Realizing and Detecting a Topological Insulator in the AIII Symmetry Class

Topological insulators in the AIII (chiral unitary) symmetry class lack experimental realization. Moreover, fractionalization in one-dimensional topological insulators has not been yet directly observed. Our work might open possibilities for both challenges. We propose a one-dimensional model realizing the AIII symmetry class which can be realized in current experiments with ultracold atomic gases. We further report on a distinctive property of topological edge modes in the AIII class: in contrast to those in the well-studied BDI (chiral orthogonal) class, they have nonzero momentum. Exploiting this feature we propose a path for the detection of fractionalization. A fermion added to an AIII system splits into two halves localized at opposite momenta, which can be detected by imaging the momentum distribution.

Figure 1

The model. (a) Schematic illustration of the model for a topological insulator in the AIII class: a one-dimensional dimerized lattice model with two sites $a$ and $b$ per unit cell, and hopping amplitudes $J^{\prime }$ and $J{e}^{i\delta }$. The phase $\delta$ leads to breaking of time reversal symmetry. (b) For periodic boundary conditions and $N$ unit cells, a total phase $N\delta$ is acquired along the whole ring.

Figure 2

Symmetries and edge modes: AIII class vs BDI class. The breaking (not-breaking) of time reversal symmetry is physically manifested in a nonzero (zero or $\pi$) momentum of the topological edge modes, characterizing the AIII (BDI) class.

Figure 3

Energy bands and bulk eigenstates. AIII class vs BDI class. (a) Energy bands for the AIII class ($\delta =-2\pi /3$, solid line) and the BDI class ($\delta =0$, dashed line), for $J^{\prime }/J=0.75$. Bulk eigenstates are fully characterized by the pair (quasimomentum $k$, isospin vector $\mathbf{n}$). (b) In the BDI class the pair is given by $[k,\mathbf{n}(k)]$. (c) In the AIII class, the correspondence between momentum and isospin is different: $[k,\mathbf{n}(k-\delta )]$, resulting in a new set of eigenstates (see text and SM [25]).

Figure 4

Topological edge modes: AIII class vs BDI class (a) Average spatial density $⟨{\widehat{n}}_x⟩$ and (b) momentum density $⟨{\widehat{a}}_k^{†}{\widehat{a}}_k⟩$ ($=⟨{\widehat{b}}_k^{†}{\widehat{b}}_k⟩$) of edge modes for $N=100$ and $J^{\prime }/J=0.9$. (a) For both classes edge modes are spatially localized at the edges of the chain. (b) For the BDI class ($\delta =0$, dashed line) both edge modes are localized at momentum $k=\pi$ (for clarity, we plot them shifted to $k=0$). For the AIII class ($\delta =-2\pi /3$, solid line), both edge modes are localized at a nonzero momentum $k=\pi +\delta$. Insets show localization lengths in position and momentum as a function of $J^{\prime }/J$.

Figure 5

Experimental scheme. (a) A double well potential with energy offset $\mathrm{\Delta }$ is created from the superposition of two standing waves, forming a one-dimensional lattice with tunneling amplitudes $j$ and $j^{\prime }$. A pair of Raman lasers is added along the $-x$ and $-y$ directions with frequency difference ${\omega }_1-{\omega }_2=\mathrm{\Delta }/\hslash$. This leads to a Hamiltonian (b), which is gauge equivalent to our model.

Figure 6

Direct observation of fractionalization in the AIII class. (a)–(d). Momentum density profiles (a), (c) for the BDI class ($\delta =0$), and (b), (d) the AIII class ($\delta =-\pi /3$), for $N=100$ and $J^{\prime }/J=0.9$. For both classes the many body state $|{\mathrm{\Phi }}_{+}⟩$ (see text) shows a flat density (a),(b). For the state $|\mathrm{\Phi }⟩$, in which a fermion is added to the flat background, the momentum density in the BDI class (c) shows a single peak of charge 1 at momentum $k=\pi$ (plotted shifted at $k=0$). In contrast, in the AIII class (d), the momentum density shows two peaks of charge $1/2$ localized at momenta $\pi \pm \delta$. (e), (f) Simultaneous fractionalization in position and momentum space in the AIII class. (e) Average spatial density and (f) momentum density for the many body state such that the upper band is empty. In the topological phase localized peaks of charge $1/2$ (at the edges of the chain, and at momenta $\pi \pm \delta$) arise.