Decorated tensor network renormalization for lattice gauge theories and spin foam models

TL;DR

This work introduces decorated tensor networks as a principled framework to coarse-grain lattice gauge theories and spin foam models while explicitly preserving gauge constraints. By decorating tensor blocks with the original gauge variables and using cube- or prism-based coarse-graining with SVD truncations, the method efficiently handles gauge redundancies and yields accessible fixed-point data. The authors develop leading-order, higher-order, and lower-cost variants and benchmark them on Abelian models (notably the 3D Ising gauge model and 2D Ising) with quantitative results close to reference simulations and direct observables, while outlining clear paths to non-Abelian and 4D extensions. They also show the approach applies to Ising-like systems beyond gauge theories, enabling direct computation of correlation functions and improved phase-transition estimates at various decoration levels. Overall, decorated tensor networks offer a flexible, gauge-respecting coarse-graining strategy with potential to bridge tensor-network methods and spin foam gravity, including possible integration with analytical decorations and Lie-group generalizations.

Abstract

Tensor network techniques have proved to be powerful tools that can be employed to explore the large scale dynamics of lattice systems. Nonetheless, the redundancy of degrees of freedom in lattice gauge theories (and related models) poses a challenge for standard tensor network algorithms. We accommodate for such systems by introducing an additional structure decorating the tensor network. This allows to explicitly preserve the gauge symmetry of the system under coarse graining and straightforwardly interpret the fixed point tensors. We propose and test (for models with finite Abelian groups) a coarse graining algorithm for lattice gauge theories based on decorated tensor networks. We also point out that decorated tensor networks are applicable to other models as well, where they provide the advantage to give immediate access to certain expectation values and correlation functions.

Figures (13)

• Figure 1: Tensor network representation of a spin foam defined on a cubical lattice. The tensors $T_e$ on the edges capture the intertwiner degrees of freedom, while the tensors $T_f$ on the faces are an auxiliary structure. For a given face $f$ they ensure that all edge tensors $T_e$ with $e \subset f$ carry the same representation label $\rho_f$ assigned to that face.
• Figure 2: Definition of a homogeneous, yet anisotropic tensor network from the vertex amplitude representation. By absorbing the auxiliary tensors in the positive quadrants into the vertex, one obtains a cubical tensor network, with tensors $T'$, with additional diagonal edges (some of the auxiliary edges can be combined with the original cubical edges, increasing the bond dimension). The regular cubical edges get equipped with additional data, namely the representation on the face.
• Figure 3: Example of a basic building block of the decorated tensor network. The amplitude tensor $A_c$ is associated to the cube, and contains the information about edge variables, $k_e$ and $I_e$ (appearing for higher order approximation) and face variables $\delta_f$ (these might arise in Abelian models for subdivides faces) and $I_f$ (appearing for higher order approximation). Bold edges represent a choice of maximal tree that represents the free variables for Abelian groups. We depict face variables as legs of the central tensor piercing through respective faces. One can think of it as one element of a tensor network decorated with additional edge variables.
• Figure 4: Full iteration of the algorithm. The initial cube is split in two ways into prisms, which are then glued back together into a rhomboid again. The following iteration is performed after a rotation into a different plane. In order to complete the iteration cycle we keep an auxiliary half-edge on two faces, which is summed over in the last step. The bold edges indicate the choice of a maximal tree, the coloured edges highlight the edges shared by the split prisms.
• Figure 5: Gluing of two prisms into a larger prisms. This gluing requires a redefinition of the tree (represented by bold edges) for one of the prisms $P_4$. After the gluing a further redefinition of the tree has been performed -- the final tree is shown in the bottom panel.