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Ising periodic paths split into two spin-chain sectors

Example

Use the actual coherent positive-FS Ising category constructed in All-label unitary Ising normal data and ribbon coherence at $κ = 1$ , $s = e^{π i / 8}$ and $β = s$ . Its simple objects are $1, η, D$ , with $η^{2} = 1$ , $η D = D η = D$ and $D^{2} = 1 \oplus η$ . In the inverse-pullback convention its complete data include $F_{D}^{D D D} = 2^{- 1 / 2} (\begin{array}{cc} 1 & 1 \\ 1 & - 1 \end{array}), R_{D D}^{0} = s^{- 1}, R_{D D}^{η} = i s^{- 1}, R_{η D}^{D} = R_{D η}^{D} = - i, R_{η η}^{0} = - 1.$ The remaining unit entries are one. Its twist is $θ_{D} = s^{- 1}$ , positive dimension $\sqrt{2}$ and FS sign $+ 1$ . This follows from the ell-four root category by the explicit associator/braiding parity conversion, rather than a vertex gauge; the original root category has negative FS and twist $s^{3}$ .

For each reference charge $D$ and length $L \geq 1$ , use the same periodic carrier $H_{L} : = ⨁_{b \in {1, η, D}} H_{D, L; b, b}$ as in Periodic Ising paths, two sectors, and the full line representation. Its cyclic path basis is the union of the fixed-boundary path bases, with every elementary fusion index retained. The full periodic path space is zero at odd length. At length $L = 2 N$ , $N \geq 1$ , it is $H \oplus \overset{ˉ}{H}$ , each sector of dimension $2^{N}$ , with no total-flux or spin-parity restriction. Encode the group labels by bits, in even positions of $H$ and odd positions of $\overset{ˉ}{H}$ . Under these orthonormal basis maps, $P_{2 j - 1} ∣_{H} = (I + X_{j}) / 2, P_{2 j} ∣_{H} = (I + Z_{j} Z_{j + 1}) / 2,$ $P_{2 j - 1} ∣_{\overset{ˉ}{H}} = (I + Z_{j} Z_{j + 1}) / 2, P_{2 j} ∣_{\overset{ˉ}{H}} = (I + X_{j + 1}) / 2.$ Indices are cyclic modulo $N$ . Put $C ∣ b_{1} \dots b_{N} ⟩ = ∣ b_{2} \dots b_{N} b_{1} ⟩$ , $Q = \prod_{j} X_{j}$ , and $K_{c, b} = 2^{- N / 2} (- 1)^{\sum_{j} c_{j} (b_{j - 1} + b_{j})}$ . The actual one-step shift and cyclic F-contracted line maps are $T = (\begin{array}{cc} 0 & C \\ I & 0 \end{array}), Y_{η} = diag (Q, Q), Y_{D} = (\begin{array}{cc} 0 & K^{†} \\ K & 0 \end{array}) .$ They intertwine all displayed projectors, including wraparound; each line commutes with $T$ and the projectors, and $Y_{D}^{2} = I + Y_{η}$ . The corresponding single-chain map instead satisfies $(K / \sqrt{2})^{2} = (I + Q) C^{- 1} / 2$ . At $N = 1$ the bond projector is $I$ and the field projector is $(I + X_{1}) / 2$ ; no three-distinct-site cyclic TL relation is asserted for length two.

Facts & Assumptions

Given: The displayed positive-FS category, its orthonormal fusion vertices, reference charge $D$ and positive periodic length.

[F1]

All-label unitary Ising normal data and ribbon coherence constructs all-label F/R data, both pentagon and hexagon identities, the positive unitary pivot and the twists for both FS signs.

[F2]

The exact ell-four Ising gauge and parity chirality conversion supplies the full ell-four root gauge and the precise parity conversion to $κ = 1$ , $β = s$ , including the inverse-pullback orientation.

[F3]

Periodic Ising paths, two sectors, and the full line representation supplies the complete periodic basis identifications, all projectors and cyclic line contractions, their wraparound, rank and Fourier identities for every $N \geq 1$ .

[F4]

Fixed-boundary anyonic fusion-path space specifies the typed path coordinates, and Local anyonic channel projector specifies F-conjugated channel projections.

Verification

technique · instantiate the complete coherent category and the full periodic representation, keeping their chirality and both sectors explicit

1.1F1givenalgebra

Take $κ = 1$ and $β = s$ . Since $s^{2} = (1 + i) / \sqrt{2}$ , the normal-data condition $β^{2} = κ (1 + i) / \sqrt{2}$ holds. Thus [F1] gives an actual coherent unitary category, not just a single sector matrix. Its full pullback R table and positive-FS Hadamard block are exactly the displayed entries, and its curl is $θ_{D} = κ / β = s^{- 1}$ .

2.1F2step 1.1algebra

The ell-four root supplier has FS $- 1$ and twist $s^{3}$ . Apply [F2]'s parity associator $(- 1)^{∣ a ∣ ∣ b ∣ ∣ c ∣}$ and odd-odd crossing factor $- i$ , reversing the odd evaluation as specified there. This produces the category of step 1.1. It proves the exact relationship to the root construction without identifying distinct pivotal or chirality conventions.

2.2F3F4step 1.1algebra

Fusion forces alternation between $D$ and a group label. Odd length is therefore impossible. For $2 N$ , choose the $N$ group labels independently in each of the two parity patterns, yielding both orthonormal bit bases and dimension $2^{N + 1}$ . Apply [F3] to the local projections specified by [F4]: the Hadamard vacuum conjugate is $(I + X) / 2$ , while a group-group endpoint pair has vacuum channel exactly when its bits agree, giving $(I + Z Z) / 2$ . These prove all four displayed projector formulas at every cyclic index.

3.1F3step 2.2algebra

With the displayed order $H \oplus \overset{ˉ}{H}$ , the path shift sends $H$ to $\overset{ˉ}{H}$ without changing the bits and sends $\overset{ˉ}{H}$ to $H$ by $C$ . Therefore it is the displayed block matrix and $T^{2} = diag (C, C)$ . [F3]'s complete cyclic F contraction at $κ = 1$ gives exactly the displayed $Y_{η}, Y_{D}$ , not an inferred fusion-only representation. Its Fourier identities $K^{†} K = K K^{†} = I + Q$ , $K Q = Q K = K$ and $K^{†} = C K = K C$ prove the line fusion relations, commutation with all projectors and $T$ , and the stated distinct single-chain square. Consequently any Hamiltonian formed from these specified projectors commutes with these line maps.

4.1F3step 3.1algebra∎

For $N = 1$ , the cyclic bond is $(I + Z_{1}^{2}) / 2 = I$ , the field remains $(I + X_{1}) / 2$ , and both two-dimensional sectors remain present. Thus the wraparound formulas include the smallest length without adding an inapplicable three-site relation or projecting away either sector.

Remarks

The earlier three- $D$ /total- $D$ Ising worked block is not a logical coherence supplier. The three A-homed results used here construct and compare the complete categories and evaluate the complete periodic representation. The displayed inverse-pullback phases have twist $e^{- π i / 8}$ ; reversing braiding and twist is a separate operation if the opposite physical chirality is desired.

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Sources

Shu-Heng Shao, What’s Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetries (standard reference, not scraped)
Chetan Nayak, Steven H. Simon, Ady Stern, Michael Freedman, and Sankar Das Sarma, Non-Abelian Anyons and Topological Quantum Computation (standard reference, not scraped)