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

TL;DR

This paper develops a lattice twisted gauging framework in (1+1)d quantum spin systems to construct non-local triality and p-ality mappings, revealing non-invertible symmetries and their implications for phase diagrams.

Contribution

It introduces a novel lattice twisted gauging method that generalizes dualities and constructs non-local mappings with non-invertible symmetries in (1+1)d systems.

Findings

Constructed triality and p-ality mappings preserving locality of symmetric operators.

Identified conditions for non-invertible symmetries to admit symmetric gapped phases.

Analyzed phase diagram constraints imposed by non-invertible symmetries.

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.