Fermionic Neural Network Field Theories

• Fermionic neural network field theories are defined by architectures with Grassmann-valued weights that satisfy fermionic anticommutation relations.
• They leverage the infinite-width limit to map neural outputs to fermionic QFT path integrals, reproducing key propagators like the Dirac propagator.
• Applications span quantum simulations, lattice field theory, and variational ansätze for gauge-fermion interactions with four-fermion and Yukawa couplings.

Fermionic neural network field theories constitute an overview of deep learning architectures, Grassmann variables, and quantum field theory (QFT) concepts, enabling the direct realization of fermionic field dynamics within neural network ensembles. Unlike their bosonic counterparts, which emerge from real-valued architectures, fermionic theories require intrinsic handling of anticommutation relations and Grassmann integration, yielding models capable of encoding not only free fermion dynamics but also richer phenomena such as four-fermion interactions, Yukawa couplings, and supersymmetric extensions. This framework rigorously relates infinite-width complex-valued neural networks with appropriately tensorized or Grassmann-valued weights to the functional integrals of free and interacting fermionic QFTs, and is finding application in quantum simulation methodologies, lattice field theory, and variational ansätze for strongly correlated systems (Huang et al., 7 Jul 2025, Frank et al., 20 Nov 2025, Albergo et al., 2021, Chen et al., 2022).

1. Neural Architectures and Mapping to Free Fermionic QFT

The construction of fermionic neural network field theories relies on complex-valued networks whose output weights are promoted to Clifford algebra or Grassmann-valued tensors. For a single-hidden-layer complex neural network with hidden width $H$, the network output at position $x\in\mathbb{R}^D$ is

$$f(Q,V\mid x) = \frac{1}{H}\sum_{h=1}^H \lambda_h(Q_h,V_h\mid x)\varphi_h$$

where $Q_h \in \mathbb{C}^D$, $V_h\in\mathbb{C}$, $\varphi_h\in\mathbb{C}$, and $\lambda_h(x)$ is a suitable exponential of network parameters and inputs. Introducing Clifford-valued output weights $\varphi_h \rightarrow \varphi_h\gamma_h$ with $\gamma_h\in \text{Cliff}(H)$ satisfying $\{\gamma_h,\gamma_{h'}\}=2\delta_{hh'}\mathbf{1}$ ensures that the emergent field variables inherit the requisite anticommutation relations for fermionic statistics (Huang et al., 7 Jul 2025).

In the infinite-width limit ($H\to\infty$), these constructions map the network output to a Grassmann-valued field

$$\psi(x) = \int_0^1 d\xi\;\lambda(Q(\xi),V(\xi)\mid x)\;\varphi(\xi)\gamma(\xi)$$

where the integral limits capture the continuum analog of the hidden layer index. This mapping directly yields, at the level of generating functionals and correlation functions, the partition function and propagators of free fermionic QFT: $$Z[\bar\eta,\eta] = \int D\psi\,D\bar\psi\,\exp\left\{i\int d^Dx\,\bar\psi(x)[i\gamma^\mu\partial_\mu - m]\psi(x) + i\int d^Dx\, [\bar\eta\psi + \bar\psi\eta]\right\}$$ The two-point function in this construction exactly reproduces the Dirac propagator in $D$ dimensions (Huang et al., 7 Jul 2025, Frank et al., 20 Nov 2025).

2. Grassmann Central Limit and Statistical Mechanics

The foundation of fermionic neural network field theories lies in a generalization of the Central Limit Theorem (CLT) to sums of independent, identically distributed Grassmann-valued random variables. For a $d$-component Grassmann vector built from $N$ i.i.d. variables,

$\psi_j = \frac{1}{\sqrt{N}}\sum_{i=1}^N X_{ij}$

higher connected moments vanish as $N^{1-r/2}$ for $r>2$, ensuring that in the infinite-width (large-$N$) limit, only the second cumulant (covariance) survives. Physically, this induces a “Grassmann-Gaussian process” whose correlation functions are those of a free fermionic field theory (Frank et al., 20 Nov 2025).

At finite width, the leading corrections arise from the four-point connected correlator, scaling as $1/N$. This corresponds, in the effective action, to a local four-fermion operator of the form $(\bar\Psi\Psi)^2$ with coupling $\lambda\propto \sigma^4/N$, where $\sigma^2$ is the Grassmann weight variance. Thus, finite-width neural architectures naturally produce weakly coupled four-fermion interactions, concretely linking finite neural resources to perturbative QFT effects (Frank et al., 20 Nov 2025, Huang et al., 7 Jul 2025).

3. Interactions: Yukawa Couplings and Supersymmetry

Interactions in these frameworks can be generated by statistical correlations between weights governing bosonic and fermionic fields. For Yukawa couplings, introducing nontrivial joint distributions among the neural weights of scalar and fermionic outputs yields a classical action term

$$S_Y = g\int d^dx\, (\bar\Psi_L\,\Phi\,\Psi_R + \bar\Psi_R\,\Phi^\dagger\,\Psi_L)$$

where $g$ is the hyperparameter controlling the deformation of the joint parameter density. The result is a neural implementation of fermion-scalar interactions analogous to those in standard quantum field theory (Frank et al., 20 Nov 2025, Huang et al., 7 Jul 2025).

Supersymmetric quantum mechanics and field theories are constructed by promoting the network input to a superspace, with parameters subject to super-affine transformations. Taking the input as $(\tau,\theta,\bar\theta)$ in 1D or $(x^m,\theta^a,\tilde\theta_{\dot a})$ in 4D, invariance under supertranslations is enforced by adopting uniform, Berezinian-unit measures over (super-)parameters. Neural network correlators constructed in this way maintain exact supersymmetry, and at infinite width the resulting fields are super-Gaussian processes. At finite width, the theory acquires nontrivial supersymmetric interactions (Frank et al., 20 Nov 2025).

4. Lattice Field Implementation and Flow-Based Sampling

Practical realization of fermionic field theories on the lattice leverages the connection between neural samplers and field-theoretic path integrals. The action for a lattice theory with bosonic field $\phi$ and Grassmann-valued fermions $\psi,\bar\psi$ is

$$S[\phi,\psi,\bar\psi] = S_B(\phi) + \bar\psi D(\phi)\psi$$

with $D(\phi)$ the Dirac operator. Integrating out Grassmann variables yields the fermion determinant, so observables require sampling from $p(\phi) \propto e^{-S_B(\phi)}\det D(\phi)$.

Normalizing flows—deep invertible neural networks—map a latent Gaussian variable $z$ to field configuration $\phi = f_\theta(z)$, with density corrections via the Jacobian determinant of the flow. Training these flows via reverse Kullback–Leibler divergence against the lattice target measure produces efficient samplers for field configurations, exact up to a Metropolis–Hastings step. For fermionic systems, the pseudofermion trick enables stochastic estimation of the fermion determinant for large lattices by introducing auxiliary complex fields. This approach has been validated on two-dimensional models with staggered fermions and Yukawa interactions, achieving high acceptance rates and observables in agreement with traditional HMC baselines (Albergo et al., 2021).

5. Neural Variational Wavefunctions and Gauge-Fermion Coupling

Neural flow-based variational wavefunctions have been constructed for compact lattice gauge theories with dynamical fermions. In the Gauge–Fermion FlowNet (GFFN) architecture, the gauge field amplitude is parameterized by a discretized normalizing flow, and the fermionic sign structure is represented by a neural net backflow of a Slater determinant. The variational ansatz is

$$\Psi_{\theta,\phi}(x) = \sqrt{p_{\theta_A}(x)}\,\exp(i\phi_{\theta_B}(x))$$

where $p_{\theta_A}(x)$ models the amplitude autoregressively and $\phi_{\theta_B}(x)$ ensures exact fermionic antisymmetry.

Sampling enforces Gauss’s law at each vertex by assigning conditional probabilities only to independent link variables, yielding uncorrelated draws from $|\Psi|^2$. Variational energies and gradients are computed over these samples. This architecture produces physically accurate results in 2+1D lattice QED, including string breaking, confinement, charge crystal transitions, magnetic flux phases, and finite density phenomena, all while respecting gauge invariance and the fermionic sign structure (Chen et al., 2022).

6. Extensions, Applications, and Outlook

Neural network field theories with fermionic content are extendable to lattice field simulations of the Standard Model, condensed matter systems, and supersymmetric models. Diagrammatic expansions in $1/H$ or $1/N$ provide Feynman rules for finite-width corrections, with Wilsonian renormalization accessible via explicit lattice cutoffs. Neural architectures can produce Wilson, staggered, or even interacting fermion discretizations via appropriate tensorizations and weight-sharing schemes.

Normalizing-flow samplers combined with pseudofermion methods provide scalable, asymptotically exact sampling for lattice theories with dynamical fermions, offering a viable alternative to traditional MCMC. The GFFN paradigm demonstrates efficient, sign-free sampling, built-in gauge invariance, and direct applicability to high-dimensional fermion-gauge systems. Generalizations to nonabelian gauge groups, higher dimensions, and richer fermion representations (e.g., via Pfaffian backflow or gauge-equivariant flows) are active directions, albeit with increasing architectural and computational complexity (Chen et al., 2022, Albergo et al., 2021).

Finite-width and superspace constructions naturally induce interacting and supersymmetric field theories, opening avenues for the simulation of Gross–Neveu, Thirring, and a broad class of supersymmetric models via parameter prescription and neural architecture design (Frank et al., 20 Nov 2025, Huang et al., 7 Jul 2025).

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References:

• (Huang et al., 7 Jul 2025) The Neural Networks with Tensor Weights and the Corresponding Fermionic Quantum Field Theory (2025)
• (Frank et al., 20 Nov 2025) Fermions and Supersymmetry in Neural Network Field Theories (2025)
• (Albergo et al., 2021) Flow-based sampling for fermionic lattice field theories (2021)
• (Chen et al., 2022) Simulating 2+1D Lattice Quantum Electrodynamics at Finite Density with Neural Flow Wavefunctions (2022)