Fractional topology in open systems

Abstract

We investigate the emergence of fractional topological invariants in a periodic Su-Schrieffer- Heeger chain subject to gain and loss, governed by the Gorini-Kossakowski-Sudarshan-Lindblad master equations. After preparing the symmetry condition for integer topological invariants, we investigate their transition to fractional ones in steady states, which can happen either by tuning parameters in jump operators or as a dynamical transition during time evolution. Moreover, we show that these fractional topological invariants no longer possess quantized topology in the conventional sense. However, by extending the Brillouin zone to cover multiple cycles, the total winding regains integer quantization. Finally, we show how such effects can be observed in long-range hopping photonic lattices with fractional fillings, via Bloch state tomography. Our results open a new pathway to understand fractional topology in open quantum systems.

Figures (6)

• Figure 1: The trajectory of $\hat{\delta}_i(k)$ on the Bloch sphere, symmetric about $k=0$ and $k=3\pi$, divides the surface into two equal parts.
• Figure 2: Topological phase transitions of nonequilibrium steady state in SSH model for period $6\pi$. Blue, red, and green lines correspond to period $2\pi$. Steady state is determined by $M_g$ (Eq. \ref{['mg1']}). $\gamma=0.5,1, {\rm and} 1.5$ for (a1), (b1), and (c1). (a2)-(c2) are the three-dimensional trajectory of $\langle\sigma_x \rangle(t)$, $\langle\sigma_y \rangle(t)$ and $\langle\sigma_z \rangle(t)$.
• Figure 3: Dynamical phase transitions in SSH model. Integral topological invariants to fractional topological invariants. (a1)-(c1) are the trajectory of $\langle\sigma_x \rangle(t)$ and $\langle\sigma_y \rangle(t)$, corresponding to time $t=0$, $t=0.1$, $t=0.2$, and nonequilibrium steady state, respectively. (a2)-(c2) are the three-dimensional trajectory of $\langle\sigma_x \rangle(t)$, $\langle\sigma_y \rangle(t)$ and $\langle\sigma_z \rangle(t)$. Here, $t_1=1$, $t_2=2$, $\gamma_1 =\gamma_2=0.5$ and $\gamma=1.2$. Under this setting, the steady state is topological trivial, corresponding to Fig. \ref{['ss']} (c1).
• Figure 4: The left figure is the trajectory of $\langle\sigma_x \rangle$ and $\langle\sigma_y \rangle$, , and nonequilibrium steady state. The right figure is the three-dimensional trajectory of $\langle\sigma_x \rangle$, $\langle\sigma_y \rangle$ and $\langle\sigma_z \rangle$. Here, $t_1=1$, $t_2=2$, $\gamma_1 =\gamma_2=0.5$ and $\gamma=0.35$. The steady state has winding number $1/3$.
• Figure 5: Sketch of the generalized Su-Schrieffer-Heeger model with long-range hopping. Solid lines represent the intra-cell hopping $t_1$ between $(n,A)$ and $(n,B)$ and the inter-cell hopping $t_2$ between site $(n,B)$ and $(n+3,A)$. Dashed lines are the fermion loss and gain, described by the dissipators $L_n^l$ and $L_n^g$.