3-Methyl­anilinium hydrogen phthalate

The asymmetric unit of the title salt, C7H10N+·C8H5O4 −, consists of two 3-methyl­phenyl­ammonium cations and two hydrogen phthalate anions. There are strong intra­molecular O—H⋯O hydrogen bonds in the virtually planar (r.m.s. deviations = 0.054 Å) phthalate anions. In the crystal, the cations and anions are connected via an extensive sytem of N—H⋯O hydrogen bonds into a corrugated layer extended parallel to (001)

Tris(hy­droxy­meth­yl)methanaminium trifluoro­acetate

In the crystal structure of the title salt, C4H12NO3 +·C2F3O2 −, N—H⋯O and O—H⋯O hydrogen bonds link the ions, forming a complex three-dimensional network

Bis(2-trifluoro­methyl-1H-benzimidazol-3-ium) naphthalene-1,5-disulfonate

The asymmetric unit of the title compound, C8H6F3N2 +·0.5C10H6O6S2 2−, consists of one 2-trifluoro­methyl-1H-benz­imidazol-3-ium cation and a half naphthalene-1,5-disulfate anion, which are linked by an N—H⋯O hydrogen bond. The anion sits across a centre of symmetry. The atoms of the benzimidazole ring are nearly coplanar (r.m.s. deviation of the fitted atoms = 0.0085 Å) and the triflouromethyl group lies out of this plane. In the crystal, the cations are linked to adjacent anions by N—H⋯O hydrogen bonds, forming a ladder structure parallel to the a axis in which the anions form the rungs. Adjacent ladders are linked by weak C—H⋯O inter­actions, forming sheets parallel to the ac plane

Intertwined fractional quantum anomalous Hall states and charge density waves

Motivated by the recent experimental breakthrough on the observation of the fractional quantum anomalous Hall (FQAH) effects in semiconductor and graphene moir\'{e} materials, we explore the rich physics associated with the coexistence of FQAH effect and the charge density wave (CDW) order that spontaneously breaks the translation symmetry. We refer to a state with both properties as "FQAH-crystal". We show that the interplay between FQAH effect and CDW can lead to a rich phase diagram including multiple topological phases and topological quantum phase transitions at the same moir\'e filling. In particular, we demonstrate the possibility of direct quantum phase transitions from a FQAH-crystal with Hall conductivity $\sigma_H = - 2/3$ to a trivial CDW insulator with $\sigma_H = 0$, and more interestingly, to a QAH-crystal with $\sigma_H= -1$.Comment: 9 page