Isometrization of Tensor Network States via Gauge Propagation

We introduce a gauge-propagation approach for approximately converting generic tensor-network states into an isometric tensor-network state form with a prescribed orthogonality center. In one dimension, this propagation is exact because the non-isometric factor produced by a QR or singular-value decomposition is supported on a single virtual bond. In higher-dimensional networks, however, a local step can have several outgoing directions, and the residual factor is generally not separable into independent single-bond contributions. We address this local obstruction by approximating a local tensor, or a contracted local cluster, by structured terms consisting of an isometric factor multiplied by a tensor product of output-leg factors. The isometric factor is retained at the current site or cluster, while the output-leg factors are absorbed into neighboring tensors along the propagation directions. This construction provides a local truncation criterion for gauge propagation and a practical route to refinement by increasing the number of retained terms or enlarging the local cluster. Benchmarks on random tensors and on the loop-gas tensor representation of the Kitaev spin liquid show that this refinement reduces both local residuals and accumulated propagation errors. For the loop-gas tensor, two structured terms reduce the local residual to numerical precision, and enlarging the local object from 2-in-2-out to 4-in-2-out and 6-in-2-out clusters lowers both local truncation errors and accumulated errors in finite honeycomb gauge propagation. These results identify propagation-compatible local decomposition as a useful building block for approximate isometrization and as a potential initializer or preconditioner for variational isoTNS algorithms.