Transcript 第五章 关联 5.1 单电子近似的理论基础 5.2 费米液体理论

第五章
关联
5.1
单电子近似的理论基础
5.2
费米液体理论
5.3
强关联体系
多电子体系(After Born-Oppenheimer 绝热近似):

2
2
2

1
1
e
1
Ze
H  
 2i  

2m
2 i  j 4 0 | r i  r j | i ,n 4 0 | r i  R n |
i

H   E
关联：电子－电子相互作用
弱：单电子近似， 电子平均场
5.1
单电子近似的理论基础
单电子近似着眼于任何用N个单电子波函数 j (q j )
构建N电子系统总体波函数
(q1 , q2 ,..., qN )
V
i
1. Hartree方程(1928)

 

 
连乘积形式：   q1 ... q N  1 ( r1 ) r2  N rN

  
 2

2
E  H     r1  
 1  V j rj  j r1 d r1
j
 2m

2
2
e2
1

 j r1
 j ' r2 d r1d r2



8 0 j ' j
r1  r2
*
j

 
按变分原理， j q j  的选取E达到极小


N


  H    j i |  j  1   0
j 1


正交归一条件
单电子方程
 j : Lagrang乘子


  (r ) j' (r )d r   jj'
*
j

2
2

e

2

  V (r ) 
 2m
4 0


( j ' j )




d r '   j ( r )   j j ( r )

r  r'

 j' (r ' )
2

2
2

e

2

  V (r ) 
 2m
4 0

动能





d r '   j ( r )   j j ( r )

r  r'

 j' (r ' )
j ' j
2
原子核对电子形成的势能 其余N-1个电子对j电子的库仑作用能
自洽求解，H2,He计算与实验相符。
26个电子的Fe原子，运算要涉及1076个数，对称简化1053个
整个太阳系没有足够物质打印这个数据表！
2. 凝胶模型(jellium model)
为突出探讨相互作用电子系统的哪些特征是区别于不计其相
互作用者，可人为地简化假定电子是沉浸在空间密度持恒的
正电荷背景之中(不考虑离子的周期性)。
正电荷的作用体现于在相互作用电子体系的Hamiltonian中出
现一个维持系统聚集的附加项 V 
金属体系，设电子波函数：

 j (r ) 
1

为体系总体积
 
exp(ik j  r )
Hartree方程中的势：
V 
j ' j
1
4 0

e
r  rj'

1
exp i k j '  rj '

d rj '
e
1 e
 V 


4 0  j '  r  r '
4 0 
j
1
 dr

2
'
j
d rj
r  rj
第二项是全部电子在r处形成的势，与 V 相抵消
第三项是须扣除的自作用，与j有关，但如取r为计算原点：
1 e d rj
1

rj drj    0


4 0  rj
 0
  
 2 2

所以对凝胶模型，Hartree方程：  
  E K  K r  0
 2m

相互作用→没有相互作用
电子＋正电荷背景→自由电子气
3. Hartree-Fock方程(1930)
Hartree方程不满足Pauli不相容原理
电子：费米子
单电子波函数f：→N电子
体系的总波函数：
f1 r1 s1   f1 rN s N 
  ( N !)
不涉及自旋－轨道耦合时：

1
2


f N r1 s1  
f j 
j

f N rN s N 
r  s 
j
N电子体系能量期待值：
  

2
ion  

E  H     ( r ) 
  U ( r )   j ( r )d r
j
 2m

e2
 2 1
' 2

j (r )
 j ' ( r ) d rd r '


8 0 jj'
r  r'
2
*
j

e2
8 0
  d rd r '
j' j
1
r  r'




 *j ( r ) j ( r ' ) *j ' ( r ' ) j' ( r ) sjsj'
j
j
1.第二项j,j'可
以相等，自相互
作用
2.自相互作用严
格相消（通过第
二，三项）
3.第三项为交换
项，同自旋电子

  H    jj'
jj '

通过变分:
   d r   
*
j
j'
jj '
 j   u jj'  j '    
么正变换:

0

j'
单电子方程：

2
2
   2  U ion r  e
 2m
40



j'

 j (r ' )
2
e2
d r' 
40
r  r'
  d rd r '
j'


 *j (r ' ) j (r ' )
r  r'



ssj, sj '  j (r )   j j (r )


与Hartree方程的差别：第三项对全体电子，第四项新增，交
换作用项。求和只涉及与j态自旋平行的j’态，是电子服从
Fermi统计的反映。
4. Koopmann定理（1934)


 j    *j (r ) j j (r )d r   N H N  N   N 1 H N 1  N 1
单电子轨道能量等于N电子体系从第j个轨道上取走一个电子
并保持N－1个电子状态不不变的总能变化值。
推广：系统中一个电子由状态j转移到态i而引起系统能量的变化
E   j
i  j
5. 交换空穴(Fermi hole)
将H-F方程改写为：
'
HF   ' 
2
 2

(
r
)


(r , r )
e




j
'


 2  U ion ( r ) 
d
r

(
r
)



(
r
)
 '
j
j j

4 0
r r
 2m

其中：
N
'

 (r )    j (r ' )
j

HF
j
 '
(r , r ) 

2

全体电子在r '位的密度




 *j ' (r ' ) j (r ' ) j (r ) *j (r )
j'
spin//
'


 (r ) j (r )
*
j
交换电荷密度
 
 '

r  r ' 时， HF
(
r
,
r
)


(
r
)
j
HF   '
 j (r , r )对全空间的积分恰好等于一个电子电量：
 

HF
'
'

(
r
,
r
)
d
r

j



j'
spin //



j'
spin //


 *j ( r ) j ' ( r )
' '
* '

(
r
)

(
r
)dr


j
j
'

*
 j ( r ) j ( r )
 *j ( r ) j ' ( r )

  jj'  1
 ( r ) j ( r )
*
j
在HF 方程中合理地扣除被研究电子
与全体电子互作用中的自身作用。 jHF 依赖于j , 难直接用于求解固体多电子问题。
1
 
引入平均交换电荷密度：  HF ( r , r ' ) 
N
 j
定性讨论：假设

N
k F6  ( 3 2  )2 ,  

Fermi hole:与某电子自
旋相同的其余邻近电子
在围绕该电子形成总量
为1的密度亏欠域
j
1
HF
j
 
(r , r ' )
 
exp(ik j  r )
 ' 
 
r  r  sin k F r  r '
 ' 6
F r r
 '
N 9 k F r  r cos k F
 HF   '
 (r , r ) 
 2
k




2
   HF
 
kF r  r '
举例：利用HF方程和Koopmann定理研究原子的电离能
6. Slater’s Xa 方法
HF方法的缺点：
•HF方程仅考虑了交换作用，没有考虑关联，许多体系不能正确描述（金属）
•对原子/分子体系计算量不算大，但对固体计算量就很大：


* 
 j (r ' )i (r ' ) 3
1
 (r ' )




[  2  V ext (r )     d 3 r ']i (r )  [  ( si , s j ) 
d
r
'
]

(
r
)



(
r
)
 
j
i i
2
| r  r '|
| r  r '|
j
Nonlocal exchange term
用平均的局域（交换）势替代HF方程中的非局域交换势：
3
V (r )  
[3 2  (r )]1/ 3 ,   0.66 ~ 1 for most systems
2
ex
X
7. 密度泛函理论(Density functional theory)
(1) Thomas-Fermi-Dirac Model
• energy as a function of the one electron density, 
• nuclear-electron attraction, electron-electron repulsion

 (1)  (2)
VNE [  ]   Z A  d J [  ]  
d 1d 2
rA
r12
A
• Thomas-Fermi approximation for the kinetic energy
T[ ]  c 
5/3
d
3
c  (3 2 ) 2 / 3
10
• Slater approximation for the exchange energy
K [  ]  c   4 / 3 d
9
c
8
1/ 3
3
 
 
  0.66 ~ 1
Thomas-Fermi Model
•
Thomas-Fermi model (semiclassical): 1927
•
Electron density of a uniform electron gas:
n
•
2
(2 )
( volume of Fermi sphere) 
3
kF
2 4 3
k F
3
(2 ) 3
Fermi wavevector and electron kinetic energy of a uniform electron gas
kF  (3 2n)1/ 3
(k )2  2 [3 2n ]2 / 3
T

2m
2m
Thomas-Fermi Model
(k )2  2 [3 2n ]2 / 3
T

2m
2m
kF
• We may assume that the kinetic energy of the electron gas depends on the
local electron density:
 2/3
 (k )
 [3n ( r )]
T (r ) 

2m
2m
2
2
2
• The total kinetic energy of electrons in the system is therefore a functional
of electron density:
 

T   T ( r )n( r )d 3r  C  [n( r )]5 / 3 d 3r
(2) The Hohenberg-Kohn Theorem
In 1964, Hohenberg and Kohn proved that
“For molecules with a nondegenerate ground state, the ground-state
molecular energy, wave function and all other molecular electronic
properties are uniquely determined by the ground-state electron
probability density  0 ( x, y, z ), namely, E0  E0 0  .”
Phys. Rev. 136, 13864 (1964)
Density functional theory (DFT) attempts to
calculate E0 and other ground-state molecular properties
from the ground-state electron density
0 .
Proof:
The electronic Hamiltonian is
n
n
1
1

2
ˆ
H     i    ri   
2 i 1
i 1
j i  j rij
Z

 ri   
 ri
Hˆ   E 
0
0
0

In DFT,  ri  is called the external potential acting on electron i, since
it is produced by charges external to the system of electrons.

Once the external potential  ri  and the number of electrons n are specified,
the electronic wave functions and allowed energies of the molecule are
determined as the solutions of the electronic Schrödinger equation.

Now we need to prove that the ground-state electron probability density  0 r 
the external potential (except for an arbitrary additive constant)
determines
the number of electrons.
 

a) Since   0 r dr  n,  0 r  determines the number of electrons.





r
b) To see that 0
determines the external potential  ri  , we suppose
that this is false and that there are two external potentials  a and b (differing
by more than a constant) that each give rise to the same ground-state electron
density
0.
Let
n
n
1
1

2
ˆ
H a     i   a ri   
2 i1
i 1
j i  j rij
n
n
1
1

2
ˆ
H b     i  b ri   
2 i1
i 1
j i  j rij
Hˆ a 0,a  E0,a 0,a
Hˆ b 0,b  E0,b 0,b
 0,a , E0,a : the exact ground-state wave function and energy of Ĥ a
 0,b , E0,b : the exact ground-state wave function and energy of Ĥ b
Since Ĥ a and
Ĥ b differ by more than a constant,
 0,a and  0 ,b
must be different functions.
Proof:
Assume  0, a
  0,b , thus
Hˆ a 0,a  E0,a 0,a 

  Hˆ a  Hˆ b  0,a  E0,a  E0,b  0,a
Hˆ b 0,a  E0,b 0,a 
 n

 


   a ri   b ri  0,a  E0,a  E0,b  0,a
 i 1

n


thus   a ri   b ri   a constant, which contradicts the given
i 1
information.
If the ground state is nondegenerate, then there is only one normalized
function, the exact ground-state wave function  0 , that gives the exact ground
state energy E0 for a given Hamiltonian
Ĥ .
According to the variation theorem, suppose that

is any normalized
well-behaved trial variation function.
If
   0 , then  Hˆ 
 E0
Now use  0 ,b as a trial function with the Hamiltonian
Hˆ a ,
E0,a   0,b Hˆ a  0,b
  0,b Hˆ a  Hˆ b  Hˆ b  0,b
  0,b Hˆ a  Hˆ b  0,b   0,b Hˆ b  0,b
Substituting Hˆ a  Hˆ b 
n







r


r
 a i b i 
i 1
n
E0,a
gives


  0,b   a ri   b ri  0,b  E0,b
i 1
then
Let Bri  be a function of the spatial coordinates xi , y i , z i of electron i,
then
n
n



  Bri     Bri d
i 1
i 1
n
2

    Bri d
i 1
2

  n Bri d
   r Br dr
Using the above result, we get
E0 , a


 
   0,b r  a r   b r dr  E0,b
Similarly, if we go through the same reasoning with a and b interchanged,
we get
E0 , b


 
   0,a r b r    a r dr  E0,a
By hypothesis, the two different wave functions give the same electron
density:  0 , a   0 ,b . Putting
yield
 0,a   0,b and adding the above two inequalities
E0,a  E0,b  E0,b  E0,a
This result is false, so our initial assumption that two different external
potentials could produce the same ground-state electron density must be false.
Hence, the ground-state electron probability density  0 determines the external
potential (to within an additive constant that simply affects the zero level of
energy) and also determines the number of electrons.
“For systems with a nondegenerate ground state, the ground-state electron

probability density  0 r  determines the ground-state wave function and energy,
and other properties”
E0  E  0 
E0   0 Hˆ  0
 T  0   VNe  0   Vee  0 
 emphasizes the dependence of E0 on

the external potential  r  , which
differs for different molecules.
n

VNe  0    0  ri  0
i 1
  
   0 r  r dr
However, the functionals T  0  and Vee  0  are unknown.
E0 is also written as
E0  E  0 
  
   0 r  r dr  F  0 
F  0   T  0   Vee  0 
The functional F  0  is
independent of the external
potential.
(3) The Hohenberg-kohn variational theorem
 

“For every trial density function  tr r  that satisfies  tr r dr  n

and
 , the following inequality holds:

tr r   0 for all r
E0  E  tr  ,
where E0 is the true ground–state energy.”
Proof:


 



r

  0 . By the

r



r
d
r

n
and
Let tr
satisfy that  tr
tr

Hohenberg-Kohn theorem,  tr r  determines the external potential  tr ,
and this in turn determines the wave function  tr that corresponds to the
density

 tr r .
Let us use the wave function  tr as a trial variation function for the molecule
with Hamiltonian
Ĥ . According to the variation theorem
n

ˆ
ˆ
ˆ
 tr H  tr   tr T  Vee   ri  tr
 E0  E  0 
i 1
Since the left hand side of this inequality can be rewritten as
 
T  tr   Vee  tr     tr r dr  E  tr 
One gets
E tr   E 0 
Hohenberg and Kohn proved their theorems only for nondegenerate ground
states. Subsequently, Levy proved the theorems for degenerate ground states.
(4) The Kohn-Sham method
沈吕九

If we know the ground-state electron density  0 r , the Hohenberg-Kohn
theorem tells us that it is possible in principle to calculate all the ground-state
molecular properties from  0 , without having to find the molecular wave
function.
1965, Kohn and Sham devised a practical method for finding

0 r  and

for finding E from 0 r  . [Phys. Rev., 140, A 1133 (1965)]. Their method
0
is capable, in principle, of yielding exact results, but because the equations of
the Kohn-Sham (KS) method contain an unknown functional that must be
approximated, the KS formation of DFT yield approximate results.
Kohn and Sham considered a fictitious reference system s of n noninteracting

electrons that each experience the same external potential  s ri  that makes

the ground-state electron probability density  s r  of the reference system



equal to the exact  0 r  of the molecule we are interested in:  s r    0 r  .
Since the electrons do not interact with one another in the reference system,
the Hamiltonian of the reference system is
n
1



2
Hˆ s    i   s ri    hˆiKS
 i 1
i 1  2
n
where ĥ KS is the one-electron Kohn-Sham Hamiltonian.
i
Thus, the ground-state wave function  s , 0 of the reference system is:
 s ,0  u1u 2  u n , ui  
KS
i

ri  i
hˆiKS iKS   iKS iKS
 i is a spin function
(either  or  )
 iKS ' s are Kohn-Sham
orbital energies.
For convenience, the zero subscript on

is omitted hereafter.
Define T   and Vee   as follows:
T    T    Ts  
T is the difference in the average
ground-state electronic kinetic energy
 
1  r1  r2   
Vee    Vee    
dr1dr2
2
r12
The quantity
1

 1  




r

r
1
2 r12 dr1dr2

2
is the classical expression (in atomic
between the molecule and the reference units) for the electrostatic interelectronic
system of noninteracting electrons.
repulsion energy.
Remember that
  
E      r  r dr  T    Vee  
With the above definitions, E   can be written as


1  r1  r2   
  
E      r  r dr  Ts    
dr1dr2  T    Vee  
2
r12
Define the exchange-correlation energy functional E XC
 
by
E XC    T    Vee  
Now we have


1  r1  r2   
  
E0  E      r  r dr  Ts    
dr1dr2  E XC  
2
r12
The fourth quantity E XC is a relatively
The first three terms on the right
small term, but is not easy to evaluate
side are easy to evaluate from 
accurately. The key to accurate KS DFT
and they make the main contributions calculation of molecular properties is to
to the ground-state energy.
get a good approximation to E XC
Now we need explicit equations to find the ground-state electron density  .
.
Since the fictitious system of noninteracting electrons is defined to have the
same electron density as that in the ground state of the molecule:
is readily proved that
n
  s   
i 1
 s  0 , it
KS 2
i
n

 Using  s   s   r  ri  s and  s  u1u2  un

i 1

n
1
1 n KS
2
Ts    s  i  s     i 1 12  iKS 1
2
2 i1
i 1
Thus E0 becomes

 r1   1 n KS
E0    Z  
dr1    i 1 12  iKS 1
r1
2 i 1

 

1  r1  r2   
 
dr1dr2  E XC  
2
r12




The Hohenberg-Kohn variational theorem tell us that we can find the

ground-state energy by varying  (subject to the constraint dr  n) so as
to minimize the functional E   . Equivalently, instead of varying
 , we
can vary the KS orbitals  iKS . Thus, the Kohn-Sham orbitals are those that
minimize the above energy expression subject to the orthonormality constraint:
hˆ KS (1) iKS (1)   iKS iKS (1)

Z
 r2  
1
hˆ KS (1)   12     
dr2   XC (1)
2
r12
 r1
with the exchange-correlation potential  XC defined by

 E  r 
 XC r   XC 
r 

(If E XC  r  is known, its functional derivative  XC is also known.)
The KS operator hˆ KS (1) is the same as the HF operator except that the
exchange operators in the HF operator are replaced by

 XC r , which handles
the effects of both exchange and electron correlation.
(5) Local density approximation
 xc

E [  ]    (r ) (  )dr
xc
LDA :
Compare to uniform electron gas result:

Jellium model : E    (  )dr
xc
xc
(High density limit)
 xc 
1
E
N
xc
4
 rs3
 3

0.916
 0.0622 ln( rs )  0.096  o(rs )
rs
The Perdew-Zunger Parameterization And the
Uniform Electron Gas Result
• Let's compare the PZ parameterization with the uniform electron gas result
for rs <<1
Uniform Electron gas result :
E xc
0.916

 0.0622 ln( rs )  0.096  o( rs ) (Ry)
N
rs
1
4 3
 rs
 3
PZ parameteri zation :
 ex (rs )  
0.916
: The uniform electron gas result
rs
 corr (rs )  0.0622 ln( rs )  0.096  [0.004 ln( rs )  0.0232]rs ;
Quantum Monte Carlo result
of Ceperley and Alder
rs  1
The Perdew-Zunger Parameterization of The
Exchange-Correlation Energy (LDA)
 xc

E [  ]    (r ) (  )dr
xc
• Perdew and Zunger’s parameterization of Ceperley and Alder result:
 xc (rs )   ex (rs )   corr (rs )
 ex (rs )  

corr
1
4 3
 rs
 3
0.916
: The uniform electron gas result
rs
0.2846
(rs )  
;
1/ 2
(1  1.0529rs  0.3334rs )
 corr (rs )  0.0622 ln( rs )  0.096  [0.004 ln( rs )  0.0232]rs ;
P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981).
rs  1
rs  1
Exchange And Correlation Energies:
LDA vs. Uniform Electron Gas
0
-0.5
-1
xc
 LDA
( rs )-1.5
( Ry )
Perdew-Zunger
parameterization
of Cepley-Alder's
QMC result
-2
-2.5
Uniform electron gas theory :
0.916

 0.0622 ln( rs )  0.096
rs
-3
0
1
2
rs
3
4
5
But electrons in real solids is not a uniform electron gas!
Flow Chat
1
(  2  V H  V xc  V ext )i (r )   ii (r )
2
Local Density Approximation:
A unexpected success
• Local density approximation is exact for a uniform electron gas
• But for any realistic material system, the charge density is far from uniform
• Kohn and Sham were careful enough to point this out
Kohn and Sham did not expect that LDA would be useful for
calculating the properties of real materials: “we do not expect
an accurate description of the chemical bonding” (by LDA).
• It is still not well understood why LDA works extremely well for a wide
variety of material systems
• The success of LDA is UNEXPECTED
Comments on the DFT methods:
(1) The KS equations are solved in a self-consistent fashion, like the HF equations.
(2) The computation time required for a DFT calculation formally scales the third
power of the number of basis functions.
(3) There is no DF molecular wave function.
(4) The KS orbitals can be used in qualitative MO discussions, like the HF orbitals.
KS
(5) Koopmans’ theorem doesn’t hold here, except  i (HOMO)   IP (HOMO)

(6) Various approximate functionals E XC  r  are used in molecular
DF calculations. The functional E XC is written as the sum of an
exchange-energy functional E X and a correlation-energy functional EC :
E XC  E X  EC
(7) Nowadays KS DFT methods are generally believed to be better than
the HF method, and in most cases they are even better than MP2
Material Properties Studied
Construct model
many-body
Hamiltonian
Thermal and
transport
properties
Phonon structure
DFT
Electron-phonon
coupling,
superconductivity
Ground state properties
(Total energy, crystal
structure, elastic
constants, etc.)
Magnetism,
Spin-orbit coupling
(relativistic effects)
Quasiparticle excitation
(GW approximation)
Optical excitation
(Bethe-Salpeter equation)
The Nobel Prize in Chemistry 1998
Walter Kohn (1923-)
“for his development of the density-functional theory "
密度泛函理论新进展
理论、方法与应用
• 理论体系
交换相关泛函、含时密度泛函、动力学平均场、
密度泛函微扰理论
• 数值方法
基组、格点、线性标度
• 应用
物理、化学、生物、材料、纳米科学、光谱学
Part I: 理论体系
交换相关泛函
+Unoccupied orbital information
jacob's ladder
+ Explicit occupied orbital information
+ Inexplicit occupied orbital information
+ Density gradient
+ Local density
Perdew PRL2003
局域密度近似(LDA)
• LDA underestimates Ec but overestimates Ex, resulting in
unexpectedly good values of Exc.
• The LDA has been applied in, calculations of band structures
and total energies in solid-state physics.
• In quantum chemistry,it is much less popular, because it fails to
provide results that are accurate enough to permit a quantitative
discussion of the chemical bond in molecules.
广义梯度近似(GGA)
• Any real system is spatially inhomogeneous, it has a
spatially varying density n(r), it would clearly be
useful to also include information on the rate of this
variation in the functional.
• In this approximation ,one tries to systematically
calculate gradient-corrections of general functions of
n(r) and ∇n(r)
E xcGGA n   d 3rf nr , nr 
• Different GGAs differ in the choice of the function
f(n,∇n).
Alex D. Becke
“一切都是合法的”
剑宗
John P. Perdew
一定的物理规律（如标度关系和渐进行
为）为基础，PBE
气宗
• GGAs used in quantum chemistry typically proceed
by fitting parameters to test sets of selected
molecules.
• Nowadays the most popular GGAs are PBE in
physics, and BLYP in chemistry.
• Current GGAs seem to give reliable results for all
main types of chemical bonds (covalent, ionic,
metallic and hydrogen bridge).
Meta-GGA
• In addition to the density and its derivatives, Meta-GGAs
depend also on the Kohn-Sham kinetic-energy density:
 (r ) 
2
2m
2
|


(
r
)
|
 i
i
• So that Exc can be written as Exc [n(r),∇n(r), τ (r)]. The
additional degree of freedom provided by τ is used to satisfy
additional constraints on Exc.
• Meta-GGAs have given favorable results, even when compared
to the best GGAs.
• The full potential of this type of approximation is only
beginning to be explored systematically.
Hybrid Functionals
• Common hybrid functional mix a fraction of Hartree-Fock
exchange into the DFT exchange functional.
E
B3
xc
E
LDA
xc
 a(E
exact
x
E
a  0.20, b  0.72, c  0.81
LDA
x
)  b E
GGA
x
 c E
(Becke, 1993)
B3PW91, B3LYP
B3LYP is the main working-horse in computational chemistry
exact
DFT
E 0xc  E DFT

a(E

E
)
xc
x
x
a  0.25
PBE0
(Perdew,1998)
GGA
c
LDA: Slater exchange
Vosko-Wilk-Nusair correlation, etc
GGA: Exchange: B88, PW91, PBE, OPTX, HCTH, etc
Correlations: LYP, P86, PW91, PBE, HCTH, etc
Hybrid GGA: B3LYP, B3PW91, B3P86, PBE0,
B97-1, B97-2, B98, O3LYP, etc
Meta-GGA: VSXC, PKZB, TPSS, etc
Hybrid meta-GGA: HCTHh, TPSSh, BMK, etc
自相互作用修正(SIC)
• Describing the behavior of many electrons interacting via
coulombs’law:
• It will vanish for one electron system because of the selfinteraction in it.
• So we have:
Fermi and Amaldi 1934(the first version of SIC)
• Now it becomes:
Excapprox,SIC [n , n ]  Excapprox [n , n ]   ( EH [ni ]  Excapprox [ni ,0])
i ,
• This correction can be applied on top of any
approximate density functional, and ensures that the
resulting corrected functional satisfies:
Excapprox , SIC [n(1) , 0]   EH [n(1) ]
for a one-electron system.
Optimized Effective Potential (OEP)
• Since orbital functionals depend on the density only
implicitly, we can not directly calculate the functional
derivative via orbitals i [n] .
• In the case of kinetic-energy functional, we use the
Kohn-Sham scheme to minimize E[n].
• In the case of orbital expressions for Exc, the
corresponding indirect scheme is known as the
optimized effective potential.
• The minimization of the orbital functional with
respect to the density is achieved by repeated
application of the chain rule for functional derivatives:
 Excorb [{i }]
v[n](r ) 
 n( r )
orb
'
''

E
[{

}]

(
r
)

v
(
r
)
3 '
3 ''
xc
i
i
s
  d r  d r [
 c.c]
'
''
i
i (r )  vs (r )  n(r )
• Further evaluation of Eq. above gives rise to an
integral equation that determines the
belonging
to the chosen orbital functional vxc [n]
Exc [{i [n]}]
• KLI(Krieger, Li and Iafrate) approximation to solve
the full OEP integral equation. The application of the
OEP methodology to the Fock term is as known as
the EXX (extra-exchange method).
GW近似
• 以自能代替密度泛函局域近似中的交换
关联能
• 固体能隙问题
• 准粒子方程
• 零级近似，plasmon-pole模型，自洽
含时密度泛函
• Runge-Gross定理
• 作用量泛函
• TDKS方程
TDDFT中的线性响应
• 外场微扰
• 一阶密度响应
• KS响应函数
TDDFT中的线性响应
• 线性响应方程
• 交换相关核，绝热局域密度近似
L(S)DA+U
• Mott绝缘体，Hubbard模型
• Anisimov et al.： Stoner I -->Hubbard U
• 轨道序：
• Dudarev et al.：惩罚泛函
动力学平均场理论
• 量子多体问题局域动力学（把点阵模型映射到
自洽的量子杂质模型）
• 冻结空间涨落，考虑局域量子涨落
• Hubbard模型哈密顿量
• 单格点动力学
• 自洽方程
• Anderson杂质模型
• DFT-DMFT
流密度泛函理论
• 处理任意强度磁场下相互作用电子体系
（1987）
• 一套规范不变且满足连续性方程的自洽方
程组
• 交换相关能量不仅依赖于电荷密度还依赖
于顺磁流密度
• 原子分子对磁场的响应，自发磁化，磁场
中的二维量子点，造新的交换相关近似
相对论性密度泛函理论
• 量子电动力学的单粒子方程：Dirac方程
• Dirac-Coulomb（DC）哈密顿量
• Dirac-Coulomb-Breit（DCB）哈密顿量
相对论性密度泛函理论
• 相对论情形的HK定理，四分量DiracKohn-Sham（DKS）方程，数值旋量基
组，缩并Gaussian型旋量基组
• 两分量准相对论方法
– Breit-Pauli近似
– ZORA近似
• 有效核势（ECP）方法
密度泛函微扰理论
• 晶格振动理论
• 线性响应-->Hessian矩阵，2n+1定理
• 冻声方法，分子动力学谱分析方法
几何Berry位相
•
•
•
•
电介质极化，介电常数
偶极矩-->宏观极化；流-->极化变化
电荷密度（波函数的模）；流（波函数的位相）
零电场情况下，任意两个晶体态之间的极化变化
对应着 一个几何量子位相
• 晶格振动、铁电、压电效应、自发极化、静态介
电张量、电子介电常数。。。
• 不如传统的微扰理论方法普适，但实现简单、计
算量小
Part II:数值方法
数值离散方法
• 基组展开
– LCAO基组（Gaussian基组、数值基组）
• 实空间网格
平面波基组：从OPW到PP
• 平面波展开
• 正交化平面波（OPW）
• 赝势（PP）方法
– 经验赝势
– 模守恒赝势
– 超软赝势
Muffin-tin势场与分波方法
• Muffin-tin势场近似
– 缀加平面波（APW）
– 格林函数方法（KKR）
• 线性化方法
– LAPW
– LMTO
• 分波方法的发展
– FP-LAPW
– third-generation MTO, NMTO, EMTO
平面波基组：从USPP到PAW
• 投影缀加波（PAW）方法
• 赝波函数空间
• USPP or PAW? (VASP, ABINIT, ...)
实空间网格
• 简单直观
• 允许通过增加网格密度系统地控制计算
收敛精度
• 线性标度
• 可以方便的通过实空间域分解实现并行
计算
• 处理某些特殊体系（带电体系、隧穿
结。。。）
有限差分
• 从微分到差分
• 提高FD方法的计算效率
– 对网格进行优化，如曲线网格（适应网格）和局部
网格优化（复合网格）
– 结合赝势方法
– 多尺度（multiscale）或预处理（preconditioning）
有限元
• 变分方法
• 处理复杂的边界条件
• 矩阵稀疏程度及带状结构往往不如有限
差分好
• 广义的本征值问题
多分辨网格上的小波基组
• 多分辨分析
• 半取样（semicardinal）基组
线性标度与量子力学中的局域性
• “近视原理”
• 局域化的Wannier函数或密度矩阵
– 绝缘体：指数衰减，能隙越大衰减越快
– 金属：零温下按幂率衰减，在有限温度下可
出现指数衰减
• 局域区域
• 线性标度系数，crossover
线性标度算法
• 分治方法
• 费米算符展开和费米算符投影方法
• 直接最小化方法
– 密度矩阵最小化
– 轨道最小化
– 优基组密度矩阵最小化
线性标度算法
• 基于格林函数的递归方法
• 脱离轨道的（orbital-free,OF）算法
• 对角化以外的线性标度
– 构造有效哈密顿量的算法
– 几何优化与分子动力学
– TDDFT
Part III:应用
物理学：强相关体系
•
•
•
•
模型哈密顿量
LDA++
电子结构：CrO2
点阵动力学: 钚
化学：弱作用体系
• 松散堆积的软物质、惰性气体、生物分
子和聚合物，物理吸附、Cl+HD反应
• 用传统的密度泛函理论处理弱作用体系
• 一个既能产生vdW相互作用系数又能产
生总关联能的非局域泛函：无缝的
（seamless）方法
• GW近似
• 密度泛函加衰减色散（DFdD）
生命科学：生物体系
• 困难（尺寸问题、时间尺度）
• QM/MM方法（饱和原子法、冻结轨道法）
• 简单势能面方法
– 线性同步过渡（LST ）
– 二次同步过渡（QST ）
• 完全的分子动力学
– 并行复制动力学（parallel replica dynamics）
– 超动力学（hyperdynamics, metadynamics）
– 温度加速的动力学（temperature accelerated
dynamics ）
– 快速蒙特卡罗（on-the-fly kineric Monte Carlo）方法
纳米和材料科学：输运性质及其他
• 输运：非平衡态第一性原理模拟
• 材料力学：运动学Monte Carlo（KMC）-> 点阵气体和元胞自动机 --> 连续方程
的有限差分有限元求解
光谱学：激发态和外场
• 系综密度泛函理论
• 考虑系统对称性，用求和方法计算多重
态激发能
• 多体微扰理论，GW近似Bethe-Salpeter方
程
• TDDFT，线性响应
一些计算软件
•
•
•
•
Gaussian, DMol3, Q-Chem, ADF, SIESTA
VASP, CASTEP,ABINIT, PWSCF, CPMD
Octopus
BigDFT
5.2 费米液体理论
1. 费米体系
费米温度：
1
TF 
kB
lim 
 : 化学势
T 0
均匀的无相互作用的三维系统，费米温度：
F
2
1
2
TF 

( 3 n) 3
kB 2mk B
费米简并系统：费米子系统的温度通常运运低于费米温度
室温下金属中的传导电子
费米温度给出了系统中元激发存在与否的标度
在费米温度以下，系统的性质由数目有限的低激发态决定。
有相互作用和无相互作用的简并费米子系统中，低激发态的
性质具有较强的对应性。
2. 费米液体
金属中电子通常是可迁移的，称为电子气， 电子动能：
3
3  2 3 2 N 2 / 3
Ek   F 
(
)  n2 / 3
5
5 2m V
2
e
电子势能：
E 
 n1 / 3
P
r
ij
ij
在高密度下，电子动能为主，自由电子气模型是较好的近
似。在低密度下，电子之间的势能或关联变得越来越重要，
电子可能由于这种关联作用进入液相甚至晶相。
较强关联下，电子系统被称为电子液体或费米液体或
Luttinger液体(1D)
无相互作用的粒子绝热方式

电子－电子相互作用
？
单电子态 

多电子本征态
从单电子态出发描述体系真正状态是否是好近似？
相互作用： (1)单电子能级分布变化(势的变化);(2)电子散射
导致某一态上有限寿命(驰豫时间)
3. 朗道费米液体理论
单电子图象不是一个正确的出发点，但只要把电子改成准
粒子或准电子，就能描述费米液体。准粒子遵从费米统计，
准粒子数守恒，因而费米面包含的体积不发生变化。

假设激发态用动量 p 表示
E  E g    ( p)n p

p
系统的熵：
n p 为准粒子分布函数，E g 为基态能量。
S   k B  [n p ln n p  (1  n p ) ln( 1  n p )]

p
在平衡态下，n 服从费米分布 n 

p
 ( p)与温度有关

p
1
e [  ( p )   ] / k BT  1
朗道费米液体理论的适用条件：
(1). 必须有可明确定义的费米面存在
在平衡态下，n p 服从费米分布
n p 
1
e [  ( p )   ] / k BT  1
 ( p )与温度有关
n( k )  Z ( k ) [    ( k )]  ( k )
( k )是一个连续变化函数。
物理图象：一个准粒子可以看作由一个裸粒子(比例为Z ( k F ))加上
周围其他元激发的贡献(比例(1  Z ( k F )), 对应的占据率为( k )
m
 正常费米液体要求：
0  Z (k F )  1
Z
Z ( k F )  0：准粒子图象失效
m  m* 
(2). 准粒子有足够长的寿命
Fermi Liquid Theory
First of all, What is Fermi Liquid ? If a system is described by
the following Hamiltonian, it is called Fermi Liquid :


 


1
H   0 ( k )n( k )   f ( k , k')n(k )n(k')
2V k , k '


k




where n( k )  C ( k )C ( k )
momentum space
is the number operator in
(1) The theory is non-trivial because of the quartic interaction
(2) However, the single-particle picture might work under some
assumptions
Consider a system with 4 fermion interaction. The most general
Hamiltonian is :











H   0 ( k )n( k ) V (ki )  (k1  k2  k3  k4 )  C (k1 )C (k2 )C (k3 )C (k4 )




ki
k
In weak coupling, some interactions are more important than
the other – by space argument. Take 2D electron gas as an
example. In the weak coupling and low temperature limit, only
electrons very close to the Fermi surface is important.

V (ki )  V (1 ,  2 , 3 ,  4 )
Due to momentum conservation, not all angles
(1 , 2 , 3 ,  4 )
are free
Take (1 , 2 ) as independent parameter
 
first. For given
k1 , k2 , the other two
 
momenta k3 , k4 are completely fixed.


However, if k1  k2  0 , the allowed
phase space is enlarged to the whole
Fermi surface. Thus, we expect this
kind of interaction would dominate
over the others.
 




( k , k')   ( k ) ( k ) ( k ) ( k )


BCS scattering
Note that this argument is only appropriate in weak coupling
limit !
On the other hand, if we take (1 , 3 ) as independent
parameter, there is another kind of interaction would
dominate the others (by similar argument)
 




F ( k , k')   ( k ) ( k ) (k ') ( k')


forward scattering
Thus, in weak coupling, we expect these two interactions are
the most important ones.
(1) For repulsive interaction, it can be shown that the BCS
interaction can be safely ignored → Fermi Liquid !
(2) For attractive interaction, the instability is triggered ……
The system is better described by BCS theory !
Now we are ready to write down the Fermi Liquid Theory in
more familiar form
In weak coupling and at low temperature, the density is not
far from the free Fermi distribution. Thus, define the
density variation





Rewrite H in terms of n( k )
n( k )  n( k )  (k F  k )
H


k





1
(kF  k )  0 ( k )   0 ( k )n( k ) 
2V
 
 

1
1
  (kF  k ') f ( k , k')n( k ) 
V 
2V
k ,k'
 




 f ( k , k ')(kF  k )(kF  k ')

k ,k'
 


 f ( k , k')n( k )n( k')

k ,k'
Bare energy : E0  

k



1
(k F  k )  0 ( k ) 
2V




 (kF  k )(kF  k ') f ( k , k')

k ,k'
It is only a constant. Will ignore it later.
quasi-particle spectrum :
H1 


 ( k )n( k )



 
1
where  ( k )   0 ( k )   f ( k , k')
V 
k ' kF
k
Finally , quasi-particle interactions
1
H2 
2V
 


 f ( k , k ')n( k )n( k ')

k ,k'
Collecting all terms, the Hamiltonian is
H


k


 ( k )n( k )  1
2V
 


 f ( k , k')n( k )n( k')

k , k'
 
Now use mean-field approximation , we can solve the quastiparticle distribution self-consistently.
H MF 


k



k


 ( k )n( k )  1
2V
 


 f ( k , k ')  n( k ')  n( k )  2

k ,k'

 


1
[ ( k )   f ( k , k ')  n( k ') ] n( k )
V 
k'
It becomes a quadratic theory and the mean-field dispersion is :
 

1
E ( k )   ( k )   f ( k , k ')  n( k ') 
V 


k'
Make use of the MF dispersion , we obtain the self-consistent
equation : 


1
1
n( k ) 
 ( k F  k ) ,  

 [ E ( k ) ]
k BT
e
1
Simple Picture for Fermi Liquid
Free theory :
Now , add in the
particle-particle
interaction.

2

k
 0 (k ) 
2m
One quasi-particle :

2
 


k
1
 (k ) 
  f ( k , k ') (kF  k')
2m V 

k'
Many quasi-particles present :
 

1
E ( k )   ( k )   f ( k , k ')  n( k ') 
V 


k'
Many interesting properties :
(1) Effective mass m* :
 
m*
1
 1  F1 , 2 N (0) f ( k , k')   Fl Pl (cos  )
m
3
l
(2) Specific heat :
Linear temperature dependence
1 *
2
Cv  m k F k B T
3
(3) Sound velocity :
(4) Spin susceptibility :
1
 1  3 F1

0 1  1 Z
0
4
 v1  1  F0


 v1 (0)  1  1 F1
3
Compare with Curie susceptibility
1
 (T ) ~ !
T
朗道费米液体理论是处理相互作用费米子体系的唯象理论。
在相互作用不是很强时，理论对三维液体正确。
二维情况下，多大程度上成立不知道。
一维情况下，不成立。luttinger液体
一维：低能激发为自旋为1/2的电中性自旋子和无自旋荷电为
 e 的波色子的激发。
非费米液体行为：与费米液体理论预言相偏离的性质
THE PHYSICS OF
LUTTINGER LIQUIDS
THE LUTTINGER LIQUID:
INTERACTING SYSTEM OF 1D ELECTRONS AT LOW ENERGIES
E (k )
EF
FERMI SURFACE HAS ONLY TWO POINTS
failure of Landau´s Fermi liquid picture
 kF
 kF
k
ELECTRONS FORM A HARMONIC CHAIN AT LOW ENERGIES
Coulomb + Pauli
interaction
collective excitations are vibrational modes
REMARKABLE PROPERTIES
Absence of electron-like quasi-particles
(only collective bosonic excitations)
Spin-charge separation
(spin and charge are decoupled and propagate with different velocities)
Absence of jump discontinuity in the momentum distribution at  kF
Power-law behavior of various correlation functions and transport
quantities. The exponent depends on the electron-electron interaction
OUTLINE
What is a Fermi liquid, and why the Fermi liquid concept breaks in 1D
The Tomonaga-Luttinger model
• The TL-Hamiltonian and its bosonization
• Diagonalization
• Bosonic fields and electron operators
• Local density of states
Tunneling into a Luttinger liquid
Luttinger liquid with a single impurity
Physical realizations of Luttinger liquids
LITERATURE
H.J. Schulz, G. Cuniberti and P. Pieri
Fermi liquids and Luttinger liquids, cond-mat/9807366
K. Flensberg
Lecture notes on the one-dimensional electron gas and the theory of
Luttinger liquids
J. von Delft and H. Schoeller
Bosonization for beginners refermionization for experts, cond-mat/9805275
J. Voit
One-dimensional Fermi liquids, Rep. Prog. Phys. 58, 977 (1995)
SHORTLY ABOUT FERMI LIQUIDS
Landau 1957-1959
Low energy excitations of a system of interacting particles described
in terms of ``quasi-particles`` (single-particle excitations)
Key point: quasi-particles have same quantum numbers as the
corresponding non-interacting system (adiabatic continuity)
Start from appropriate noninteracting system
Renormalization of a set of parameters (e.g. effective mass)
Also collective excitations occur (e.g. zero sound) at finite energies
FERMI LIQUIDS II
Effect of Coulomb interactionis to induce a finite life-time 


k q
k

k'
kz
 
k ' q
ky
Pauli exclusion principle
kx
only states within kT around Fermi sphere available
quasiparticle states near Fermi sphere scatter only weakly
 k1  0 when k  k F
QUASI-PARTICLE PICTURE IS APPLICABLE IN 3D
3D
FERMI LIQUIDS III
 / EF
collective 3
excitations
2
(plasmons)
1
0
0
DISPERSION OF EXCITATIONS IN 3D
single-particle
excitations
1
2
3
4
q / kF
Finite jump in momentum distribution
nointeracting
n p
T=0
Z quasi-particle weight
Z
pF
p
LIFETIME OF ``QUASI-PARTICLES´´
Fermi´s golden rule yields for the lifetime 
2

1
k

screened Coulomb interaction
2 1
V (q)

2 2
 ( k   k '   k q   k ' q )
 

  k 'q  (q ,  k q   k )
energy conservation
spin

 nF ( k ' )[1  nF ( k q )][1  nF ( k ' q )]  [1  nF ( k ' )]nF ( k q )nF ( k ' q )
k   k  
scattering out of state k
scattering into state k
In 3D an integration over angular dependence takes care of -function
V 
1
k 

2
V  k3 k2   , k  k
k
F
d 0 d ' θ(k     ' )  2
T=0
3 0

2
LIFETIME OF ``QUASI-PARTICLES´´ II
What about the lifetime  in 1D?
In 1D k, k´ are scalars. Integration over k´ yields

1
k
2 dq V1 (q)
 
nF ( k q )[1  nF ( k q )][1  2nF ( k )]
 q  (q)
formally, it diverges
at small q
but we can insert a
small cut-off
At small T
nF ( )[1  nF ( )]  k BT (   F )
1  2nF ( )  (   F ) / 2k BT
1/ (   F ), q  0
 /(   F )  
q  2k F
const,
1
i.e., this ratio cannot be
made arbitrarily small
as in 3D
BREAKDOWN OF LANDAU THEORY IN 1D
(   F )
no longer diverges at kF

(no angular integration over direction of k ' as in 3D )
collective excitations are plasmons with 1D  v k
 / EF
DISPERSION OF EXCITATIONS IN 1D (RPA)
3
2
single particle
1
gapless
plasmon 0
1
2
q / kF
3
4
COLLECTIVE AND SINGLE-PARTICLE EXCITATION NON DISTINCT
THE TOMONAGA-LUTTINGER MODEL
EXACTLY SOLVABLE MODEL FOR INTERACTING
1D ELECTRONS AT LOW ENERGIES
E (k )
Assumptions:
 kF
kF k
Only small momenta exchanges are included
Dispersion relation is linearized near EF
(both collective and single-particle excitations have linear dispersion)
Model becomes exact when linearized branches extend from (, )
TOMONAGA-LUTTINGER HAMILTONIAN
H  H 0  H int
interaction
free part
Free part
H 0   k ck ck ,
k   k    ( k  kF )vF
k
ck , ck fermionic annihilation/creation operators
c , c  
k

k'
k ,k '


, ck , ck '   0, ck , ck'  0
Introduce right moving k > 0, and left moving k < 0 electrons
ck  cRkθ(k )  cLk θ(-k )


H 0  vF  k (cRk
cRk cLk
cLk )  ( N R  N L )kF vF
k
TL HAMILTONIAN II
H  H 0  H int
Interactions
H int
free part
interaction
1
 

V
(
q
)
c

k ck 'ck '  q ck  q  H 2  H 4
L k ,k ',q0
backscattering
g1
forward
g2
umklapp
g3
forward
g4
BOSONIZATION
BOSONIZATION: EXPRESS FERMIONIC HAMILTONIAN
IN TERMS OF BOSONIC OPERATORS
construct bosonic Hamiltonian with the same spectrun
(a) and (b) have
same spectrum but
different ground
state
EF
EF
(a)
(b)
(c)
(d)
EXCITED STATE CAN BE WRITTEN IN TERMS OF CHARGE
EXCITATIONS, OR BOSONIC ELECTRON-HOLE EXCITATIONS
STEP 1
WHICH OPERATORS DO THE JOB?
Introduce the density operators (create excitation of momentum q)
 R (q)   ck ck q ,
k 0
 L (q)   ck ck q
k 0
and consider their commutation relations
Lq
[  R (q ),  R (q ' )]   q ,q '
2
Lq
[  L (q ),  L ( q ' )]   q ,q '
2
[  R (q ),  L (q ' )]  0
nearly bosonic
commutation relations
STEP 1: PROOF
Consider e.g. [ R (q), R (q)]
[  R (q ),  R (q)] 



[
c
c
,
c
 k k q k 'ck ' q ]
k ,k '0




(
c
c
c
c

c
c
c
 k k q k ' k ' q k ' k ' q k ck q )
k ,k '0
algebra of
fermionic operators



(
c
c


c
 k k ' q k q,k ' k 'ck q k ,k ' q )
k ,k '0
  θ(k  q )(ck ck  ckq ck q )
k 0
nk  ck cc
occupation operator
nk  nk  1
if
k  kF
  θ(k  q )( nk  nk q )
k 0
q
L
Lq

dk 

2 0
2
STEP 2
Examine now [ H 0 ,  R (q)]  qvF  R (q)
[ H 0 ,  L (q)]  qvF  L (q)
STATES CREATED BY  R ARE EIGENSTATES OF H 0WITH ENERGY
vF q
BOSONIZED HAMILTONIAN
H0 
 v F
v F
2
2
(

(

q
)

(
q
)


(

q
)

(
q
))

(
N

N

R
R
L
L
R
L)
L q 0
L
and
H 2   g 2 (q)(  R (q)  L (q)   L (q)  R (q))
q 0
H 4   g 4 (q)(  L (q)  L (q)   R (q)  R (q))
q 0
interactions
STEP 2: PROOF
Example:
[ H 0 ,  R (q)]  qvF  R (q)
[ H 0 ,  R (q )]  v F
 v F


k
[
c
c
,
c
 k k k 'ck ' q ]
k ,k '0
 v F


k
(
c
c


c
 k k ' q k 'k k 'ck k ,k ' q )
k ,k '0


k
(
c
c

c
 k k q k qck θ(k  q))
k ,k '0
 v F q R (q )  v F

(
k

q
)
c

k ck  q   qv F  R ( q )
0 k   q
STEP 3
Introduce the bosonic operators
aq 
2
 R (q ), aq 
qL
2
 R (q)
qL
bq 
2
 L ( q ), bq 
qL
2
 L (q)
qL
[aq , aq ]  [bq , bq ]   q ,q '
[aq , aq ' ]  [bq , bq ' ]  0
yielding
g4
1


H  v F  q (1 
(q ))( aq aq  bq bq )   qg 2 (q )(aqbq  aqbq )
v F
 q 0
q 0
DIAGONALIZATION
E (q)  qv(q), v  vF
2

g 4 (q)   g 2 (q) 
1 
  

 2vF   2vF 
2
SPIN-CHARGE SEPARATION
If we include spin, it gets slightly more complicated ... and interesting
H0 
 v F
L
(   (q )   (q )    (q )   (q ))


,q  0
R
R
L
L
and interaction (satisfying SU2 symmetry)
V
H int   (  R (q)  L ' (q)  R  L)
L  ',q
Introduce the spin and charge densities     ,       
Hamiltonian decouple in two independent spin and charge parts,
with excitations propagating with velocities v  , and v  v F
SPACE REPRESENTATION
Long wavelength limit (interactions g4 (q)  0, g2 (q)  V2 )
Appropriate linear combinations ,  of the field (x) can be defined.
Then one finds
v
1
2
H
dx
[
g

(
x
)



(
x
)
]
x

2
g
2
where
1  V2 / vF
g
,
1  V2 / vF
v   vF 1  (V2 / vF )2
Luttinger parameter g <1 repulsive interaction
BOSONIC REPRESENTATION OF 
Fermionic operator
Ψ ( x) 
1
ikx
c
e
 Ψ R ( x)  Ψ L ( x)

k
L k
Where e.g. Ψ R
• decreases the number of electrons by one
• displaces the boson configuration for that state
Express  in the form of a bosonic displacement operator B
  R, L
• from [Ψ ( x),  ( x' )]   ( x-x')
[eB ,  ( x' )]  [ B,  ( x' )]eB
if [ B,  ] a c-number
BOSONIZATION IDENTITY
Ψ R , L ( x)  U R , L e
 ikF x  iR ,L ( x )
e
U ladder operator,  bosonic
LOCAL DENSITY OF STATES
i) Local density of states at x = 0



1
i t / 

 ( ) 
dt
e
Ψ
(
0
,
t
),
Ψ
(0,0) 

 
 density of states of non-interacting system
vF
 ( ) 
v(( g  1 / g ) / 2)


c
1
( g 1/ g 2 )
2
at T = 0
 c cut-off energy
 gamma function
ii) Local density of states at the end of a Luttinger liquid
vF
end ( ) 
vg / 2

 (1/ g 1)
c
at T = 0
MEASURING THE LDOS
Measurement of the local density of states by tunneling
I
V
H  H1  H2  HT
coupling
system 2
system 1
* 
H T   (tkp c2p c1k  tkp
c1k c2 p )  H 21  H12
kp
See e.g. carbon nanotube experiment by Bockrath et al. Nature, 397, 598 (1999)
MEASURING THE LDOS II
Tunneling current can be evaluated by use of Fermi´s golden rule
2
21 

I 21  e(21  12 )
ij
tunneling rate i to j
I
t kp constant

f 21
2
f 21 HT i  ( Ei  E f21 )eEi
f 21  c2 c1
1
d [nF (  1 )  nF (   2 )] 1 ( )  2 ( )

eR
R   /( 2 t  1 2e 2 )
2
V
1
( g 1/ g 1)
I
d


(

)

(

)

V
LL
LL
LL to LL
eR 0
V
I
1
( g 1/ g ) / 2
d


(

)

V
LL
eR 0
LL to metal
SINGLE IMPURITY
Weak link
x=0
Again tunneling current can be evaluated by use of Fermi´s golden rule
However, now is tunneling from the end of a LL
V
1
( 2 / g 1)
I
d


(

)

(

)

V
end
end
eR 0
end to end
Charge density wave is pinned at the impurity
PHYSICAL REALIZATIONS
Semiconducting quantum wires
Edge states in fractional quantum Hall effect
Energy
Single-walled metallic carbon nanotubes
EF
k
metallic 1D conductor with 2 linear bands
5.3
强关联体系
1. 窄能带现象
金属与绝缘体之分：
(1)能带框架下的区分：
导带
导带
价带
价带
(2)无序引起的Anderson 转变：
EF
局域态 扩展态
局域态 局域态
EF
扩展态
局域态
(3)电子间关联导致的Mott金属－绝缘体转变
(a).MnO:5个3d未满3d带；O2- 2p是满带不与3d能带重叠
能带论MnO的3d带将具有金属导电性
实际上，MnO是绝缘体！
(b).ReO3:能带论绝缘体。实际上是金属。
(c). 一些过渡金属氧化物当温度升高时会从绝缘体金属
f电子或d电子波函数的分布范围是否和近邻产生重叠，是电
子离域还是局域化的基本判据
l壳层体积与Winger-Seitz元胞体积的比值：
4f最小，5f次之，3d,4d,5d…
多电子态的局域化强度的顺序：
4f>5f>3d>4d>5d
______________能带宽度上升
另外，从左往右穿过周期表，部分填充壳层的半径逐步
降低，关联重要性增加。
Y
Sc
4f，5f元素和3d,4d,5d元素的壳层体积与
Winger-Seitz元胞体积的比值
Smith和Kmetko准周期表
局域性
窄带区域
重费米子
强铁磁性
超导体
离域性
另一类窄带现象：来自能带中的近自由电子与溶在晶格中具
有3d,5f或4f壳层电子的溶质原子相互作用
Friedel与Anderson
稀土元素或过渡金属化合物中的能隙不可能仅用“电荷转移
能”、“杂化能隙”、“有效库仑相关能”三者之一来描述，
而应该说三者同时发挥作用。
稀土化合物部分存在混价“mixed valence”。混价的作用导
致在Fermi面附近存在非常窄的能带(部分填充f能带或f能
级），电子可以在4f能级和离域化能带之间转移，对固体基
态性质产生显著影响。
2. 窄能带现象的理论模型
选择经验参数的模型Hamilton量方法
Hubbard模型和Anderson模型
The Hubbard Model
From simple quantum mechanics to
many-particle interaction in solids
-a short introduction
Historical facts
• Hubbard Model was first
introduced by John
Hubbard in 1963.
• Who was Hubbard? He
was born in 1931 and died
1980. Theoretician in solid
state physics, field of work:
Electron correlation in
electron gas and small
band systems. He worked
at the A.E.R.E., Harwell,
U.K., and at the IBM
Research Labs, San José,
USA.
Picture taken from: Physics Today, Vol. 34, No4, 1981
What, in general, is the HM?
• Hubbard model is a quantum theoretical
model for many-particle interaction in
and with a periodic lattice
• It is based on an interaction Hamitonian,
some transformations and assumptions to
be able to treat certain problems (e.g.
magnetic behaviour and phase
transitions) with solid state theory
Quantum mechanics
• Basics:
– Schrödinger equation
– Expectation values
 2 2
 

 
  V ( r )   i   E
2m
t




H
* ˆ
ˆ
A   A  or A    Adu
– Orthonormality and closure relation


3

 
* 

(
r
,
t
)

(
r
,
t
)
d
r 1
 k r , t  l r , t    kl ( r   r )

– The bra-ket notation  ( r , t )  r  ( t )
k
k






real space fct .
state fct .
k l   kl
Basis transformation, mathematically
• A basis transformation can be simply
performed:
A  1 A
  k k A

• An equation is transformed the same way:
A
 A1
 A k k 
  k A k

Single particle equations
• Particle in a potential:
 2 2
 

 
  V ( r )   i   E
2m
t





H
– Periodic potentials:



V ( r )  V ( r  nR)
• Solution for weak coupling to potential:
Bloch wave



1
 k ( r ) 
exp ikr uk ( r )
V
 
Single particle equations
2 2


• Dispersion relation for free electrons (dashed line):  ( 0 ) ( k )  k
2m
•
Dispersion
relation for
Bloch electrons
(quasifree)(solid line):
The
 energies at
k  G are no
longer
degenerated.
Two
eigenenergies
at those points.
Graph from Gerd Czycholl, „Theoretische Festkörperphysik“, Vieweg-Verlag
Single particle equations
• Stronger lattice potential: coupling to lattice
points occurs; a modified Bloch wave is used,
e.g. Wannier states resulting from the TightBinding-Model:

1

 nk ( r ) 
N

 
 exp(ikR)wn (r  R)
R
• Wannier states produce an orthonormal base of
localized states; atomic wavefunctions would also
be localized, but they are not orthonormal.
Comparison between the two new
wavefunctions
 



1


 k (r ) 
exp ikr uk ( r )
V
Bloch wavefunction

1

 nk ( r ) 
N

 
 exp(ikR)wn (r  R)
R
Graph from Gerd Czycholl, „Theoretische Festkörperphysik“, Vieweg-Verlag
Graph from Gerd Czycholl, „Theoretische Festkörperphysik“, Vieweg-Verlag
Wannier wavefunction (w-part)
Wavefunction for many particles
•
Wavefunction is not simply the product of
all single particle wavefunctions;
1. Particles can not be differed
2. Fermions must obey Pauli principle
•
Ansatz: Slaterdeterminante
 k ... k 1... N e  
1
Ne
 k1 1 

1
det 

1

Ne!
 kNe


Ne 




Ne  

k Nen

k1
Second Quantization for Fermions
• Creation and distruction operators create or destroy
states:
ck 1 0  0 1  0

ck 1 0 1 1   1 0 0 1


ck 1 0  0 1   1 01 1

ck 1 0 1 1  0


Second Quantization
• The operators fulfill the commutator relation:
c
c
k
k
, ck
,c


 ck ck  ck ck  0  c , c
 

k 


k



k 
1 for   

0 else
• This is a must, otherwise one would disturb
closure relation and orthonormality of
wavefunctions described by second quantization
Hamiltonian for many particles
• Summation over all single particles Hamiltonians +
interaction Hamiltonian:
2

 
pi
H  H 0  H1 with H 0  
 V ri  and H1   u ri  rj 
i 1 2m
i j
Ne
• interaction potential u is the repulsive Coulomb
interaction
 
e2
uri  r j    
ri  r j
Operators in second quantization
Ne

A   A ri 
1
1
i 1


i  i 

1 
1 
A   k
k A ri  k 
k
i 1  1
 1



Ne
i  i 
1
A
1 
Ne

  k

i  1  1


 
, 1
1
i  i 
 
k A ri  k 
1 
 1

k A r  k 
1 
Ne
 k
i  i 
k
i 1


 c k c k 
A
1 


1 

A
c
 k ,k k ck
 ,  1
i  i 
k
Operators in second quantization
A 2  
A
2 



A 2 

1
2   
ri , rj 
A

2 i j
 j  j
1
i  i 
2   
ri , rj  k
k
k
k
k
A






2 i  j  ,  , ,
 j  i 
 j
1
i 
2   
k
k
k
k  A ri , r j  k


2 i  j  ,  , ,
 j  j
1
i 
i 
2 
Ak k  ,k k  k
k
k
k

2  ,  , ,
i j
1  2 
Ak k  ,k k ck ck ck ck

2  ,  , , 1
 j  j
 j
k
k k 
i   j 
i  i 
k
i 
k
k
Hamiltonian in second quantization
• Is transformed like the one-particle operator A(1) and
the two-particle operator A(2)

 
pi2
H 
 V ( ri )   uri  r j 
2m i  j
i 1 
Ne
t
1
 
H   t k k  c c   uk k  ,k k ck ck ck ck
2 k k  k k
k k 

k k 
Coming closer to Hubbard...
Evaluation of matrix elements with Wannier wave functions:
1
 
H   t k k c c   uk k ,k k ck ck ck ck
2 k k k k
k k 

k k 
H 

 
nR , nR ,
t nR ,nR cnR  cnR 
1
     un1 R1 n2 R 2 ,n3 R 3 n4 R4 cn1 R1 cn2 R 2  cn3 R 3  cn4 R4
2   n1 R1 n4 R4 
Final Assumptions
• Now: only direct neighbor interactions, restriction to
one band.
E 0  t R , R ; t  t R , R  
 2  2 e²
  w ( r ) w ( r )   d 3 rd 3 r 
U u
r  r
 2 e²


2
V  uR R   , R   R   w ( r ) w ( r    )   d 3 rd 3 r 
r  r




e²
2
*






X  uRR , R R     w ( r )w ( r   ) w ( r )   d 3 rd 3 r 
r  r
 
RR , RR
Meaning of matrix elements
• t: single particle hopping
• U: Hubbard-U, describes onsite-Coulomb
interaction
• V: Nearest-neighbor (density) interaction
• X: conditional hopping interaction
The Hubbard Models
• simple Hubbard model
H 
tc c
 
R n . n .

R
 
R  
U
c c c c





R R R R
R
• extended Hubbard model
H ext  H  V 
c c c

1
2
R  

R

R


R  R 
c
• and any combination of matrix elements...
Mott-Hubbard transition, insulating (Mott)
phase
Case 1: Strong coupling, U/t >> 1: Mott insulating
state for a half-filled system. The density of states
(available states for adding or removing particle) consits
of 2 “Hubbard bands” at E0 and E0+U. The system is
insulating if Efermi is between the bands. This phase is
antiferromagnetic, remember the Heisenberg term.
Mott-Hubbard transition, metallic phase
Case 2: t/U>>1, weak coupling: Gap disappears, density of
states unchanged to simple tight-binding; the Fermi energy
now lies in the band middle and the system is metallic. This
transition from insulating to metallic due to changes in U/t is
called Mott-Hubbard transition.
Mott-Hubbard transition
T , 下Hubbard 带的e  上Hubbard 带，形成电子－空穴对激子：
V 
e2
4 0 r
T  ，上Hubbard 带上有足够多电子，屏蔽作用使电子－空穴对减为：
V 
e2
4 0 r
e  k0 r
当电子浓度到达临界值，使屏蔽长度1 / k 0 短于电子－空穴对的尺度a 0 :
a0 k0  1
时，束缚解除，电导率急剧增加，过渡到金属态
Some Examples...
Let‘s look at the following case:
2D square lattice, the band we restrict to is half filled
t, U  0
Antiferromagnetism for half-filling
H 
tc
c

 
R n . n .

R
 
R  
U
c
c
c
c





R R R R
R
U/t>>1, strong coupling: Spin-spin interaction expected
(direct exchange interaction, RKKY interaction, superexchange interaction): virtual hopping is introduced, treated
as perturbation. Calculation and operator relations yield as
only dynamical part
 
 
2t ²
S   J
S 
H 
S
S


R R 
R R 


U
R
 R
0
This is exactly the Heisenberg Hamiltonian for
antiferromagnetic exchange coupling with
coupling constant J.
Dependence of phases on U/t and n (where n=number of
electrons/lattice site)
• The following graph is shown without any warranty:
(Perturbation theory can not be applied in the mid
region of U/t)
Graph from P. Fazekas, Electron correlation and magnetism
Limits of the model
• The Hamiltonian is in principle applicable for every
solid state problem; often, a screened potential instead
of the unscreened Coulomb potential is used
• Up to now, the problem is to find calculable
wavefunctions; the problem is often not analytically
solvable. The advantage of the Hamiltonian, not to be
restricted to very special conditions, is the disadvantage
during the calculation
Conclusions
• Hubbard Model is derived from manyfermion Hamiltonian
• Is a powerful model to describe phases in
terms of interactions
Graphs taken from:
Theoretische Festkörperphysik / Gerd Czycholl.
- Braunschweig ; Wiesbaden : Vieweg, 2000
Lecture notes on electron correlation and magnetism / Patrik Fazekas.
- Singapore : World Scientific, 1999
Thanks to Gerd Czycholl for writing the book „Theoretische Festkörperphysik“
Hubbard处理干净系统的，Anderson模型则被用来处理包含
杂质的系统。近藤Hamilton量：
H 
k

 

E ( k )nk  JS  s

s   ck   'ck ' '

S : 杂质的局域自旋
为自由电子在杂质附近的局域自旋密度
kk '
 
J为S与s 间的交换作用积分。

这一理论对S的存在未加论证。
引入在位相关能U , 并为描述杂质原子引入附加轨道d，它处于自由
电子的Fermi 面附近，其占据数为nd，产生数符为cd，
Anderson
H 
k
Hamilton :

E ( k )nk  Unˆ d  nˆ d   E d ( nˆ d   nˆ d  )   Vdk (C d C k  c .c .)
k
d  k隧穿项Vdk : 使局域轨道 d 通过离心力势垒的隧穿过程进入Friedel共振态
由于穿过离心力势垒的隧道效应所引起的d电子共振
共振宽度：＝ (Vdk2 )  ( E d )
为态密度。
如E d  E F ，  / U  ，共振态将分裂
磁性区
局域态 d 与被散射的自由电子态相混合：
虽不存在局域束缚电子态，却存在局域磁距。
Anderson与Kondo等效：
J
2

(
1
1

)
Ed Ed  U