Subsystem Non-Invertible Symmetry Operators and Defects

TL;DR

<3-5 sentence high-level summary>This paper introduces subsystem non-invertible symmetry by gauging a subsystem $\mathbb{Z}_2$ in (2+1)d and developing a subsystem Kramers-Wannier (KW) duality framework. It constructs KW duality operators and defects in both (1+1)d and (2+1)d settings, derives non-invertible fusion rules that mix co-dimension-1 operators with subsystem defects, and demonstrates mobility of defects in the full two-dimensional plane. The analysis shows that subsystem KW duality symmetry can be anomalous, adapting established 1+1d anomaly arguments to the subsystem context via gauging and SSPT considerations. The work opens pathways to generalize KW-type dualities to broader subsystem symmetries and motivates potential applications in SSPT, lattice gauge theories, and quantum information platforms.

Abstract

We explore non-invertible symmetries in two-dimensional lattice models with subsystem $\mathbb Z_2$ symmetry. We introduce a subsystem $\mathbb Z_2$-gauging procedure, called the subsystem Kramers-Wannier transformation, which generalizes the ordinary Kramers-Wannier transformation. The corresponding duality operators and defects are constructed by gaugings on the whole or half of the Hilbert space. By gauging twice, we derive fusion rules of duality operators and defects, which enriches ordinary Ising fusion rules with subsystem features. Subsystem Kramers-Wannier duality defects are mobile in both spatial directions, unlike the defects of invertible subsystem symmetries. We finally comment on the anomaly of the subsystem Kramers-Wannier duality symmetry, and discuss its subtleties.

Figures (10)

• Figure 1: $\mathbb{Z}_2$ symmetry defects (blue) and $\mathbb{Z}_2$ twist operators (red) on an infinite chain and on a ring. The black line represents the space, and the orthogonal direction represents time. In the first row, acting a twist operator on the left half chain creates a $\mathbb{Z}_2$ defect along the time direction at the origin. In the second row, in order to use the twist operator to create a symmetry defect, one needs to lift $S^1$ up to $\mathbb{R}$ and act the twist operator on the intervals $\cup_{k\in \mathbb{Z}}[2kL,(2k+1)L]$.
• Figure 2: KW duality defects and twist operators. The left figure is the KW duality defect along the time direction, localized at $i=\frac{1}{2}$. Such a defect can be created, in the Hamiltonian formalism, by acting the original system with a KW duality twist operator ${\mathcal{N}}^t_0$, as shown in the right top figure. In resulting Hilbert space is represented in the right bottom figure, where the vertical slashes represent where the spins live, and they occupy half-integer links to the left of the origin and integer sites to the right. The empty circles are sites to the left of the origin where no spins are supported.
• Figure 3: Examples of subsystem $\mathbb Z_2$ symmetry operators, defect operators and twist operators.
• Figure 4: Mapping of symmetry-twist sectors.
• Figure 5: Fusion between two subsystem KW duality operators gives rise to a grid operator, where the grid is along the space direction.