Gauge-Invariant PEPS: Tensor Network Approach

Updated 14 April 2026

Gauge-invariant PEPS are tensor network states that incorporate local gauge symmetry constraints directly into vertex and edge tensors.
They are constructed by transforming globally symmetric PEPS using controlled entangling operations to impose local gauge invariance across various gauge groups.
These methods offer efficient variational ansätze that bypass sign problems and yield accurate benchmarks for ground state and correlation properties in lattice gauge theories.

Gauge-invariant Projected Entangled Pair States (PEPS) are a class of tensor network states in which local gauge symmetry constraints are imposed at the level of the constituent tensors. These constraints encode the local Gauss’s law of a quantum gauge theory directly into the virtual and physical indices of the PEPS, yielding global many-body wavefunctions that are by construction invariant under arbitrary local gauge transformations. Gauge-invariant PEPS provide a unifying language and flexible variational ansatz for non-perturbative studies of lattice gauge theories, accommodating both Abelian and non-Abelian gauge groups, bosonic and fermionic matter, and permitting hybrid schemes such as variational Monte Carlo in high-dimensional settings (Kelman et al., 2024). Their formalism has enabled controlled analytic and numerical investigations of ground state physics, topological order, screening, confinement, and other key phenomena in gauge field models.

1. Mathematical Structure of Gauge-Invariant PEPS

A gauge-invariant PEPS is built from local tensors placed on the vertices and edges of a spatial graph (typically a $d$ -dimensional lattice). The construction proceeds as follows (Zohar et al., 2015, Canals et al., 2024, Blanik et al., 2024):

Vertex tensors $A_v$ : Each vertex $v$ (matter site) is assigned a rank- $(1 + 2d)$ tensor, with one physical index (matter state) and $2d$ virtual indices (one per incident link, incoming or outgoing). These tensors are intertwiners for the gauge group $G$ , satisfying local invariance constraints.
Edge/link tensors $B_e$ or maximally entangled states: Each edge $e$ (gauge field link) is assigned a bond tensor (or a maximally entangled state) connecting the virtual indices of its two endpoints, possibly with a physical gauge field degree of freedom.
Local gauge invariance constraint: For each vertex $v$ and group element $g \in G$ ,

$A_v$ 0

here $A_v$ 1 acts on the matter index, and $A_v$ 2 on the virtual leg associated with link $A_v$ 3.

Global PEPS state: The many-body wavefunction is obtained by contracting all virtual indices on the network, typically by projecting onto maximally entangled bonds on each link.

The fundamental theorem established in (Blanik et al., 2024) states that a PEPS is gauge-invariant for $A_v$ 4 if and only if all local tensors $A_v$ 5 are $A_v$ 6-intertwiners and the irrep labels of virtual spaces are matched across edges. In more general terms, this realizes the physical Hilbert space as the kernel of the local Gauss operators, guaranteeing that the PEPS lies entirely within the gauge-invariant sector (Emonts et al., 2018, Zohar et al., 2015).

2. Construction Methods: From Global to Local Symmetry and Minimal Coupling

The construction of gauge-invariant PEPS can follow several routes, but one influential approach starts from a PEPS ansatz with global (on-site) symmetry and “gauges” it to local symmetry using a systematic mapping (Haegeman et al., 2014, Williamson et al., 2014). The key steps are:

Start with a globally symmetric PEPS: This involves tensors invariant under the action of a global group $A_v$ 7 (e.g., $A_v$ 8, $A_v$ 9).
Introduce link Hilbert spaces: Each edge receives an auxiliary degree of freedom, typically labeled by an element of $v$ 0 or an irrep label, forming a local Hilbert space for the gauge degrees of freedom.
Apply controlled entangling operations: At each vertex, controlled unitaries entangle the matter site with its incident gauge fields, implementing the minimal coupling principle at the tensor level. These operations encode the “minimal coupling” prescription of Kogut–Susskind and transfer the global symmetry into a local, or gauge, symmetry (Kelman et al., 2024, Emonts et al., 2018).
Impose the Gauss law on each vertex, either by direct projection or by engineering the tensor structure (e.g., through Clebsch–Gordan coefficient contractions for non-Abelian groups) so that the resulting PEPS is automatically “in the kernel” of the lattice divergence operator.

This prescription applies to both bosonic and fermionic matter fields, and is compatible with various types of gauge groups, including both Abelian (e.g., $v$ 1, $v$ 2) and non-Abelian ( $v$ 3) settings (Zohar et al., 2015, Zohar et al., 2016, Canals et al., 2024). The detailed algebraic structure relies on intertwiner theory and representation decompositions, which are encoded directly in the local tensors.

3. Physical and Numerical Properties

Gauge-invariant PEPS offer several distinct advantages for both theoretical and computational studies of lattice gauge theories (Kelman et al., 2024, Wu et al., 26 Mar 2025, Roose et al., 2024):

Exact local symmetry: Gauge invariance is encoded at the level of each tensor, guaranteeing global physical invariance under arbitrary local group actions without the need for explicit projectors.
Modularity and extensibility: Any globally symmetric PEPS can be systematically “gauged” to obtain a gauge-invariant PEPS, facilitating the study of both pure gauge and gauge-matter systems.
Absence of the sign problem: For families such as gauged Gaussian fermionic PEPS, the resulting probability distributions for Monte Carlo sampling over gauge field configurations are positive definite and real, removing the sign problem in computations (Roose et al., 2024, Kelman et al., 2024).
Truncation and computational cost: The bond dimension $v$ 4 of the PEPS determines the possible gauge field representations and fluxes included on each link (truncated Hilbert spaces), with computational complexity scaling rapidly with $v$ 5. Systematic extrapolation in $v$ 6 is required for continuum limits (Wu et al., 26 Mar 2025, Canals et al., 2024).
Efficient contraction and optimization: By working in a canonical or gauge-fixed form, variational optimization becomes more stable and robust, and local contractions can exploit the block-diagonal structure imposed by irreducible representations (Tang et al., 14 Aug 2025, Haghshenas et al., 2019).
Variational flexibility: The local parameterization of tensors (e.g., via Clebsch–Gordan tensor networks or gauge-covariant parametrizations) enables compact representational power and systematic exploration of phase diagrams, including confining, deconfined, and screened regimes (Wu et al., 26 Mar 2025, Zohar et al., 2016).

4. Example Implementations and Symmetry Sector Decomposition

Explicit PEPS ansätze have been constructed for a variety of lattice gauge theories, including:

Abelian $v$ 7 and $v$ 8 gauge theories: Vertex tensors enforce charge conservation, e.g.,

$v$ 9

with charge sectors defining the block structure of tensors (Zohar et al., 2015, Wu et al., 26 Mar 2025, Canals et al., 2024).

Non-Abelian $(1 + 2d)$ 0 gauge theory: Local tensors are built from Clebsh–Gordan trees, encoding the fusion rules and intertwiner structure required for proper transformation under local $(1 + 2d)$ 1 rotations. Gauge fields are truncated to representations $(1 + 2d)$ 2 for feasible computational complexity (Zohar et al., 2016).
Gauged Gaussian fermionic PEPS: For staggered fermion models, the “gauged” PEPS wavefunction is constructed by integrating virtual fermionic contractions with group actions entangling physical gauge field basis states, employing covariance matrix techniques for efficient contraction (Kelman et al., 2024, Roose et al., 2024).
Gauge-canonical forms and optimization: Gauge-canonical forms isolate all variational parameters into matter tensors, enabling highly efficient variational Monte Carlo algorithms and sector-by-sector evaluation of observables. Bond-dimension sectorization further reduces the computational load (Wu et al., 26 Mar 2025).
Topological order and classification: In systems with symmetry-protected topological (SPT) order, gauging symmetric PEPS with nontrivial third cohomology (“twisted injectivity”) yields PEPS whose parent Hamiltonians realize Dijkgraaf–Witten topological phases, with ground state degeneracy and anyon content controlled by group cohomology data (Williamson et al., 2014, Buerschaper, 2013).

5. Variational Monte Carlo and Benchmark Results

Gauge-invariant PEPS serve as variational ansätze for ground-state and low-energy properties of lattice gauge theories, with the following computational techniques and results observed (Wu et al., 26 Mar 2025, Roose et al., 2024, Kelman et al., 2024):

Variational Monte Carlo (VMC): The positive-definite property of certain PEPS families (e.g., gauged Gaussian PEPS) enables direct stochastic sampling using single-layer amplitude networks and avoids sign problems. Gradients are evaluated via stochastic reconfiguration.
Observables: Energy densities, Wilson and ’t Hooft loops, and screening order parameters (e.g., Fredenhagen–Marcu horseshoe) are all evaluated directly using the PEPS representation, with computational costs scaling as $(1 + 2d)$ 3 (with $(1 + 2d)$ 4 samples and lattice size $(1 + 2d)$ 5) (Wu et al., 26 Mar 2025).
Accuracy benchmarks: For $(1 + 2d)$ 6D $(1 + 2d)$ 7, $(1 + 2d)$ 8, $(1 + 2d)$ 9, and odd-$2d$0 gauge theories, energies computed with PEPS at moderate bond dimension ($2d$1--$2d$2) match unbiased quantum Monte Carlo results to $2d$3--$2d$4 per site, resolving first- and second-order deconfinement transitions and providing precise measurements of ground state and correlation observables (Wu et al., 26 Mar 2025).
Fermionic extensions: PEPS formalism naturally extends to gauge theories coupled to fermionic matter fields, via graded tensor constructions and intertwined Gaussian and non-Gaussian blocks (Roose et al., 2024, Kelman et al., 2024, Zohar et al., 2015).

6. Gauge-Fixing, Canonical Forms, and Optimization

The PEPS representation contains inherent “virtual gauge” degrees of freedom, corresponding to invertible transformations on the virtual bonds that leave the physical state unchanged (Tang et al., 14 Aug 2025, Haghshenas et al., 2019). These gauge freedoms must be controlled for robust energy optimization and accurate contraction benchmarks:

Gauge-fixing by minimal canonical form (MCF): By choosing a unique representative in each gauge orbit (e.g., the tensor with minimal Frobenius norm), one removes gauge-induced artifacts during variational optimization. Constraints of the form $2d$5 for generators $2d$6 are imposed at each optimization step, and parameter updates are projected to the gauge-fixed tangent space (Tang et al., 14 Aug 2025).
Blocking and canonicalization: The canonical form for a PEPS (analogous to MPS canonical forms) is realized by column-wise QR decompositions and absorption of isometric network blocks, exposing the entanglement structure and enabling block-diagonal optimizations for gauge-invariant states (Haghshenas et al., 2019).
Numerical stability and scaling: Gauge-fixed optimization procedures suppress unphysical parameter drifts and ensure the measured energy remains within reliable bounds at each iterative update (Tang et al., 14 Aug 2025).

7. Limitations and Future Directions

Despite their advantages, gauge-invariant PEPS face several notable limitations:

Bond-dimension truncation: The finite bond dimension restricts the range of virtual charges, necessitating careful extrapolation for continuum gauge theories or to access regimes with large fluctuations (Zohar et al., 2015, Zohar et al., 2016).
Computational complexity: Exact contraction in two (or higher) dimensions is $2d$7-hard; all practical algorithms rely on approximations such as boundary MPS compression, corner transfer matrix, or tensor renormalization methods.
Parent Hamiltonian structure: While local and frustration-free at zero coupling, time-evolved or filtered PEPS generically yield Hamiltonians with longer-ranged interactions, requiring further study for their continuum correspondence (Haegeman et al., 2014, Blanik et al., 2024).
Scaling and non-Abelian generalizations: Extending these constructions efficiently to fully non-Abelian gauge groups and to three or more spatial dimensions remains computationally intensive (Zohar et al., 2016, Canals et al., 2024).

Ongoing research explores increasing bond dimensions, integrating Clifford/circuit techniques for efficient preparation (Canals et al., 2024), advancing gauge-invariant contraction algorithms, and mapping the full phase structures of interacting gauge–matter systems.

For comprehensive, state-of-the-art technical treatments see (Kelman et al., 2024, Wu et al., 26 Mar 2025, Blanik et al., 2024, Zohar et al., 2015, Zohar et al., 2016, Roose et al., 2024, Tang et al., 14 Aug 2025, Haghshenas et al., 2019), and references therein. These works collectively formalize gauge-invariant PEPS as a systematic and robust platform for the non-perturbative study of quantum lattice gauge theories and their emergent phenomena.