Open spin chain realization of topological defect on 1d Ising model and boundary and bulk symmetry

TL;DR

The paper investigates how topological defects in a critical 1D Ising chain with open boundaries can be absorbed into boundary states, yielding edge degrees of freedom and boundary RG phenomena. By combining BCFT concepts (Cardy/Ishibashi states, fusion rules) with lattice realizations of defects (including the KW duality defect), it shows that defects act to generate Graham-Watts boundary states and that multiple defects produce robust boundary DOF such as $2^{n}(|+\rangle+|−\rangle)$ or $2^{n}|\mathrm{free}\rangle$. The work analyzes RG behavior via the $g$-factor and boundary/bulk perturbations, demonstrating selective robustness of certain edge states under bulk perturbations and RG flows to disordered or free boundaries. It also connects Ising BCFT to fermionic (Kitaev) BCFT, highlighting dualities and potential generalizations to other $Z_N$ and anyonic chains. Overall, the results reveal nontrivial edge physics induced by defects and provide a lattice-BCFT framework for understanding boundary states and their RG evolution in topological defect-rich 1D systems.

Abstract

We study the realizations of topological defects in 1d quantum Ising model with open boundary condition at criticality. Applying the construction discussed in [M. Hauru, G. Evenbly, W. W. Ho, D. Gaiotto, and G. Vidal, Phys. Rev. B 94, 115125 (2016)], we prove that the Ising model on an open chain with multiple topological defects can be transformed to the same model with boundary magnetic fields and noninteracting boundary degrees of freedom. This results in the appearance of linear combination of Cardy states [J. L. Cardy, Nucl. Phys. B 324, 581 (1989)], which can be interpreted as an edge state of the spin or fermion chain. We show that this edge state with the large boundary entropy can be protected under bulk perturbation whereas it is fragile to a boundary perturbation. Our formulation suggests an existence of nontrivial edge physics under the existence of topological defects and opens many interesting questions for future analysis related to boundary and bulk physics.