Separating Matter and Gauge Fields: An Efficient Tensor Network Construction for Lattice Gauge Theories

Tensor network (TN) methods have been highly successful in studying low-dimensional, strongly interacting many-body systems, but lattice gauge theories (LGTs) remain challenging due to the local Gauss's law constraints. While specialized ansätze have been developed [1-3], they usually require additional gauge fields [1] and combined matter and gauge field Hilbert spaces [1-2]. Furthermore, available methods typically involve specialized implementations [2-3] that are laborious to integrate with standard TN libraries.

We present a construction based on the quantum link formulation (QLM) [4], which avoids the need for additional degrees of freedom and, crucially, keeps the local Hilbert spaces of matter and gauge fields separate. It builds only on standard Abelian symmetry-preserving tensors, making implementations on top of existing libraries simple.

We demonstrate the efficacy of this approach by presenting Density Matrix Renormalization Group (DMRG) results for the lattice realization ground states of the U(1) QED in (1+1)D (Multi-Flavor Schwinger Model) and (2+1)D, based on the TeNPy library [5].

[1] Silvi et al. New J. Phys. 2014

[2] Buyens et al. Phys. Rev. Lett. 2014

[3] Canals et al. ArXiv 2412.16961 2024

[4] Brower et.al Phys. Rev. D. 1999

[5] Hauschild, Pollmann SciPost 2018