It is known that there are two mechanisms for localizing a bulk fermion on a brane, one is the well-known Yukawa coupling and the other is the new coupling proposed in [Phys. Rev. D 89, 086001 (2014)]. In this paper, we investigate localization and resonance spectrum of a bulk fermion on the same branes with the two localization mechanisms. It is found that both the two mechanisms can result in a volcano-like effective potential of the fermion Kaluza-Klein modes. The left-chiral fermion zero mode can be localized on the brane and there exist some discrete massive fermion Kaluza-Klein modes that quasilocalized on the brane (also called fermion resonances). The number of the fermion resonances increases linearly with the coupling parameter.

The "complexity = action" duality states that the quantum complexity is equal to the action of the stationary AdS black hole within the Wheeler-DeWitt patch at late time approximation. We compute the action growth rates of the neutral and charged black holes in massive gravity and the neutral, charged and Kerr-Newman black holes in f (R) gravity to test this conjecture. Besides, we investigate the effects of the massive graviton terms, higher derivative terms and the topology of the black hole horizon on the complexity growth rate.

In this paper, we investigate the braneworld scenario in f (T ) gravity with a K-field as the background field. We consider various different specific forms of f (T ) gravity and K-field, and find a general way to construct the braneworld model. Based on our solutions, the split of branes is investigated. Besides, the stability of the braneworld is studied by investigating the tensor perturbation of the vielbein. I. INTRODUCTIONOne of the most well-known and earliest extra dimension theories was first proposed by T. Kaluza [1] and O. Klein [2] to unify Einstein's general relativity and Maxwell's electromagnetism in the 1920s. The extra dimension theories drew wide attention with the work of N. Arkani-Hamed, S. Dimopoulos, and G. R. Dvali [3] and the works of L. Randall and R. Sundrum [4, 5] in the end of the 20th century. Later, various braneworld scenarios were developed such as the Gregory-Rubakov-Sibiryakov (GRS) model [6], the Dvali-Gabadadze-Porrati (DGP) model [7], the thick brane model [8-17], the universal extra dimension model[18], etc. Among these theories, one important model is the thick brane theory originated from the domain wall model proposed by V. A. Rubakov and M. E. Shaposhnikov [19] in 1983.In the thick brane model, the brane could be generated by scalar fields [11,[20][21][22][23][24][25][26], as well as vector fields and spinor fields [12,27,28]. In addition, there are also braneworld models without matter fields [14,29,30]. The standard model fields in the bulk can be localized near the brane [6, 11,[31][32][33][34].The braneworld was also studied in different modified gravity theories, for example, the scalar-tensor gravity theory [35][36][37][38][39][40][41][42][43][44], the metric f (R) gravity theory [45][46][47] and the Palatini f (R) theory [48,49]. As f (T ) gravity theory came up as an alternative to dark energy for the explanation of the acceleration of the universe [50], it was then widely investigated [51][52][53][54][55]. Braneworld models in f (T ) theories were studied in Refs. [56][57][58]. In the previous works [56,57], the solutions of braneworld scenarios in f (T ) gravity with the form of f (T ) = T + αT n were investigated by the first-order formalism, i.e., the superpotential way. Besides, the split of the brane in f (T ) gravity was given in Ref. [56]. Furthermore, the tensor perturbation of the braneworld was also studied in Ref. [58] and it was shown that the solutions to the f (T ) braneworld were stable. The localization of matter fields was also investigated in Ref. [56]. *