Topological phase transitions for interacting finite systems

In this paper, we investigate signatures of topological phase transitions in interacting systems. We show that the key signature is the existence of a topologically protected level crossing, which is robust and sharply defines the topological transition, even in finite-size systems. Spatial symmetries are argued to play a fundamental role in the selection of the boundary conditions to be used to locate topological transitions in finite systems. We discuss the theoretical implications of these results, and utilize exact diagonalization to demonstrate its manifestations in the Haldane-Fermi-Hubbard model. Our findings provide an efficient way to detect topological transitions in experiments and in numerical calculations that cannot access the ground-state wave function.

Figure 1

(a) CDW structure factor $S_{\mathrm{CDW}}$ and Chern number $C$, (b) ground-state energy $E_0$ and first three excited state energies ($E_1$, $E_2$, and $E_3$) with an inset that shows a closeup view of the avoided level crossing at $V=V_C$, (c) excitation gaps ${\Delta }_{\mathrm{ex}}^{(1,2)}$ and a single-particle gap ${\Delta }_{\mathrm{sp}}$, and (d) fidelity metric $g(V,\delta V)$ with $\delta V={10}^{-4}$ as a function of the interaction strength $V/t_1$ for the $24C$ cluster [see the inset of (d)], and parameters $t_1=1.0$, $t_2=0.8$, and $\phi =\pi /2$. Note that this cluster does not possess all of the symmetries present in the infinite system.

Figure 2

(a) CDW structure factor $S_{\mathrm{CDW}}$ and Chern number $C$, (b) ground-state energy $E_0$ and first three excited state energies ($E_1$, $E_2$, and $E_3$) with an inset that shows a closeup view of the level crossing at $V=V_C=V_T$, (c) excitation gaps ${\Delta }_{\mathrm{ex}}^{(1,2)}$ and a single-particle gap ${\Delta }_{\mathrm{sp}}$, and (d) fidelity metric as a function of the interaction strength $V/t_1$ for the $24D$ cluster [see the inset of (d)] and the same parameters as Fig. 1.

Figure 3

$t_2$-$V$ phase diagram for the (a) $12A$, (b) $18C$, (c) $24C$, and (d) $24D$ clusters with parameters $t_1=1.0$ and $\phi =\pi /2$. The red squares indicate the onset of CDW order, and the blue circles indicate the topological transition. The shaded region marks the coexistence of CDW and topological order. All points to the vertical line are dominated by strong finite-size effects. Illustrations of the $12A$ and $18C$ clusters are shown as insets in the corresponding panel.