In this thesis we consider the properties of a class of Z2 topological phases on a two-dimensional square lattice. The ground states of Z2 topological order are generally degenerate on a periodic lattice, characterized by certain global Z2 quantum numbers. This property is important for application in quantum computing as the global quantum numbers can be used as protected qubits. It is therefore instrumental to construct and study Z2 topological order from a general framework.
Our results in this thesis provide such a framework. It is based on the simplest and most illustrative Z2 topological order: the Toric Code (TC), which contains static and non-interacting anyonic quasiparticles e, m and ε. Building on this interpretation, in the first part of the thesis two exact mappings are presented from the spin Hilbert space to the Hilbert space of (e,m) and (e,ε). The mappings are derived on infinite, open, cylindrical and periodic lattices respectively. Mutual anyonic statistics as well as the effect of the global Z2 quantum numbers are taken into account. Due to the mutual anyonic statistics of the elementary excitations, the mappings turn out to be highly non-local. In addition, it is shown that the mapping to e and ε anyons can be carried over directly to the honeycomb lattice, where the anyons become visons and Majorana fermions in the Kitaev honeycomb model.
The mappings allow one to rewrite any spin Hamiltonians as Hamiltonians of anyons. In the second part of the thesis, we construct a series of spin models which are mapped to Hamiltonians of free anyons. In particular, a series of Z2 topological phases `enriched by lattice translation symmetry' are constructed which are also topological superconductors of ε particles. Their properties can be analyzed generally using the duality and then the theory of topological superconductivity. In particular, their ground state degeneracy on a periodic lattice may depend on lattice size. For these phases a classification scheme is proposed, which generalizes classification by the integer Chern number. Some of the conclusions are then verified directly by exact solutions on the spin lattice.
The emergent anyon statistics of e-particles in these phases is also analyzed by computing numerically the Berry phase of their motion on top of the superconducting vacua. For phases with C=0 yet still topologically non-trivial, we discover examples of `weak symmetry breaking': the e-lattice splits into two inequivalent sublattices which are exchanged by lattice translations. The e-particles on the two sublattices acquire mutual anyonic statistics. In topological phases with non-zero C, the mutual braiding of e is confirmed explicitly. In addition, the Berry phase due to background flux of each square unit cell is quantized depending on the underlying topology of the phases. This quantity is related to properties of the vison band in Kitaev materials.
Lastly, the ZN (N>2) extension of Z2 topological order is discussed. Constructing the duality to `parafermions' in this case is much more complex. The difficulties of deriving such a mapping are pointed out and we only present exact solutions to certain Hamiltonians on the spin lattice.
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:83028 |
| Date | 24 January 2023 |
| Creators | Rao, Peng |
| Contributors | Moessner, Roderich, Sodemann, Inti, Vojta, Matthias, Technische Universität Dresden, Max-Planck-Institut für Physik komplexer Systeme |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | info:eu-repo/semantics/publishedVersion, doc-type:doctoralThesis, info:eu-repo/semantics/doctoralThesis, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
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