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The Ohio State University

1. Lu, Fuyan. Topological Phases with Crystalline Symmetries.

Degree: PhD, Physics, 2018, The Ohio State University

In this dissertation, we focus on topological phases protected or enforced by crystalline symmetries. The topological phase can appear in the ground state of fermion and interacting boson or in the excitation bands of the free boson such as phonon and magnon. Those topics are covered in three parts. In the second chapter, we study the Glide Symmetry Protected Topological (GSPT) phases of interacting bosons and fermions in three spatial dimensions with certain on-site symmetries. They are crystalline Symmetry Protected Topological (SPT) phases, which are distinguished from a trivial product state only in the presence of non-symmorphic glide symmetry. We classify these GSPT phases with various on-site symmetries such as U(1) and time reversal, and show that they can all be understood by stacking and coupling two-dimensional short-range-entangled phases in a glide-invariant way. Using such a coupled layer construction we study the anomalous surface topological orders of these GSPT phases, which gap out the two-dimensional surface states without breaking any symmetries. While this framework can be applied to any non-symmorphic SPT phase, we demonstrate it in many examples of GSPT phases including the non-symmorphic topological insulator with "hourglass fermion" surface states.In the third chapter, we discuss the non-symmorphic crystalline symmetry enforced Quantum Spin Hall effect (QSHE). In the generic classification frame of SPT phases, the classification is \mbz2 for the ground state of fermion in 2D with charge conservation and time-reversal (TR) symmetry. The ground state can be in either trivial state or QSHE state, which depends on the specific model. However, we prove that the gapped ground state must be QSHE other than trivial as long as given non-symmorphic symmetry present. We also propose a crystal structure as one realization, and make an effort on material searching based on the space group. This work can supply a new principle of QSHE materials searching, which is purely depending on the symmetry instead of electronic details.Last but not least, in the fourth chapter, we discuss the topological phases of the bands of magnon. The crystalline symmetries or its combination with TR symmetry are proved to play a crucial role in magnon bands topology classification. In spite of flourishing studies on the topology of spin waves, a generic framework to classify and compute magnon band topology in non-collinear magnets is still missing. We provide such a theory framework, by mapping an arbitrary linear spin wave into a local free-fermion Hamiltonian with exactly the same spectrum, symmetry implementation and band topology, which allows for a full classification and calculation on any topological properties of magnon bands under different symmetry groups. We apply this fermionization approach to honeycomb Kitaev magnet α-RuCl3. The crystalline symmetry protected band touchings will be discussed in detail and the evolution of its magnon band topology is also investigated under an external magnetic… Advisors/Committee Members: Lu, Yuan-Ming (Advisor).

Subjects/Keywords: Physics; Theoretical Physics; Symmetry Protected Topological Phases; Topological Magnon

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APA (6th Edition):

Lu, F. (2018). Topological Phases with Crystalline Symmetries. (Doctoral Dissertation). The Ohio State University. Retrieved from

Chicago Manual of Style (16th Edition):

Lu, Fuyan. “Topological Phases with Crystalline Symmetries.” 2018. Doctoral Dissertation, The Ohio State University. Accessed October 19, 2018.

MLA Handbook (7th Edition):

Lu, Fuyan. “Topological Phases with Crystalline Symmetries.” 2018. Web. 19 Oct 2018.


Lu F. Topological Phases with Crystalline Symmetries. [Internet] [Doctoral dissertation]. The Ohio State University; 2018. [cited 2018 Oct 19]. Available from:

Council of Science Editors:

Lu F. Topological Phases with Crystalline Symmetries. [Doctoral Dissertation]. The Ohio State University; 2018. Available from: