Topological phenomena in tensor network states of quantum spin systems

Iqbal, Mohsin; Schuch, Norbert (Thesis advisor); We├čel, Stefan (Thesis advisor)

Aachen (2018)
Dissertation / PhD Thesis


The focus of our investigations in this thesis is quantum spin systems in two dimensions. We examine phase transitions between topologically distinct phases of quantum matter using the framework of tensor network states. We study the phenomena of anyon condensation and confinement in the context of tensor network states where these two notions offer a robust probe to characterize the universal features of quantum phase transitions. We do a comprehensive study of the numerical methods enabled by the tensor network formalism for the study of quantum phase transitions. We map out the phase diagram of certain exotic phases of quantum matter (namely the $D(\mathbb{Z}_4)$ 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.