In this thesis, we study the response function or electronic properties in condensed matter system with impurities or an edge. It is well-know that the magnetic impurities will induce the Ruderman-Kittel-Kasuya-Yosida exchange coupling, mediated by free itinerant carries. However, it is less study what electronic properties of the itinerant carries will reshape the famous RKKY interaction. Here, we show that the exchange coupling induced by a magnetic impurity depends on the Fermi surface topology of the itinerant carrier in the two-dimensional free electron gases with the Rashba type spin-orbital interaction. By both numeric and analytic methods, we clearly demonstrate that the Fermi surface topology greatly alters the property of mediated exchange coupling in two-dimensional Rashba gas system. In addition, inclusion of finite spin relaxation always makes the non-collinear spiral exchange interaction dominant. Meanwhile, this exploration encourages us to build up a trilayer magnetic junction for application.

Next, switching attention to edge physics. In condensed matter systems, the physical properties at the edge are often tied up with related bulk properties.
Andreev edge state in a superconductor, for instance, is tied up with the pairing symmetry in the bulk. Inspire of that, we study the Andreev edge states with different pairing symmetries and boundary topologies on semi-infinite triangular lattice, and hope to shad light on determining the pairing symmetry of Na$_x$CoO$_2$$\cdot y$H$_2$O. By developing a general mapping from the two dimensional lattice to the one dimensional tight-binding model, we show that the phase diagram of the Andreev edge states depends on the pairing symmetry and also the boundary topology. We also compute the momentum-resolved local density of states near the edge which is helpful to predict the hot spots which are measurable in Fourier transformed scanning tunneling spectroscopy.

The general methods also help us to calculate the spin-wave excitation near an edge. Interestingly, we obtain a single branch of relativistic ferromagnetic magnon near the zigzag edge of graphene due to the presence of the open boundary. Note that magnons in antiferomagnets appear in pairs, while the single branch magnon in ferromagnets does not have relativistic dispersion. Thus, the magnon near the zigzag edge of graphene is a hybrid of both, signaling its intrinsic property as a boundary excitation that must be embedded in a higher dimensional bulk system.

In the end, we will focus on the electronic properties in graphene with a point defect. It is generally believed that a point defect in graphene gives rise to an impurity state at zero energy and causes a sharp peak in the local density of states near the defect site. We revisit the defect problem in graphene and find the general consensus incorrect. By both analytic and numeric methods, we show that the contribution to the local density of states from the impurity state vanishes in the thermodynamic limit. Instead, the pronounced peak of the zero-bias anomaly is a power-law singularity $1/|E|$ from infinite resonant peaks in the low-energy regime induced by the defect. Our finding shows that the peak shall be viewed as a collective phenomenon rather than a single impurity state in previous studies.

[1] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Moln ́ar,
M. L. Roukes, A. Y. Chtchelkanova, and D. M. Treger, Science 294, 1488
(2001).
[2] I. Zutic, J. Fabian and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004).
[3] A. H. MacDonald, P. Schiffer and N. Samarth, Nat. Mat. 4, 195 (2005).
[4] S.-J. Sun and H.-H. Lin, Phys. Lett. A 327, 73 (2004).
[5] S.-J. Sun and H.-H. Lin, Eur. Phys. J. B 49, 403 (2006).
[6] S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1989).
[7] S. Murakami, N. Nagaosa and S.-C. Zhang, Science 301, 1348 (2003).
[8] J. Sinova et al., Phys. Rev. Lett. 92, 126603 (2004).
[9] S.-J. Sun, S.-S. Cheng and H.-H. Lin, Appl. Phys. Lett. 84, 2862 (2004).
[10] J. Luo, H. Munekata, F.F. Fang and P.J. Stiles, Phys. Rev. B. 38, R10142
(1988).
[11] B. Das, D.C. Miller, S. Datta, R. Reifenberger, W.P. Hong, P.K. Bhat-
tacharya, J. Singh and M. Jaffe, Phys. Rev. B. 39, 1411 (1989).
[12] J. Nitta, T. Akazaki, H. Takayanagi and T. Enoki, Phys. Rev. Lett. 78, 1335
(1997).
[13] G. Engels, J. Lange, Th. Sch ̈apers and H. L ̈uth, Phys. Rev. B. 55, R1958
(1997).
[14] J.P. Heida, B.J. van Wees, J.J. Kuipers, T.M. Klapwijk and G. Borghs, Phys.
Rev. B 57, R11911(1998).
[15] I.L. Aleiner and V.I. Fal’ko, Phys. Rev. Lett. 87, 256801 (2001).
[16] H. Imamura, P. Bruno and Y. Utsumi, Phys. Rev. B 69, 121303(R) (2004).
[17] G. Lommer, F. Malcher and U. R ̈ossler, Phys. Rev. Lett. 60, 728 (1988).
[18] G. E. W. Bauer, Y. Tserkovnyak, D. Huertas-Hernando and A. Brataas, Phys.
Rev. B 67, 094421 (2003).
[19] E.B. Myers et al., Science 285,867(1999).
[20] A. A. Tulapurkar et al., Nature 438, 339 (2005).
[21] P. Sharma, Science 307, 531 (2005).
[22] D. D. Awschalom, M. E. Flatt ́e, Nature Phys. 3, 153 (2007).
[23] E. I. Rashba, Sov. Phys. Solid State 2, 1109 (1960).
[24] S. Francoeur, M.-J. Seong, A. Mascarenhas, S. Tixier, M. Adamcyk and T.
Tiedje, Appl. Phys. Lett. 82, 3874 (2003).
[25] B. Fluegel, S. Francoeur, A. Mascarenhas, S. Tixier, E. C. Young and T.
Tiedje, Phys. Rev. Lett. 97, 067205 (2006).
[26] C. R. Ast, J. Henk, A. Ernst, L. Moreschini, M. C. Falub, D. Pacil ́e, P. Bruno,
K. Kern, and M. Grioni, Phys. Rev. Lett. 98, 186807 (2007).
[27] C. R. Ast, D. Pacil ́e, L. Moreschini, M. C. Falub, M. Papagno, K. Kern,
M. Grioni, J. Henk, A. Ernst, S. Ostanin and P. Bruno, Phys. Rev. B 77,
081407(R) (2008).
[28] M. I. Dyakonov and V. I. Perel, Sov. Phys. Solid State 13, 3023 (1972).
[29] C. Grimaldi, Phys. Rev. B 72, 075307 (2005).
[30] E. I. Rashba, Phys. Rev. B 70, 201309(R) (2004).
[31] O. V. Dimitrova, Phys. Rev. B 71, 245327 (2005).
[32] C. Grimaldi, E. Cappelluti and F. Marsiglio, Phys. Rev. B 73, 081303(R)
(2006).
[33] A. V. Chaplik and L. I. Magarill, Phys. Rev. Lett. 96, 126402 (2006).
[34] A. G. Galstyan and M. E. Raikh, Phys. Rev. B 58, 6736 (1998).
[35] C. Grimaldi, Phys. Rev. B 77, 113308 (2008).
[36] E. Cappelluti,C. Grimaldi and F. Marsiglio, Phys. Rev. Lett. 98, 167002
(2007).
[37] W.-M. Huang, C.-H. Chang, and H.-H. Lin, Phys. Rev. B 73, 241307(R)
(2006).
[38] W.-M. Huang, H.-H. Lai, C.-H. Chang, and H.-H. Lin, Int. J. Mod. Phys. B
22, 88 (2008).
[39] N. J. Craig, J. M. Taylor, E. A. Lester, C. M. Marcus, M. P. Hanson, and A.
C. Gossard, Science 304, 565 (2004).
[40] P. Simon, R. L ́opez, and Y. Oreg , Phys. Rev. Lett. 94, 086602 (2005).
[41] J. Simonin, Phys. Rev. Lett. 97, 266804 (2006).
[42] C.-H. Lin, H.-H. Lin and T.-M. Hong Appl. Phys. Lett. 89, 032503 (2006).
[43] H. Bruus and K. Flensberg, Many-Body Quantum Theory in Condensed Mat-
ter Physics – An Introduction (Oxford University Press, 2004).
[44] G. D. Mahan, Many-Particle Physics (springer, 3nd edition, 2007).
[45] A. A. Abricosov, L. P. Gorkov, and I. E. Dzyaloshinski, Quantum Field The-
oretical Method in Statistical Mechanics (Pergamon, New York, 1965).
[46] V.K. Dugaev, V.I. Litvinov, and P.P. Petrov, Superlattices Microstruct. 16,
413 (1994).
[47] V.I. Litvinov, and V. K. Dugaev, Phys. Rev. B 58, 3584 (1998).
[48] J. W. Negele, H. Orland, Quantum Many-particle Systems (Westview Press,
1998)
[49] A. Auerbach, Interacting Electrons and Quantum Magnetism (Springer Ver-
lag, 1994).
[50] K. Takada, H. Sakurai, E. Takayama-Muromachi, F. Izumi, R.A. Dilanian,
and T. Sasaki, Nature 422, 53 (2003).
[51] M. Ogata, J. Phys.: Condens. Matter 19, 145282 (2007).
[52] F. C. Chou, J. H. Cho, P. A. Lee, E. T. Abel, K. Matan, and Y. S. Lee, Phys.
Rev. Lett. 92, 157004 (2004).
[53] Y. Ihara, K. Ishida, C. Michioka, M. Kato, K. Yoshimura, K. Takada, T.
Sasaki, H. Sakurai, and E. Takayama-Muromachi, J. Phys. Soc. Jpn. 73,
2069 (2004).
[54] R. E. Schaak, T. Klimczuk, M. L. Foo, and R. J. Cava, Nature 424, 527
(2003).
[55] M. L. Foo, Y. Wang, S. Watauchi, H. W. Zandbergen, T. He, R. J. Cava, and
N. P. Ong, Phys. Rev. Lett. 92, 247001 (2004).
[56] T. Fujimoto, G. Q. Zheng, Y. Kitaoka, R. L. Meng, J. Cmaidalka, and C. W.
Chu, Phys. Rev. Lett. 92, 047004 (2004).
[57] K. Ishida, Y. Ihara, Y. Maeno, C. Michioka, M. Kato, K. Yoshimura, K.
Takada, T. Sasaki, H. Sakurai, and E. Takayama-Muromachi, J. Phys. Soc.
Jpn. 72, 3041 (2003).
[58] G.-q. Zheng, K. Matano, R. L. Meng, J. Cmaidalka, and C. W. Chu, J. Phys.
Condens. Matter 18, L63 (2006).
[59] H. D. Yang, J.-Y. Lin, C. P. Sun, Y. C. Kang, C. L. Huang, K. Takada, T.
Sasaki, H. Sakurai, and E. Takayama-Muromachi, Phys. Rev. B 71, 020504(R)
(2005).
[60] A. Kanigel, A Keren, L. Patlagan, K. B. Chashka, P. King, and A. Amato,
Phys. Rev. Lett. 92, 257007 (2004).
[61] W. Higemoto, K. Ohishi, A. Koda, S. R. Saha, R. Kadono, K. Ishida, K.
Takada, H. Sakurai, E. Takayama-Muromachi, and T. Sasaki, Phys. Rev. B
70, 134508 (2004).
[62] Y. Kobayashi, H. Watanabe, M. Yokoi, T. Moyoshi, Y. Mori, and M. Sato, J.
Phys. Soc. Jpn. 74, 1800 (2005).
[63] Y. Ihara, K. Ishida, H. Takeya, C. Michioka, M. Kato, Y. Itoh, K. Yoshimura,
K. Takada, T. Sasaki, H. Sakurai, and E. Takayama-Muromachi, J. Phys. Soc.
Jpn. 75, 013708 (2006).
[64] M. Kato, C. Michioka, T. Waki, Y. Itoh, K. Yoshimura, K. Ishida, H. Saku-
rai, E. Takayama-Muromachi, K. Takada, and T. Sasaki, J. Phys. Condens.
Matter 18, 669 (2006).
[65] G.-q. Zheng, K. Matano, D. P. Chen, and C. T. Lin, Phys. Rev. B 73,
180503(R) (2006).
[66] T. Shimo jima, K. Ishizaka, S. Tsuda, T. Kiss, T. Yokoya, A. Chainani, S.
Shin, P. Badica, K. Yamada, and K. Togano, Phys. Rev. Lett. 97, 267003
(2006).
[67] Y.-J. Chen, C.-J. Liu, J.-S. Wang, J.-Y. Lin, C. P. Sun, S. W. Huang, J. M.
Lee, J. M. Chen, J. F. Lee, D. G. Liu, and H. D. Yang, Phys. Rev. B 76,
092501 (2007).
[68] I. I. Mazin, M. D. Johannes, Nature Physics 1, 91 (2005).
[69] G. Baskaran, Phys. Rev. Lett. 91, 097003 (2003).
[70] B. Kumar, and B. S. Shastry, Phys. Rev. B 68, 104508 (2003).
[71] Q.-H. Wang, D.-H. Lee, and P. A. Lee Phys. Rev. B 69, 092504 (2004).
[72] H. Ikeda, Y. Nisikawa, and K. Yamada, J. Phys. Soc. Jpn. 73, 17 (2003).
[73] Y. Tanaka, Y. Yanase, and M. Ogata J. Phys. Soc. Jpn. 73, 319 (2003).
[74] B. J. Powell and R. H. McKenzie, Phys. Rev. Lett. 98, 027005 (2007).
[75] A. Tanaka and X. Hu, Phys. Rev. Lett. 91, 257006 (2003).
[76] Q. Han and Z. D. Wang, Phys. Rev. B 70, 184504 (2004).
[77] M. Mochizuki, Y. Yanase, and M. Ogata, Phys. Rev. Lett. 94, 147005 (2005).
[78] K. Kuroki ,Y. Tanaka , and R. Arita, Phys. Rev. Lett. 93, 077001 (2004).
[79] Q. Han, Z. D. Wang, Q. H. Wang, and T. L. Xia, Phys. Rev. Lett. 92, 027004
(2004).
[80] J.-X. Li and Z. D. Wang, Phys. Rev. B 70, 212512 (2004).
[81] B. Braunecker, P. A. Lee and Z. Wang, Phys. Rev. Lett. 95, 017004 (2005).
[82] T. Pereg-Barnea and H.-H. Lin, EuroPhys. Lett. 69, 791 (2005).
[83] C.-R. Hu, Phys. Rev. Lett. 72, 1526 (1994).
[84] J. E. Hoffman, K. McElroy, D.-H. Lee, K. M Lang, H. Eisaki, S. Uchida, and
J. C. Davis, Science 297, 1148 (2002).
[85] K. McElroy, R. W. Simmonds, J. E. Hoffman, D.-H. Lee, J. Orenstein, H.
Eisaki, S. Uchida and J. C. Davis, Nature 422, 592 (2003).
[86] Georg Junker, Supersymmetric Methods in Quantum and Statistical Physics
(Springer-Verlag Berlin Press, 1996).
[87] M. Oshikawa, Phys. Rev. Lett. 84, 1535 (2000).
[88] G. Refael and H.-H. Lin, Phys. Rev. B 72, 073109 (2005).
[89] P. G. de Gennes, Superconductivity Of Metals And Al loys (Westview Press,
1999).
[90] D. J. Singh, Phys. Rev. B 61, 13397 (2000).
[91] T. Valla, P. D. Johnson, Z. Yusof, B. Wells, Q. Li, S. M. Loureiro, R. J. Cava,
M. Mikami, Y. Mori, M. Yoshimura and T. Sasaki, Nature 417, 627 (2002).
[92] Hsiu-Hau Lin, Phys. Rev. B 58, 4963 (1998).
[93] B.-L. Huang, S.-T. Wu, and C.-Y. Mou, Phys. Rev. B 70, 205408 (2004).
[94] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V.
Dubonos, V. Grigorieva and A. A. Firsov, Science 306, 666 (2004).
[95] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I.
V. Grigorieva, S. V. Dubonos and A. A. Firsov Nature 438, 197 (2005).
[96] Y. Zhang, Y.-W. Tan, H. L. Stormer and P. Kim, Nature 438, 201 (2005).
[97] A. K. Geim and K. S. Novosselov, Nature Mat. 6, 183 (2007) and references
therein.
[98] A. Rycerz, J. Tworzydlo and C. W. J. Beenakker, Nature Phys. 3, 172 (2007).
[99] B. Trauzettel, D. S. Bulaev, D. Loss and G. Burkard, Nature Phys. 3, 192
(2007).
[100] Y.-W. Son, M. L. Cohen and S. G. Louie, Nature 444, 347 (2006).
[101] M. Fujita, K. Wakabayashi, K. Nakada and K. Kusakabe, J. Phys. Soc. Jpn.
65, 1920 (1996).
[102] K. Nakada, M. Fujita, G. Dresselhaus and M. S. Dresselhaus, Phys. Rev. B
54, 17954 (1996).
[103] K. Wakabayashi, M. Fujita, H. Ajiki and M. Sigrist, Phys. Rev. B 59, 8271
(1999).
[104] S. Okada and A. Oshiyama, Phys. Rev. Lett. 87, 146803 (2001).
[105] T. Hikihara, X. Hu, H.-H. Lin and C.-Y. Mou, Phys. Rev. B 68, 035432
(2003).
[106] X. G. Wen, Phys. Rev. Lett. 64, 2206 (1990).
[107] S. Y. Zhou, G.-H. Gweon, J. Graf, A. V. Fedorov, C. D. Spataru, R. D.
Diehl, Y. Kopelevich, D.-H. Lee, Steven G. Louie and A. Lanzara, Nature
Phys. 2, 595 (2006).
[108] G. W. Semenoff, Phys. Rev. Lett. 53, 2449(1984).
[109] F. D. M. Haldane, Phys. Rev. Lett. 61, 2015(1988).
[110] M. I. Katsnelson, K. S. Novoselov and A. K. Geim, Nat. Phys. 2, 620 (2006).
[111] N. Tombros, C. Jozsa, M. Popinciuc, H. T. Jonkman and B. J. van Wees,
Nature 448, 571 (2007).
[112] A. Bostwick, T. Ohta, T. Seyller, K. Horn and E. Rotenberg, Nat. Phys. 3,
36 (2007).
[113] C.-H. Park, L. Yang, Y.-W. Son, M. L. Cohen, S. G. Louie, Nat. Phys. 4,
213 (2008).
[114] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov and A. K.
Geim, arXiv:0709.1163.
[115] S.-H. Dong, X.-W. Hou ,and Z.-Q. Ma, Phys. Rev. A 58, 2160 (1998).
[116] V. M. Pereira, F. Guinea, J. M. B. Lopes dos Santos, N. M. R. Peres, and
A. H. Castro Neto, Phys. Rev. Lett. 96, 036801(2006).
[117] P. O. Lehtinen, A. S. Foster, Y. Ma, A. V. Krasheninnikov, and R. M.
Nieminen, Phys. Rev. Lett. 93, 187202 (2004).
[118] M. A. H. Vozmediano, M. P. L ́opez-Sancho, T. Stauber and F. Guinea, Phys.
Rev. B 72, 155121(2005).
[119] T. O. Wehling, A. V. Balatsky, M. I. Katsnelson, A. I. Lichtenstein, K.
Scharnberg, and R. Wiesendanger, Phys. Rev. B 75, 125425(2007).
[120] O. V. Yazyev and L. Helm, Phys. Rev. B 75, 125408(2007).
[121] V. M. Pereira, J. M. B. Lopes dos Santos, and A. H. Castro Neto, Phys.
Rev. B 77, 115109 (2008).
[122] E. H. Lieb, Phys. Rev. Lett. 62, 1201 (1989).
[123] J.-C. Charlier, X Blase, and S. Roche, Rev. Mod. Phys. 79, 677 (2007).