Fractional topological insulators: From sliding Luttinger liquids to Chern-Simons theory

The sliding Luttinger liquid approach is applied to study fractional topological insulators (FTIs). We show that a FTI is the low energy fixed point of the theory for realistic spin-orbit and electron-electron interactions. We find that the topological phase pertains in the presence of an interaction that breaks the spin invariance, and its boundaries are even extended by those terms. Finally we show that the one-dimensional chiral anomaly in the Luttinger liquid leads to the emergence of topological Chern-Simons terms in the effective gauge theory of the FTI state.

Figure 1

Wire arrangement considered for the construction of fractional Hall states and FTIs with time reversal. In the former case a constant magnetic field $B\widehat{z}$ is assumed. For the FTI construction, the presence of Rashba interaction and a special external confining potential is assumed (see text).

Figure 2

Dispersion relations $E_j\left(k\right)$ for the different wires. Due to the Rashba term, the energies of different eigenstates are displaced to opposite sides around zero momentum. The ever increasing Rashba terms induce a different momentum shift for each wire. The arrows in the top of each parabola indicate the corresponding spin projection. Around the Fermi level, tunneling processes are allowed (see the main text) and they are represented by red (green) arrows for the $+ (-)$ spin states.

Figure 3

Renormalization flows for Eqs. (56) and (57). The different lines indicate the different initial strengths of $g/u_{\rho }$. The thick dashed line represents the separatrix of the BKT transition. When the initial $g/u_{\rho }$ is above $g^{*}/u_{\rho }$, the system flows to the strong coupling regime (fractional topological phase), while for initial values below $g^{*}/u_{\rho }$ the system flows to weak coupling regime (sliding LL). This plot is made for $c=0.1$.

Figure 4

Various separatrix lines with various interaction strengths $c$. (a) For a generic electron interaction (finite $c$) the critical point $\mathrm{\Delta }$ shifts to the right with increasing $c$. (b) The slope of the separatrix lines decreases as a function of $c$; that means that the region where the system flows to strong coupling (i.e the topological phase) is enlarged.

Figure 5

Arrangement of concentric wires appropriate to realize a FQHE with disk geometry.