Topological Phenomena in Tensor Network States of Quantum Spin Systems

quantum double, the toric code, and the double semion model) and identify the order and the universality classes of the phase transitions between these distinct topological phases. We analyze the phase boundaries of the toric code and the double semion model which exhibit first and second order phase transitions. We investigate the class of

\\mathbb{Z}_2

-invariant tensor network states while taking in to account the spin rotation and lattice symmetries. The resulting tensor network states allow us to map the phase diagram of

\\mathbb{Z}_2

spin liquids. The behavior of the system at the phase boundaries of

\\mathbb{Z}_2

spin liquid is governed by the condensation of spinons and visions. We also present our preliminary findings regarding the

\\mathbb{Z}_4

-invariant tensors with

SU(2)

symmetry. This approach enables us to study the spin liquid states of the toric code and the double semion model in a unified framework. We give a local tensor description for the approximate ground states of the Heisenberg antiferromagnet on the kagome lattice by using the variational manifolds of just three and five parameters in the gapped

\\mathbb{Z}_2

spin liquid phase. The approximation of the ground states we construct have an energy density that is remarkably close to the results from the state-of-the-art density matrix renormalization group and the exact diagonalization method. By analyzing the deconfinement of anyonic excitations, we also present our findings regarding the vicinity of the variational ground state in gapped

\\mathbb{Z}_2

spin liquid phase.

DOI: 10.18154/rwth-2018-224210

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D(\\mathbb{Z}_4)

\\mathbb{Z}_2

-invariant tensor network states while taking in to account the spin rotation and lattice symmetries. The resulting tensor network states allow us to map the phase diagram of

\\mathbb{Z}_2

spin liquids. The behavior of the system at the phase boundaries of

\\mathbb{Z}_2

spin liquid is governed by the condensation of spinons and visions. We also present our preliminary findings regarding the

\\mathbb{Z}_4

-invariant tensors with

SU(2)

\\mathbb{Z}_2

\\mathbb{Z}_2

spin liquid phase.

D(\mathbb{Z}_4)

\mathbb{Z}_2

-invariant tensor network states while taking in to account the spin rotation and lattice symmetries. The resulting tensor network states allow us to map the phase diagram of

\mathbb{Z}_2

spin liquids. The behavior of the system at the phase boundaries of

\mathbb{Z}_2

spin liquid is governed by the condensation of spinons and visions. We also present our preliminary findings regarding the

\mathbb{Z}_4

-invariant tensors with

SU(2)

\mathbb{Z}_2

\mathbb{Z}_2

spin liquid phase.

R=\sqrt{6}

at $\ensuremath{\lambda}=\ifmmode\pm\else\textpm\fi{}\sqrt{3}$ and

R=\sqrt{8/3}

at $\ensuremath{\lambda}=0$. From the complete transfer operator spectra, the finite-size spectra of the CFTs for the critical eight-vertex model are obtained, and the topological sectors of anyonic excitations are yielded as well. Furthermore, for the QCP at $\ensuremath{\lambda}=0$, no anyon condensation occurs, but the emerged symmetries of the matrix product operators significantly enrich the topological sectors of the CFT spectra. Finally, we provide our understanding of the (2+0)-dimensional conformal quantum criticalities and their possible connection with the generic (2+1)-dimensional CFTs for quantum topological phase transitions.

D=3

and two tunable parameters. We find that the time-reversal (TR) symmetric system exhibits three distinct phases: (i) an SET toric code phase in which anyons transform nontrivially under TR, (ii) a toric code phase in which TR does not fractionalize, and (iii) a topologically trivial phase that is adiabatically connected to a product state. We characterize the different phases using the topological entanglement entropy and a membrane order parameter that distinguishes the two SET phases. Along the phase boundary between the SET toric code phase and the toric code phase, the model has an enhanced

U(1)

symmetry and the ground state is a quantum critical loop gas wavefunction whose squared norm is equivalent to the partition function of the classical

O(2)

model. By duality transformations, this tensor-network solvable model can also be used to describe transitions between SET double-semion phases and between ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}^{T}$ symmetry protected topological phases in two dimensions.

G

, the so-called

G

-injective PEPS, and on global on-site symmetries coming from a finite group

Q

: as a result of the classification we have a finite number of phases closely related to the theory of group extensions. We also provide the maps that appear in group extension theory with physical meaning and we characterize their actions on the ground subspace and on the anyons. We construct a representative of each class of our classification. Moreover, we connect our construction with an interesting physical phenomenon, the so-called anyon condensation, which describes topological phase transitions. We have also proposed a family of gauge invariant quantities and their corresponding order parameters (in the form of expectation values of a local operator in 2D) to detect the corresponding quantum phases, in particular their symmetry fractionalization patterns. We conclude this thesis by touching on some mathematical open problems in PEPS.

{\mathbb{Z}}_{4}

symmetry, which contains

{\mathbb{Z}}_{4}

quantum double, toric code, double semion, and trivial phases. We find a diverse phase diagram which exhibits a variety of different phase transitions of both first and second order which we comprehensively characterize, including direct transitions between the toric code and the double semion phase.