Computing Classification of Interacting Fermionic Symmetry-Protected Topological Phases Using Topological Invariants

In recent years,great success has been achieved on the classification of symmetry-protected topological(SPT)phases for interacting fermion systems by using generalized cohomology theory.However,the explicit calculation of generalized cohomology theory is extremely hard due to the difficulty of computing obstruction functions.Based on the physical picture of topological invariants and mathematical techniques in homotopy algebra,we develop an algorithm to resolve this hard problem.It is well known that cochains in the cohomology of the symmetry group,which are used to enumerate the SPT phases,can be expressed equivalently in different linear bases,known as the resolutions.By expressing the cochains in a reduced resolution containing much fewer basis than the choice commonly used in previous studies,the computational cost is drastically reduced.In particular,it reduces the computational cost for infinite discrete symmetry groups,like the wallpaper groups and space groups,from infinity to finity.As examples,we compute the classification of two-dimensional interacting fermionic SPT phases,for all 17 wallpaper symmetry groups.

Insight into relationship between composition,thickness and electrochemistry behavior of oxide film formed on TA15under different potentials in 0.5 mol·L^(-1)H_(2)SO_(4)

The oxide film formed on TA15 alloy in0.5 mol·L^(-1)H_(2)SO_(4)at different applied potentials is investigated by evaluating the film formation current density,film thickness,surface film composition,valence state,and the electrochemical characteristics.The relative quantitative spectral analysis revealed by X-ray photoelectron spectroscopy(XPS)and X-ray absorption spectra(XAS)shows that the increasing applied potential promotes the formation of thick and high-valence state films.By contrasting with XAS of the reference titanium oxides,it is found that TiO_(2)within the oxide film changes from anatase type to rutile type with the potential increases to 2.00 V.The oxide film formed at passive region has better protection performance,and with the increase of applied potential,the corrosion resistance of passive film increases,whereas the ND(donor density)values obtained from the Mott-Schottky(MS)results decrease with the increasing applied potential,indicating a better uniformity and protective effect of the oxide film.Based on above facts,the model of the oxide films formed on TA15 at different applied potentials in passive region and dissolve region is proposed.Besides,the existence forms of all the TA15alloying elements under the most corrosion resistant condition are confirmed.