Multi-kink topological terms and charge-binding domain-wall condensation induced symmetry-protected topological states: Beyond Chern-Simons/BF theory

TL;DR

This work introduces a hydrodynamic framework to realize bosonic Abelian SPT states beyond conventional Chern-Simons/BF theories via multi-kink domain-wall condensation, where intersections carry symmetry charges and drive nontrivial SPT order. It develops two bulk descriptions—the U^k(1) nonlinear sigma model with a multi-kink term (featuring an anomalous U(1) but anomaly-free Z_N^k) and a beyond-Abelian gauge theory—then shows their low-energy equivalence and reproduces SPT invariants under symmetry twists. The authors compute explicit SPT invariants in 1+1D and 2+1D for Z_{N1}×Z_{N2} and Z_{N1}×Z_{N2}×Z_{N3} SPTs (including type III), and extend the construction to 3+1D with quad-kink terms, along with edge theories and a DW dual description. The framework clarifies how decorated domain-wall condensation yields nontrivial SPTs and provides practical methods to extract bulk invariants, partition functions, and edge degeneracies, enriching the landscape of topological phases beyond traditional gauge theories. The results have potential implications for engineered SPTs, their boundary physics, and dualities with Dijkgraaf-Witten-type gauge theories.

Abstract

Quantum-disordering a discrete-symmetry breaking state by condensing domain-walls can lead to a trivial symmetric insulator state. In this work, we show that if we bind a 1D representation of the symmetry (such as a charge) to the intersection point of several domain walls, condensing such modified domain-walls can lead to a non-trivial symmetry-protected topological (SPT) state. This result is obtained by showing that the modified domain-wall condensed state has a non-trivial SPT invariant -- the symmetry-twist dependent partition function. We propose two different kinds of field theories that can describe the above mentioned SPT states. The first one is a Ginzburg-Landau-type non-linear sigma model theory, but with an additional multi-kink domain-wall topological term. Such theory has an anomalous $U^k(1)$ symmetry but an anomaly-free $Z_N^k$ symmetry. The second one is a gauge theory, which is beyond Abelian Chern-Simons/BF gauge theories. We argue that the two field theories are equivalent at low energies. After coupling to the symmetry twists, both theories produce the desired SPT invariant.

Figures (8)

• Figure 1: (a) Disordering a $U(1)$-symmetry breaking superfluid with an action by condensing the vortices, e.g., tuning some coupling constant U to increase the charge repulsion. Fisher-LeeDasgupta-HalperinNelson. (b) Disordering a discrete-symmetry breaking state by condensing the domain walls. The gray region qualitatively indicates the phase transition region, such as a critical point or a different phase. (c) In this work, we generalize the previous process by condensing domain walls with a multi-kink topological terms. We obtain nontrivial SPT states with SPT invariants listed in Table \ref{['table:field_theory']}.
• Figure 2: (a) Symmetry twist along the boundary $\prt R$ is generated by the symmetry transformation that act only within $R$. (b) The symmetry twist $h_x,h_y$ on torus gives rise to the twisted ground state ${|\Psi_{(h_x,h_y)}\>}$.
• Figure 3: $\hat{S}$-move is $90^\circ$ rotation.
• Figure 4: $\hat{T}$-move is the Dehn twist followed by a symmetry transformation $h_x$ in the shaded area.
• Figure 5: A closed orbit in the $(h_x,h_y)$ space.