^{3}.

Hawking’s calculation is unable to predict the final stage of the black hole evaporation. When effects of quantum gravity are taken into account, there is a minimal observable length. In this paper, we investigate fermions’ tunnelling from the charged and rotating black strings. With the influence of the generalized uncertainty principle, the Hawking temperatures are not only determined by the rings, but also affected by the quantum numbers of the emitted fermions. Quantum gravity corrections slow down the increases of the temperatures, which naturally leads to remnants left in the evaporation.

Hawking radiation is a quantum tunnelling phenomenon of particles across black holes’ horizons. To describe this phenomenon, the semiclassical tunnelling method, which relies on calculating the imaginary part of emission particle’s action, was put forward [

The null geodesic method and the Hamilton-Jacobi method are usual methods employed to derive the imaginary part [

Various theories of quantum gravity predict the existence of a minimal observable length [

This modification plays an important role in quantum gravity. With considerations of modifications, the cosmological constant problem was discussed and the finiteness of the constant was derived in [

In this paper, we extend this work to anti-de Sitter spacetime and investigate the tunnelling radiation of fermions from black strings, where effects of quantum gravity are taken into account. Black strings are cylindrically symmetric solutions of the Einstein-Maxwell equations with a negative cosmological constant. The solutions are asymptotically anti-de Sitter in transverse direction and along the axis. There are three Killing vectors,

The rest is organized as follows. In the next section, using operators of position and momentum defined in [

Here we adopt the modified fundamental commutation relation put forward in [

The 4-dimensional neutral black string solutions to Einstein-Maxwell equations with a negative cosmological constant were derived in [

For a spin-1/2 fermion, there are two states corresponding to spin-up and spin-down. Here we only investigate the state with spin-up. The investigation of the state with spin-down is parallel and the same result can be obtained. To describe the motion of a charge fermion, we suppose that the wave function takes on the form

It is shown that the corrected temperature appears and is lower than the standard one. The correction is not only determined by the mass and charge of the black string, but also affected by the quantum number (mass, charge, and energy) of the emitted fermion. Quantum gravity correction slows down the increase of the Hawking temperature caused by the evaporation. Finally, the black string is in a balance state. At this state, the evaporation stops and the remnant is produced.

It is of interest to discuss the corrected area entropy. The entropy can be derived by the first law of thermodynamics with the corrected temperature (

In this section, we investigate uncharged fermions’ tunnelling from the event horizon of a rotating black string. Therefore, effects of the electromagnetic field in the generalized Dirac equation are not taken into account here. The rotating black string solution in a spacetime asymptotically anti-de Sitter in the radial direction was derived by Lemos and Zanchin [

To find the temporal contribution, we use the Kruskal coordinates

In this paper, taking into account the influence of quantum gravity, we modified the Dirac equation in curved spacetime by the modified fundamental commutation relations put forward in [

When

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by the National Natural Science Foundation of China with Grant no. 11205125.