Topological phenomena in periodically driven quantum systems

TOPOLOGICAL PHENOMENA IN PERIODICALLY DRIVEN QUANTUM SYSTEMS DEREK Y H HO BSc (Hons), NUS A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN SCIENCE DEPARTMENT OF PHYSICS NATIONAL UNIVERSITY OF SINGAPORE 2014 Declaration I hereby declare that this thesis is my original work and it has been written by me in its entirety I have duly acknowledged all the sources of information which have been used in the thesis This thesis has also not been submitted for any degree in any university previously Derek Yew Hung Ho May 2014 i ii Dedicated to my parents iii iv Acknowledgements I would like to first thank my supervisor, A/Prof Gong Jiang Bin Throughout my PhD years, he has given me lots of encouragement and indispensable guidance I have greatly benefited from his intellectual openness and patience in discussing at length all kinds of physics ideas and questions His extensive physics knowledge, his positive energy, his creativity and his ability to bring out the best in people have frequently inspired me I shall always be grateful to him for everything he has taught me I am also deeply grateful to him for continuing to always be there for me as my supervisor in spite of the great tragedy that happened I would also like to thank fellow group members Adam, Hailong, Qifang, Yon Shin, Long Wen, Gao Yang and Da Yang for many productive and enjoyable discussions My thanks especially to Adam from whom I have learned a lot of physics through discussions and also for his generous help with typesetting issues I thank Jose-Garcia Palacios and Zhang Qi for educational discussions during the early stage of my PhD I also thank Wayne Lawton for illuminating discussions on Chern number calculations I thank fellow PhD students in block S10 Bijay and Juzar for their friendship and good company Last but not least, I would like to express my profound gratitude to my father Ho Kwok Yew, mother Ho May Chee Ruby, and my brothers Victor and David for their love and support over the years I am especially grateful to my mother who has always been a pillar of support for me I also thank my fiancée Sim Hui Shan who has made my days in graduate school so much happier with her companionship and encouragement I am v very grateful and fortunate to have them all in my life vi Contents Introduction 1.1 Topology in Quantum Physics 1.2 Periodically Driven Systems: A New Playground 1.3 Outline of Thesis Mathematical Preliminaries and Background 2.1 The Floquet Formalism for Driven Systems 2.2 The Rotator and Lattice Hilbert Space Formalisms 2.3 Method for Diagonalizing of Periodic Floquet Operators 12 2.4 Introduction to the Chern Number Invariant 14 Topological Phenomena in Quantum Physics: Some Examples 3.1 The Integer Quantum Hall effect 29 30 3.1.1 Topological Quantization of Hall Conductance: Weak Modulation 31 3.1.2 Topological Quantization of Hall Conductance: Strong Modulation 39 3.2 Hofstadter’s Butterfly: History and Recent Developments 43 3.3 The Kicked Harper Model 47 3.4 Topology and Quantum Chaos 51 Quantized Transport in Momentum Space 4.1 Introduction to the ORDKR Model vii 57 57 Contents 4.2 Topological Characterization in the ORDKR 69 4.3 Quantized Transport in Momentum Space 73 4.3.1 Theoretical Background: Quantized Transport in Position Space 73 4.3.2 Quantized Transport in Momentum Space in the ORDKR Model 80 4.3.3 Stability of Quantized Transport to Perturbations 90 Further Numerical Experiments 93 4.4.1 The Seven Band Case 93 4.4.2 Initial and Final Distributions in Momentum Space 95 Summary and Discussion 95 4.A Numerical Evaluation of Matrix Elements in Eq (4.15) 96 4.4 4.5 Topological Equivalence of the ORDKR and phase-shifted KHM 5.0.1 Motivation and Notations 99 99 5.1 Numerical Findings 102 5.2 Proof of Topological Equivalence 104 5.2.1 5.3 Derivation of Eq (5.7) 108 Summary and Discussion 115 Topological Edge States in Driven Quantum Systems 6.1 Introduction 117 6.1.1 6.2 117 Novelty of Driven Systems 118 Theoretical Background Knowledge for Driven Topological Systems 119 6.2.1 The Tenfold Classification 119 6.2.2 Bulk-Boundary Correspondence 125 6.2.3 Review of Previous Studies on Topologically Driven Systems 128 6.2.4 Topological Quantum Random Walk Studies 135 6.3 Mapping from the Rotator to Particle Hopping on a Lattice 147 6.4 Bulk-Boundary Correspondence of the Double Kicked Lattice and Kicked Harper Lattice Models 156 viii ¯ 6.A Details for extraction of Heff (k) from UDKL in Eq.(6.67) The 2-band case for the 1D DKL model was found to be able to host an arbitrarily large number of topologically protected and π-quasienergy edge modes, while the 1D KHL model was unable to host any topological edge modes in the 2-band case The large numbers of protected edge modes in the 1D DKL might have potential applications in quantum information The 3-band cases for the 2D DKL and KHL models were also studied in detail We saw that in spite of the matching Chern numbers of the two models, the KHL model possesses anomalous counter-propagating chiral edge modes [132] which are absent in the DKL model The presence of such counter-propagating modes have yet to be successfully associated with bulk topological invariants This remains an open problem 6.A ¯ Details for extraction of Heff (k) from UDKL in Eq.(6.67) ¯ ¯ We first rewrite the different parts of UDKL in terms of the k, A and k, B defined in Eq.(6.65) Note that summations over j in what follows refers to summing over ¯ j = 0, 1, · · · , S − for each sublattice, and summations over k refer to summing over ¯ k = −π, −π + 2π/S, · · · , π − 2π/S e±i nx π ˆ (|j, A j, A| ± i |j, B j, B|) = j ¯ ¯ k, A ± i k, B ¯ ( k, A = ¯ k, B ) ¯ k   1 = ¯ k  0 ¯ ⊗ k ¯ k , (6.81) ±i where we define × matrices as describing the A-B sublattice degree of freedom Namely,   a b   c d  ≡ a |A A| + b |A B| + c |B A| + d |B B| 180 (6.82) Chapter Topological Edge States in Driven Quantum Systems Similarly, J2 cos(q) = J2 J2 = (|n, B n, A| + |n + 1, A n, B| + h.c) n ¯ ¯ ¯ k, A + eik k, A ¯ ( k, B ¯ k  J2 = ¯ k, B + h.c) ¯ k   1+ 1+ ¯ e−ik ¯ eik   ¯ ⊗ k ¯ k (6.83) Entirely analogous steps also yield  J1 cos(q + α) = J1 e−iα ¯ k   eiα + ¯ e−i(k+α) + ¯ ei(k+α)   ¯ ⊗ k ¯ k (6.84) The exponentials of the previous two expressions may be reduced to × matrices using the standard formulae for dealing with exponentials of sums of Pauli matrices, e−ian·σ = cos(a)1 − i sin(a)n · σ, where |n| = Hence, we are now able to write UDKL as a multiplication of five simple matrices We then combine these using direct matrix multiplication and rewrite the resulting matrix using the aforementioned formulae as UDKL = ¯ k ¯ ¯ e−iHeff (k) ⊗ k ¯ k = e−i ¯ k ¯ ¯ Heff (k)⊗|k 181 ¯ k| ¯ , where Heff (k) is a × matrix ¯ 6.A Details for extraction of Heff (k) from UDKL in Eq.(6.67) 182 Chapter Conclusion and Outlook 7.1 Conclusion We now summarize what has been presented in this thesis We began in chapter by introducing the Floquet formalism for studying driven quantum systems as well as the mathematical details of topological invariants We then reviewed in Chapter the famous result that the Integer Quantum Hall Effect (IQHE) attains its precise yet robust quantization due to topological reasons We also reviewed the historical development of the Hofstadter butterfly problem and its recent experimental progress We also went through an early work applying topological considerations towards understanding quantum-classical correspondence In chapter 4, we introduced the on-resonance double kicked rotor (ORDKR), a driven model which was periodic in momentum space and also possessed non-trivial topology We saw how this combination of periodicity in momentum space and topological nontriviality led to the phenomenon of quantized transport in momentum space, an effect similar to the IQHE but taking place in momentum rather than position space We uncovered in Chapter a novel connection between the ORDKR and kicked harper model (KHM), which explained why the two seemingly different models possessed identical topological numbers In chapter 6, we moved on to the study of topologically protected boundary states in lattice models After a rather detailed review of the most recent 183 7.2 Outlook studies regarding topologically non-trivial driven systems, we applied these methods to study the topologically protected boundary modes in lattice analogues of the ORDKR and KHM We found that in spite of the close connection between the ORDKR and KHM models, their edge state behaviour are still very different We found that the ORDKR surprisingly is able to host very large numbers of topological and π quasienergy modes, a discovery which might be useful for quantum information applications The KHM, on the other hand, is unable to host any of such modes We showed that the lattice analogue of KHM is instead able to host novel counter-propagating edge modes that are absent from the lattice analogue of ORDKR 7.2 Outlook In this thesis, we unveiled the novel concept of quantized transport in momentum space This was done in a single particle picture Hence, it would be interesting to see how many-body effects might affect the robustness of this phenomenon In the case of the IQHE, a single-particle phenomenon, adding strong inter-particle interactions leads to the fractional quantum hall effect (FQHE) It is thus possible that we might see some similar fractionalization effect of the quantized momentum space transport if interactions were to be added Secondly, we also showed the ability of the ORDKR model to host an arbitrarily large number of topological edge states It is worth looking into whether this might be harnessed for quantum information processing in some way Lastly, the problem of topologically classifying periodically driven systems in the same precise manner as has been achieved for static systems is still an open one The insights gained from studying the bulk-boundary correspondence in the ORDKR and KHM lattice models may be useful for making future contributions towards solving it Generally speaking, new insights into the role played by topology in quantum mechanics are still being found on a regular basis in the research community We believe that there is still plenty left to learn about topological effects within quantum mechanics 184 Bibliography [1] D J Thouless, M Kohmoto, M P Nightingale, and M den Nijs, Phys Rev Lett 49, 405 (1982) [2] K v Klitzing, G Dorda, and M Pepper, Phys Rev Lett 45, 494 (1980) [3] D C Tsui, H L Stormer, and A C Gossard, Phys Rev Lett 48, 1559 (1982) [4] X G Wen, Phys Rev B 40, 7387 (1989) [5] X G Wen, Int J Mod Phys B 4, 239 (1990) [6] X G Wen and Q Niu, Phys Rev B 41, 9377 (1990) [7] C L Kane and E J Mele, Phys Rev Lett 95, 226801 (2005) [8] M König, S Wiedmann, C Brüne, A Roth, H Buhmann, L W Molenkamp, X.-L Qi, and S.-C Zhang, Science 318, 766 (2007) [9] M Z Hasan and C L Kane, Rev Mod Phys 82, 3045 (2010) [10] A Kitaev, Ann Phys (N Y.) 303, (2003) [11] C Nayak, S H Simon, A Stern, M Freedman, and S Das Sarma, Rev Mod Phys 80, 1083 (2008) [12] S Ryu, A P Schnyder, A Furusaki, and A W W Ludwig, New J Phys 12, 065010 (2010) 185 Bibliography [13] X Chen, Z.-X Liu, and X.-G Wen, Phys Rev B 84, 235141 (2011) [14] N Regnault and B A Bernevig, Phys Rev X 1, 021014 (2011) [15] G Casati and B V Chirikov, Quantum Chaos: Between Order and Disorder (Cambridge University Press, New York, 1995) [16] F Haake, Quantum Signatures of Chaos (Springer, 2010) [17] P Leboeuf, J Kurchan, M Feingold, and D P Arovas, Phys Rev Lett 65, 3076 (1990) [18] P Leboeuf, J Kurchan, M Feingold, and D P Arovas, Chaos 2, 125 (1992) [19] G Casati, B V Chirikov, F M Izraelev, and J Ford, Lecture Notes in Phys 93, 334 (1979) [20] D Y H Ho and J B Gong, Phys Rev Lett 109, 010601 (2012) [21] D Loss and K Mullen, J Phys A: Math Gen 25, L235 (1992) [22] T Boness, S Bose, and T S Monteiro, Phys Rev Lett 96, 187201 (2006) [23] V Balachandran and J B Gong, Phys Rev A 77, 012303 (2008) [24] N Goldman, Quantum Transport and Phase Transitions in Lattices Subjected to External Gauge Fields, Ph.D thesis (2009) [25] D Thouless, Phys Rep 110, 279 (1984) [26] B A Bernevig, Topological Insulators and Topological Superconductors (Princeton University Press, 2013) [27] D Xiao, M.-C Chang, and Q Niu, Rev Mod Phys 82, 1959 (2010) [28] N W Ashcroft and N D Mermin, Solid State Physics (Saunders College Publishing, 1976) [29] D R Hofstadter, Phys Rev B 14, 2239 (1976) 186 Bibliography [30] M A Paalanen, D C Tsui, and A C Gossard, Phys Rev B 25, 5566 (1982) [31] K von Klitzing, Rev Mod Phys 58, 519 (1986) [32] M Janssen, O Viehweger, U Fastenrath, and J Hajdú, Introduction to the Theory of the Integer Quantum Hall Effect (Wiley-VCH, 1994) [33] U Rössler and M Suhrke, Adv Solid State Phys 40, 35 (2000) [34] C R Dean, L Wang, P Maher, C Forsythe, F Ghahari, Y Gao, J Katoch, M Ishigami, P Moon, M Koshino, W K Taniguchi, T., K L Shepard, J Hone, and P Kim, Nature 497, 598 (2013) [35] L A Ponomarenko, R V Gorbachev, G L Yu, D C Elias, R Jalil, A A Patel, A Mishchenko, A S Mayorov, C R Woods, J R Wallbank, M MuchaKruczynski, B A Piot, M Potemski, I V Grigorieva, K S Novoselov, F Guinea, V I FalŠko, , and A K Geim, Nature 497, 594 (2013) [36] B Hunt, J D Sanchez-Yamagishi, A F Young, M Yankowitz, B J LeRoy, K Watanabe, T Taniguchi, P Moon, M Koshino, P Jarillo-Herrero, and R C Ashoori, Science 340, 1427 (2013) [37] M Aidelsburger, M Atala, M Lohse, J T Barreiro, B Paredes, and I Bloch, Phys Rev Lett 111, 185301 (2013) [38] H Miyake, G A Siviloglou, C J Kennedy, W C Burton, and W Ketterle, Phys Rev Lett 111, 185302 (2013) [39] J Zak, Phys Rev 134, A1602 (1964) [40] J Zak, Phys Rev 134, A1607 (1964) [41] P G Harper, Proc Phys Soc Lond A 68, 874 (1955) [42] R Kubo, J Phys Soc Jpn 12, 570 (1957) [43] J M Luttinger, Phys Rev 84, 814 (1951) 187 Bibliography [44] B Mandelbrot, The Fractal Geometry of Nature (Henry Holt and Company, 1983) [45] A Avila and S Jitomirskaya, Ann Math 170, 303 (2009) [46] C Chin and E J Mueller, Physics 6, 118 (2013) [47] D Jaksch and P Zoller, New J Phys 5, 56 (2003) [48] S Aubry and G André, Ann Isreal Phys Soc 3, 133 (1980) [49] R Artuso, G Casati, F Borgonovi, L Rebuzzini, and I Guarneri, Int J Mod Phys B 8, 207 (1994) [50] R Lima and D Shepelyansky, Phys Rev Lett 67, 1377 (1991) [51] R Ketzmerick, K Kruse, and T Geisel, Phys Rev Lett 80, 137 (1998) [52] I I Satija, Phys Rev E 66, 015202 (2002) [53] I I Satija, Phys Rev E 71, 056213 (2005) [54] J Wang and J B Gong, Phys Rev A 77, 031405 (2008) [55] J Wang, A S Mouritzen, and J B Gong, J Mod Opt 56, 722 (2009) [56] I Dana, Phys Lett A 197, 413 (1995) [57] G A Kells, J Twamley, and D M Heffernan, Phys Rev E 70, 015203 (2004) [58] T Geisel, R Ketzmerick, and G Petschel, Phys Rev Lett 67, 3635 (1991) [59] R Ketzmerick, G Petschel, and T Geisel, Phys Rev Lett 69, 695 (1992) [60] R Artuso, G Casati, and D Shepelyansky, Phys Rev Lett 68, 3826 (1992) [61] J Zak, Phys Rev 168, 686 (1968) [62] M Kohmoto, Ann Phys (N Y.) 160, 343 (1985) [63] D P Arovas, R N Bhatt, F D M Haldane, P B Littlewood, and R Rammal, Phys Rev Lett 60, 619 (1988) 188 Bibliography [64] J Kurchan, P Leboeuf, and M Saraceno, Phys Rev A 40, 6800 (1989) [65] K Husimi, Proc Phys Math Soc Jpn 22, 264 (1940) [66] D R Grempel, R E Prange, and S Fishman, Phys Rev A 29, 1639 (1984) [67] D L Shepelyanskii and F M Izrailev, Theor Math Phys 43, 553 (1980) [68] J M McCaw, e-print arXiv:math-ph/0503032 (2005) [69] B V Chirikov, F M Izrailev, and S D L., Sov Sci Rev Sect C 2, 209 (1981) [70] F L Moore, J C Robinson, C F Bharucha, B Sundaram, and M G Raizen, Phys Rev Lett 75, 4598 (1995) [71] H Ammann, R Gray, I Shvarchuck, and N Christensen, Phys Rev Lett 80, 4111 (1998) [72] B G Klappauf, W H Oskay, D A Steck, and M G Raizen, Phys Rev Lett 81, 1203 (1998) [73] M B d’Arcy, R M Godun, M K Oberthaler, D Cassettari, and G S Summy, Phys Rev Lett 87, 074102 (2001) [74] M B d’Arcy, R M Godun, M K Oberthaler, G S Summy, K Burnett, and S A Gardiner, Phys Rev E 64, 056233 (2001) [75] C Ryu, M F Andersen, A Vaziri, M B d’Arcy, J M Grossman, K Helmerson, and W D Phillips, Phys Rev Lett 96, 160403 (2006) [76] C Zhang, J Liu, M G Raizen, and Q Niu, Phys Rev Lett 92, 054101 (2004) [77] G J Duffy, A S Mellish, K J Challis, and A C Wilson, Phys Rev A 70, 041602 (2004) [78] J F Kanem, S Maneshi, M Partlow, M Spanner, and A M Steinberg, Phys Rev Lett 98, 083004 (2007) 189 Bibliography [79] J Chabé, G Lemarié, B Grémaud, D Delande, P Szriftgiser, and J C Garreau, Phys Rev Lett 101, 255702 (2008) [80] I Talukdar, R Shrestha, and G S Summy, Phys Rev Lett 105, 054103 (2010) [81] A Ullah and M D Hoogerland, Phys Rev E 83, 046218 (2011) [82] I Dana and D L Dorofeev, Phys Rev E 73, 026206 (2006) [83] I Dana, V Ramareddy, I Talukdar, and G S Summy, Phys Rev Lett 100, 024103 (2008) [84] D Steck, Quantum Chaos, Transport, and Decoherence in Atom Optics, Ph.D thesis (2001) [85] J B Gong and J Wang, Phys Rev E 76, 036217 (2007) [86] J Wang and J B Gong, Phys Rev E 78, 036219 (2008) [87] I Dana, E Eisenberg, and N Shnerb, Phys Rev E 54, 5948 (1996) [88] P H Jones, M M Stocklin, G Hur, and T S Monteiro, Phys Rev Lett 93, 223002 (2004) [89] J B Gong and P Brumer, Phys Rev E 70, 016202 (2004) [90] J B Gong and P Brumer, Phys Rev Lett 97, 240602 (2006) [91] E Lundh and M Wallin, Phys Rev Lett 94, 110603 (2005) [92] M Sadgrove, M Horikoshi, T Sekimura, and K Nakagawa, Phys Rev Lett 99, 043002 (2007) [93] C L Kane and E J Mele, Phys Rev Lett 95, 226801 (2005) [94] D J Thouless, Phys Rev B 27, 6083 (1983) [95] H Wang, D Y H Ho, W Lawton, J Wang, and J B Gong, Phys Rev E 88, 052920 (2013) 190 Bibliography [96] W Lawton, A S Mouritzen, J Wang, and J B Gong, J Math Phys 50, 032103 (2009) [97] K Drese and M Holthaus, Phys Rev Lett 78, 2932 (1997) [98] K Osterloh, M Baig, L Santos, P Zoller, and M Lewenstein, Phys Rev Lett 95, 010403 (2005) [99] J P Dahlhaus, J M Edge, J Tworzydło, and C W J Beenakker, Phys Rev B 84, 115133 (2011) [100] L.-J Lang, X Cai, and S Chen, Phys Rev Lett 108, 220401 (2012) [101] F Mei, S.-L Zhu, Z.-M Zhang, C H Oh, and N Goldman, Phys Rev A 85, 013638 (2012) [102] Y E Kraus, Y Lahini, Z Ringel, M Verbin, and O Zilberberg, Phys Rev Lett 109, 106402 (2012) [103] I Dana, Phys Rev E 52, 466 (1995) [104] I Dana, M Feingold, and M Wilkinson, Phys Rev Lett 81, 3124 (1998) [105] A Bohm, A Mostafazadeh, H Koizumi, Q Niu, and J Zwanziger, The Geometric Phase in Quantum Systems (Springer, 2003) [106] H P Breuer and M Holthaus, Phys Lett A 140, 507 (1989) [107] A Dranov, J Kellendonk, and R Seiler, J Math Phys 39, 1340 (1998) [108] A Tanaka, J Phys Soc Japan 80, 125002 (2011) [109] A Tanaka and M Miyamoto, Phys Rev Lett 98, 160407 (2007) [110] D W Hone, R Ketzmerick, and W Kohn, Phys Rev A 56, 4045 (1997) [111] S Fishman, I Guarneri, and L Rebuzzini, J Stat Phys 110, 911 (2003) [112] L Jiang, T Kitagawa, J Alicea, A R Akhmerov, D Pekker, G Refael, J I Cirac, E Demler, M D Lukin, and P Zoller, Phys Rev Lett 106, 220402 (2011) 191 Bibliography [113] Y T Katan and D Podolsky, Phys Rev Lett 110, 016802 (2013) [114] N H Lindner, G Refael, and V Galitski, Nat Phys 7, 490 (2011) [115] M C Rechtsman, J M Zeuner, Y Plotnik, Y Lumer, D Podolsky, F Dreisow, S Nolte, M Segev, and A Szameit, Nature 496, 196 (2013) [116] C L Kane and E J Mele, Phys Rev Lett 95, 146802 (2005) [117] Q.-J Tong, J.-H An, J B Gong, H.-G Luo, and C H Oh, Phys Rev B 87, 201109 (2013) [118] Y Niu, S B Chung, C.-H Hsu, I Mandal, S Raghu, and S Chakravarty, Phys Rev B 85, 035110 (2012) [119] D Sticlet and F Piéchon, Phys Rev B 87, 115402 (2013) [120] T Kitagawa, E Berg, M Rudner, and E Demler, Phys Rev B 82, 235114 (2010) [121] M S Rudner, N H Lindner, E Berg, and M Levin, Phys Rev X 3, 031005 (2013) [122] A P Schnyder, S Ryu, A Furusaki, and A W W Ludwig, Phys Rev B 78, 195125 (2008) [123] A P Schnyder, S Ryu, A Furusaki, and A W W Ludwig, AIP Conf Proc 1134, 10 (2009) [124] A Kitaev, AIP Conf Proc 1134, 22 (2009) [125] M R Zirnbauer, J Math Phys 37, 4986 (1996) [126] A Altland and M R Zirnbauer, Phys Rev B 55, 1142 (1997) [127] T Kitagawa, M A Broome, A Fedrizzi, M S Rudner, E Berg, I Kassal, A Aspuru-Guzik, E Demler, and A G White, Nat Commun 3, 882 (2012) [128] T Kitagawa, Quantum Inf Process 11, 1107 (2012) 192 Bibliography [129] S Ryu and Y Hatsugai, Phys Rev Lett 89, 077002 (2002) [130] M Miyamoto and A Tanaka, Phys Rev A 76, 042115 (2007) [131] J K Asbóth and H Obuse, Phys Rev B 88, 121406 (2013) [132] L Mahmoud, I I Satija, and E Zhao, e-print arXiv: 1307.3569 (2013) [133] T Kitagawa, M S Rudner, E Berg, and E Demler, Phys Rev A 82, 033429 (2010) [134] J K Asbóth, Phys Rev B 86, 195414 (2012) [135] A Y Kitaev, Phys Usp 44, 131 (2001) [136] J K Pachos, Introduction to Topological Quantum Computation (Cambridge, 2012) [137] M Thakurathi, A A Patel, D Sen, and A Dutta, Phys Rev B 88, 155133 (2013) [138] Y H Wang, H Steinberg, P Jarillo-Herrero, and N Gedik, Science 342, 453 (2013) [139] Y Aharonov, L Davidovich, and N Zagury, Phys Rev A 48, 1687 (1993) [140] T Oka, N Konno, R Arita, and H Aoki, Phys Rev Lett 94, 100602 (2005) [141] R A Williams, S Al-Assam, and C J Foot, Phys Rev Lett 104, 050404 (2010) [142] M Aidelsburger, M Atala, S Nascimbène, S Trotzky, Y.-A Chen, and I Bloch, Phys Rev Lett 107, 255301 (2011) [143] K Jiménez-García, L J LeBlanc, R A Williams, M C Beeler, A R Perry, and I B Spielman, Phys Rev Lett 108, 225303 (2012) 193 Bibliography [144] J Struck, C Ölschläger, M Weinberg, P Hauke, J Simonet, A Eckardt, M Lewenstein, K Sengstock, and P Windpassinger, Phys Rev Lett 108, 225304 (2012) [145] P Hauke, O Tieleman, A Celi, C Ölschläger, J Simonet, J Struck, M Weinberg, P Windpassinger, K Sengstock, M Lewenstein, and A Eckardt, Phys Rev Lett 109, 145301 (2012) [146] Y E Kraus and O Zilberberg, Phys Rev Lett 109, 116404 (2012) [147] S Ganeshan, K Sun, and S Das Sarma, Phys Rev Lett 110, 180403 (2013) [148] D E Liu, A Levchenko, and H U Baranger, Phys Rev Lett 111, 047002 (2013) 194 ... already-known insights and experience gained from studying driven systems in the quantum chaos context to look for interesting topological effects in order to contribute to the growing understanding of topological. .. studying topologically protected boundary modes in driven systems We begin the chapter by reviewing the most recent findings by other authors in the study of topological effects in driven quantum systems, ... settings In this thesis, we study novel topological effects occurring in periodically driven quantum systems We first report an intriguing connection between the topology and dynamics of a periodically