The quantum statistics mechanism is very powerful for investigating the equilibrium states and the phase transitions in complex spin disorder systems. The spin disorder systems act as an interdisciplinary platform for solving the optimum processes in computer science. In this work, I determined the lower bound of the computational complexity of knapsack problems. I investigated the origin of nontrivial topological structures in these hard problems. It was uncovered that the nontrivial topological structures arise from the contradictory between the three-dimensional character of the lattice and the two-dimensional character of the transfer matrices used in the quantum statistics mechanism. I illustrated a phase diagram for the non-deterministic polynomial (NP) vs polynomial (P) problems, in which a NP-intermediate (NPI) area exists between the NP-complete problems and the P-problems, while the absolute minimum core model is at the border between the NPI and the NP-complete problems. The absolute minimum core model of the knapsack problem cannot collapse directly into the P-problem. Under the guide of the results, one may develop the best algorithms for solving various optimum problems in the shortest time (improved greatly from O(1.3^N) to O((1+ε)^N) with ε→0 and ε≠1/N) being in subexponential and superpolynomial. This work illuminates the road on various fields of science ranging from physics to biology to finances, and to information technologies.

We report the photovoltaic effects of n-type topological insulator (TI) Bi₂Te₃ films grown on p-type Si substrates by chemical vapor deposition (CVD). The films containing large nanoplates with a smooth surface formed on p-Si exhibit good p–n diode characteristics under dark and light illumination conditions and display a good photovoltaic effect under the broadband range from ultraviolet (UV) to near infrared (NIR) wavelengths. Under the light illumination with a wavelength of 1, 000 nm, a short circuit current (I_SC) of 19.2 μA and an open circuit voltage (V_OC) of 235 mV are achieved. The maximum fill factor (FF) increases with a decrease in the wavelength or light density, achieving a value of 35.6% under 600 nm illumination. The photoresponse of the n-Bi₂Te₃/p-Si device can be effectively switched between the on and off modes in millisecond time scale. These findings are important for both the fundamental understanding and solar cell device applications of TI materials.