Topological Effects in Neural Network Field Theory

Christian Ferko; James Halverson; Vishnu Jejjala; Brandon Robinson

hep-thApr 2, 2026

Topological Effects in Neural Network Field Theory

Authors: Christian Ferko, James Halverson, Vishnu Jejjala, Brandon Robinson

Organizations: The NSF Institute for Artificial Intelligence and Fundamental Interactions (IAIFI) and Department of Physics, Northeastern University, Boston, MA 02115, USA · Mandelstam Institute for Theoretical Physics, School of Physics, and NITheCS, University of the Witwatersrand, Johannesburg, WITS 2050, South Africa · Institute of Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, Netherlands

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.

Figures & tables

Figure 1: Spin-wave correlator

C(r)

below

T_{c}

(log-log). Solid lines: data. Dashed lines: predicted power law

r^{-b^{2}}

$b$	$\eta_{\text{exp}}=b^{2}$	$\eta_{\text{fit}}$	$R^{2}$
0.050	0.00250	0.00252	0.99196
0.089	0.00791	0.00796	0.99196
0.158	0.02500	0.02517	0.99196
0.281	0.07906	0.07963	0.99198
0.500	0.25000	0.25204	0.99204

Table 1: Below-

T_{c}

power law exponents.

\eta_{\text{fit}}

agrees with

b^{2}

to better than

1\%

throughout, confirming ( 44 ).

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 ( 48 ).

R^{2}

values of the per-point fit ( 48 ) are annotated. Dashed curve: BKT essential singularity fit ( 49 ) (

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.

Figure 6: Vortex–antivortex pair correlation

g_{+-}(r)

. At large

T

b\to\infty

and

g_{+-}=1

for all

r

, since the unbinding makes it an uncorrelated Coulomb gas (dashed line). At lower temperatures, we expect excess (deficit) probability at low (large)

r

, which is reflected in the indigo (

b=0.60

) and blue (

b=0.75

) curves, due to the logarithmic potential in

P_{\text{vort}}

. The noisy statistics is due to a paucity of vortices.

Figure 7: Renormalized stiffness

K_{R}

vs.

b

. Gray dashed: bare coupling

K_{0}=1/(2\pi b^{2})

. Blue dotted: universal value

2/\pi

. Red dashed:

b_{c}=1/2

. Below

b_{c}

K_{R}\approx K_{0}

; above

b_{c}

, vortex screening drives

K_{R}

sharply downward.

Figure 8: Helicity modulus

\Upsilon_{R}=2\pi b^{2}K_{R}

vs. temperature

T=2\pi b^{2}

. Blue dashed: Nelson–Kosterlitz line

\Upsilon_{R}=2T/\pi

. Red dashed:

T_{c}=\pi/2

. The data cross the NK line near

T_{c}

as predicted, and

\Upsilon_{R}

collapses toward zero in the disordered phase.

Observable	$b<b_{c}$ (ordered)	$b=b_{c}$ (critical)	$b>b_{c}$ (disordered)
$\eta_{\text{sw}}$	$b^{2}<1/4$	$1/4$	$b^{2}>1/4$
$G_{2}(r)$	$\sim r^{-b^{2}}$	$\sim r^{-1/4}$	$\sim r^{-b^{2}}e^{-r/\xi}$
$\xi$	$\infty$	$\infty$	finite, $\sim e^{c/\sqrt{b^{2}-b_{c}^{2}}}$
$\rho_{v}$	$\approx 0$	onset	$>0$ , increasing
$K_{R}$	$\approx K_{0}=1/(2\pi b^{2})$	$2/\pi$	$\ll 1/(2\pi b^{2})$
$\Upsilon_{R}$	above NK line	on NK line	below NK line

Table 2: Observable signatures across the BKT transition in NN-FT.

Test	NN-FT data	Exact statement	Numerical signal
Circle duality	Oscillator sampler + discrete $(n,w)$ labels + zero-mode	$R\leftrightarrow\alpha^{\prime}/R$ , $n\leftrightarrow w$ , $p_{L}$ fixed, $p_{R}\to-p_{R}$	Lattice weights, monodromies, and zero-mode slopes agree under duality

Buscher on $T^{2}$	Vector-valued oscillator covariance + compact lattice with $G,B$ dependence	Buscher map on $(G,B)$ , compact charges, and the dilaton invariant	Transformed weights and marginals match; oscillator covariance tracks $G^{-1}$ ; $e^{-2\Phi}\sqrt{\det G}$ is preserved

Vertex operators	Chiral finite-mode sampler with explicit $X_{L},X_{R}$ and zero-mode averaging	Neutrality rules and dual matching of momentum/winding correlators	Non-neutral channels are suppressed; dual two-point functions and fitted dimensions agree

Table 3: Summary of the T-duality tests realized in the NN-FT mixed ensemble.

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Thomas R. Harvey

The NSF AI Institute for Artificial Intelligence and Fundamental Interactions. Center for Theoretical Physics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA.

High-Energy Physics

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One aim of neural network field theory is to put a quantum or effective field theory on a computer, with the network ensemble itself as the theory. We ask how far that aim can be pushed for a function class regular enough to be computed with. Our main result is a no-go theorem with assumptions that hold for standard network architectures. We use it to separate four versions of neural network field theory, according to whether the defining object is the finite width ensemble or its infinite width limit, and whether the target we want to compute is a quantum or an effective field theory. Neither finite width interpretation is straightforwardly consistent. For finite width ensembles with finite variance at each point, the QFT interpretation fails reflection positivity, while the EFT interpretation establishes no scale separation by which the positivity violation can be placed outside its domain of validity. Of the two limit versions, one can be simulated in full and the other only in part, as only its smeared correlators are computable with a controlled error. As such, at the level of a controlled numerical computation, the QFT and EFT versions cannot be distinguished. One dimension escapes the obstruction, yet reflection positivity is shown to still fail there at every finite width for the cosine network. Two escapes from the theorem remain, giving up either finite variance at a point or exact rotation invariance, and we discuss both of these possibilities.

Abstract

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