We propose a general formula for the group of invertible topological phases on a space $Y$ , possibly equipped with the action of a group $G$ . Our formula applies to arbitrary symmetry types. When $Y$ is Euclidean space and $G$ a crystallographic group, the term `topological crystalline phases' is sometimes used for these phases of matter.We propose a general formula for the group of invertible topological phases on a space $Y$, possibly equipped with the action of a group $G$. Our formula applies to arbitrary symmetry types. When $Y$ is Euclidean space and $G$ a crystallographic group, the term `topological crystalline phases' is sometimes used for these phases of matter.

arXiv:1901.06419Advances in Theoretical and Mathematical PhysicsJan 1, 202021 citations

Topological Matter

Dario Bercioux, J. Cayssol, Maia G. Vergniory +1 more

Springer series in solid-state sciencesJan 1, 201815 citations

INTERPLAY OF REAL SPACE AND MOMENTUM SPACE TOPOLOGIES IN STRONGLY CORRELATED FERMIONIC SYSTEMS

Григорий Ефимович Воловик

We discuss the momentum-space topology of 3+1 and 2+1 strongly correlated fermionic systems. For the 3+1 systems the important universality class is determined by the topologically stable Fermi points in momentum space. In the extreme limit of low energy the condensed matter system of this universality class acquires all the symmetries, which we know today in high energy physics: Lorentz invariance, gauge invariance, general covariance, etc. The chiral fermions as well as gauge bosons and gravity field arise as fermionic and bosonic collective modes of the system. This introduces the conceptual similarity between such condensed matter and the quantum vacuum of Standard Model, which also contains the Fermi point. For the 2+1 fermionic systems the momentum space topology gives rise to the quantization of different physical parameters, such as Hall conducivity and spin-Hall conductivity, and to quantum phase transitions with the abrupt change of the momentum-space topology. The combined p-space and r-space topology determines in particular the topology of the energy spectrum of fermion zero modes living within the topological defects; fermionic charges of the topologically nontrivial extended objects; axial anomaly; etc.We discuss the momentum-space topology of 3+1 and 2+1 strongly correlated fermionic systems. For the 3+1 systems the important universality class is determined by the topologically stable Fermi points in momentum space. In the extreme limit of low energy the condensed matter system of this universality class acquires all the symmetries, which we know today in high energy physics: Lorentz invariance, gauge invariance, general covariance, etc. The chiral fermions as well as gauge bosons and gravity field arise as fermionic and bosonic collective modes of the system. This introduces the conceptual similarity between such condensed matter and the quantum vacuum of Standard Model, which also contains the Fermi point. For the 2+1 fermionic systems the momentum space topology gives rise to the quantization of different physical parameters, such as Hall conducivity and spin-Hall conductivity, and to quantum phase transitions with the abrupt change of the momentum-space topology. The combined p-space and r-space topology determines in particular the topology of the energy spectrum of fermion zero modes living within the topological defects; fermionic charges of the topologically nontrivial extended objects; axial anomaly; etc.

arXiv:cond-mat/0012195Topology of Strongly Correlated SystemsAug 1, 20011 citation

Topological phases for extended objects: Semiclassical phase-space approach with tensorial coordinates

Giandomenico Palumbo

It has long been established that certain higher-dimensional topological phases of matter support extended objects like quasistrings and quasimembranes in their bulk states. In this study, we investigate the physics of these topological systems using a phase-space approach in the semiclassical regime, incorporating tensorial coordinates related to the extended objects. Specifically, we explore the semiclassical currents associated to open quasistrings in topological phases in three spatial dimensions, considering the presence of both coordinate-space Kalb-Ramond fields and momentum-space tensor Berry connections. Our results show that the currents associated to the endpoints of open quasistrings on the system’s boundary exhibit a quantum Hall response. This formalism accurately reproduces the boundary response as predicted by an Abelian BF theory proving a novel way to study extended objects in topological matter.It has long been established that certain higher-dimensional topological phases of matter support extended objects like quasistrings and quasimembranes in their bulk states. In this study, we investigate the physics of these topological systems using a phase-space approach in the semiclassical regime, incorporating tensorial coordinates related to the extended objects. Specifically, we explore the semiclassical currents associated to open quasistrings in topological phases in three spatial dimensions, considering the presence of both coordinate-space Kalb-Ramond fields and momentum-space tensor Berry connections. Our results show that the currents associated to the endpoints of open quasistrings on the system’s boundary exhibit a quantum Hall response. This formalism accurately reproduces the boundary response as predicted by an Abelian BF theory proving a novel way to study extended objects in topological matter.

arXiv:2406.16534Physical review. D/Physical review. D.Jan 13, 20253 citations

Relativity of topology and dynamics

David Finkelstein, Ernesto Rodriguez

International Journal of Theoretical PhysicsNov 1, 198414 citations

We propose a general formula for the group of invertible topological phases on a space

Y

, possibly equipped with the action of a group

G

. Our formula applies to arbitrary symmetry types. When

Y

is Euclidean space and

G

a crystallographic group, the term `topological crystalline phases' is sometimes used for these phases of matter.