The Topology of the Quantum Vacuum

Abstract

Topology in momentum space is the main characteristics of the ground states of a system at zero temperature, the quantum vacua. The gaplessness of fermions in bulk, on the surface or inside the vortex core is protected by topology, and is not sensitive to details of the microscopic physics (atomic or trans-Planckian). Irrespective of the deformation of the parameters of the microscopic theory, the energy spectrum of these fermions remains strictly gapless. This solves the main hierarchy problem in particle physics: for fermionic vacua with Fermi points the masses of elementary particles are naturally small. The quantum vacuum of Standard Model is one of the representatives of topological matter alongside with topological superfluids and superconductors, topological insulators and semi-metals, etc. There is a number of of topological invariants in momentum space of different dimensions. They determine universality classes of the topological matter and the type of the effective theory which emerges at low energy. In many cases they also give rise to emergent symmetries, including the effective Lorentz invariance, and emergent phenomena such as effective gauge and gravitational fields. The topological invariants in extended momentum and coordinate space determine the bulk-surface and bulkvortex correspondence. They connect the momentum space topology in bulk with the real space. These invariants determine the gapless fermions living on the surface of a system or in the core of topological defects (vortices, strings, domain walls, solitons, monopoles, etc.). The momentum space topology gives some lessons for quantum gravity. In effective gravity emerging at low energy, the collective variables are the tetrad field and spin connections, while the metric is the composite object of tetrad field. This suggests that the Einstein-Cartan-Sciama-Kibble theory with torsion field is more relevant. There are also several scenarios of Lorentz invariance violation governed by topology, including splitting of Fermi point and development of the Dirac points with quadratic and cubic spectrum. The latter leads to the natural emergence of the Hořava-Lifshitz gravity.

Topological Node-Line Semimetal and Dirac Semimetal State in AntiperovskiteCu3PdN

“…In the present work, based on first-principles calculations and effective model analysis, we demonstrate that the antiperovskite Cu 3 PdN is a new candidate for realizing NLS and "drumhead"-like surface flat bands [12,[39][40][41][42][43], which may open an important route to achieving high-temperature superconductivity [44][45][46]. Strong SOC will drive NLS in Cu 3 PdN into Dirac semimetal state with three pairs of Dirac points, leading to exotic surface Fermi arcs which can be observed on various surfaces of this material.…”

“…The protection of a nodal line can be inferred from the topological number of the form [39][40][41][42][43] …”

Model Hamiltonian and time reversal breaking topological phases of antiferromagnetic half-Heusler materials

“…Different Weyl points can be related to each other by symmetries: K 1 and K 2 (or K 3 and K 4 ) are related by Π (110) ≡ π (110) or T a 3 π (110) and therefore dubbed a 'Π pair', K 1 and K 3 (or K 2 and K 4 ) are related by Π (110) S and called a 'ΠS pair', and finally K 1 and K 4 (or K 2 and K 3 ) are related by S and named a 'S pair'. It is known [64][65][66] that a Weyl fermion can carry topological charge or chirality, which can be extracted from Chern number (CN) on a small spherical surface surrounding the Weyl point. Two Weyl points related by mirror symmetry have opposite Chern numbers while time reversal operation leaves Chern number of a Weyl point unchanged.…”

Multiple Lorentz Groups—A Toy Model for Superluminal Muon Neutrinos

“…This framework leads to a modified neutrino dispersion relation depending on the momentum of the neutrino.  The neutrino velocity can be modified by Fermi point splitting (for a recent review see [18]), which removes the degeneracy of zeros of the fermionic energy spectrum [19].  The neutrino dispersion relation can change because of environmental effects caused by fields that accumulate at the position of the Earth.…”