2006
DOI: 10.5488/cmp.9.2.283
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Homotopy in statistical physics

Abstract: In condensed matter physics and related areas, topological defects play important roles in phase transitions and critical phenomena. Homotopy theory facilitates the classification of such topological defects. After a pedagogic introduction to the mathematical methods involved in topology and homotopy theory, the role of the latter in a number of mainly low-dimensional statistical-mechanical systems is outlined. Some recent activities in this area are reviewed and some possible future directions are discussed.

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Cited by 30 publications
(20 citation statements)
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“…On the other hand, in the XY model in the square lattices, the deterministic calculations by using the higher-order tensor renormalization group (HOTRG) method [8] provided the computation of the leading Fisher zeros for the system sizes up to L = 128. Although the WL approach with energy space binning [9,10] reported the leading-zero computation performed for up to L = 200, the transition temperature estimate was T BKT ≈ 0.70, which deviated from the known value T BKT ≈ 0.89 [16][17][18]. In contrast, the HOTRG calculation [8] for the power-law leading-zero trajectory was consistent with the known value of the BKT transition temperature.…”
Section: Introductionmentioning
confidence: 94%
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“…On the other hand, in the XY model in the square lattices, the deterministic calculations by using the higher-order tensor renormalization group (HOTRG) method [8] provided the computation of the leading Fisher zeros for the system sizes up to L = 128. Although the WL approach with energy space binning [9,10] reported the leading-zero computation performed for up to L = 200, the transition temperature estimate was T BKT ≈ 0.70, which deviated from the known value T BKT ≈ 0.89 [16][17][18]. In contrast, the HOTRG calculation [8] for the power-law leading-zero trajectory was consistent with the known value of the BKT transition temperature.…”
Section: Introductionmentioning
confidence: 94%
“…On the other hand, the situations are very different in the XY model where the specific heat does not diverge [16]. The radius R ∼ L −d/2 decreases much faster than the imaginary part of the leading zero that is expected to scale as Im[β 1 ] ∼ [ln(bL)] −q withq = 1 + 1/ν [8].…”
Section: Uncertainty Of Finding the Leading Zero Under Stochastimentioning
confidence: 99%
“…The importance of homotopy theory accentuates on its applications of the low-diemnsion statisticalmechanical systems, singularities in liquid crystal, and phase transition in physics [13], while studying warped products and Differential topology approaches in mathematical physics effectively relevant in general relativity [11,[18][19][20]. Particularly, the space-time homology is one of the main apparatus for quantum gravity [17,25].…”
Section: Main Results With Their Motivationsmentioning
confidence: 99%
“…There are many applications of the singularity structure at liquid crystals, at statistical mechanics considering low dimensions, as well as at physical phase transitions ( [13]). additionally, the General relativity includes warped product manifolds as the kind of space-times.…”
Section: Conclusion Remarkmentioning
confidence: 99%
“…Topology is the appropriate mathematical tool for the study of shapes and spaces (without a notion of distance) which can be continuously deformed into each other, continuous deformations mean twisting and stretching but not tearing or puncturing. For example, a sphere is topologically equivalent to a cube and a square is topologically equivalent to a circle [6]. In general relativity, space-time exists as a manifold which opens the door to the use of topological concepts and methods.…”
Section: Introductionmentioning
confidence: 99%