PhD thesis in English

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$Edge excitations of paired fractional quantum Hall states$

2. Diagonalization of Transition Amplitudesnential growth in the size of analytic expressions for the effective potential with theincrease of the level p, as discussed in Ref. [44]. Therefore, the required CPU timefor construction of the matrix to be diagonalized in the presented approach growsexponentially with the level p, while in other methods the construction of such amatrix does not require a significant amount of time. However, the time for exactdiagonalization far outweighs the time needed for construction of even large matriceswith moderate levels p of the order 10-20. The significant benefit of practicallyeliminating errors associated with the time of propagation therefore fully justifiesthe use of the effective action approach. Of course, in practical applications one hasto study the complexity of the algorithm and to choose the optimal level p whichwill sufficiently reduce the errors, while keeping the complexity of the calculation onthe acceptable level.2.6 Conclusions and outlookIn this Chapter, we have dealt with the thorough understanding and optimizationof the method of the calculation of the properties of quantum systems based on thediagonalization of transition amplitudes, previously introduced in Ref. [9]. First,we have focused on analyzing the errors associated with real-space discretizationand finite size effects. In particular, we have shown that within this calculationscheme spatial discretization leads to a universal and non-perturbatively small discretizationerror. This highly outperforms the usual polynomial behavior of errorsin approaches based on the diagonalization of space-discretized Hamiltonians. Akey problem in practical applications of this approach - accurate calculation of transitionamplitudes, matrix elements of the space-discretized evolution operator, hasbeen resolved using recently introduced effective action approach [44], which givessystematic short-time expansion of the evolution operator.The derived analytical estimates for all types of errors, including errors due to theapproximative calculation of transition amplitudes, provide us with a way to chooseoptimal discretization parameters and to reduce overall errors in energy eigenvaluesand eigenstates for many orders of magnitude, as was demonstrated for several oneandtwo-dimensional models. We have shown that numerical diagonalization of thespace-discretized evolution operator can be successfully applied for studies of manyinteresting lower dimensional models. The approach allows exact numeric calculationof a large number of energy eigenvalues and eigenstates of the system. Due to55
2. Diagonalization of Transition Amplitudesthe superior behavior of discretization and other errors in this method comparedto methods based on diagonalization of the discretized Hamilton operator and relatedmethods, the presented approach is a method of choice for numerical studiesof lower-dimensional physical systems. An application of the approach for numericalinvestigation of properties of fast rotating Bose-Einstein condensates is givenin Chapter 3. Further interesting line of research would be to combine the presentmethod with the density matrix renormalization group (DMRG) approach [48, 67].56
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UNIVERSITY OF BELGRADEFACULTY OF PH
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Thesis advisor, Committee member:Dr
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lence for Computer Modeling of Comp
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dobijanje kondenzata odabrani su at
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Uticaj slabih interakcija na fenome
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Abstract of the doctoral dissertati
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highly accurate information on ener
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Keywords: cold quantum gases, Bose-
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CONTENTS3.4.2 Time-of-flight graphs
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NomenclatureRoman Symbolsagk BLMNn(
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Chapter 1Introduction1.1 ForewordTh
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Page 26 and 27: Summations in the last expression c
Page 28 and 29: Figure 1.1: The hallmark of the Bos
Page 30 and 31: we discuss in some detail the exper
Page 32 and 33: In the first papers [3, 4], the TOF
Page 34 and 35: where a BG is the off-resonant scat
Page 36 and 37: system given by( ) ǫ Bog ⃗k =
Page 38 and 39: Having the efficient numerical meth
Page 40 and 41: Chapter 2Properties of quantum syst
Page 42 and 43: 2. Diagonalization of Transition Am
Page 66 and 67: ||2. Diagonalization of Transition
Page 74: 2. Diagonalization of Transition Am
Page 79 and 80: magnetic field ⃗ B = 2M ⃗ Ω.3.
Page 81 and 82: 3. Rotating ideal BECof the rotatio
Page 83 and 84: 3. Rotating ideal BEC3.1 Numerical
Page 85 and 86: 3. Rotating ideal BEC350300SC appro
Page 87 and 88: 3. Rotating ideal BEC3.2 Finite num
Page 89 and 90: 3. Rotating ideal BECN - N 03.0·10
Page 91 and 92: 3. Rotating ideal BEC3.3 Global pro
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Page 95 and 96: 3. Rotating ideal BECT c [nK]120110
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Page 105 and 106: Chapter 4Mean-field description of
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Chapter 5Nonlinear BEC dynamics by
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5. BEC excitation by modulation of
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where n = 1, 2, 3, . . . is an inte
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Chapter 6SummarySince the first exp
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short-range interactions.The excita
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of the ground state. Imaginary-time
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(A.13). With another boundary condi
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Appendix B Time-dependent variation
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List of papers by Ivana VidanovićT
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References[1] S. N. Bose, Plancks g
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REFERENCES[21] W. Ketterle, D. S. D
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REFERENCES[45] A. Bogojević, A. Ba
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REFERENCES[70] M. R. Matthews, B. P
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REFERENCES[94] M.-O. Mewes, M. R. A
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REFERENCES[116] K. Staliunas, S. Lo
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CURRICULUM VITAE - Ivana Vidanović
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