*VOLOVIK GRIGORY E.*

- Published in print:
- 2009
- Published Online:
- January 2010
- ISBN:
- 9780199564842
- eISBN:
- 9780191709906
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199564842.003.0011
- Subject:
- Physics, Condensed Matter Physics / Materials, Particle Physics / Astrophysics / Cosmology

This chapter discusses the momentum space topology of 2+1 systems. In the D = 2 space the possible manifolds of gap nodes in the quasiparticle energy are point nodes and nodal lines. The nodal lines ...
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This chapter discusses the momentum space topology of 2+1 systems. In the D = 2 space the possible manifolds of gap nodes in the quasiparticle energy are point nodes and nodal lines. The nodal lines are described by the same invariant as Fermi surfaces, while point nodes are typically marginally stable: they may be topologically protected being described by the Z2 topological charge. The chapter focuses on topologically non-trivial fully gapped vacua — vacua with fully non-singular Green's function. The topological invariant for the gapped 2+1 systems is introduced either in terms of Hamiltonian (where the relevant topological object in momentum space is the p-space skyrmion) or in terms of Green's function (the invariant is obtained by dimensional reduction from the invariant describing the point nodes in 3+1 space). Examples are provided by p-wave and d-wave superfluids/superconductors. Topological quantum phase transitions are discussed at which the integer topological invariant changes abruptly. Topological transition occurs via the intermediate gapless state, and the process represents the diabolical point — analog of magnetic monopole — the termination point of Dirac string at which the Berry phase has singularity. The chapter also discusses broken time reversal symmetry, families (generations) of fermions in 2+1 systems, and Dirac vacuum as marginal state with fractional topological charge.Less

This chapter discusses the momentum space topology of 2+1 systems. In the *D* = 2 space the possible manifolds of gap nodes in the quasiparticle energy are point nodes and nodal lines. The nodal lines are described by the same invariant as Fermi surfaces, while point nodes are typically marginally stable: they may be topologically protected being described by the *Z*_{2} topological charge. The chapter focuses on topologically non-trivial fully gapped vacua — vacua with fully non-singular Green's function. The topological invariant for the gapped 2+1 systems is introduced either in terms of Hamiltonian (where the relevant topological object in momentum space is the p-space skyrmion) or in terms of Green's function (the invariant is obtained by dimensional reduction from the invariant describing the point nodes in 3+1 space). Examples are provided by p-wave and d-wave superfluids/superconductors. Topological quantum phase transitions are discussed at which the integer topological invariant changes abruptly. Topological transition occurs via the intermediate gapless state, and the process represents the diabolical point — analog of magnetic monopole — the termination point of Dirac string at which the Berry phase has singularity. The chapter also discusses broken time reversal symmetry, families (generations) of fermions in 2+1 systems, and Dirac vacuum as marginal state with fractional topological charge.

*Paula Tretkoff*

- Published in print:
- 2016
- Published Online:
- October 2017
- ISBN:
- 9780691144771
- eISBN:
- 9781400881253
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691144771.003.0002
- Subject:
- Mathematics, Geometry / Topology

This chapter deals with topological invariants and differential geometry. It first considers a topological space X for which singular homology and cohomology are defined, along with the Euler number ...
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This chapter deals with topological invariants and differential geometry. It first considers a topological space X for which singular homology and cohomology are defined, along with the Euler number e(X). The Euler number, also known as the Euler-Poincaré characteristic, is an important invariant of a topological space X. It generalizes the notion of the cardinality of a finite set. The chapter presents the simple formulas for computing the Euler-Poincaré characteristic (Euler number) of many of the spaces to be encountered throughout the book. It also discusses fundamental groups and covering spaces and some basics of the theory of complex manifolds and Hermitian metrics, including the concept of real manifold. Finally, it provides some general facts about divisors, line bundles, and the first Chern class on a complex manifold X.Less

This chapter deals with topological invariants and differential geometry. It first considers a topological space *X* for which singular homology and cohomology are defined, along with the Euler number *e*(*X*). The Euler number, also known as the Euler-Poincaré characteristic, is an important invariant of a topological space *X*. It generalizes the notion of the cardinality of a finite set. The chapter presents the simple formulas for computing the Euler-Poincaré characteristic (Euler number) of many of the spaces to be encountered throughout the book. It also discusses fundamental groups and covering spaces and some basics of the theory of complex manifolds and Hermitian metrics, including the concept of real manifold. Finally, it provides some general facts about divisors, line bundles, and the first Chern class on a complex manifold *X*.