TY - JOUR
T1 - Quantum walks in external gauge fields
AU - Cedzich, C.
AU - Geib, T.
AU - Werner, A. H.
AU - Werner, R. F.
PY - 2019
Y1 - 2019
N2 - Describing a particle in an external electromagnetic field is a basic task of quantum mechanics. The standard scheme for this is known as "minimal coupling" and consists of replacing the momentum operators in the Hamiltonian by the modified ones with an added vector potential. In lattice systems, it is not so clear how to do this because there is no continuous translation symmetry, and hence, there are no momenta. Moreover, when time is also discrete, as in quantum walk systems, there is no Hamiltonian, but only a unitary step operator. We present a unified framework of gauge theory for such discrete systems, keeping a close analogy to the continuum case. In particular, we show how to implement minimal coupling in a way that automatically guarantees unitary dynamics. The scheme works in any lattice dimension, for any number of internal degrees of freedom, for walks that allow jumps to a finite neighbourhood rather than to nearest neighbours, is naturally gauge invariant, and prepares possible extensions to non-abelian gauge groups.
AB - Describing a particle in an external electromagnetic field is a basic task of quantum mechanics. The standard scheme for this is known as "minimal coupling" and consists of replacing the momentum operators in the Hamiltonian by the modified ones with an added vector potential. In lattice systems, it is not so clear how to do this because there is no continuous translation symmetry, and hence, there are no momenta. Moreover, when time is also discrete, as in quantum walk systems, there is no Hamiltonian, but only a unitary step operator. We present a unified framework of gauge theory for such discrete systems, keeping a close analogy to the continuum case. In particular, we show how to implement minimal coupling in a way that automatically guarantees unitary dynamics. The scheme works in any lattice dimension, for any number of internal degrees of freedom, for walks that allow jumps to a finite neighbourhood rather than to nearest neighbours, is naturally gauge invariant, and prepares possible extensions to non-abelian gauge groups.
UR - http://www.scopus.com/inward/record.url?scp=85060791018&partnerID=8YFLogxK
U2 - 10.1063/1.5054894
DO - 10.1063/1.5054894
M3 - Journal article
AN - SCOPUS:85060791018
SN - 0022-2488
VL - 60
JO - Journal of Mathematical Physics
JF - Journal of Mathematical Physics
IS - 1
M1 - 012107
ER -