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.

Logarithmic finite-size scaling correction to the leading Fisher zeros in the p -state clock model: A higher-order tensor renormalization group study

“…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.…”

“…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].…”

Applications of stable currents and homology groups in CR-warped products of complex space forms

“…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].…”

“…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.…”

Existence of deformations and dimensional reduction in wormholes and black holes

“…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.…”