Pseudospectral computational methods for the time-dependent Dirac equation in static curved spaces

Pseudospectral numerical schemes for solving the Dirac equation in general static curved space are derived using a new pseudodifferential representation of the Dirac equation along with a simple Fourier-basis technique. Owing to the presence of non-constant coefficients in the curved space Dirac equation, convolution products usually appear when the Fourier transform is performed. To circumvent this problem, a strategy based on operator splitting and pseudodifferential operators allows for using an ordinary fast Fourier transform algorithm. The resulting numerical methods are efficient and have spectral convergence. Simultaneously, wave absorption at the boundary can be easily included using absorbing layers to cope with some potential unwanted effects of periodic conditions inherent to spectral methods on bounded domains. The numerical schemes are first tested on simple systems to verify the convergence, and are then applied to the dynamics of charge carriers in strained graphene.

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