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Entanglement Spectrum of One Dimensional Ising Model in Transverse Field

Safaei, Alireza | 2011

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 42176 (04)
  4. University: Sharif University of Technology
  5. Department: Physics
  6. Advisor(s): Langari, Abdollah
  7. Abstract:
  8. Entanglement spectrum is the set of eigenvalues of reduced density matrix for a definite state. It contains more information than the entanglement entropy. This spectrum can show entanglement measure, quantum phase transition and topological order in a system. In recent years, the entanglement spectrum for several models has been studied. The entanglement spectrum shows a correspondence to the energy spectrum within some specific regions, which are related to the edge state of the system; It has been verified for the Heisenberg ladder [1] and fractional quantum Hall effect [3]. Moreover, the topological order can be seen in the entanglement spectrum. There exists an acceptable adaption between the edge spectrum and the entanglement spectrum which is related to the topological characteristics of the system. In addition, the range of validity of model for a specific system, can be axamined via the entanglement spectrum. For instance, the validity of Moor-Read model for the fractional quantum Hall effect is determined by the entanglement spectrum. In this thesis, we studied the entanglement spectrum of the Ising model in Transverse Field (ITF) within two approaches: exact diagonalization and quantum renormalization group (QRG). we compare the spectrum of exact diagonalization with the corresponding one of the QRG method. we address the validity region of QRG. Although the less important degrees of freedom are integrated out in the QRG approach, the low energy levels are the most important ones to reconstruct an acceptable antanglement spectrum. Thus, QRG can be implemented to get the entanglement spectrum of those models which suffer from an analytic solution. Moreover, we found that the entanglement spectrum has a wide distribution at the quantum critical point which highlights the entanglement spectrum to determine the quantum phase transition
  9. Keywords:
  10. Quantum Phase Transition ; Topological Order ; Entanglement Spectrum ; Ising Model in Transverse Field (ITF) ; Topological phase

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