Dirac electron in a chiral space-time crystal created by counterpropagating circularly polarized plane electromagnetic waves

Abstract

The family of solutions to the Dirac equation for an electron moving in an electromagnetic lattice with the chiral structure created by counterpropagating circularly polarized plane electromagnetic waves is obtained. At any nonzero quasimomentum, the dispersion equation has two solutions which specify bispinor wave functions describing electron states with different energies and mean values of momentum and spin operators. The inversion of the quasimomentum results in two other linearly independent solutions. These four basic wave functions are uniquely defined by eight complex scalar functions (structural functions), which serve as convenient building blocks of the relations describing the electron properties. These properties are illustrated in graphical form over a wide range of quasimomenta. The superpositions of two basic wave functions describing different spin states and corresponding to (i) the same quasimomentum (unidirectional electron states with the spin precession) and (ii) the two equal-in-magnitude but oppositely directed quasimomenta (bidirectional electron states) are also treated.

Localized solutions of the Dirac equation in free space and electromagnetic space-time crystals

“…where g 4d (n 1 , n 2 , n 3 , n 4 ) = max{|n 1 | + |n 2 | + |n 3 |, |n 4 |} and g max is the integer specifying the domain size and hence the accuracy of such approximations [10][11][12].…”

“…Substitution of A ′ (2) and Ψ (7) in Eq. ( 5) results in the infinite system of homogeneous matrix equations relating bispinors c(n) [8,11,12]. In the general case, each amplitude c(n) enters in 13 different equations of this system.…”

“…The fundamental solution S has been expressed in terms of an infinite series of projection operators calculated by a recurrent process based on a fractal approach. This technique has been detailed and applied to various ESTCs in [8][9][10][11][12].…”

“…In a different context, the terms "time crystal" and "space-time crystal" have been used in the recent discussion [2][3][4][5][6] around the question of whether time-translation symmetry might be spontaneously broken in a time-independent, conservative classical system and a closed quantum mechanical system. In [7][8][9][10][11][12], we presented the fundamental solution of the Dirac equation for the ESTC created by six plane waves with the same frequency ω 0 and the four-dimensional wave vectors…”

“…In [7][8][9][10][11][12], we presented the fundamental solution of the Dirac equation for the ESTC created by six plane waves with the same frequency ω 0 and the four-dimensional wave vectors…”