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Entanglement, Quantum Phase Transition and Topological Order

Kargarian, Mehdi | 2009

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 39837 (04)
  4. University: Sharif University of Technology
  5. Department: Physics
  6. Advisor(s): Langari, Abdollah
  7. Abstract:
  8. We are interested in the relation between entanglement and quantum phase transitions. We have presented a formulation on how the entanglement of a very large the system is related to the entanglement of a small part of system. The renormalization of entanglement is an indicator of the critical behavior of the models. The framework of our approach is introduced about the Ising model. In particular, we show that derivative of entanglement developes a minimum close to the critical point. We further clarify that this minimum point approaches to the exact critical point of the system as the system becomes thermodynamic. This phenomenon is governed by an exponent that is closely related to the diverging of the correlation length. Then, we generalize the approach to other models wich fall into the same universality calss. First, we emphasize on the small block of models and investigated the role of different tuning interactions on the general feature of the entanglement. Then, the thermodynamic limit is appropriately treated by using RG. In all, the entanglement corresponds to the crtitical behavior of the models. The conection to the critical exponents, gap exponent and universality classes are fully discussed. Then, we turn on to the topological models. They are some exotic states of matters, an immune place for storing and processing information which is free from errors. Topological color codes (TCC) are a whole class of models that provide an instance of an interdisciplinary subject between quantum information and the physics of quantum many-body Systems. We investigate the underlying topological order and its connection to entanglement. We also address the question of whether the color code model is robust against temperature. In particular, we show that for finite size system, the topological order survives even at finite temperatures. Then, we put next step to connect this model with physical models with 2-body interactions. To do so, we introduce a new quantum lattice Hamiltonian. We discuss it at the non-perturbative level. The particular structure of the 2-body Hamiltonian provides a fruitful interpretation in terms of mapping to bosons coupled to effective spins. We show that high energy excitations of the model have fermionic statistics. They form three families of high energy excitations each of one color. Furthermore, we show that they belong to a particular family of topological charges. The emergence of invisible charges related to the string-net structure of the model.
  9. Keywords:
  10. Entanglement ; Quantum Phase Transition ; Renormalization Group ; Topological Order ; Anyon

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