A new class of (2+1)(2+1)-d topological superconductor with Z8\mathbb{Z}_8 topological classification

The classification of topological states of matter depends on spatial dimension and symmetry class. For non-interacting topological insulators and superconductors the topological classification is obtained systematically and nontrivial topological insulators are classified by either integer or $Z_2$. The classification of interacting topological states of matter is much more complicated and only special cases are understood. In this paper we study a new class of topological superconductors in $(2+1)$ dimensions which has time-reversal symmetry and a $\mathbb{Z}_2$ spin conservation symmetry. We demonstrate that the superconductors in this class is classified by $\mathbb{Z}_8$ when electron interaction is considered, while the classification is $\mathbb{Z}$ without interaction.Comment: 5 pages main text and 3 pages appendix. 1 figur

Topological States of Matter in Frustrated Quantum Magnetism

Frustrated quantum magnets may exhibit fascinating collective phenomena. The main goal of this dissertation is to provide conclusive evidence for the emergence of novel phases of matter like quantum spin liquids in local quantum spin models. We develop novel algorithms for large-scale Exact Diagonalization computations. Sublattice coding methods for efficient use of lattice symmetries in the procedure of diagonalizing the Hamiltonian matrix are proposed and suggest a randomized distributed memory parallelization strategy. Benchmarks of computations on various supercomputers with system size up to 50 spin-1/2 particles have been performed. Results concerning the emergence of a chiral spin liquid in a frustrated kagome Heisenberg antiferromagnet are presented. The stability and extent of this phase are discussed. In an extended Heisenberg model on the triangular lattice, we establish another chiral spin liquid phase. We discuss the special case of the Heisenberg $J_1$-$J_2$ model and present a scenario where the critical point of phase transition from the 120-degree N\'eel to a putative $\mathbf{Z}_2$ spin liquid is described by a Dirac spin liquid. A generalization of the SU(2) Heisenberg model with SU(N) degrees of freedom on the triangular lattice with an additional ring-exchange term is discussed. We present our contribution to the project and the final results that suggest a series of chiral spin liquid phases in an extended parameter range. Finally, we present preliminary data from a Quantum Monte Carlo study of an SU(N) version of the J-Q model on a square lattice for N=2,...,10, and multi-column representations. We establish the phase boundary between the N\'eel ordered phase and the disordered phases. The disordered phase in the four-column representation is expected to be a two-dimensional analog of the Haldane phase for the spin-1 Heisenberg chain.Comment: Ph.D. thesis, 161 page

Topological states of non-Hermitian systems

Recently, the search for topological states of matter has turned to non-Hermitian systems, which exhibit a rich variety of unique properties without Hermitian counterparts. Lattices modeled through non-Hermitian Hamiltonians appear in the context of photonic systems, where one needs to account for gain and loss, circuits of resonators, and also when modeling the lifetime due to interactions in condensed matter systems. Here we provide a brief overview of this rapidly growing subject, the search for topological states and a bulk-boundary correspondence in non-Hermitian systems.Comment: Invited short review for the special issue "Topological States of Matter: Theory and Applications

A road to reality with topological superconductors

Topological states of matter are a source of low-energy quasiparticles, bound to a defect or propagating along the surface. In a superconductor these are Majorana fermions, described by a real rather than a complex wave function. The absence of complex phase factors promises protection against decoherence in quantum computations based on topological superconductivity. This is a tutorial style introduction written for a Nature Physics focus issue on topological matter.Comment: pre-copy-editing, author-produced version of the published paper: 4 pages, 2 figure

Symmetry-enriched topological states of matter in insulators and semimetals

Topological states of matter are a novel family of phases that elude the conventional Landau paradigm of phase transitions. Topological phases are characterized by global topological invariants which are typically reflected in the quantization of physical observables. Moreover, their characteristic bulk-boundary correspondence often gives rise to robust surface modes with exceptional features, such as dissipationless charge transport or non-Abelian statistics. In this way, the study of topological states of matter not only broadens our knowledge of matter but could potentially lead to a whole new range of technologies and applications. In this light, it is of great interest to find novel topological phases and to study their unique properties. In this work, novel manifestations of topological states of matter are studied as they arise when materials are subject to additional symmetries. It is demonstrated how symmetries can profoundly enrich the topology of a system. More specifically, it is shown how symmetries lead to additional nontrivial states in systems which are already topological, drive trivial systems into a topological phase, lead to the quantization of formerly non-quantized observables, and give rise to novel manifestations of topological surface states. In doing so, this work concentrates on weakly interacting systems that can theoretically be described in a single-particle picture. In particular, insulating and semi-metallic topological phases in one, two, and three dimensions are investigated theoretically using single-particle techniques