Solitonic, Topological, and Boundary Sectors

A kink or boundary-condition-changing sector is characterized by different asymptotic vacua on different sides of space. Such a representation cannot agree with one vacuum on the entire complement of a bounded region: the left component approaches one phase and the right component another. The correct object is therefore an interpolating representation, often localized relative to wedges or half-lines, with composition governed by matching boundary labels. Topological and boundary sectors require their own selection data rather than a forced DHR interpretation.

Required background. Sector Selection, Localization, and Transportability supplies the compact criterion that fails; BF Sectors, Spacelike Cones, and Massive Charges supplies unbounded localization.

Helpful background. Braided Sectors, Anyonic Statistics, and Low-Dimensional Nets explains low-dimensional exchange; Finite-Energy Boundary Data, Charge, and Stability supplies classical boundary conditions; Operators, Boundaries, and Relative Topological Theories explains boundary-changing operators.

Interpolating representations

Consider a $1+1$-dimensional net with inequivalent translation-invariant vacuum states $\omega_a$ and $\omega_b$, with GNS representations $\pi_a$ and $\pi_b$. A kink state $\omega_{ab}$ is an energy-positive covariant state whose representation approaches $\pi_a$ on observables translated far to the left and $\pi_b$ far to the right. Algebraically, one demands equivalence to $\pi_a$ on one wedge algebra and to $\pi_b$ on the opposite wedge, together with a positive-energy translation representation. The ordered pair $(a,b)$ is part of the sector label.

Composition is only defined when endpoints match:

$$(a,b)\otimes(b,c)\longrightarrow(a,c),$$

possibly with several fusion channels. Reversing spatial orientation sends a kink to an antikink $(b,a)$. A closed kink–antikink composite can contain the vacuum sector $(a,a)$, but a single $(a,b)$ object with $a\ne b$ has no DHR representative relative to either vacuum.

For algebraic $P(\phi)_2$ models, Schlingemann defines interpolating kink states in §4, proves an existence criterion for extendible dynamics in §5, and verifies the criterion for $P(\phi)_2$ dynamics in §6 Schlingemann 1996, §§4–6, pp. 18–30. The construction uses a doubled theory and the split property to splice two vacuum states across a bounded transition region while retaining positive-energy dynamics.

The broken-phase φ⁴₂ kink

Take a two-dimensional $\phi^4$ model in a broken $\mathbb Z_2$ phase with vacuum expectations $\langle\phi\rangle_+=v$ and $\langle\phi\rangle_-=-v$. A classical kink tends to $-v$ as $x\to-\infty$ and $+v$ as $x\to+\infty$. Its algebraic state $\omega_{-+}$ has the same asymptotic pattern: for a local observable $A$,

$$\lim_{x\to-\infty}\omega_{-+}(\alpha_x(A))=\omega_-(A), \qquad \lim_{x\to+\infty}\omega_{-+}(\alpha_x(A))=\omega_+(A),$$

under the clustering assumptions needed for these limits. The topological charge records the ordered boundary data, not a Noether eigenvalue localized at one point. The dynamical and classical interpretation is developed in Kinks and Domain Walls.

The licensed statement and its boundary

Given two admissible vacua, an extendible positive-energy dynamics, and the split/separation properties used in the construction, the theorem licenses a covariant kink state interpolating them. It does not assert uniqueness, completeness of all kink sectors, an isolated one-particle mass shell, or compact localization. Extra spectral analysis is needed before interpreting the kink as a stable particle.

The converse also fails. Merely specifying two boundary expectation values does not construct a positive state on the quasilocal algebra or a positive-energy representation. Likewise, an interface category or fusion rule does not determine the bulk net on either side.

Adversarial failure: a false DHR equivalence

Assume $\omega_{-+}$ were DHR-localized relative to $\omega_+$. For any proposed double cone $O$, translate a local order-parameter observable far to the left so that it lies in $O'$. Its expectation tends to $-v$ in the kink state but $+v$ in the $+$ vacuum. Thus the exterior representations remain distinguishable for every $O$. The same argument relative to $\omega_-$ uses a far-right observable. No bounded region can hide two different asymptotic vacua.

Independent checks

Measure the order parameter separately at both spatial ends. Reverse the kink and check that boundary labels swap. Compose a kink with its antikink and verify that asymptotic labels match before claiming a vacuum channel. Finally inspect the translation spectrum; finite interpolation energy alone does not prove an isolated mass shell.

Exercises

1. Partial composition. Why is $(a,b)\otimes(c,d)$ not canonically composable when $b\ne c$?

Solution

The right asymptotic vacuum of the first interface must equal the left asymptotic vacuum of the second. If $b\ne c$, gluing leaves an unresolved infinite region with incompatible boundary phase, so no finite-energy composite is specified.

2. Antikink. What are the asymptotic limits of the reflected state $x\mapsto-x$ applied to $\omega_{-+}$?

Solution

Reflection exchanges left and right, producing $\omega_{+-}$: it approaches $\omega_+$ at $x\to-\infty$ and $\omega_-$ at $x\to+\infty$.

3. DHR test. Construct the exterior discriminator when $O$ is fixed.

Solution

Choose a translate of a bounded order-parameter observable supported far left of $O$. It belongs to $\mathcal A(O')$ and has limiting expectations $-v$ and $+v$ in the kink and plus-vacuum representations, respectively, so exterior equivalence fails.

References

• Fröhlich, Jürg. “New Super-Selection Sectors (‘Soliton-States’) in Two-Dimensional Bose Quantum Field Models.” Communications in Mathematical Physics 47 (1976): 269–310. DOI.
• Schlingemann, Dirk. “Kink States in $P(\phi)_2$-Models (An Algebraic Approach).” arXiv:hep-th/9604075 (1996). Open PDF.