The Ising dual-reflection interface: $\mathbb{Z}_4$ symmetry and Majorana strong zero modes

Abstract

We investigate an interface in the transverse field quantum Ising chain connecting an ordered ferromagnetic phase and a disordered paramagnetic phase that are Kramers-Wannier duals of each other. Unlike prior studies focused on non-invertible defects, this interface exhibits a symmetry that combines Kramers-Wannier transformation with spatial reflection. We demonstrate that, under open boundary conditions, this setup gives rise to a discrete $\mathbb{Z}_4$ symmetry, encompassing the conventional $\mathbb{Z}_2$ Ising parity as a subgroup, while in a closed geometry a non-invertible symmetry emerges. Using the Jordan-Wigner transformation, we map the spin chain onto a solvable quadratic Majorana fermion system. In this formulation, the $\mathbb{Z}_4$ symmetry is realized manifestly as a parity-dependent reflection with respect to a Majorana site, in contrast to the conventional reflection which mirrors with respect to the central link of the Majorana chain. Additionally, we construct Majorana strong zero modes that retain the $\mathbb{Z}_4$ symmetry, ensure degeneracies of all energy eigenstates, and are robust under generic local symmetry-preserving perturbations of the fermion model, including interactions. Finally, we develop quantum circuit realizations of our model paving the way towards the creation of exact Majorana strong zero modes with digital quantum hardware.

Figures (6)

• Figure 1: An interface between ferromagnetic and paramagnetic Ising regions that are Kramers-Wannier duals of each other: The strengths of the Ising couplings (horizontal wavy lines) and transverse field couplings (vertical straight lines) are color-coded. By virtue of the duality, the Ising and transverse field couplings on opposite sides of the interface have the same strength. The interface Hamiltonian is a sum of the central link-Ising coupling and the transverse field acting on the spin to the right of the interface, $H_{\text{int}} = -J_0\, (X_{n}X_{n+1} + Z_{n+1}) \,.$
• Figure 2: Schematic action of the dual-reflection symmetry $S$ on the Hamiltonian \ref{['H_int']} of the dual-reflection interface: In the first step, the Kramers-Wannier transformation locally interchanges the nearest-neighbor Ising coupling with the external field strength. The second step is a spatial reflection. Given an even number of sites, we reflect about the central link.
• Figure 3: Schematic action of the Kramers-Wannier transformation combined with spatial reflection on the Ising dual-reflection interface in its fermion formulation: The empty circles represent Majorana sites, and the coupling strengths between them are colour-coded. The transformation of an individual Majorana operator $\eta_a$ under $\mathcal{S}=\mathcal{R} \mathcal{U}_{KW}$ amounts to a reflection about the Majorana site $N+1$, which exchanges $\eta_a$ and $\eta_{\widetilde{a}}\,,$ and multiplication by $\pm i \mathcal{P}$.
• Figure 4: Decomposition of the open quadratic Majorana chain with ${\mathcal{S}}$-symmetry: The fermion model \ref{['H_f']}, illustrated in the upper part of the figure, splits into two Majorana chains with the Hamiltonians $\mathcal{H^+}$ and $\mathcal{H}^-$ which decouple from each other (see Eq. \ref{['eq:H_two_Majorana']}). To reveal this, we combine the Majoranas $\eta_a$ and $\eta_{\widetilde{a}}$ into Majoranas $\xi_a^\pm$ which obey $\mathcal{S}\xi_a^\pm \mathcal{S}^{-1}=\pm i\mathcal{P}\, \xi_a^\pm$. They form the two separate chains depicted in the lower part. The Majorana $\eta_1=\xi^+_1$ belongs to the $\xi^+$ chain, but is not connected to other Majoranas in the open geometry due to locality; the interface Majorana $\eta_{N+1}=\xi_{N+1}^-$ is part of the $\xi^-$ chain.
• Figure 5: The two decoupled Majorana chains corresponding to the closed chain Hamiltonian \ref{['H_Majorana_closed']}: The link between site 1 and site 2 of the $\xi^+$ chain represents the second (antiperiodic) interface with the coupling $J'_0$. The mapping \ref{['eq:map_close']} indicated with blue arrows exchanges $\xi_a^p$ and $\xi_{N+2-a}^{-p}$ Majoranas such that the couplings change roles as $J\leftrightarrow h,\, J_0\leftrightarrow J_0^\prime\,.$