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Instanton constituents in the O(3) model at finite temperature

Falk Bruckmann

TL;DR

The paper shows that at finite temperature the instantons of the two dimensional $O(3)$ model dissociate into two static constituents carrying fractional topological charges governed by a holonomy parameter $\omega$. The solution is constructed analytically via a holonomy modified function $u(z;\omega)$ on a cylinder $z\sim z+i\beta$, yielding constituent masses $\omega/\beta$ and $(1-\omega)/\beta$ and a separation that grows with $\lambda$. In the limit $\beta\to\infty$ the constituents merge, recovering the usual unit charge instanton, while finite temperature reveals a caloron-like picture with fractional charges tied to boundary conditions. These results establish a concrete link between the $O(3)$ model and YM calorons, with implications for understanding finite temperature dynamics and potential fermion localization in related models such as CP$(N)$.

Abstract

It is shown that instantons in the O(3) model at finite temperature consist of fractional charge constituents and the (topological) properties of the latter are discussed.

Instanton constituents in the O(3) model at finite temperature

TL;DR

The paper shows that at finite temperature the instantons of the two dimensional model dissociate into two static constituents carrying fractional topological charges governed by a holonomy parameter . The solution is constructed analytically via a holonomy modified function on a cylinder , yielding constituent masses and and a separation that grows with . In the limit the constituents merge, recovering the usual unit charge instanton, while finite temperature reveals a caloron-like picture with fractional charges tied to boundary conditions. These results establish a concrete link between the model and YM calorons, with implications for understanding finite temperature dynamics and potential fermion localization in related models such as CP.

Abstract

It is shown that instantons in the O(3) model at finite temperature consist of fractional charge constituents and the (topological) properties of the latter are discussed.

Paper Structure

This paper contains 6 sections, 16 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The model and its solitons on $R^2$
  3. The case of finite temperature
  4. Constituent properties
  5. Topology
  6. Conclusions

Figures (3)

  • Figure 1: Logarithm of the topological density of solitons with different size and holonomy parameter (plugging (\ref{['eqn_cal_u']}) into (\ref{['eqn_inst_Qu']}) and cut off below $e^{-5}$). From left to right $\lambda=1,\,10,\,100$ (with locations growing like $\ln \lambda$ according to Eq. (\ref{['eqn_const_locs']})). From top to bottom the periodic case $\omega=0$ with one lump (the massless constituent being infinitely spread like for the Harrington-Shepard caloron), an intermediate case, $\omega=1/3$, and the antiperiodic case $\omega=1/2$ with identical constituents.
  • Figure 2: The image of the function $u_{\rm const}(z;\omega)$, Eq. (\ref{['eqn_const_1']}), of a single constituent covers a fraction $\omega$ of the complex plane.
  • Figure 3: The distribution of the southern and northern hemisphere for a large (top) and small (bottom) instanton and the contributions to the topological charge $Q$ from the oriented equator lines. To get the picture for the individual constituents, one simply cuts the upper plot vertically in the middle.