1. Real-Space Renormalization Approach to the Kubo- Greenwood Formula Chumin Wang Universidad Nacional Autónoma de México Universidad Nacional Autónoma de México

2. Outlines - Introduction 1.- Macroscopic scale analysis 2.- Kubo-Greenwood formula 3.- Real-space renormalization 4.- Convolution technique - Electronic Transport in Quasicrystals 1.-Transparent states in mirror Fibonacci chains 2.-Power-law localization nature 3.- Quantized electrical conductance - Conclusions

3. Solid State Physics SystemCrystalQuasicrystal Amorphous Structure Symmetry Translational Long range order Short range order Wave Function Extended CriticalLocalized A typical macroscopic sample contains 10 23 atoms, the possibility of solving all the quantum equations in full detail is truly remote.

4. Macroscopic Scale Analysis 1. Reciprocal Space - Bloch theorem: - First Brillouin zone: contains all the quantum states. 2. Real-Space Renormalization - Consists in reducing the degree of freedom of the system, but conserving their exact participation in the final solution of the problem. There are essentially two options:

5. Fibonacci Sequence The simplest quasiperiodic lattice: …-A-B-A-A-B-A-B-A-… Fibonacci Chain 1.Projection method: sites in the window 2.Inflation method: A A-B and B A, for example: A-B A-B-A A-B-A-A-B … 3.Addition method: f(n) = f(n-1)  f(n-2), for instance: f(1)=A, f(2)=A-B f(3)=A-B-A.  Equivalence: Given  a linear operator of inflation, i.e.,  (B)=A,  (A)=A-B, f(1)=  (B) and f(2)=   (B)=  (B))=A-B, therefore,  n (B)=  n-1 (A)=  n-2 (AB)=  n-2 (A)  n-2 (B)=  n-1 (B)  n-2 (B). …-B-A-A-B-A-A-B-A-A-B-… Mirror Fibonacci Chain

6. Bond Problem Let us consider an s-band tight-binding Hamiltonian:

7. Renormalization of the DOS where D 1 (z,n)= D 1 (z,n-1)+[  1 (z,n)] 2 [D 1 (z,n-2)+ D 2 (z,n-1)-1]+  1 (z,n)D 3 (z,n-1), D 2 (z,n)= D 2 (z,n-2)+[  2 (z,n)] 2 [D 1 (z,n-2)+ D 2 (z,n-1)-1]+  2 (z,n)D 3 (z,n-2), D 3 (z,n)=  2 (z,n)D 3 (z,n-1)+  1 (z,n) D 3 (z,n-2)+2  2 (z,n)  1 (z,n)[D 1 (z,n-2)+D 2 (z,n-1)-1], D 4 (z,n)= D 4 (z,n-1)+D 4 (z,n-2)+  0 (z,n)[ D 1 (z,n-2)+D 2 (z,n-1)-1],  0 (z,n)=[z - E R (z,n-1) - E L (z,n-2)] -1  1 (z,n)= t(z,n-1)  0 (z,n),  2 (z,n)= t(z,n-2)  0 (z,n). The density of states (DOS) can be written as

8. where Kubo-Greenwood Formula V. Sánchez, L.A. Pérez, R. Oviedo-Roa, and C. Wang, Phys. Rev. B 64, 174205 (2001).

9. Green’s Function The Dyson’s equation:

10. Convolution Theorem W.A. Schwalm and M.K. Schwalm, Phys. Rev. B 37, 9524 (1988).

11. Computing Efficiency Computing time of the Kubo-Greenwood formula for tapes of 4  N atoms, being N a Fibonacci number, through 3 methods: by direct calculation of the Green's function (open squares), using 1D inversion+convolution (open triangles) and by means of the renormalization+convolution method (open circles). The numerical computations are performed on a Silicon Graphics O2 workstation with a MIPS R12000 microprocessor.

12. Transparent States Fibonacci chain contains 433494438 atoms with semi-infinite leads, t A /t B =0.9, Im(E)=10 -13 |t|

13. Power-Law Localization

14. Quantized Conductance

15. Quantization in Nano Systems (a) R. de Picciotto, et al., Nature 411, 51 (2001). (b) A. Urbina, et al., Phys. Rev. Lett. 90, 106603 (2003).

16. Conclusions The real-space renormalization method seems to be a very efficient manner to study non periodic systems. The results reveal interesting features of the electronic transport, such as quantized conductance, multiple transparent states in mirror Fibonacci lattices and the power-law localization nature. For cubic-like structures, combining with the convolution theorem, the Kubo-Greenwood formula can be evaluated in an exact way for multidimentional quasiperiodic lattices.

18. Método de Corte y Proyección Sea un operador de proyección: Como y mRmRm R m -R m-1 1 2A 3B 4A

19. Estado Transparente donde Transmitancia del sistema [ ] es uno cuando x = 1, es decir, para generaciones n = 1, 4, 7,...

20. Soluciones Analíticas - Red cuadrada periódica: - Cadena periódica:

21. Generador de Armónicos Multicapas cuasiperiódicas de ferroeléctricos (LiTaO 3 ) S. Zhu, et al., Science 278, 843 (1997).