Boundary-induced topological and mid-gap states in charge conserving one-dimensional superconductors: Fractionalization transition

We investigate one-dimensional charge conserving, spin-singlet (SSS), and spin-triplet (STS) superconductors in the presence of boundary fields. In systems with open boundary conditions (OBC) it has been demonstrated that STS display a fourfold topological degeneracy, protected by the ${\mathbb{Z}}_2$ symmetry, which reverses the spins of all fermions, whereas SSS are topologically trivial. In this paper we show that it is not only the type of the bulk superconducting instability that determines the eventual topological nature of a phase, but rather the interplay between bulk and boundary properties. In particular we show by means of the Bethe ansatz technique that SSS may as well be in a ${\mathbb{Z}}_2$-protected topological phase provided suitable “twisted” open boundary conditions $\widehat{\mathrm{OBC}}$ are imposed. More generally, we find that depending on the boundary fields, a given superconductor, either SSS or STS, may exhibit several types of phases such as topological, mid-gap, and trivial phases; each phase being characterized by a boundary fixed point, which we determine. Of particular interest are the mid-gap phases, which are stabilized close to the topological fixed point. They include fractionalized phases where spin-$1/4$ bound states are localized at the two edges of the system and unfractionalized phases where a spin-$1/2$ bound state is localized at either the left or the right edge.

Figure 1

Qualitative phase diagram of charge conserving superconductors in the presence of a boundary field. The coupling $g_{\perp }$ controls the interaction in the bulk and $\epsilon$ is a twist angle parametrizing the boundary conditions. In the $U\left(1\right)$ Thirring model under study this corresponds to $g_{\parallel }>0$ in (1) and ${\epsilon }_{\mathcal{R}}={\epsilon }_{\mathcal{L}}=\epsilon$ for the boundary conditions (9). A bulk SSS (STS) instability corresponds to $g_{\perp }>0$ ($g_{\perp }<0)$. ${\mathbb{Z}}_2$ symmetric OBC ($\widehat{\mathrm{OBC}}$) corresponds to $\epsilon =0$ ($\epsilon =\pi /2$). For a fixed boundary condition twist $\epsilon =(0,\pi /2)$, the system undergoes a quantum phase transition from a trivial phase (blue line) to a topological phase (red line) at $g_{\perp }=0$, which is Luttinger liquid (LL) phase (dark-red line). Fixing $g_{\perp }>0(g_{\perp }<0)$ as one varies the boundary twist $\epsilon$, the ${\mathbb{Z}}_2$ symmetry is broken, and one may go from a trivial (topological) phase when $\epsilon =0$ ($\epsilon =\pi /2$) to a topological (trivial) phase at $\epsilon =\pi /2$ ($\epsilon =0$). In the process, before reaching the topological phase, one enters a mid-gap region whose physics is controlled by a topological fixed point.

Figure 2

The figure shows various phases corresponding to the topological region. There are four low-lying states in the blue region representing phases $A_j$, and they form a representation of ${\mathbb{Z}}_2$. There are two low-lying states in the red region, which represents phases $B_j$, and there is only one low-lying state, which is the ground state in the yellow and green regions representing phases $C_j$. Tables 1, 2, 3 summarize the energies of low-lying states in the phases $A_j$, $B_j$, and $C_j$ respectively. Fractionalization-nonfractionalization phase transition occurs at the boundary between $A_j$ and $B_j$ or $C_j$. At boundaries within the blue region and within the red regions, as the signs of ${\epsilon }_i^{\prime }$ change, level crossings occur between the low-lying states leading to first-order phase transitions.

Figure 3

In the subphases $A_1,A_2,A_3,A_4$ spin fractionalization occurs. In the rest of the sub-phases spin fractionalization does not occur. The ground state is denoted by $g.s$ and the mid-gap energies are denoted by ${}^{\prime }{m}_{\mathcal{R}}^{\prime }{,}^{\prime }{m}_{\mathcal{L}}^{\prime }$ where $m_{\mathcal{R}}=msin({\epsilon }_{\mathcal{R}}^{\prime }\pi /2)$ and $m_{\mathcal{L}}=msin({\epsilon }_{\mathcal{L}}^{\prime }\pi /2)$. The subscripts on the states, which denote the phase are suppressed.