Kennedy-Tasaki transformation and non-invertible symmetry in lattice models beyond one dimension

TL;DR

This work extends the explicit operator construction of Kramers-Wannier duality and the Kennedy-Tasaki transformation from 1D to higher-dimensional lattice models with subsystem symmetries. By formulating KW duality as a unitary circuit composed with a symmetry-projection, and KT as a circuit sandwich around a cluster entangler, the authors map between spontaneous symmetry breaking and symmetry-protected topological phases in 2D and higher, for both Z2 and ZN cases. They demonstrate that the higher-dimensional KW operators are non-invertible, square to translations times projections, and act via light-cone/membrane-like structures on Pauli operators, establishing a comprehensive duality web between SSPT, SSSB, and gauged fermionic counterparts. The results provide explicit circuits for KW and KT transformations, discuss measurement-based implementations, and open pathways to explore higher-form and non-invertible symmetries in more general SSPT/SSSB contexts with potential experimental relevance. Overall, the paper offers a unified framework for dualities linking SSPT and SSB in higher dimensions under subsystem symmetries.

Abstract

We give an explicit operator representation (via a sequential circuit and projection to symmetry subspaces) of Kramers-Wannier duality transformation in higher-dimensional subsystem symmetric models generalizing the construction in the 1D transverse-field Ising model. Using the Kramers-Wannier duality operator, we also construct the Kennedy-Tasaki transformation that maps subsystem symmetry-protected topological phases to spontaneous subsystem symmetry breaking phases, where the symmetry group for the former is either $\mathbb{Z}_2\times\mathbb{Z}_2$ or $\mathbb{Z}_2$. This generalizes the recently proposed picture of one-dimensional Kennedy-Tasaki transformation as a composition of manipulations involving gauging and stacking symmetry-protected topological phases to higher dimensions.

Figures (10)

• Figure 1: Quantum circuit representation of the operator $\tilde{\mathbf{D}}$ given in Eq. \ref{['eq:Dtilde']}. The time ordering of the circuit is from top to bottom. The top layer denotes the input sites $1$,$2$,...,$L$. The bottom layer represents the output.
• Figure 2: (a) KT transformation from SPT to SSB; (b) KT transformation from SSPT to SSSB.
• Figure 3: The Hamiltonian term $\prod_{v \in \partial c} Z_v$ in the transverse-field hypercube Ising model in three dimensions.
• Figure 4: (a) Each site resides a red qubit, and each face resides a blue qubit. (b, c) $\text{KT}^{(2)}$ transformation acting on a single $Z$ operator is given by a product of Pauli $X$ operators in a light cone.
• Figure 5: Hamiltonian terms in the $\mathbb{Z}_2 \times \mathbb{Z}_2$ cluster state on the three-dimensional lattice. Here, we draw the blue lattice as a background. The red lattice sites reside at the centers of the blue cubes. (left) $X_{v^r}\prod_{v^b\in \partial c^b}Z_{v^b}$, (right) $X_{v^b}\prod_{v^r\in \partial c^r}Z_{v^r}$.