − (εk−q − εk )2
k
For high frequency and long wave length (ω À vF q),
The integral over time is
Z ∞
dtei(ω+iδ)t ei(εk−q −εk )t =
i
.
ω
+
iδ
+
(ε
k−q − εk )
0
(49)
The positive infinitesimal δ is added to ensure the convergence of the exponential at t = ∞.
Finally,
χ0ρ (q, ω) =
2 X f (εk−q ) − f (εk )
,
V0
ω + iδ + (εk−q − εk )
(50)
k
χ0ρ (0, ω) '
q2 2 X
q2 n
f (εk ) =
,
2
mω V0
mω 2
where n is the particle density. Therefore,
²(0, ω) = 1 −
ωp2
,
ω2
(51)
k
(59)
where ωp2 = 4πne2 /m is the plasma frequency.
Notice that
q→0 ω→0
4πe2 2 X f (εk−q ) − f (εk )
.
q 2 V0
ω + iδ + (εk−q − εk )
(58)
k
lim lim ²(q, ω) 6= lim lim ²(q, ω).
and
²(q, ω) = 1 −
(57)
k
ω→0 q→0
(60)
That is, the dielectric function is not analytic at (q, ω) =
(0, 0).
This is called the Lindhard dielectric function.
III. CURRENT RESPONSE AND CONDUCTIVITY
1. Low frequency limit
For frequency as low as ω ¿ vF q, the ω in the denominator can be neglected, and
χ0ρ (q, 0) '
2 X f (εk−q ) − f (εk )
.
V0
εk−q − εk
(52)
k
For general wave length (in 3-dim), it can be shown
that
µ
¶
q
0
χρ (q, 0) ' −D(εF )F
,
(53)
2kF
where D(εF ) is the density of states at the Fermi energy,
and
¯
¯
1 1 − x2 ¯¯ 1 + x ¯¯
F (x) = +
ln ¯
(54)
2
4x
1 − x¯
is the Lindhard function (see Sec. II.A).
At long wavelength,
µ
¶
2 X
∂fk
0
χρ (q, 0) ' −
−
= −D(εF ).
V0
∂ε
(55)
In this limit, the dielectric function is
kT2 F
,
q2
where VL is the one-body interaction, and Vee is the electron interaction. In general, the external electric field
depends on both scalar and vector potentials,
E(r, t) = −∇φ −
(56)
where kT2 F = 4πe2 D(εF ) is the Thomas-Fermi wave
vector.
1 ∂A
.
c ∂t
(62)
For a static field, it is common to use E = −∇φ, as in
Eq. (15). A static and uniform field then has φ(r) =
−E · r. A disadvantage of this scalar potential is that it
is not bounded at infinity. To avoid such a problem, one
can choose a gauge such that
E(r, t) = −
k
²(q, 0) = 1 +
In this section, we consider the generation of electron
current caused by an external electric field. Before perturbation,
Z
p2
H0 = dvψ † (r)
ψ(r) + VL + Vee ,
(61)
2m
1 ∂A
.
c ∂t
(63)
In this case, a static and uniform field has A(t) = −cEt.
After applying the electric field, the Hamiltonian becomes
¡
¢2
Z
p + ec A
†
H =
dvψ (r)
ψ(r) + VL + Vee (64)
2m
Z
¡
¢
e
= H0 +
dv ψ † p · Aψ + ψ † A · pψ
2mc
Z
e2
+
dvA2 ψ † ψ,
2mc2
5
where H0 refers to the parts that do not depend on A.
The particle current density operator J is related to the
variation of the Hamiltonian as follows,
Z
e
δH =
dvJ · δA,
(65)
c
Therefore, for the electric current density Je = −eJ, one
has
Z
p
hJαe (r, ω)i = dv 0 σαβ
(r, r0 ; ω)Eβ (r0 , ω).
(76)
where
e2
χαβ (r, r0 ; ω).
(77)
ω
Since the conductivity in general is a non-local quantity,
the current density at point r would not only depend on
the electric field at r, but also on neighboring electric
field.
For a homogeneous material,
¡
¢ ¤
1 £ †
J≡
ψ ∇ψ − ∇ψ † ψ +
|2mi
{z
}
paramagnetic current Jp
σαβ (r, r0 ; ω) = i
e
Aψ † ψ.
mc
| {z }
diamagnetic current JA
(66)
We would like to find out the connection between hJi
and A (to first order). After perturbation, a manybody
state
0
0
1
|n i → |ni ' |n i + |n i,
(67)
where |n1 i is of order A. Therefore,
hn|J|ni = hn0 |JA |n0 i + hn0 |Jp |n1 i + hn1 |Jp |n0 i + O(A2 ).
(68)
We have assumed, of course, that the equilibrium state
carries no current, hn0 |Jp |n0 i = 0.
After taking the thermal average, the first term becomes
e
hJA i =
A(r, t)hρ(r)i.
(69)
mc
The other two terms are evaluated using the Kubo formula in Eq. (12), with the following replacement,
A
B
→
→
f
→
Jαp ,
Jp ,
e
A.
c
This gives us (recall that x = (r, t))
Z
e
hJαp (x)i =
dx0 χpαβ (x, x0 )Aβ (x0 ),
c
(70)
(71)
where
D
E
χpαβ (x, x0 ) = −iθ(t − t0 ) [Jαp (x), Jβp (x0 )] .
ρ(x)
χαβ (x, x ) = δαβ δ(x − x )
+ χpαβ (x, x0 ).
m
0
σαβ (r, r0 ; ω) = σαβ (r − r0 ; ω).
(78)
We can then apply the convolution theorem to the space
variable and get
hJαe (q, ω)i = σαβ (q, ω)Eβ (q, ω),
(79)
where
·
¸
e2
ρ(q, ω)
p
σαβ (q, ω) = i
δαβ
+ χαβ (q, ω) ,
ω
m
(80)
in which (Cf. eq. (27))
Z
0
i
p
χαβ (q, ω) = −
dtθ(t−t0 )eiω(t−t ) h[Jα (q, t), Jβ (−q, t0 )]i.
V0
(81)
Notice that the diamagnetic part diverges as ω → 0. For
usual conductors and insulators, this divergence would
be cancelled by part of the paramagnetic term, so that
the DC conductivity remains finite. In a superconductor
(which is a perfect diamagnet), the paramagnetic term
vanishes in the DC limit, and the conductivity is purely
imaginary,
e2
ρ(q, ω)
δαβ
.
(82)
ω
m
A purely imaginary conductivity leads to inductive behavior, and would not cause energy dissipation.
SC
σαβ
(q, ω) = i
(72)
After combining with the diamagnetic term in Eq. (69),
the response function for the total current is
0
The conductivity tensor is
(73)
Since H0 is time independent, the response function
χαβ (x, x0 ) = χαβ (r, r0 ; t − t0 ). Applying the convolution
theorem to the time variable (see Eq. (21)), one has
Z
e
dv 0 χαβ (r, r0 ; ω)Aβ (r0 , ω).
(74)
hJα (r, ω)i =
c
The vector potential is related to the electric field as follows,
ω
E(ω) = i A(ω).
(75)
c
A. Conductivity for non-interacting electrons
We would like to start from a formulation that does
not presume spacial homogeneity:
Z ∞
h
i
i
χpαβ (r, r0 ; ω) = −
dteiωt h Jαp (r, t), Jβp (r0 , 0) i,
V0 0
(83)
where Jαp (r, t) = eiH0 t Jαp (r)e−iH0 t . Therefore,
Iαβ (r, t, r0 , 0)
X e−βEn
≡
hn|Jαp (r, t)Jβp (r0 , 0)|ni
Z
n
=
X e−βEn
n,m
Z
(84)
ei(En −Em )t hn|Jαp (r, 0)|mihm|Jβp (r0 , 0)|ni,
6
P
in which we have inserted a complete set
m |mihm|,
and used e−iH0 t |mi = e−iEm t |mi (see Sec. II.C).
The current density operator can be written as (see
Chap 1)
X
Jαp (r) =
hµ|Jα(1) (r)|νia†µ aν ,
(85)
µν
(1)
where Jα is a one-body operator to be specified later.
From now on, assume the electrons are non-interacting.
Substitute Jαp (r) into Eq. (84), we get terms with the
form
hn|a†1 a2 |mihm|a†3 a4 |ni,
(86)
where 1, 2 · · · are simplified notations for single-particle
state labels µ, ν. For this type of term to be non-zero,
the single-particle states have to satisfy (1 = 4, 2 = 3),
or (1 = 2, 3 = 4). They both lead to En − Em = ε1 − ε2
(the second case has ε1 = ε2 ).
The summation over m can now be removed, and
X
(1)
Iαβ (r, t, r0 , 0) =
ei(ε1 −ε2 )t h1|Jα(1) |2ih3|Jβ |4i
(87)
1,2,3,4
× ha†1 a2 a†3 a4 i(δ14 δ23 + δ12 δ34 ).
ha†1 a1 a†2 a2 i
(We have re-written 1, 2 as µ, ν.)
The denominator can be decomposed as
µ
¶
1
1
Ω
=
1∓
,
²µν ± Ω
εµν
εµν ± Ω
(94)
where εµν ≡ εµ − εν . Substitute this to Eq. (93), then
the first term of the decomposition would cancel with the
diamagnetic term, because of the following f -sum rule:
1 X
hµ|pα |νihν|pβ |µi
(fµ − fν )
= −mρδαβ .
V0 µν
εµν
(95)
As a result,
σαβ (0, ω) =
e2 X
hµ|vα |νihν|vβ |µi
,
(fµ − fν )
iV0 µν
εµν (Ω + εµν )
(96)
2. Uniform and static, Hall conductivity
= f1 f2 .
Iαβ (r, t, r0 , 0) − Iβα (r0 , 0, r, t)
(89)
X
(1)
i(ε1 −ε2 )t
(1)
=
e
h1|Jα |2ih2|Jβ |1i(f1 − f2 ).
12
Therefore,
0
χP
(90)
αβ (r, r , ω)
Z ∞
i
= −
dteiωt [Iαβ (r, t, r0 , 0) − Iβα (r0 , 0, r, t)]
V0 0
(1)
(1)
h1|Jα (r)|2ih2|Jβ (r0 )|1i
1 X
(f1 − f2 )
,
=
V0 12
Ω + ε1 − ε2
Finally, we would like to consider the DC Hall conductivity as an example. According to Eq. (96), it can be
re-written as
e2 X hµ|vα |νihν|vβ |µi − hµ|vβ |νihν|vα |µi
.
σα6DC
fµ
=β =
iV0 µν
ε2µν
(97)
If the single-particle states are Bloch states (µ → nk),
hr|nki = eik·r unk (r), where unk (r) is the cell-periodic
function, then one can show that
µ¿
À ¿
˦
e2 X
1
∂unk ∂unk
∂unk ∂unk
DC
σα6=β =
fnk
|
−
|
,
V0
i
∂kα ∂kβ
∂kβ ∂kα
nk
|
{z
}
Berry curvature Ωγ
nk
where Ω ≡ ω + iδ
If the material is homogeneous, then
(1)
(1)
h1|Jα (q)|2ih2|Jβ (−q)|1i
1 X
,
(f1 − f2 )
V0 12
Ω + ε1 − ε2
(91)
The one-body operator is (see Chap 1)
χP
αβ (q, ω) =
¢
1 ¡
pα e−iq·r + e−iq·r pα
2m
σαβ (0, ω)
(93)
#
"
hµ|pα |νihν|pβ |µi
ie2
1 X
ρ(ω)
=
δαβ
+ 2
(fµ − fν )
.
ω
m
m V0 µν
Ω + εµ − εν
(88)
where f is the Fermi distribution function. As a result,
one can show that
Jα(1) (q) =
For the uniform case (q = 0), the conductivity is
where vα = pα /m. This is sometimes called the KuboGreenwood formula.
The thermal averages are (see Eq. (45))
ha†1 a2 a†2 a1 i = f1 (1 − f2 ),
1. Uniform limit
(92)
(98)
where α, β, and γ are cyclic. We will call this as the
TKNdN formula (see Ref. 4).
In 2-dim, for a filled band,
Z
1
(n)
d2 kΩznk
(99)
C1 ≡
2π
filled BZ
must be an integer (see Sec. II.B of Ref. 5). Therefore,
the Hall conductivity from filled bands is quantized,
DC
σxy
=
e2 X (n)
C1 ,
h
filled n
(100)
7
where we have put back the ~ explicitly. The quantized
(topological) nature of such an integral is first pointed
out by D.J. Thouless to explain the quantum Hall
effect.
Prob. 1 Derive the f -sum rule,
1 X
hµ|pα |νihν|pβ |µi
(fµ − fν )
= −mρδαβ .
V0 µν
εµν
(101)
Prob. 2 Start from Eq. (97), derive the TKNdN formula
in Eq. (98).
References
[1] Chap 6 of H. Bruss and K. Flensberg, Many-body quantum theory in condensed matter physics, Oxford University
Press, 2004.
[2] Sec. I.5 of T. Giamarchi, A. Iucci, and C. Berthod, Introduction to Many body physics, on-line lecture notes.
[3] Chap 6: Electron Transport, by P. Allen, in Conceptual
Foundations of Materials: A standard model for groundand excited-state properties, ed. by S.G. Louie and K.L.
Choen, Elsevier Science 2006.
[4] D.J. Thouless, M. Kohmoto, P. Nightingale, and M. de
Nijs, Phys. Rev. Lett. 49, 405 (1982).
[5] D. Xiao, M.C. Chang, and Q. Niu, Rev. Mod. Phys. 82,
1959 (2010).