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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1 One- and two-dimensional Anderson model with long- range correlated-disorderAnderson model-IntroductionEntanglement in 1D2D Entanglement2D conductance2D transmission2D magnetoconductance2Anderson model-Introd
2、uctionWhat 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) topologicaldisorderdisorder3 diagonal disorder off-diagonal disorder complete disorder Lo
3、calization prediction：an electron, when placed in a strong disordered lattice, will be immobile 1 P.W.Anderson, Phys.Rev.109 ,1492(1958). Anderson model-IntroductionBy P.W.Anderson in 195814Anderson model-IntroductionIn 1983 and 1984 John extended the localization concept successfully to the classic
4、al 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 1983 28 6358; Jo
5、hn S 1984 Phys.Rev.Lett. 53 21692Schwartz Tal, Bartal Guy, Fishman Shmuel and Segev Mordechai 2007 Nature 446 525Anderson model-open problemsAbrahans et al.s scaling theory for localization in 19791（ 3000 citations ，one of the most important papers in condensed matter physics） Predictions（1）no metal
6、-insulator transition in 2d disordered systems Supported by experiments in early 1980s. （2） （dephasing time ）Results of J.J.Lin in 19872 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 (1987); J.J. Lin an
7、d J.P. Bird, J. Phys.: Condes. Matter 14, R501 (2002). 6Results of J.J.Lin in 19872dephasing time7Work 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，物理学报，56(5)，5394（2
8、007）；马松山，徐慧，等，物理学报， 56(9)，5394（2007）。8Anderson 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,localization-deloc
9、alization 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 of entanglement：vo
10、n 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). 9Anderson model- new points of view3.new applications（1
11、）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.Phys.76 ,195(2004
12、)； 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 structure10Entanglement in one-dimensional Anderson model with long-range correlated disorder one-dimensional n
13、earest-neighbor tight-binding model Concurrence:von Neumann entropy 11Left. 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-
14、law correlation for =3.0 and at the bigger W range. A jumping from the upper band to the lower band is shown 122D 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 latt
15、ice13Left. 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 von Newmann entropy of the Anderson model with power-law correlation as the function of disorder degree W and
16、 for various . A band structure is demonstrated.14Lonczos method15Entanglement in DNA chain guanine (G), adenine (A), cytosine(C), thymine (T) Qusiperiodical modelR-S model to generate the qusiperiodical sequence with four elements (G,C,A,T) .The inflation(substitutions) rule is GGC;CGA;ATC;TTA. Sta
17、rting with G (the first generation), the first several generations are G,GC,GCGA,GCGAGCTC, GCGAGCTC GCGATAGA .Let Fi the element (site) 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 g
18、eneration , the site number is 2048. 16The 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. 17Spin Entanglement of non-interacting multiple particles:Greens function methodFinite temp
19、erature two body GreenFinite temperature two body Greens functions functionOne particle density matrix and One body GreenOne particle density matrix and One body Greens functions functionTwo particle density matrixTwo particle density matrixwhere,HF approx.18 Ifandwhere&whereGeneralized Werner State
20、thenInbasisSeparability criterion=PPT= always satisfied since19;.Conductance 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 conductanc
21、e in the localized regime1d:clear Gaussian2d: unclearMethod to calculating the conductance :Greens function and Kubo formula20Fig.1Fig.2aFig.2b21Fig.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 compare
22、d 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 different Fermi energies(a) Gaussian correlated disorder, solid: Ef=0;dash: Ef=1.5;short
23、 dash: Ef=-1.5;dash dot dot: Ef=2.5;dot: Ef=-2.5 (b)power-law correlated disorder ,solid: Ef=0;dash: Ef=1.5; dot: Ef=2.5 (c) disorder with uniform distribution, solid: Ef=0;dash: Ef=1.5; dot: Ef=2.522(2)Transmittance of the two-dimensional quantum dot systems with Gaussian correlated disorder Effects of leads23(3)magnetoconductance Related with quantum chaos24Thank you !25
