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A nontrivial Z2 topological defect in finite quantum mechanics

Physical scope

An exact finite-dimensional 0+1-dimensional quantum-mechanical transfer model with hbar=1 and zero Hamiltonian; no continuum QFT or empirical claim is made.

Example

Let $H = C^{2}$ with its standard positive inner product, and let the Hamiltonian be the zero operator $H = 0 \in End (H)$ . This defines a finite-dimensional $0 + 1$ -dimensional quantum-mechanical transfer model on the Euclidean-time interval $[0, T]$ , $T > 0$ , with $ℏ = 1$ : propagation over every length $s \geq 0$ is $e^{- s H} = I$ . Here $H$ has units of energy, and $T$ and $s$ have units of inverse energy. Its combined background and boundary data consist here of a density matrix $ρ$ on $H$ , meaning a positive semidefinite matrix with $Tr ρ = 1$ . At each time $t_{i} \in (0, T)$ an ordinary local observable insertion is any matrix $O_{i} \in End (H)$ . The times $t_{i}$ have units of inverse energy, and the matrices $O_{i}$ are dimensionless by convention. The complete admissible input is any finite list of such insertions with distinct ordered times $0 < t_{1} < \dots < t_{n} < T$ , together with $ρ$ ; its normalized correlator is

$C_{ρ} ((t_{1}, O_{1}), \dots, (t_{n}, O_{n})) = Tr (ρ O_{n} \dots O_{1}) .$

The empty-list correlator is $Tr ρ = 1$ . Let $G = Z_{2} = {0, 1}$ with addition modulo $2$ , and define $U_{0} = I$ and $U_{1} = (\begin{array}{cc} 1 & 0 \\ 0 & - 1 \end{array})$ . The matrices $U_{g}$ are dimensionless by convention. A point defect $D_{τ, g}$ at $τ \in (0, T)$ is the additional insertion of the matrix $U_{g}$ in the same ordered product; $τ$ has inverse-energy units. This declares the correlator for every finite list of other insertions and every allowed boundary datum, not only for a selected pair of observables.

The complete input domain also admits any finite number of point defects at distinct times, using their matrices $U_{g}$ as factors in the same ordered trace; there are no additional junction strata in this one-dimensional model. The defect data act on every insertion matrix, including other defect factors, by conjugation. Since $Z_{2}$ is Abelian, $U_{g} U_{h} U_{g}^{- 1} = U_{h}$ , so the allowed defect labels are preserved. In particular they act on observables by $α_{g} (O) = U_{g} O U_{g}^{- 1}$ and on the boundary density matrix by $β_{g} (ρ) = U_{g} ρ U_{g}^{- 1}$ . The $g = 1$ insertion is a nontrivial invertible $Z_{2}$ symmetry defect of this finite model.

Example references

Gaiotto, Kapustin, Seiberg, and Willett, §1, PDF pp. 1–3

Carqueville, §2.1, PDF pp. 5–8

Facts & Assumptions

Given: The finite transfer model, its matrix observable algebra, and the insertion $U_{g}$ specified in the Example.

[A1]

The only propagation factors are $e^{- s H}$ with $H = 0$ , so they equal $I$ for every interval length $s \geq 0$ ; the correlator is the stated trace against the boundary density matrix.

[F1]

Topological defect insertion in a specified model: admissible complete model inputs include their boundary data, ordinary insertions, and decorated defects; a defect is topological when every allowed isotopy preserving its full decoration and avoiding other insertions preserves every declared correlator.

[F2]

Topological interface between specified models: fusion requires convergence of every declared correlator with the pair in the model's specified value-space topology; an invertible symmetry also requires a two-sided inverse and an induced observable/boundary-data automorphism preserving correlators.

Verification

technique · direct

Given: [A1], [F1], [F2], and the explicitly defined matrices in the Example.

1.1givenalgebra

The matrices satisfy $U_{0} = I$ , $U_{1}^{†} = U_{1}$ , and $U_{1}^{2} = I$ . Hence $g \mapsto U_{g}$ is a unitary representation of $Z_{2}$ . For $X = (\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array})$ , direct multiplication gives $U_{1} X U_{1}^{- 1} = - X$ , so this action is nontrivial.

1.2givenalgebra

For every $O \in End (H)$ and density matrix $ρ$ , $Tr (β_{g} (ρ) α_{g} (O_{n}) \dots α_{g} (O_{1})) = Tr (U_{g} ρ O_{n} \dots O_{1} U_{g}^{- 1}) = Tr (ρ O_{n} \dots O_{1})$ by cyclicity of the finite-dimensional trace. The same cancellation applies when any factors $O_{i}$ are admitted defect matrices $U_{h}$ , which are fixed under conjugation. Thus the induced insertion and boundary-data automorphisms preserve every declared complete correlator.

1.3A1givenalgebra

Fix any admissible finite list of other insertion times and move $τ$ through an allowed decorated isotopy. Since the motion crosses no other insertion or boundary, its relative order with every insertion is fixed. Every propagation factor is $I$ by [A1], so the ordered matrix product, and therefore its trace against every fixed $ρ$ , is unchanged. This verifies topologicality for the full declared input set.

2.1A1F2step 1.1algebra

For two group defects separated by $δ > 0$ , the propagation between them is $e^{- δ H} = I$ by [A1], so their ordered product is $U_{h} U_{g} = U_{h + g}$ , independent of $δ$ . With any fixed finite list of other insertions outside their shrinking collar, every declared correlator is unchanged when this adjacent pair is replaced by $U_{h + g}$ . Hence the fusion limit exists in the usual topology of $C$ for all such correlators, and $U_{1} U_{1} = U_{0} = I$ gives a two-sided inverse.

3.1F1F2step 1.1step 1.2step 1.3step 2.1∎

By [F1] and step 1.3, the point insertion is topological. Step 1.1 gives its nontrivial observable automorphism, step 1.2 verifies correlator covariance including the boundary data, and step 2.1 with [F2] gives group-like fusion and a two-sided inverse. It is therefore an invertible symmetry defect in this finite model only.

A nontrivial Z2 topological defect in finite quantum mechanics

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 1 level

Sources