Gaussian Fermionic PEPS (GFPEPS)

• GFPEPS are tensor network states defined by local tensors and virtual Majorana modes that efficiently represent fermionic Gaussian states in quantum many-body systems.
• They offer a sign-problem-free approach for simulating both free-fermion systems and interacting phases, enabling robust variational Monte Carlo algorithms.
• Their gauged extensions (GGFPEPS) incorporate local gauge symmetries via controlled unitaries and additional gauge fields, streamlining non-perturbative lattice gauge theory studies.

Gaussian Fermionic Projected Entangled Pair States (GFPEPS) are a class of tensor network states tailored for simulating quantum many-body systems with fermionic degrees of freedom. Defined by local tensors that are constrained to generate fermionic Gaussian states, GFPEPS provide an efficient, sign-problem-free representation for both free-fermion and certain interacting gapped and gapless phases in two and higher spatial dimensions. Their gauged extensions (GGFPEPS) further enable sign-free, variational Monte Carlo simulations of lattice gauge theories with dynamical fermions coupled to compact gauge groups, providing a numerically tractable ansatz for non-perturbative regimes of lattice gauge theories in two and three dimensions (Gomelski et al., 15 Dec 2025, Kelman et al., 2024, Emonts et al., 2022, Roose et al., 2024).

1. Formal Definition and Covariance Structure

A GFPEPS is constructed on a lattice (typically d-dimensional), where each site carries physical fermionic modes $\psi_m(x)$ and 4 "legs" of virtual Majorana modes $a_{i\mu m}(x)$, with $i$ labeling the leg (e.g., corresponding to directions on the lattice), and $\mu, m$ labeling internal indices such as flavor or color. The local site operator is a fermionic Gaussian creator,

$$A(x) = \exp\Big[ T_{\alpha\beta} \Psi^\dagger_\alpha(x) \Psi^\dagger_\beta(x) \Big],$$

where $\Psi^\dagger_\alpha$ collects all physical and virtual creation operators (Gomelski et al., 15 Dec 2025).

Neighboring sites are entangled by applying Gaussian projectors $w(x,k)$ on each link, forming maximally entangled bond states between the virtual degrees of freedom (Mortier et al., 2020, Li et al., 2022). The global GFPEPS state is represented as

$$|\Psi\rangle = \langle \Omega_v | \prod_{x,i} w^\dagger(x,i) \prod_x A(x) | \Omega_p \rangle.$$

All physical properties of the GFPEPS, as a pure Gaussian state, are encoded in its antisymmetric covariance matrix,

$$\Gamma_{ij} = \frac{i}{2} \langle [\gamma_i, \gamma_j] \rangle,$$

where $\gamma_i$ are Majorana operators spanning all physical and virtual fermion spaces. For translationally invariant systems, covariance matrices reduce to momentum-space blocks, greatly facilitating analytic and numerical studies (Mortier et al., 2020, Emonts et al., 2023).

In practical terms, the full PEPS tensor reduces, in the Gaussian case, to three real matrices per site, typically denoted $a_{i\mu m}(x)$0, encoding the local Gaussian map, with the output covariance constructed via a Schur complement: $a_{i\mu m}(x)$1

2. Gauging Procedure and Extension to Lattice Gauge Theories

To construct gauged GFPEPS (GGFPEPS), a global symmetry group $a_{i\mu m}(x)$2 (such as $a_{i\mu m}(x)$3, $a_{i\mu m}(x)$4, or $a_{i\mu m}(x)$5) is promoted to a local gauge symmetry. On each lattice link, a Hilbert space for the gauge field is introduced, and controlled-unitary operators $a_{i\mu m}(x)$6 act to entangle the virtual fermions with the physical gauge-field registers. The total gauged state on a 2D lattice is written as

$a_{i\mu m}(x)$7

where $a_{i\mu m}(x)$8 is the bare gauge-field vacuum (Gomelski et al., 15 Dec 2025, Kelman et al., 2024).

The physical content of the GGFPEPS is found by integrating over all gauge configurations, decomposing the state as

$a_{i\mu m}(x)$9

where $i$0 is a pure fermionic Gaussian state for fixed gauge-field background $i$1. Expectation values of gauge-invariant observables reduce to Gaussian integrals over the covariance matrices, enabling the elimination of the sign problem inherent to conventional Monte Carlo algorithms for fermions (Emonts et al., 2022, Roose et al., 2024).

3. Variational Monte Carlo Framework and Algorithmic Aspects

The ground state of a gauge theory Hamiltonian is variationally optimized within the GGFPEPS manifold using Markov chain Monte Carlo over the gauge field configurations. The weight for each configuration $i$2 is the normalized squared norm of the (Gaussian) fermion wavefunction $i$3: $i$4 where $i$5 is the normalization. Local updates are proposed and accepted with Metropolis probability $i$6 (Gomelski et al., 15 Dec 2025).

Key computational bottlenecks—the evaluation of overlaps and observables—are polynomial in system size, as the Gaussian overlap (and hence the PEPS contraction) reduces to the computation of submatrices' Pfaffians or determinants. For local updates,

$i$7

and Woodbury/matrix-determinant lemma techniques bring the update cost to $i$8 per step for local changes (Gomelski et al., 15 Dec 2025, Emonts et al., 2022, Kelman et al., 2024).

Algorithmic studies find that updating $i$925–50% of the links at each Monte Carlo step yields optimal error convergence per wall clock time. Gauge fixing, while reducing the configuration space, can slow the convergence rate due to induced nonlocality in updates; partial “chessboard” gauge fixing avoids this problem, providing a favorable trade-off (Gomelski et al., 15 Dec 2025).

A summary of key algorithmic strategies is shown below:

Algorithmic Parameter	Optimal Choice/Effect	Reference
Update size	$\mu, m$025–50% links per step	(Gomelski et al., 15 Dec 2025)
Gauge fixing	Avoid maximal trees; chessboard/isolate	(Gomelski et al., 15 Dec 2025)
Averaging (energy terms)	Magnetic: average all; Electric: one	(Gomelski et al., 15 Dec 2025)
Cost per MC step	$\mu, m$1	(Gomelski et al., 15 Dec 2025Emonts et al., 2022)

4. Physical Realizations and Applications

GFPEPS and their