Realizing triality and $p$-ality by lattice twisted gauging in (1+1)d quantum spin systems

TL;DR

This work develops a lattice realization of twisted gauging in (1+1)d spin systems and shows it equals first applying an SPT entangler and then untwisted gauging. It constructs and analyzes non-local, triality- and p-ality-like mappings for $ ext{Z}_N imes ext{Z}_N$ and $ ext{Z}_p imes ext{Z}_p$ symmetries, preserving locality of symmetric operators while mapping charges to nonlocal excitations. The authors provide both lattice (Pauli-stabilizer) and quantum-process (Kraus-operator) formulations, and connect these mappings to non-invertible fusion category symmetries arising at self-dual points, with explicit conditions for invariant gapped phases. They further relate the constructions to bulk SymTFT and anyon-permutation symmetries, offering group-theoretical fusion-category frameworks for Tri (triality) and P ($p$-ality) structures and outlining paths to classify symmetric gapped phases and their fiber functors.

Abstract

In this paper, we study the twisted gauging on the (1+1)d lattice and construct various non-local mappings on the lattice operators. To be specific, we define the twisted Gauss law operator and implement the twisted gauging of the finite group on the lattice motivated by the orbifolding procedure in the conformal field theory, which involves the data of non-trivial element in the second cohomology group of the gauge group. We show the twisted gauging is equivalent to the two-step procedure of first applying the SPT entangler and then untwisted gauging. We use the twisted gauging to construct the triality (order 3) and $p$-ality (order $p$) mapping on the $\mathbb{Z}_p\times \mathbb{Z}_p$ symmetric Hamiltonians, where $p$ is a prime. Such novel non-local mappings generalize Kramers-Wannier duality and they preserve the locality of symmetric operators but map charged operators to non-local ones. We further construct quantum process to realize these non-local mappings and analyze the induced mappings on the phase diagrams. For theories that are invariant under these non-local mappings, they admit the corresponding non-invertible symmetries. The non-invertible symmetry will constrain the theory at the multicritical point between the gapped phases. We further give the condition when the non-invertible symmetry can have symmetric gapped phase with a unique ground state.