Influence of topological constraints and topological excitations: Decomposition formulas for calculating homotopy groups of symmetry-broken phases

Higashikawa, Sho; Ueda, Masahito

doi:10.1103/physrevb.95.134520

Cited by 3 publications

(4 citation statements)

References 88 publications

(209 reference statements)

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“…Note that H denotes the equivalence of path homotopy between two homotopic paths in the topological space. 1]) and β G : G → G 1 is a bijection in (Y, τ Y ) then there are oriented homotopy paths in p 0 :…”

Section: Proofmentioning

confidence: 99%

“…The symmetry breaking in any system involves a wide variety of topological excitations and it contains the associated topological constraints. The preparations of homotopy groups and related decompositions provide meaningful insights into the phases of a system after the symmetry is broken [1]. The homotopy decomposition in the classifying space is constructed considering that the space contains torsion-free groups [2].…”

Section: Introductionmentioning

confidence: 99%

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Analysis of Homotopy Decomposition Varieties in Quotient Topological Spaces

Bagchi

2020

Symmetry

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The fundamental groups and homotopy decompositions of algebraic topology have applications in systems involving symmetry breaking with topological excitations. The main aim of this paper is to analyze the properties of homotopy decompositions in quotient topological spaces depending on the connectedness of the space and the fundamental groups. This paper presents constructions and analysis of two varieties of homotopy decompositions depending on the variations in topological connectedness of decomposed subspaces. The proposed homotopy decomposition considers connected fundamental groups, where the homotopy equivalences are relaxed and the homeomorphisms between the fundamental groups are maintained. It is considered that one fundamental group is strictly homotopy equivalent to a set of 1-spheres on a plane and as a result it is homotopy rigid. The other fundamental group is topologically homeomorphic to the first one within the connected space and it is not homotopy rigid. The homotopy decompositions are analyzed in quotient topological spaces, where the base space and the quotient space are separable topological spaces. In specific cases, the decomposed quotient space symmetrically extends Sierpinski space with respect to origin. The connectedness of fundamental groups in the topological space is maintained by open curve embeddings without enforcing the conditions of homotopy classes on it. The extended decomposed quotient topological space preserves the trivial group structure of Sierpinski space.

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Section: Proofmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Analysis of Homotopy Decomposition Varieties in Quotient Topological Spaces

Bagchi

2020

Symmetry

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“…The number of times the m −dimensional sphere (S m ) wraps around itself is the integer topological number that defines the topological charge of the defect [14,30,31]. Importantly, since the group of integers  is an additive group [6], the homotopy groups π m (S m ) are necessarily Abelian [14,32], i.e., the rule for combining π m (S m ) defect elements is that their topological numbers add.…”

Section: Su(2) Systemsmentioning

confidence: 99%

SU(2) orientational ordering in restricted dimensions: evidence for a Berezinskiĭ-Kosterlitz-Thouless transition of topological point defects in four dimensions

Gorham

Laughlin

2018

J. Phys. Commun.

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We employ the theory of topological phase transitions, of the Berezinskiĭ-Kosterlitz-Thouless (BKT) type, in order to investigate orientational ordering in four spatial dimensions that is characterized by a quaternion n−vector (i.e., n=4) order parameter. Due to the dimensionality of the quaternion n −vector order parameter, the development of orientational order for systems that exist in four spatial dimensions must be viewed within the context of ordering in restricted dimensions. At finite temperatures, despite the development of a well-defined amplitude of the n−vector order parameter within separate regions, ordered systems that exist in restricted dimensions are prevented from developing global orientational phase-coherence as a consequence of misorientational fluctuations throughout the system. In four dimensions, this gas of misorientational fluctuations takes the form of spontaneously generated topological point defects that belong to the third homotopy group. These topological point defects must become topologically ordered in order to obtain a ground state of aligned order parameters at zero Kelvin; we argue that this topological transition belongs to the BKT universality class. We use standard 'Metropolis' Monte Carlo simulations to estimate the thermodynamic response functions, susceptibility and heat capacity, in the vicinity of the transition towards the ground state of perfectly aligned order parameters in our four-dimensional quaternion n−vector ordered model system. On lowering the temperature below a critical value, we identify a transition that results from the minimization of misorientations in the scalar phase angles throughout the system. The thermodynamic response functions obtained by Monte Carlo simulations show characteristic behavior of a topological ordering phase transition.

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“…Topological complexity is either a natural product of physical evolution (as in quantum turbulence [27]) or an artifact for possible applications [28]. Relationships between topological properties of defects and existence and stability of such structures is currently an area of intense, active research [29][30][31][32]. New evidence based on the Hermiticity properties of the governing Hamiltonian [33,34], for instance, shows that topological properties of these systems may influence stability.…”

Section: Introductionmentioning

confidence: 99%

Zero helicity of Seifert framed defects

Sumners¹,

Cruz-White²,

Ricca³

2021

J. Phys. A: Math. Theor.

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Line defects are one-dimensional phase singularities (forming knots and links) that arise in a variety of physical systems. In these systems, isophase surfaces (Seifert surfaces) have the phase defects as boundary, and these Seifert surfaces define a framing of the normal bundle of each link component. We define the individual helicity for each component of a link singularity, and prove that each individual helicity is zero if and only if there exists a Seifert framing for the link. We extend these results to multi-armed defects. We prove that under anti-parallel reconnection of defect strands total twist is conserved.

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Influence of topological constraints and topological excitations: Decomposition formulas for calculating homotopy groups of symmetry-broken phases

Cited by 3 publications

References 88 publications

Analysis of Homotopy Decomposition Varieties in Quotient Topological Spaces

Analysis of Homotopy Decomposition Varieties in Quotient Topological Spaces

SU(2) orientational ordering in restricted dimensions: evidence for a Berezinskiĭ-Kosterlitz-Thouless transition of topological point defects in four dimensions

Zero helicity of Seifert framed defects

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