Interacting topological mirror excitonic insulator in one dimension

We introduce the topological mirror excitonic insulator as a new type of interacting topological crystalline phase in one dimension. Its mirror-symmetry-protected topological properties are driven by exciton physics, and it manifests in the quantized bulk polarization and half-charge modes on the boundary. And the bosonization analysis is performed to demonstrate its robustness against strong correlation effects in one dimension. Besides, we also show that Rashba nanowires and Dirac semimetal nanowires could provide ideal experimental platforms to realize this new topological mirror excitonic insulating state. Its experimental consequences, such as quantized tunneling conductance in the tunneling measurement, are also discussed.

Figure 1

Schematics of two coupled nanowires for the topological mirror excitonic insulator. The left, red wire is $n$-type and the right, blue wire is $p$-type. An in-plane magnetic field $\vec{B}$ is applied along the $\widehat{x}$ direction. The top panel shows electron or hole dispersions for $n$-type (left) and $p$-type (right) wires in Eq. (1), respectively, with only one left ($L$)-moving and one right ($R$)-moving electron or hole at a chemical potential $\mu$ illustrated by the dashed horizontal line. A topological mirror excitonic insulator may be realized when the interaction between two nanowires is introduced, and half-charge modes are localized at the ends.

Figure 2

The Zak phase in different states. (a) Both mirror and chiral symmetries are preserved. (b) Mirror symmetry is broken but chiral symmetry is preserved. (c) Mirror symmetry is preserved but chiral symmetry is broken. (d) Both mirror and chiral symmetries are broken.

Figure 3

Phase diagrams. (a) Three phases: the exciton condensation (EC) insulating phase when $K_{-}>1$ and $K_{+}<1$; the topological superconducting phase (TSC) phase for $K_{\pm }>1$; and the gapless Luttinger liquid (${\mathrm{LL}}_{a,b}$) phase when $K_{-}<1$. (b) Given by $K_{-}>1$ and $K_{+}<1$, the phase diagrams for EC as functions of ${\alpha }_1$ and ${\alpha }_2$. When ${\alpha }_1<0$, the EC insulating phase breaks mirror symmetry $M_x$; otherwise, it is mirror symmetric. The expectation values of order parameters $[{\mathrm{\Delta }}_{\mathrm{ex}}^{\left(1\right)},{\mathrm{\Delta }}_{\mathrm{ex}}^{\left(2\right)}]$ are shown in each quarter.

Figure 4

Transport conductance $G_{i,j}$ in units of $e^2/h$ between leads. In (a, c) trivial and (b, d) topological phases, the bulk is fully gapped, whereas the quantized $G_{1,2}$ and $G_{3,4}$ identify the half-charge end modes in the topological mirror excitonic phase. Resonant quantization occurs when the energy goes to 0 in (b) but at a finite energy in (d), due to the breaking of chiral symmetry. Parameters used are $m_0=1,v=1,\delta \mu =0,\mu =0$, and ${\mathrm{\Delta }}_0=0.2$ in (a)–(d); $\delta m=0$ in (a) and (b) and $\delta m=0.1$ in (c) and (d); and $h=0.1$ in (a) and (c) and $h=0.4$ in (b) and (d). The barrier potential between leads and nanowires is $V_0=1.5m_0$.