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Emergent Photons and New Transitions in the O(3) Sigma Model with Hedgehog Suppression

Olexei I. Motrunich, Ashvin Vishwanath

TL;DR

The paper shows that suppressing hedgehog defects in the 2+1D $O(3)$ sigma model yields a magnetically disordered but topologically nontrivial phase with emergent spinons and a gapless photon, described by the noncompact $CP^1$ model. Through Monte Carlo simulations and duality arguments, it identifies a continuous deconfined critical point distinct from the Heisenberg transition and demonstrates a self-dual easy-plane variant with robust plasma-like and XY-related finite-temperature behavior. The work provides solid numerical and analytical support for a fractionalized Coulomb phase in a bosonic system with SU(2) symmetry and outlines potential higher-D generalizations and experimental signatures, including thermal KT transitions and dipolar spin-chirality correlations.

Abstract

We study the effect of hedgehog suppression in the O(3) sigma model in D=2+1. We show via Monte Carlo simulations that the sigma model can be disordered while effectively forbidding these point topological defects. The resulting paramagnetic state has gauge charged matter with half-integer spin (spinons) and also an emergent gauge field (photons), whose existence is explicitly demonstrated. Hence, this is an explicit realization of fractionalization in a model with global SU(2) symmetry. The zero temperature ordering transition from this phase is found to be continuous but distinct from the regular Heisenberg ordering transition. We propose that these phases and this phase transition are captured by the {\it noncompact} $CP^1$ model, which contains a pair of bosonic fields coupled to a noncompact U(1) gauge field. Direct simulation of the transition in this model yields critical exponents that support this claim. The easy-plane limit of this model also displays a continuous zero temperature ordering transition, which has the remarkable property of being self-dual. The presence of emergent gauge charge and hence Coulomb interactions is evidenced by the presence of a finite temperature Kosterlitz-Thouless transition associated with the thermal ionization of the gauge charged spinons. Generalization to higher dimensions and the effects of nonzero hedgehog fugacity are discussed.

Emergent Photons and New Transitions in the O(3) Sigma Model with Hedgehog Suppression

TL;DR

The paper shows that suppressing hedgehog defects in the 2+1D sigma model yields a magnetically disordered but topologically nontrivial phase with emergent spinons and a gapless photon, described by the noncompact model. Through Monte Carlo simulations and duality arguments, it identifies a continuous deconfined critical point distinct from the Heisenberg transition and demonstrates a self-dual easy-plane variant with robust plasma-like and XY-related finite-temperature behavior. The work provides solid numerical and analytical support for a fractionalized Coulomb phase in a bosonic system with SU(2) symmetry and outlines potential higher-D generalizations and experimental signatures, including thermal KT transitions and dipolar spin-chirality correlations.

Abstract

We study the effect of hedgehog suppression in the O(3) sigma model in D=2+1. We show via Monte Carlo simulations that the sigma model can be disordered while effectively forbidding these point topological defects. The resulting paramagnetic state has gauge charged matter with half-integer spin (spinons) and also an emergent gauge field (photons), whose existence is explicitly demonstrated. Hence, this is an explicit realization of fractionalization in a model with global SU(2) symmetry. The zero temperature ordering transition from this phase is found to be continuous but distinct from the regular Heisenberg ordering transition. We propose that these phases and this phase transition are captured by the {\it noncompact} model, which contains a pair of bosonic fields coupled to a noncompact U(1) gauge field. Direct simulation of the transition in this model yields critical exponents that support this claim. The easy-plane limit of this model also displays a continuous zero temperature ordering transition, which has the remarkable property of being self-dual. The presence of emergent gauge charge and hence Coulomb interactions is evidenced by the presence of a finite temperature Kosterlitz-Thouless transition associated with the thermal ionization of the gauge charged spinons. Generalization to higher dimensions and the effects of nonzero hedgehog fugacity are discussed.

Paper Structure

This paper contains 13 sections, 27 equations, 8 figures.

Table of Contents

  1. Introduction
  2. The Hedgehog Free O(3) Sigma Model
  3. The Model
  4. The Disordered Phase
  5. Emergent Photons in $P^*$
  6. The Transition
  7. Deformations of the NC$CP^1$ Model. Easy Plane, Zeeman Field and Effects of Finite Temperature.
  8. Easy Plane Deformation
  9. Effect of Zeeman Field
  10. Finite Temperatures
  11. Towards D=3+1 and Layered Phases
  12. Conclusions
  13. Acknowledgements

Figures (8)

  • Figure 1: a) Decorated cubic lattice used in the simulations. Spins live on the lattice points shown, and the monopole number is defined within each cube. b) The only spin configurations accepted in the simulation are those that are either hedgehog free, or have hedgehogs that can be paired uniquely into isolated nearest neighbor hedgehog-antihedgehog pairs. A schematic depiction of such a pairing is shown here in a vertical section through an isolated pair.
  • Figure 2: Magnetization per spin, $m = \langle |{\bf M}| \rangle / N_{\rm spin}$, with ${\bf M} = \sum_i {\bf n}_i$, as a function of $J$ for different system sizes (we show the data for both sweep directions). Inset shows the product $m N_{\rm spin}^{1/2}$; in the magnetically disordered phase, we expect the measured $\langle |{\bf M}| \rangle \sim N_{\rm spin}^{1/2}$ (for completely uncorrelated spins, the numerical coefficient is close to $1$)
  • Figure 3: Spin-spin correlations for $J = 0.0$, measured for spins at the vertices of the cubic lattice separated by a distance $r$ along the $\hat{z}$ direction. The system size is $L=16$, so the measurements are done for $r\leq 8$. Note the logarithmic scale for the vertical axis; the lower cutoff is roughly the limit of what can be reliably measured in our Monte Carlo.
  • Figure 4: Chirality-chirality (flux) correlations measured along the $\hat{z}$ direction for the same system as in Fig. \ref{['spincorr']}. Note the logarithmic scale for both axes. We also show a $\sim 1/z^3$ line to indicate the observed power law falloff.
  • Figure 5: Finite-size scaling plots for the cumulant ratio (left vertical axis) and magnetization (right axis), corresponding to the scaling forms (\ref{['gscale']}) and (\ref{['mscale']}). We used $J_c=0.725$, $\nu=1.0$, and $\beta/\nu = 0.80$; the range of the horizontal axis corresponds roughly to $J \in [0.40, 1.05]$ for $L=8$ (compare with Fig. \ref{['magn']}).
  • ...and 3 more figures