Topics In Anyons And Quantum Spin Systems

Torbjörn Einarsson

Shiing-Shen Chern, C. W. Chu, C. S. Ting

Lucjan Jacak, Piotr Sitko, Konrad Wieczorek +1 more

Abstract The book presents the wide range of topics in two-dimensional physics of quantum Hall systems, especially fractional quantum Hall states. It starts with the fundamental problems of quantum statistics in two dimensions and the corresponding braid group formalism. The braid group formalism of anyons (previously known) is developed for composite fermions. The effective formalism used in many-body quantum Hall theories — the Chern–Simons theory is also presented. The Chern–Simons theory of anyons (particles obeying fractional statistics) and composite fermions (related to Hall systems) is given, in detail. Numerical studies, which play the important role in quantum Hall analyses, are presented for spherical systems (Haldane sphere). The composite fermion theory is tested in numerical studies. The concept of the hierarchy of condensed states of composite fermion excitations is introduced (in analogy to the Haldane hierarchy). The hierarchies of odd-denominator states and even-denominator states are presented. The BCS paired Hall state is also discussed. An introduction to multi-component quantum Hall systems and spin quantum Hall systems is sketched.

Michele Burrello

This thesis deals with the study of topological quantum computation and the possible realization of non-Abelian anyons in cold atomic gases. Two main topics are investigated: the first subject is the quantum hashing technique to approximate unitary operators by braiding non-Abelian anyons, the second one is the analysis of systems of multicomponent ultracold atoms in the presence of an effective non-Abelian gauge potential giving rise to a quantum Hall regime. The common frame of these topics is the emergent study of topological phases of matters, driven by the necessity to overcome the Landau-Ginzburg paradigm to describe strongly correlated quantum systems such as the quantum Hall ones. To achieve this goal it is crucial to involve seemingly distant branches of knowledge such as conformal field theories, topological field theories, integrable models, knot theory, tensor category theory but also quantum information and computation, in order to deepen our understanding of the new and exciting experimental and numerical results given by the analysis of different systems sharing these topological properties.

Alberto Lerda A, S. Sciuto

Kyusung Hwang

Transitions between quantum spin liquids (QSLs) are fundamental problems lying beyond the Landau paradigm and requiring a deep understanding of the entanglement structures of QSLs called topological orders. The novel concept of anyon condensation has been proposed as a theoretical mechanism, predicting various possible transitions between topological orders, but it has long been elusive to confirm the mechanism in quantum spin systems. Here, we introduce a concrete spin model that incarnates the mechanism of the anyon condensation transition. Our model harbors two topological QSLs in different parameter regions, a non-Abelian Kitaev spin liquid (KSL) bilayer state and a resonating valence bond (RVB) state. The bilayer-KSL-to-RVB transition indeed occurs by the mechanism of anyon condensation, which we identify by using parton theories and exact diagonalization studies. Moreover, we observe “anyon confinement” phenomena in our numerical results, akin to the quark confinement in high-energy physics. Namely, non-Abelian Ising anyons of the bilayer KSL are confined in the transition to the RVB state. Implications and extensions of this study are discussed in various aspects such as (i) anyon-condensed multilayer construction of Kitaev's 16-fold way of anyon theories, (ii) an additional vison condensation transition from the RVB to a valence bond solid in the Kitaev bilayer system, (iii) dynamical anyon condensation in a non-Hermitian Kitaev bilayer, (iv) generalizations of our model to other lattice geometries, and (v) experimental realizations. This work puts together the two fascinating QSLs that are extensively studied in modern condensed matter and quantum physics into a concrete spin model, offering a comprehensive picture that unifies the anyon physics of the Kitaev spin liquids and the resonating valence bonds. Published by the American Physical Society 2024