一维和二维关联无序安德森模型

1、One- and two-dimensional Anderson model with long- range correlated-disorder一维和二维关联无序安德森模型 One- and two-dimensional Anderson model with long- range correlated-disorderAnderson model-IntroductionEntanglement in 1D2D Entanglement2D conductance2D transmission2D magnetoconductanceAnderson model-Introduc

2、tionWhat is a disordered system? No long-range translational orderTypes of disorder (a)crystal(a)crystal(b) Component (b) Component disorderdisorder(c) position (c) position disorderdisorder(d) topological(d) topologicaldisorderdisorder diagonal disorder off-diagonal disorder complete disorder Local

3、ization prediction：an electron, when placed in a strong disordered lattice, will be immobile 1 P.W.Anderson, Phys.Rev.109 ,1492(1958). jitiiHNijijNiNii11Anderson model-IntroductionBy P.W.Anderson in 19581Anderson model-IntroductionIn 1983 and 1984 John extended the localization concept successfully

4、to the classical waves, such as elastic wave and optical wave 1. Following the previous experimental work ,Tal Schwartz et al. realized the Anderson localization with disordered two-dimensional photonic lattices2.1John S,Sompolinsky H and Stephen M J 1983 Phys.Rev.B27 5592; John S and Stephen M J 19

5、83 28 6358; John S 1984 Phys.Rev.Lett. 53 21692Schwartz Tal, Bartal Guy, Fishman Shmuel and Segev Mordechai 2007 Nature 446 52Anderson model-open problemsAbrahans et al.s scaling theory for localization in 19791（ 3000 citations ，one of the most important papers in condensed matter physics） Predictio

6、ns（1）no metal-insulator transition in 2d disordered systems Supported by experiments in early 1980s. （2） （dephasing time ）Results of J.J.Lin in 19872 0T 1 E.Abrahans,P.W.Anderson, D.C.Licciardello and T.V. Ramakrisbnan, Phys.Rev.Lett. 42 ,673(1979)2 J.J. Lin and N. Giorano, Phys. Rev. B 35, 1071 (19

7、87); J.J. Lin and J.P. Bird, J. Phys.: Condes. Matter 14, R501 (2002). 0Tc Results of J.J.Lin in 19872dephasing timeWork of Hui Xu et al.on systems with correlated disorder :刘小良，徐慧，等，物理学报，55(5)，2493（2006）；刘小良，徐慧，等，物理学报，55(6)，2949（2006）；徐慧,等，物理学报， 56(2)，1208（2007）；徐慧,等，物理学报， 56(3)，1643（2007）；马松山，徐慧，等

8、，物理学报，56(5)，5394（2007）；马松山，徐慧，等，物理学报， 56(9)，5394（2007）。Anderson model-new points of view1。Correlated disorderCorrelation and disorder are two of the most important concepts in solid state physicsPower-law correlated disorder Gaussian correlated disorder 2。Entanglement1：an index for metal-insulator,l

9、ocalization-delocalization transition”entanglement is a kind of unlocal correlation”（MPLB19,517,2005）.Entanglement of spin wave functions:four states in one site:0 spin; 1up; 1down; 1 up and 1 downEntanglement of spatial wave functions (spinless particle) :two states:occupied or unoccupiedMeasures o

10、f entanglement：von Newmann entropy and concurrence1Haibin Li and Xiaoguang Wang, Mod. Phys. Lett. B19,517(2005)；Junpeng Cao, Gang Xiong, Yupeng Wang, X. R. Wang, Int. J.Quant. Inform.4 , 705(2006). Hefeng Wang and Sabre Kais, Int. J.Quant. Inform.4 , 827(2006). Anderson model- new points of view3.ne

11、w applications（1）quantum chaos（2）electron transport in DNA chainsThe importance of the problem of the electron transport in DNA1 （3）pentacene2（并五苯）Molecular electronicsOrganic field-effect-transistorspentacene:layered structure, 2D Anderson system1R. G. Endres, D. L. Cox and R. R. P. Singh,Rev.Mod.P

12、hys.76 ,195(2004)； Stephan Roche, Phys.Rev.Lett. 91 ,108101(2003). 2 M.Unge and S.Stafstrom, Synthetic Metals,139(2003)239-244;J.Cornil,J.Ph.Calbert and J.L.Bredas, J.Am.Chem.Soc.,123,1520-1521(2001). DNA structureEntanglement in one-dimensional Anderson model with long-range correlated disorder one

13、-dimensional nearest-neighbor tight-binding model Concurrence: 1)(121NiiMCijijiiijtEvon Neumann entropy 0ncn011nNnnNnncn00)1 (11nnnnnnnzz)1 (log)1 (log22nnnnvnzzzzENnvnvENE1134567890.020.040.060.080.100.120.140.160.18 (10-4)W 1.5 1.7 2.0 2.05 2.1 3.0 3.5 4.0 5.0power-law correlated91011121314150.010

14、.020.030.040.050.060.070.08 (10-4)W power-law correlatedLeft. The average concurrence of the Anderson model with power-law correlation as the function of disorder degree W and for various .A band structure is demonstrated.Right. The average concurrence of the Anderson model with power-law correlatio

15、n for =3.0 and at the bigger W range. A jumping from the upper band to the lower band is shown 2D entanglementMethod:taking the 2D lattice as 1D chain1 Longyan Gong and Peiqing Tong,Phys.Rev.E 74 (2006) 056103.；Phys.Rev.A 71 ,042333(2005). Quantum small world network in 1 square lattice051015200.00.

16、51.01.52.02.53.03.5 (10-4)W 1.75 2.0 2.5 3.5 4.0 4.5 5.5L=30051015200.20.30.40.50.60.7 von Neumann entropyW 1.75 2.0 2.5 3.5 4.0 4.5 5.5L=30Left. The average concurrence of the Anderson model with power-law correlation as the function of disorder degree W and for various . A band structure is demons

17、trated.Right. The average von Newmann entropy of the Anderson model with power-law correlation as the function of disorder degree W and for various . A band structure is demonstrated.Lonczos methodEntanglement in DNA chain guanine (G), adenine (A), cytosine(C), thymine (T) Qusiperiodical modelR-S mo

18、del to generate the qusiperiodical sequence with four elements ( G , C , A , T ) . T h e i n f l a t i o n ( s u b s t i t u t i o n s ) r u l e i s GGC;CGA;ATC;TTA. Starting with G (the first generation), the first several generations are G,GC,GCGA,GCGAGCTC, GCGAGCTC GCGATAGA .Let Fi the element (s

19、ite) number of the R-S sequence in the ith generation, we have Fi+1=2Fi for i=1 . So the site number of the first several generations are 1,2,4,8,16, , and for the12th generation , the site number is 2048. 01230.00.20.40.60.81416 (10-4)W uncorrelated uniformdistribution =1.5 =2.0 =3.0 =5.0power-law

20、correlatedThe average concurrence of the Anderson model for the DNA chain as the function of site number. The results are compared with the uncorrelated uniform distribution case. Spin Entanglement of non-interacting multiple particles:Greens function methodFinite temperature two body Greens functio

21、nFinite temperature two body Greens functionOne particle density matrix and One body Greens functionOne particle density matrix and One body Greens functionTwo particle density matrixTwo particle density matrixwhere,HF approx. Ifandwhere&whereGeneralized Werner StatethenInbasisSeparability crite

22、rion=PPT= 31palways satisfied sinceConductance and magnetoconductance of the Anderson model with long-range correlated disorder (1)Static conductance of the two-dimensional quantum dots with long-range correlated disorder Idea:the distribution function of the conductance in the localized regime1d:cl

23、ear Gaussian2d: unclearMethod to calculating the conductance :Greens function and Kubo formulaln gN10HHH.).(,1, 1,0cccctcctccHjijiijyjijijiijxjijijiijxwteVH)cos(1jijijiccx, ,xHvx)(Im)(Im)()(22EgvEgvTrheGxx)()(21)(ImEgEgiEgARIgHI)()()(21EEEE),4(21)(2EiEE-4-20240246810 ConductanceFermi energy Referenc

24、e 02468100246 ConductanceWGaussian corellation distribution Ef=2.5 Ef=-2.5 Ef=0 Ef=1.5 Ef=-1.502468100123456 ConductanceWUniform distribution Ef=0 Ef=1.5 Ef=2.5Fig.1Fig.2aFig.2bFig.1 Conductance as the function of Fermi energy for the systems with power-law correlated disorder (W=1.5 ) for various exponent .The results are compared to the reference of that of a uniform random on-site energy distribution. solid: uniform distribution reference; dash:; dash dot: ;dash dot dot: ;short dash: Fig .2 Conductance changes with disorder degree for differ