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We present a generalization of the geometric phase to pure and thermal states in $\mathcal{PT}$-symmetric quantum mechanics (PTQM) based on the approach of the interferometric geometric phase (IGP). The formalism first introduces the parallel-transport conditions of quantum states and reveals two geometric phases, $\theta^1$ and $\theta^2$, for pure states in PTQM according to the states under parallel-transport. Due to the non-Hermitian Hamiltonian in PTQM, $\theta^1$ is complex and $\theta^2$ is its real part. The imaginary part of $\theta^1$ plays an important role when we generalize the IGP to thermal states in PTQM. The generalized IGP modifies the thermal distribution of a thermal state, thereby introducing effective temperatures. \textcolor{red}{At certain critical points, the generalized IGP may exhibit discrete jumps at finite temperatures, signaling a geometric phase transition. We illustrate the IGP of PTQM through two examples and compare their differences}. 

Stepwise deprotonation of truxene with alkali metals affords truxenyl anions with different charges, exhibiting core curvature dependence on charge and metal binding. UV-vis and PL studies reveal charge-dependent optical properties, which is further supported by DFT calculations.

 

A series of dibenzannulated phenyl-annulated [4,2] peri -acenoacenes have been synthesized in three straightforward steps from 4,10-dibromoanthanthrone (vat orange 3). The phenyl bisannulation of [4,2] peri -acenoacene provides extra stability by increasing the overall aromatic character of the molecules, and allows for a 45–80% increase of the molar extinction coefficient ( ε ) compared to their [5,2] peri -acenoacene isomers. Depending on the substituents attached to the π-conjugated core, some derivatives exhibit strong aggregation in the solid state with association constant ( K a ) up to 255 M −1 , resulting in a significant broadening of the absorption spectrum and a substantial decrease of the bandgap value (more than 0.3 V) from solution to the solid state. One [4,2] peri -acenoacene derivative was doubly reduced using cesium and the crystal structure of the resulting salt has been obtained. Field-effect transistors showing a temperature-dependent hole mobility have been tested.