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Topological Effects in Neural Network Field Theory

Christian Ferko, James Halverson, Vishnu Jejjala, Brandon Robinson

Abstract

Neural network field theory formulates field theory as a statistical ensemble of fields defined by a network architecture and a density on its parameters. We extend the construction to topological settings via the inclusion of discrete parameters that label the topological quantum number. We recover the Berezinskii--Kosterlitz--Thouless transition, including the spin-wave critical line and the proliferation of vortices at high temperatures. We also verify the T-duality of the bosonic string, showing invariance under the exchange of momentum and winding on $S^1$, the transformation of the sigma model couplings according to the Buscher rules on constant toroidal backgrounds, the enhancement of the current algebra at self-dual radius, and non-geometric T-fold transition functions.

Topological Effects in Neural Network Field Theory

Abstract

Neural network field theory formulates field theory as a statistical ensemble of fields defined by a network architecture and a density on its parameters. We extend the construction to topological settings via the inclusion of discrete parameters that label the topological quantum number. We recover the Berezinskii--Kosterlitz--Thouless transition, including the spin-wave critical line and the proliferation of vortices at high temperatures. We also verify the T-duality of the bosonic string, showing invariance under the exchange of momentum and winding on , the transformation of the sigma model couplings according to the Buscher rules on constant toroidal backgrounds, the enhancement of the current algebra at self-dual radius, and non-geometric T-fold transition functions.

Paper Structure

This paper contains 48 sections, 139 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Neural network field theory
  3. From networks to random fields:
  4. Infinite width and the emergence of free fields:
  5. Engineering propagators via random feature architectures:
  6. Interactions as finite width effects and non-Gaussianities:
  7. Symmetries and targets:
  8. Summary:
  9. Berezinskii--Kosterlitz--Thouless transition
  10. NN-FT architecture
  11. Spin-wave sector and random Fourier features:
  12. Vortex sector --- the Coulomb gas:
  13. Analytic matching
  14. Numerical results
  15. Spin-wave correlator below $T_c$:
  16. ...and 33 more sections

Figures (8)

  • Figure 1: Spin-wave correlator $C(r)$ below $T_c$ (log-log). Solid lines: data. Dashed lines: predicted power law $r^{-b^2}$.
  • Figure 2: Spin-wave correlator above $T_c$ (log-log). The correlator remains power law because the RFF field is Gaussian by construction: the spin-wave sector alone has no disordered phase.
  • Figure 3: Correlation length $\xi$ extracted from the full two-point function via (\ref{['eq:G2_xi_fit']}). $R^2$ values of the per-point fit (\ref{['eq:G2_xi_fit']}) are annotated. Dashed curve: BKT essential singularity fit (\ref{['eq:xi_fit']}) ($R^2=0.98$).
  • Figure 4: Vortex density $\rho_v$ vs. $b$. The vertical dashed line marks $b_c=1/2$. Below $T_c$ the vortex density is consistent with zero; above $T_c$ it rises steeply as vortices unbind.
  • Figure 5: Vortex angle field $\theta_v(x)\;\text{mod}\;2\pi$ for nine values of $b$. Red (blue) markers denote vortices (antivortices). At small $b$ the field is featureless; as $b$ increases past $b_c=1/2$, vortex--antivortex pairs appear and eventually form a dense plasma.
  • ...and 3 more figures