Beyond Bond Gauge: Exact Tensor-Network Tangent Spaces at Weighted Graph States

This paper investigates the structural properties of tensor network representations, specifically focusing on the first-order differential of the contraction map at weighted graph states (WGS). The author asks two fundamental questions: which tensor variations leave the physical state unchanged (the kernel) and which physical state changes are impossible to reach (the image). By evaluating these at WGS points—where the physical and bond dimensions are equal—the author computes the exact tangent space and kernel, providing a complete characterization of the representation's redundancy.

The study establishes that the non-bond kernel is generated by overlapping closed neighborhoods of vertices, with specific motifs—triangles, chordless squares, and shared-edge diamonds—acting as the minimal regions for these relations. The author derives a face-order formula that counts these relations for any periodic regular-polygon tiling. A key result is that for thirty-three of the thirty-nine families studied, generic tensors are gauge-complete, meaning the WGS non-bond nullity is a specific rank drop at the WGS point rather than a universal defect. Furthermore, the paper identifies that certain physical directions, such as nearest-neighbor XX and YY operators on honeycomb lattices, are fundamentally missing from the tangent space, a result that holds regardless of gauge fixing or regularization.

These findings provide a rigorous foundation for understanding the geometry of tensor network varieties and the limitations of variational methods like the time-dependent variational principle (TDVP). By separating the kernel (redundancy) from the image (expressivity), the paper clarifies that kernel removal regularizes the reduced linear system without affecting physical velocities, whereas missing directions require an explicit expansion of the bond dimension. The results offer a controlled benchmark for testing regularization strategies and bond-expansion algorithms in tensor network evolution.