Quantum graphs with spin Hamiltonians

Abstract

The article surveys quantization schemes for metric graphs with spin. Typically quantum graphs are defined with the Laplace or Schrödinger operator which describe particles whose intrinsic angular momentum (spin) is zero. However, in many applications, for example modeling an electron (which has spin-1/2) on a network of thin wires, it is necessary to consider operators which allow spin-orbit interaction. The article presents a review of quantization schemes for graphs with three such Hamiltonian operators, the Dirac, Pauli and Rashba Hamiltonians. Comparing results for the trace formula, spectral statistics and spin-orbit localization on quantum graphs with spin Hamiltonians.1991 Mathematics Subject Classification. Primary 34B45; Secondary 81Q10, 81Q50.

Trace formulae for quantum graphs

“…This can, e.g., be a Paulior a Dirac-operator, describing spin-orbit coupling on a quantum graph. In that case the quantum graph Hilbert space is L 2 (Γ)⊗C n , and the construction of self adjoint realisations of the operators is largely similar to the case of the Laplacian [BH03], see also [Har07]. Functions on the graph now consist of n-component functions on the edges, where the n components reflect the presence of of the additional spin degree of freedom.…”

Dirac and magnetic Schrödinger operators on fractals

“…Among the problems where fractal spaces seem to appear naturally we would like to mention, in particular, the spaces of fractional dimension appearing in quantum gravity [6,69,82, and references therein]. Besides that, our motivation is coming from the theory of quantum graphs [30,31,35,42,59,60,61,62,63,64,76, and references therein]; from the spectral theory on fractals [8,27,41,57,58,67,72,73,75,84,85, and references therein]; form some questions of non-commutative analysis [23,24,18,19,20,50, and references therein] and the theory of spectral zeta functions [29,68,90,97]; and from the localization problems [1,74,77,86,95, and references therein].…”

“…A preceding article dealing with index theorems on quantum graphs is [35], and a related much earlier reference for Dirac operators is [17]. Different quantization schemes are reviewed in [42].…”

Quantum statistics on graphs

“…There one looks for systems with a degenerate ground state spanned by distinct quasiparticle configurations, in which the only (easily) realizable evolutions are, up to a phase, a discrete set of unitary transformations generated by the (adiabatic) exchange of quasiparticles. By introducing spin (Harrison 2008), quantum graphs might also provide models in which to investigate the role of quantum statistics in the quantum spin Hall effect and topological insulators (Hasan & Kane 2010, Qi & Zheng 2010.…”