Entanglement content of kink excitations

TL;DR

This work addresses how kink excitations in ordered 1D quantum systems imprint entanglement differently from local quasiparticles. By combining explicit lattice calculations with a field-theoretic framework based on twist fields and semilocal disorder operators, it derives universal Rényi-entropy predictions for one- and two-kink states, including tripartite geometries, and shows that the entropy difference between kink states and symmetry-broken vacua is model-independent in the large-region limit. The study further develops an algebraic approach to Kramers-Wannier duality, distinguishing entanglement of algebras under duality and introducing dual twist fields, explaining why region entanglement is not self-dual. Overall, the paper clarifies the non-local entanglement structure of kinks, provides a universal semiclassical formula for the one-kink entropy, and offers a robust framework to explore entanglement in symmetry-broken phases and dual theories.

Abstract

Quantum one-dimensional systems in their ordered phase admit kinks as elementary excitations above their symmetry-broken vacua. While the scattering properties of the kinks resemble those of quasiparticles, they have distinct locality features that are manifest in their entanglement content. In this work, we study the entanglement entropy of kink excitations. We first present detailed calculations for specific states of a spin-1/2 chain to highlight the salient features of these excitations. Second, we provide a field-theoretic framework based on the algebraic relations between the twist fields and the semilocal fields associated with the excitations, and we compute the Rényi entropies in this framework. We obtain universal predictions for the entropy difference between the excited states with a finite number of kinks and the symmetry-broken ground states, which do not depend on the microscopic details of the model in the limit of large regions. Finally, we discuss some consequences of the Kramers-Wannier duality, which relates the ordered and disordered phases of the Ising model, and we explain why, counterintuitively, no explicit relations between those phases are found at the level of entanglement.

Figures (3)

• Figure 1: Sketch of a two-kink state. Top: the two kinks are bound together and they behave as a single particle. Bottom: A kink and an antikink are deconfined along the system, and they interpolate between the two vacua.
• Figure 2: Commutation relations between the twist field $\mathcal{T}$ and the disordered operator in the $j$th replica ($\mu^j$). Top: The disorder line, in red, is deformed around the branch cut; the lower part of the line above crosses the branch cut, and it generates an additional line in the replica $j+1$. In this way, the operator $\mu^{j}\mu^{j+1}\cdot \mathcal{T}$ is obtained. Bottom: the whole disorder line is moved slightly above the branch cut and, yielding $\mu^{j+1}$.
• Figure 3: Schematic structure of the correlation functions between disorder operators and twist fields in the $n$th replica model ($n=2$). The blue/pink lines correspond to the left/right boundary points ($x=0,L$) associated with boundary conditions $(b_+)^{\otimes n}$ and $(b_-)^{\otimes n}$ respectively. The dashed black line is the branch cut and the disorder lines are depicted in red. a) Contribution coming from $\mu^1$ and $\mu^2$ inserted in $B_1 = [0,\ell_1]$. Here, after the contraction the disorder lines cancel each other and they give rise to $^2\bra{+}\mathcal{T}(\ell_2)\tilde{\mathcal{T}}(\ell_2) \ket{+}^2$. b) Contribution from $\mu^1$ and $\mu^2$ inserted at $A = [\ell_1,\ell_2]$. c) Contribution from $\mu^1$ and $\mu^2$ inserted at $B_2 = [\ell_2,L]$. d) Vanishing term associated with the insertion of $\mu^1$ in $B_1$ and $\mu^2$ in $B_2$. Here, a pair of composite twist fields is generated after the contraction and the corresponding expectation value is zero (in the large volume limit).