Non-invertible symmetries and LSM-type constraints on a tensor product Hilbert space

TL;DR

This paper constructs and analyzes an exact non-invertible Kramers-Wannier symmetry on a 1+1d lattice with a tensor-product Hilbert space, showing that its associated defect D blends with lattice translations and is not captured by fusion categories on the lattice. It derives multiple equivalent operator and defect descriptions, including a matrix product operator (MPO) form for D and defect fusion rules, and clarifies how the lattice theory flows to continuum fusion categories with different Frobenius-Schur indicators depending on the deformation. A key result is a lattice-level LSM-type constraint: any D-preserving Hamiltonian must be gapless or exhibit a gapped phase with a ground-state degeneracy that is a multiple of 3, illustrated by tricritical Ising physics. The work also elucidates the differences between lattice and continuum symmetries, showing that the FS indicator becomes meaningful only in the continuum, while the lattice hosts emergent (emanant) topological data captured by F-symbols and lattice quantum dimensions. Overall, the paper sets a framework for understanding non-invertible lattice symmetries, their defects, and the implications for RG flows and phase structure in 1+1d systems.

Abstract

We discuss the exact non-invertible Kramers-Wannier symmetry of 1+1d lattice models on a tensor product Hilbert space of qubits. This symmetry is associated with a topological defect and a conserved operator, and the latter can be presented as a matrix product operator. Importantly, unlike its continuum counterpart, the symmetry algebra involves lattice translations. Consequently, it is not described by a fusion category. In the presence of this defect, the symmetry algebra involving parity/time-reversal is realized projectively, which is reminiscent of an anomaly. Different Hamiltonians with the same lattice non-invertible symmetry can flow in their continuum limits to infinitely many different fusion categories (with different Frobenius-Schur indicators), including, as a special case, the Ising CFT. The non-invertible symmetry leads to a constraint similar to that of Lieb-Schultz-Mattis, implying that the system cannot have a unique gapped ground state. It is either in a gapless phase or in a gapped phase with three (or a multiple of three) ground states, associated with the spontaneous breaking of the lattice non-invertible symmetry.

Figures (5)

• Figure 1: The phase diagram at the vicinity of the tricitical Ising CFT point (purple point). The continuous black line is a second order line ending at the critical Ising lattice model (red point). The black solid line flows to the Ising CFT. The black dashed line is a first order transition. Along that line, the theory is gapped and has three low-lying states.
• Figure 2: Fusion of topological defects vs. the algebra of conserved operators. In these spacetime diagrams, time runs upward. Figure \ref{['fig:defect']} on the left denotes a local unitary operator $\lambda_{\mathcal{A} \otimes \mathcal{B}}$ implementing the fusion of $\mathcal{A}$ with $\mathcal{B}$ -- it conjugates the defect Hamiltonian $H_{\mathcal{A};\mathcal{B}}$ to $H_{\mathcal{C}_1 \oplus \cdots \oplus \mathcal{C}_d} = H_{\mathcal{C}_1} \otimes {\left| {1} \right>} {\left< {1} \right|} + \cdots + H_{\mathcal{C}_d} \otimes {\left| {d} \right>}{\left< {d} \right|}$. In the special case when $\cal B$ is the trivial defect, this fusion operation reduces to the movement operator for $\cal A$ in \ref{['movementintro']}. Figure \ref{['fig:op']} on the right indicates the fusion of the symmetry operators $\mathsf{A}$ and $\mathsf{B}$.
• Figure 3: Construction of symmetry operator $\mathsf{D}_{1 \oplus \eta}$ from the symmetry defect $\mathcal{D}$. Recall from \ref{['fusion.intro']} that $\mathcal{D} \otimes \mathcal{D} = \mathcal{T}^- \oplus \mathcal{T}^- \eta$. In the first and the last row, we represent $\mathcal{T}^- \oplus \mathcal{T}^- \eta$ by $1\oplus\eta$, but with one site less compared to the other rows.
• Figure 4: The Hilbert space in each step of the half-gauging between site 0 and site $J$ on a closed, periodic Ising chain with $L$ sites. In the first line, the black dots stand for the original qubits with Pauli operators $X_j,Z_j$ on each site $j=1,2,\cdots, L$. The white dots stand for the qubits on the links with Pauli operators $\tilde{X}_{j-\frac{1}{2}},\tilde{Z}_{j-\frac{1}{2}}$ for the $\mathbb{Z}_2$ gauge fields. In the second line we impose the Gauss law constraints, and the local operators are now generated by $X_j,Z_j$ for $j=J,\cdots,L$ (black dots) and $\widehat{X}_{j-\frac{1}{2}} ,\widehat{Z}_{j-\frac{1}{2}}$ for $j=1,\cdots, J$ (white dots). In the third line we rename the qubits on the links to the sites by $\widehat{X}_{j-\frac{1}{2}}\to X_j, \widehat{Z}_{j-\frac{1}{2}} \to Z_j$ for $j=1,2,\cdots, J-1$ (black dots), and $\widehat{X}_{J-\frac{1}{2}} \to X_{(J-1,J)}, \widehat{Z}_{J-\frac{1}{2}} \to Z_{(J-1,J)}$ (white dot). In the end we find a duality defect $\mathcal{D}$ on the link $(L,1)$ and its dual defect $\mathcal{D}^*$ on the link $(J-1,J)$. The latter defect involves an extra qubit labeled by the white dot.
• Figure 5: Construction of the lattice translation symmetry operator $T^{-1}$ from the topological defect $\mathcal{T}^-$. The diagram represent $T^{-1}$ as a sequence of unitary operators implementing pair creation of $\mathcal{T}^{\pm}$ defects, moving $\mathcal{T}^-$ around the chain and finally the fusion of $\mathcal{T}^-$ with $\mathcal{T}^+$.