Geometry, Topology and Physics

Nakahara, Mikio

doi:10.1887/0750306068

Cited by 1,033 publications

(1,164 citation statements)

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“…This has a structure reminiscent of a mathematical index theorem 54 that relates a topological index to an analytical index that counts the number of zero modes. 41,45,46,[55][56][57][58][59] In this paper we will not attempt to prove the index theorem.…”

Section: ͑11͒mentioning

confidence: 99%

Topological defects and gapless modes in insulators and superconductors

2010

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We develop a unified framework to classify topological defects in insulators and superconductors described by spatially modulated Bloch and Bogoliubov de Gennes Hamiltonians.We consider Hamiltonians H(k,r) that vary slowly with adiabatic parameters r surrounding the defect and belong to any of the ten symmetry classes defined by time-reversal symmetry and particle-hole symmetry. The topological classes for such defects are identified and explicit formulas for the topological invariants are presented. We introduce a generalization of the bulk-boundary correspondence that relates the topological classes to defect Hamiltonians to the presence of protected gapless modes at the defect. Many examples of line and point defects in three-dimensional systems will be discussed. These can host one dimensional chiral Dirac fermions, helical Dirac fermions, chiral Majorana fermions, and helical Majorana fermions, as well as zero-dimensional chiral and Majorana zero modes. This approach can also be used to classify temporal pumping cycles, such as the Thouless charge pump, as well as a fermion parity pump, which is related to the Ising non-Abelian statistics of defects that support Majorana zero modes. We develop a unified framework to classify topological defects in insulators and superconductors described by spatially modulated Bloch and Bogoliubov de Gennes Hamiltonians. We consider Hamiltonians H͑k , r͒ that vary slowly with adiabatic parameters r surrounding the defect and belong to any of the ten symmetry classes defined by time-reversal symmetry and particle-hole symmetry. The topological classes for such defects are identified and explicit formulas for the topological invariants are presented. We introduce a generalization of the bulk-boundary correspondence that relates the topological classes to defect Hamiltonians to the presence of protected gapless modes at the defect. Many examples of line and point defects in three-dimensional systems will be discussed. These can host one dimensional chiral Dirac fermions, helical Dirac fermions, chiral Majorana fermions, and helical Majorana fermions, as well as zero-dimensional chiral and Majorana zero modes. This approach can also be used to classify temporal pumping cycles, such as the Thouless charge pump, as well as a fermion parity pump, which is related to the Ising non-Abelian statistics of defects that support Majorana zero modes. Disciplines Physical Sciences and Mathematics | Physics

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Section: ͑11͒mentioning

confidence: 99%

Topological defects and gapless modes in insulators and superconductors

2010

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“…Note that the last equality follows from (18). Finally, H X is positive definite since it is a direct sum of positive definite Hamiltonians (see formula (15)).…”

Section: Proof Of Theorem 28 -Stepmentioning

confidence: 99%

“…Moreover α is a connection form on P such that ω is its curvature . That is dα = p * ω where p : P → M is the fiber bundle projection from P to M ( See [18] and [24] chapter 8 as a source on connections and curvature of principal bundles). This means that for every a ∈ P we have the smooth decomposition…”

Section: Proof Of Theorem 28 -Stepmentioning

confidence: 99%

The Geometry of Partial Order on Contact Transformations of Prequantization Manifolds

Simon

2010

Arithmetic and Geometry Around Quantization

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In this paper we find connection between the Hofer's metric of the group of Hamiltonian diffeomorphisms of a closed symplectic manifold, with an integral symplectic form, and the geometry, defined in [12], of the quantomorphisms group of its prequantization manifold. This gives two main results: First, we calculate, partly, the geometry of the quantomorphisms groups of a prequantization manifolds of an integral symplectic manifold which admits certain Lagrangian foliation. Second, for every prequantization manifold we give a formula for the distance between a point and a distinguished curve in the metric space associated to its group of quantomorphisms. Moreover, our first result is a full computation of the geometry related to the symplectic linear group which can be considered as a subgroup of the contactomorphisms group of suitable prequantization manifolds of the complex projective space. In the course of the proof we use in an essential way the Maslov quasimorphism.

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“…In (1 þ 1) dimensions spacetime is conformally flat for all metrics; that is, one can always choose a coordinate system ðt; xÞ such that the metric tensor takes the form g ¼ e 2 , where is a scalar potential and is the Minkowski metric tensor [19]. By working in a coordinate system in which the metric takes this form, we will show that the field equation (7) and the particle equation of motion (8) can be reinterpreted as describing a point particle coupled to a scalar field in flat spacetime.…”

Section: Flat Spacetime Interpretationmentioning

confidence: 99%

Nordström gravity coupled to point particles in (1+1) dimensions

Boozer

2010

Phys. Rev. D

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We consider a (1 þ 1)-dimensional model of general relativity that is based on a geometric theory of gravity due to Nordström. We show that the theory is formally equivalent to scalar field theory in flat spacetime, and we exploit this equivalence to solve the initial value problem for the system for the case of point particle sources. We illustrate our results by obtaining several example solutions to the model.

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Geometry, Topology and Physics

Cited by 1,033 publications

References 0 publications

Topological defects and gapless modes in insulators and superconductors

Topological defects and gapless modes in insulators and superconductors

The Geometry of Partial Order on Contact Transformations of Prequantization Manifolds

Nordström gravity coupled to point particles in (1+1) dimensions

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