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Classification of Quanutm Phases by Tensor Network Formalism

Haghshenas, Reza | 2015

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 48859 (04)
  4. University: Sharif University of Technology
  5. Department: Physics
  6. Advisor(s):
  7. Abstract:
  8. Tensor network states have been appeared as an alternative promising framework to address the long-standing problems of many-body systems. They offer a conceptual representation of the phases of matter—based on the structure of entanglement—and also provide a(powerful) variational ansatz to explore new physics in strongly entangled many-body systems. The range of their applicability is very vast, and encompassing almost all of interesting today’s systems. In addition, tensor network states are not limited to only numerical approaches, but, have some analytical application. For instance, they could be utilized to uniquely distinguish the phases of matter, where in one-dimensional case based on matrix product state a “full classification” of the phases has been proposed, referred to “symmetry fractionalization”. Our main objective in this thesis is to study basic principle of tensor network states and symmetry fractionalization and to investiage nature of phases of some specific spin models. We study topological quantum phase transitions of the bond-alternating spin-1=2 Heisenberg model by using the symmetry fractionalization mechanism and infinite matrix-product state. We show that the phases belong to symmetry-protected topological order, so that their labels are quite similar to that of Haldane phase of spin-1 Heisenberg model. We then propose this question, if well-known models of Kitaev’s honeycomb and Toric code is being considered on quasi one-dimensional lattice, what happens to the nature of their intrinsic topological orders? We show that Toric code model (on ladder geometry) represents a symmetry-protected topological order protected by Z2 * Z2 symmetry. We study the stability of the phases under Ising-like perturbations (especially in presence of frustration). Furthermore, we utilize degenerate perturbation theory to obtain an effective model for each phase of Kitaev’s model (on ladder geometry), clarifying the nature of the phases. An important result of this characterization is that the associated quantum phase transitions are accompanied by an explicit symmetry breaking, and thus a local-order parameter conclusively identifies the phase diagram of the underlying model. Finally, we introduce renormalized central charge based on ‘entanglement RG flow’ to answer this principal well-known problem, How to distinguish gapped and gapless phases? We benchmark the validity/accuracy of our criteria for several spin models different types of quantum phase transitions
  9. Keywords:
  10. Topological Phase Transition ; Quantum Phase Transition ; Symmetry Fractionalization Theory ; Tensor Network States ; Quantum Phases Classification

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