Quantum Mechanical Phases and Topological Numbers in Modern Physics

Topology plays an important role in modern solid state physics describing intriguing quantum states such as topological insulators. It is an intrinsically non-local property and therefore challenging to access, often studied only via the resulting edge states. Here, we measure the topological index directly from the far-from-equilibrium dynamics of the bulk. We use the mapping of the Chern number to the linking number of dynamical vortex trajectories appearing after a quench to the Hamiltonian of interest. We thereby map out the topological phase diagram of quantum gases in optical lattices via a purely dynamical response. Such relations between two topological indices in static and dynamical properties could be also an important approach for exploring topology in the case of interactions.Topology plays an important role in modern solid state physics describing intriguing quantum states such as topological insulators. It is an intrinsically non-local property and therefore challenging to access, often studied only via the resulting edge states. Here, we measure the topological index directly from the far-from-equilibrium dynamics of the bulk. We use the mapping of the Chern number to the linking number of dynamical vortex trajectories appearing after a quench to the Hamiltonian of interest. We thereby map out the topological phase diagram of quantum gases in optical lattices via a purely dynamical response. Such relations between two topological indices in static and dynamical properties could be also an important approach for exploring topology in the case of interactions.

Sep 4, 201714 citations

Topological Phases of Quantum Matter

Wei-Feng Tsai, Hsin Lin, Arun Bansil

Springer series in solid-state sciencesJan 1, 20181 citation

Quantum Phase Transitions

Mike Guidry, Yang Sun

Written for use in teaching and for self-study, this book provides a comprehensive and pedagogical introduction to groups, algebras, geometry, and topology. It assimilates modern applications of these concepts, assuming only an advanced undergraduate preparation in physics. It provides a balanced view of group theory, Lie algebras, and topological concepts, while emphasizing a broad range of modern applications such as Lorentz and Poincaré invariance, coherent states, quantum phase transitions, the quantum Hall effect, topological matter, and Chern numbers, among many others. An example based approach is adopted from the outset, and the book includes worked examples and informational boxes to illustrate and expand on key concepts. 344 homework problems are included, with full solutions available to instructors, and a subset of 172 of these problems have full solutions available to students.Written for use in teaching and for self-study, this book provides a comprehensive and pedagogical introduction to groups, algebras, geometry, and topology. It assimilates modern applications of these concepts, assuming only an advanced undergraduate preparation in physics. It provides a balanced view of group theory, Lie algebras, and topological concepts, while emphasizing a broad range of modern applications such as Lorentz and Poincaré invariance, coherent states, quantum phase transitions, the quantum Hall effect, topological matter, and Chern numbers, among many others. An example based approach is adopted from the outset, and the book includes worked examples and informational boxes to illustrate and expand on key concepts. 344 homework problems are included, with full solutions available to instructors, and a subset of 172 of these problems have full solutions available to students.

Cambridge University Press eBooksMar 17, 20220 citations

Topological Phases in the Quantum Theory.

Tom D. Imbo

Jan 1, 198839 citations

Fermionic fractional quantum Hall states: A modern approach to systems with bulk-edge correspondence

Yoshiki Fukusumi, Bo Yang

In contemporary physics, especially in condensed-matter physics, fermionic topological order and its protected edge modes is one of the most important objects. In this work, we propose a systematic construction of the cylinder partition corresponding to the fermionic fractional quantum Hall effect and a general mechanism for obtaining the candidates of the protected edge modes. In our construction, when the underlying conformal field theory has ${Z}_{2}$ duality defects corresponding to the fermionic ${Z}_{2}$ electric particle, we show that the fractional quantum Hall partition function has a fermionic T duality. This duality is analogous to (hopefully the same as) the dualities in the dual resonance models, typically known as supersymmetry, and it provides a renormalization-group (RG) theoretical understanding of the topological phases. We also introduce a modern understanding of bulk topological degeneracies and topological entanglement entropy. This understanding is based on the traditional tunnel problem and the recent conjecture of correspondence between bulk RG flow and boundary conformal field theory. Our formalism provides an intuitive and general understanding of the modern physics of topologically ordered systems in the traditional language of RG and fermionization, and it may serve as a complement to more mathematical physical frameworks such as fermionic category theories.In contemporary physics, especially in condensed-matter physics, fermionic topological order and its protected edge modes is one of the most important objects. In this work, we propose a systematic construction of the cylinder partition corresponding to the fermionic fractional quantum Hall effect and a general mechanism for obtaining the candidates of the protected edge modes. In our construction, when the underlying conformal field theory has ${Z}_{2}$ duality defects corresponding to the fermionic ${Z}_{2}$ electric particle, we show that the fractional quantum Hall partition function has a fermionic T duality. This duality is analogous to (hopefully the same as) the dualities in the dual resonance models, typically known as supersymmetry, and it provides a renormalization-group (RG) theoretical understanding of the topological phases. We also introduce a modern understanding of bulk topological degeneracies and topological entanglement entropy. This understanding is based on the traditional tunnel problem and the recent conjecture of correspondence between bulk RG flow and boundary conformal field theory. Our formalism provides an intuitive and general understanding of the modern physics of topologically ordered systems in the traditional language of RG and fermionization, and it may serve as a complement to more mathematical physical frameworks such as fermionic category theories.

arXiv:2212.12993Physical review. B./Physical review. BAug 17, 20234 citations

{Z}_{2}

duality defects corresponding to the fermionic

{Z}_{2}

electric particle, we show that the fractional quantum Hall partition function has a fermionic T duality. This duality is analogous to (hopefully the same as) the dualities in the dual resonance models, typically known as supersymmetry, and it provides a renormalization-group (RG) theoretical understanding of the topological phases. We also introduce a modern understanding of bulk topological degeneracies and topological entanglement entropy. This understanding is based on the traditional tunnel problem and the recent conjecture of correspondence between bulk RG flow and boundary conformal field theory. Our formalism provides an intuitive and general understanding of the modern physics of topologically ordered systems in the traditional language of RG and fermionization, and it may serve as a complement to more mathematical physical frameworks such as fermionic category theories.