A tensor network formulation of Lattice Gauge Theories based only on symmetric tensors

TL;DR

This paper introduces a new tensor network approach for lattice gauge theories using only symmetric tensors, simplifying numerical simulations and revealing a duality with globally symmetric systems.

Contribution

It presents a purely symmetric tensor network formulation of gauge-invariant states, enabling easier implementation and theoretical insights into gauge-spin dualities.

Findings

Enables implementation with existing tensor network libraries

Reveals a duality between gauge theories and globally symmetric systems

Provides a new perspective on gauge invariance and topological states

Abstract

The Lattice Gauge Theory Hilbert space is divided into gauge-invariant sectors selected by the background charges. Such a projector can be directly embedded in a tensor network ansatz for gauge-invariant states as originally discussed in [Phys. Rev. B 83, 115127 (2011)] and in [Phys. Rev. X 4, 041024 (2014)] in the context of PEPS. The original ansatz is based on sparse tensors, though parts of them are not explicitly symmetric, and thus their actual implementation in numerical simulations has been hindered by the complexity of developing ad hoc libraries. Here we provide a new PEPS tensor network formulation of gauge-invariant theories purely based on symmetric elementary tensors. The new formulation can be implemented in numerical simulation using available state-of-the-art tensor network libraries but also holds interest from a purely theoretical perspective since it requires embedding the original gauge theory with gauge symmetry G into an enlarged globally symmetric theory with symmetry GxG. By revisiting the original ansatz in the modern landscape of i) duality transformations between gauge and spin systems, ii) finite depth quantum circuits followed by measurements that allow generating topologically ordered states, and iii) Clifford enhanced tensor networks, we show that such a new formulation provides a novel duality transformation between lattice gauge theories and specific sectors of globally invariant systems.