Finite-lattice realization of the Ising non-invertible defect

Fix an integer $L\ge 2$ and the periodic site set $\mathbb{Z}\mathrm{/}L\mathbb{Z}$. Let ${\mathcal{H}}_L={\mathrm{\ell }}^2({\mathbb{Z}}_2^L)$ with orthonormal basis $\mathrm{\mid }{h}_1,\dots ,h_L⟩$, where every $h_j\in {\mathbb{Z}}_2$ and $h_0=h_L$. Define Pauli operators by $Z_j\mathrm{\mid }\mathbf{h}⟩=(-1{)}^{h_j}\mathrm{\mid }\mathbf{h}⟩$ and $X_j\mathrm{\mid }\mathbf{h}⟩=\mathrm{\mid }\mathbf{h}+{\mathbf{e}}_j⟩$. Adopt the finite quantum Ising model with Hamiltonian

where $Z_{L+1}=Z_1$, $J$ has units of energy, and $\mathrm{\hslash }=1$. Euclidean propagation through a time interval $s\ge 0$ is the exact finite matrix $e^{-s{H}_L}$, with $s$ in inverse-energy units. Correlators are matrix elements between arbitrary boundary vectors of finite ordered lists of observable matrices and these propagation factors.

Let ${\widehat{\mathcal{H}}}_L={\mathrm{\ell }}^2({\mathbb{Z}}_2^L)$ have basis $\mathrm{\mid }{\widehat{h}}_{1\mathrm{/}2},\dots ,{\widehat{h}}_{L-1\mathrm{/}2}⟩$, with dual Pauli operators $\widehat{Z}$ and $\widehat{X}$ defined by the same formulas on half-integer sites. Define the dual Hamiltonian, with half-integer indices periodic modulo $L$, by

Define the original-to-dual Kramers–Wannier map $K_L:{\mathcal{H}}_L\to {\widehat{\mathcal{H}}}_L$ by

The reverse-oriented map is $K_L^{†}:{\widehat{\mathcal{H}}}_L\to {\mathcal{H}}_L$. To give a single endodefect with the fusion convention below, use the explicit reflection identification $R_L:{\widehat{\mathcal{H}}}_L\to {\mathcal{H}}_L$ sending $\mathrm{\mid }\widehat{\mathbf{h}}⟩$ to $\mathrm{\mid }\mathbf{r}⟩$ with $r_j={\widehat{h}}_{1\mathrm{/}2-j}$ (all indices are periodic), and set $D_L=R_L{K}_L$. This is an isometry of the two spin bases and identifies the dual Hamiltonian with $H_L$. The matrix of $D_L$ is real symmetric: its exponent is ${\sum }_j{r}_{1-j}(h_{j-1}+h_j)$, whose coefficient matrix is symmetric in $\mathbf{r}$ and $\mathbf{h}$. Thus $D_L^2=K_L^{†}{K}_L$. The original-to-dual map and its reverse also have the typed fusion $K_L^{†}{K}_L=I+P_L$ and $K_L{K}_L^{†}=I+{\widehat{P}}_L$. A repeated fixed half-site identification is not used for this square; it would retain a spatial translation. A defect line is a wrapping insertion of $D_L$ on a Euclidean time slice. Its allowed position changes are slides within gaps between fixed observable insertions. In this finite lattice model, topologicality means invariance under the declared local line slides; re-association of junction channels is a separate linear recoupling identity; no smooth continuum isotopy claim is made. A finite defect network has labels $\mathbf{1},\psi ,\sigma$, represented on the chain by $I,P_L,D_L$, where $P_L={\prod }_j{X}_j$. Its fusion spaces are one-dimensional precisely for

and zero otherwise. Choose a basis vector in each allowed one-dimensional fusion space and define the associator in those bases as follows. Write $ϵ(\mathbf{1})=0$ and $ϵ(\psi )=1$; then

and every other allowed associator coefficient is $1$. In the abstract channel category, the unit maps $\mathbf{1}\otimes a\to a$ and $a\otimes \mathbf{1}\to a$ have coefficient $1$. Each simple label is self-dual; its left and right evaluation and coevaluation maps have coefficient ${\lambda }_a$ in the unit channel, where ${\lambda }_{\mathbf{1}}={\lambda }_{\psi }=1$ and ${\lambda }_{\sigma }=2^{1\mathrm{/}4}$, with zero coefficient in the $\psi$ channel for $\sigma \otimes \sigma$. These are categorical maps, distinct from the lattice triangle normalizations below.

For local lattice tensors, heights $h,h^{\mathrm{\prime }},\tilde{h}$ range over ${\mathbb{Z}}_2$, with addition modulo $2$; height $0$ denotes the label $\mathbf{1}$, height $1$ denotes $\psi$, and a vertex marked $\sigma$ denotes an empty site. Set $Q_{h{h}^{\mathrm{\prime }}}={\delta }_{h^{\mathrm{\prime }},h+1}$. The two nontrivial height placements of a trivalent junction of two $\sigma$ lines and one $\psi$ line have the following weights: when the vertex between the two $\sigma$ lines has ordinary height $h$ and the other vertices are empty, the weight is $T_{\sigma \sigma \psi }(h)$; when that vertex is empty and the other two have ordinary heights $h,h^{\mathrm{\prime }}$, the weight is $T_{\sigma \psi \sigma }(h,h^{\mathrm{\prime }})$. Explicitly,

The identity-defect termination tensors have the following four orientations: the first two represent the two local orientations of the $\sigma \otimes \sigma \to \mathbf{1}$ unit channel, and the last two represent the two local orientations of the $\psi \otimes \psi \to \mathbf{1}$ unit channel. In each case the dashed third edge has label $\mathbf{1}$ and the two sloping defect edges have the indicated common label:

• For defect label $\sigma$, when the upper vertex has height $h$ and the two lower vertices are empty sites, the tensor is $2^{-1\mathrm{/}4}$.
• For defect label $\sigma$, when the upper vertex is empty and the lower vertices have heights $h,h^{\mathrm{\prime }}$, the tensor is ${\delta }_{h{h}^{\mathrm{\prime }}}$.
• For defect label $\psi$, when all three site vertices are empty, the tensor is $2^{-1\mathrm{/}4}$.
• For defect label $\psi$, when the upper vertex has dual height $\tilde{h}$ and the lower vertices have heights $h,h^{\mathrm{\prime }}$, the tensor is $Q_{h\tilde{h}}{\delta }_{h{h}^{\mathrm{\prime }}}$.

In the last orientation, $\tilde{h}$ is the upper dual-basis height and $h,h^{\mathrm{\prime }}$ are the lower ordinary heights, all in ${\mathbb{Z}}_2$. The mixed index $Q_{h\tilde{h}}$ uses the same $h$ as ${\delta }_{h{h}^{\mathrm{\prime }}}$; there is no second independent dual height. Reversed orientations use adjoint tensors; every fusion-forbidden label or height assignment has weight zero, and all components outside the stated domains are zero. The other allowed channels use their unit basis maps of coefficient $1$. These local triangles and the abstract channel maps above are the adopted junction data of this finite network model, not consequences of the fusion multiplicities alone.

The complete admissible input domain is declared as follows. Each input is a finite decorated lattice tensor network with boundary vectors or boundary tensors, arbitrary finite ordinary insertion matrices, propagation tensors $e^{-s{H}_L}$ (or their dual versions), and any finite number of the wrapping defects and local trivalent junction tensors above, with their oriented site orderings. Local height edges have index set $\{0,1\}$, empty-site edges have the singleton label $\sigma$, and row edges have index set ${\mathbb{Z}}_2^L$. Matching edges are contracted only when their index sets and model sides agree; reversal uses the stated adjoint tensor. Boundary vectors and all uncontracted indices are part of the combined boundary data. The correlator is the unnormalized complex number

Here the finite graph $\mathrm{\Gamma }$ records all sites, incidences, side labels, tensor orientations, and the external boundary contraction; ${\mathcal{T}}_t$ is exactly the specified matrix component or local triangle component at $t$. The product over $v$ runs over local patch sites whose factors are not already absorbed into a row tensor; row matrices have their displayed normalization. Every site weight is included once; if a site is split among patches, its positive factors are split with product $d_v$. Unit defect quadrilaterals carry $2^{-1\mathrm{/}4}{\delta }_{a\alpha }{\delta }_{b\beta }$, and a duality quadrilateral carries $2^{-1\mathrm{/}2}(-1{)}^{ab}$ at its two ordinary corners; forbidden assignments vanish. These give a fully finite contraction prescription, including the source's site-weight convention, rather than inferring network correlators from fusion multiplicities. On a row with no extra local junction patch it is exactly the ordered matrix element already specified. Reflection identifications permute the row and boundary indices consistently and do not change a contraction. The value space is $\mathbb{C}$ with its usual topology. A declared line slide moves an entire $I$, $P_L$ or $D_L$ slice within a gap containing no other tensor insertion, leaving all boundary tensors fixed. Junction recouplings compare the two local contractions with the same external height indices, including site factors; equality or a specified linear combination must hold before the rest of the graph is contracted. No arbitrary continuum network or additional support deformation is asserted.

For this selected finite model, $D_L$ is topological under the declared defect slides, its exact operator fusion is $D_L^2=I+P_L$, $P_L^2=I$, and $D_L{P}_L=P_L{D}_L=D_L$, and the junction spaces and associators are the finite semisimple unitary Ising fusion data above. In particular $D_L$ is non-invertible. The claim is confined to this finite lattice model and asserts no continuum limit or empirical fact.

Aasen, Mong, and Fendley, §§2–4

Fröhlich, Fuchs, Runkel, and Schweigert, critical-Ising defect discussion

Simon, Topological Quantum, §§9.2–9.3

Carqueville, Lecture Notes on 2-Dimensional Defect TQFT, §2.1

The square records the exact local intertwiner used in step 1.1. Its commutativity follows from the displayed kernel calculation; no proof relies on the diagram.