童培庆 - 兰州大学物理学院计算物理与复杂系统研究所
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Transcript 童培庆 - 兰州大学物理学院计算物理与复杂系统研究所
一维非均匀系统的电子性质和
量子相变
童培庆
南京师范大学物理
科学与技术学院
[email protected]
Contents
﹡一维无序和非均匀系统中的电子能谱和波
函数性质
﹡一维无序和非均匀量子自旋链的量子相变
能带论
单电子
理论
Born-Oppenheimer近似
单电子近似（Hartree-Fock）
周期势场中的单电子Schrodinger方程
2 2
H (r ) [
V (r )] (r ) (r )
2m
V (r Rn ) V (r )
Bloch定理（扩展态）
ik Rn
(r Rn ) e (r )
ik r
(r ) e u(r ) ; u(r Rn ) u(r )
Anderson Localization （局域化）
Cite 4325+ (from APS publication)
Anderson Localization （局域化）
Scaling Theory of Localization
Cite 3057+ (from APS publication)
d>2, 扩展态、局域态
迁移率边（mobility edge）
无序引起的金属绝缘体相变
metal-insulator transition
d=2, 临界情况（是否存在扩展态？）
d=1, 所有的本征态都是局域态
mobility
edge
1D Nonuniform Systems
紧束缚近似（二次量子化形式）
H Vncn cn
n
c
t
(
c
c
h
.
c
.)
n n 0 Vn n tmn m n
mn m n
mn
m
n
一维均匀势场中的电子能谱和本征态
V0 n t ( n 1 n 1 ) n
V0 2t cos(k ) n
1
N
一维非均匀势场中的电子能谱和本征态方程
Vn n tn,n1 n1 tn,n1 n1 n
问题：如果不是完全无序的一维
系统中是否存在扩展态或
其他形式的本征态？
ikn
e
k
Example 1: Harper or Aubry-Andre Model
S.Aubry & C Andre, Proc. Israel Phys. Soc 1979
Harpermodel
Vn cos(2n)
irrational
t n ,n 1 t n ,n 1 1
2 扩展态
2 临界态（分形）
2 局域态
1159+ (from APS publication)
Example 2：Slowly Varying Potential
Vn cos(n )
Vn cos(n )
当1 0 时，
2，所有态是扩展的；
2, Ec （2 - ）
2
等价于无序势，所有态是局域态
1 0 存在扩展态和mobilityedge
Example 2：Slowly Varying Potential
Example 3：General Quasiperiodic Potential
局域态
Vn tanh(2n) / tanh A
扩展态
( 5 1) / 2
局域态
当A较小时，随增大，
先从带中间出现局域态
Example 4：Random Dimer Model
420+ (from APS publication)
P.Phillips & H.-L. Wu, “Localization and its absence:
a new metallic state for conducting polymers”
Science 252, 1805 (1991)
V2 n V2 n 1
a
b
2 a b 2
概率p
概率1 p
N 个态是扩展的
a b 2
波包是正常扩散的
a b 2
波函数是局域的
Example 5：Long-Range Correlated Disorder
203+ (from APS publication)
213+ (from APS publication)
2 (1 )
2nk
Vn k
k )
cos(
N
N
k 1
S (k ) 1 / k
N /2
VnVnk 02 (k )
1/ 2
RG和数值计算发现
2 存在扩展态和迁移率边
应用二维映射，在Vn 1情形下
l
1
02 ( )
; ( ) 1 2 ( k ) cos(2 k )
8 sin 2
k 1
对Harper模型和Slowly Varying
Potential
模型，得到一致的结果
构造具有迁移率边的势场
Example 6：Fibonacci Binary Potential
712+ (from APS publication)
434+ (from APS publication)
通过求解重整化群的
T raceMap
不动点，
分析了能谱和
M l 1 M l M l 1
波函数的标度性质，发
1
xl TrM l
2
xl 1 2 xl xl 1 xl 2
现波函数既不是扩展的，
rl 1 T (rl ) (2xl xl 1 xl 2 , xl , xl 1 )
也不是 局域的，而是临
界的（Critical）
Experiments: Quasicrystals and Its Discovery
完整晶体
周期性
对称性
不存在5度对称轴
3157+ (from APS publication)
Dan Shechtman
2011 Chemistry
Experiments: Quasicrystals and Its Discovery
Linus Carl Pauling
1954 Chemistry
1962 Peace
郭可信
Tasi An-Pang
Quasicrystal
准晶
or
Twinned Crystal？ 孪晶
长程序
准晶
没有周期性
Penrose Tiles
1D Quasiperiodic and aperiodic sequence
Fibonacci sequence
Fn1 Fn Fn1
F0 F1 1
1,1,2,3,5,8,13,21,
Fn1
5 1
0.618...
Fn
2
A AB
BA
ABAABABA......
S n1 {S n , S n1}
S0 B, S1 A
Quasiperiodic and Aperiodic Sequences
General Fibonacci
sequence
A Ak B l
BA
Fn1 kFn lFn1
Thue-Morse
sequence
A AB
B BA
Fn 2n
Period-doubling
sequence
A AB
B AA
Fn 2n
Winding Number and
the classification of
sequences
N
M
N
1
A
2
A
N
N
1
B
2
B
ln 2
ln 1
0 1st class
0 Marginal
0 2nd class
1D Quasiperiodic and Aperiodic Superlattices
581+ (from APS publication)
1D 3-component Aperiodic Superlattices
1D k-component Aperiodic Superlattices
1D Frenkel-Kontorowa model
Y. Frenkel
基态 标准映射
K Kc
Invariancecircle
K Kc
Cantorus
K c 0.9716, Aubry T ransition
K Kc
Harper
K K c Fibonacci
Energy Spectra: Trace Maps
Vn n n1 n1 n
n 1
n
M (n)
n
n 1
Vn 1
M (n)
0
1
M ( N ) M ( N ) M ( N 1) M (1)
det M ( N ) 1
M l M ( Fl )
M l 1 M l M l 1
M l12 M l1M l 1
M l 1 M l12 ( M l M l1 ) M l 1
M l M l1 (TrM l ) I
TrM l TrM l1
1
x l TrM l
2
xl 1 2 xl xl 1 xl 2
1
xl TrM l 1
2
Energy Spectra: degenerate perturbation theory
Energy Spectra: Degenerate Perturbation Theory
Wave Functions
描述波函数局域性的物理量
T ransmission coefficient
and Landauer resistance
Participation Ratio: PR n4
n
T huolessExponentor
LyapnuovExponent:
N
1
'
( i )
ln i j
N 1 j 1
n Aeikn Beikn
n 1
n Ceikn Deikn
2 cosk
n N 1
TN S 1M ( N )S
Fractal and Multifractal
Quantum Information
etc
B C
TN
A D
12 2
N
T
Fractals 分形
Cantor 集合
G. Cantor
B. Mandelbort
Self-similarity
自相似性
非整数的Hausdroff维数
Box counting
F.Hausdroff
Von Koch
Koch曲线
Multifractals 复分形
Arnold
tongue
1688+ (from APS publication)
Devil’s Staircase
Fractals Energy Spectra and Wave Functions
Fibonacci model
Energy spectra
427+ (from APS publication)
Centre of Band
HarperModel
at 2
Concurrence and Von Neumann Entropy
Von Neumann entropy
Concurrence
Harper
Slowly
Varying
potential
Dynamics and Diffusions
2 (t ) t
Fibonacci
Harper
Thue-Morse
Hyperdiffusion
2 (t ) n2 n t
2
n
0
1 0
1
2 1
2
2
locaizat ion
subdiffusi on
Vn 0
n L, L
normaldiffusion
superdiffusion
Period
ballistic
hyperdiffusion
Vn 0 n N , L L, N
Split-operator method
Fibonacci
Disorder
Applications and Related Issuses
＊其他的一维结构（分形结构、hierarchical结构）
＊能级统计和量子混沌
＊光学系统（矢量波）
＊声学系统（声子晶体，张量波）
＊相互作用的影响（Hubbard-Anderson模型）
＊BEC和非线性相互作用
＊拓扑超导体和量子计算
＊生物系统（DNA）
＊ ‥‥‥
结论：这些一维的非均匀系统提供了研究非均匀
性对系统物理及其他性质影响的“标准模型”。
正如：两体模型、谐振子模型、Ising模型等
Quantum Phase Transitions
＊Two kinds of phase transitions:
T >0, thermodynamical phase transition,
critical temperature
T=0, quantum phase transition,
parameters of the systems
＊ Mermin-Wigner Theorem:
there is no long-range order for
one-dimensional system at T>0
＊ One-dimensional quantum spin systems
a) Quantum Ising chain in a transverse field,
corresponding to 2D classical Ising model,
b) XY chain, pure quantum model,
no classical correspondence
c) Heisenberg model, general models
S.Sachdev
Quantum Ising and Anisotropic XY Chains
H
1
{J n (1 ) nx nx1 J n (1 ) ny ny1 hn nz }
2 n
1 quantum Ising model
0
quantum XX model
0 1 anisotropic XY model
Jordan
Wigner
Jordan-Wigner transformation
an
1 x
1
( n i ny ), an ( nx i ny )
2
2
n 1
cn exp(i a a j )an
j 1
j
H [cn Anm cm
n,m
nx a n a n ,
1
i
ny (a n a n ),
nz 2a n a n 1
n 1
c a exp(i a j a j )
n
n
j 1
1
(cn Bnm cm h.c.)]
2
N (cN c1 cN c1 cN c1 cN c1 )[exp(i) 1]
1
( nz 1)
2
Quantum Ising and Anisotropic XY Chains
h1
1J
2 1
A
0
1 J N
2
1
J1
2
0
h2
1
J2
2
1
J2
2
h3
0
0
1
JN
2
0
0
hN
Equations of eigenvalues and eigenvectors
N
0
2
J1
0
2
JN
2
J1
0
2
0
2
J2
J2
0
0
0
1
[(k ,n k ,n )cn (k ,n k ,n )cn ]
2 n
1
k [(k ,n k ,n )cn (k ,n k ,n )cn ]
2 n
k
Bogoliubov transformation
B
1
H k ( k k )
2
k 1
JN
2
0
0
0
Quantum Ising and anisotropic XY Chains
v
Order parameters: Correlation function,
Magnetization
G1, 2
G1,3
C Lx GS 1x 1x L GS
GL,2
Mz
G1, L 1
G L ,3 G L , L 1
1
Gn , n
N n
Gi , j k ,i k ,i
k
Ising h 2 J , 0 1
Anisotropic 0, 2 h 2
Disorder and Quasiperiodic Quantum Ising Chains
无序和非均匀的结构对相变的影响
Harris判据：
•
满足Harris判据，无序不影响原系统的相变行为、无序是 RG下是
不相关的变量、物理量是自平均的。 3d经典Heisenberg模型
•
不满足Harris判据：
a, 系统在所有尺度上都是不均匀的，但不均匀度趋于有限的值，相变
对应于有限的无序强度的不动点、物理量是非自平均 。
3d经典Ising模型
（均匀）
（无序）
b, 系统中无序的相对强度在粗粒后无限地增大、相变对应于无限的无
序强度的不动点（被称为infinite-randomness critical points)，在临界
点是指数标度，而不是Power-law的。物理量的分布是非常broad、
因此平均值是由rare events决定。
McMoy-Wu模型、1d无序的量子自旋链、1d无序的contact processes
Disorder Quantum Ising and XX Chains: RG
Energy Scale
202+ (from APS publication)
284+ (from APS publication)
345+ (from APS publication)
Quasiperiodic and aperiodic
quantum Ising chains
Luck-Harris criteria
J.M. Luck J. Stat. Phys.72 417 (1993)
0 uniformquantum Ising chain
0 disorder quantum Ising chain
Periodic anisotropic XY chains
周期2：J 2n J , J 2n1 J
Quasiperiodic Anisotropic XY Chains
J A J B J A J A J B J A J B J A
J A J , J B J
Quantum Lee-Yang Zeros
Concurrence and Quantum Phase Transition
Nature 416 608 (2002)
Cite: 1137
659+ (from APS publication)
Concurrence
Anisotropic
transition
Period
Von Neumann Entropy and Quantum Phase Transition
c:central
charge
809+ (from APS publication)
122+ (from APS publication)
random Ising
random Heisenberg
Ising c c 1 / 2
XX
c c 1
Period
Loschmidt Echos
198+ (from APS publication)
1st class
2nd class
disorder
Three-site Interaction
DM Interaction
Long-range Interaction
Nature Physics 1 53
Majarona Fermion and Topological Superconductor
n V 2 cos(2n)
谢 谢!
Thank You