Continuum limit of gauged tensor network states
Gertian Roose, Erez Zohar
TL;DR
The paper addresses the challenge of representing gauge theories non-perturbatively directly in the continuum by constructing gauged continuous tensor-network states (gCPEPS) that incorporate gauge invariance and Gauss-law constraints from the outset. It establishes a well-defined continuum limit of gauged lattice tensor networks, showing that gauged PEPS reduce to gauged CPEPS in the continuum, and that in one spatial dimension this framework recovers gauge-invariant gauged CMPS. The authors provide explicit proofs of gauge invariance and Gauss-law closure, derive the continuum limit by promoting global symmetries to local gauge symmetries and coupling to link gauge fields, and demonstrate that the resulting continuum state takes a kinetic form with a generating Lagrangian $-\int d^D x\; \bar{D}_i \bar{\chi} D^i \chi + V(\phi,\chi)$. This continuum, gauge-invariant tensor-network class offers a promising non-perturbative tool for studying gauge theories directly in the continuum and may serve as a scalable numerical ansatz, especially in low dimensions, with potential extensions to dualities and gauged operator formalisms.
Abstract
It is well known that all physically relevant states of gauge theories lie in the sectors of the Hilbert space which satisfy the Gauss law. On the lattice, the manifeslty gauge invariant subspace is known to be exactly spanned by gauged tensor networks. In this work, we demonstrate that the continuum limit of certain types of gauged tensor networks is well defined and leads to a new class of states that may be helpful for the non-perturbative study of gauge theories directly in the continuum.