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Register a new userDirac particles on periodic quantum graphs
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We consider the Dirac equation on periodic networks (quantum graphs). The self-adjoint quasi periodic boundary conditions are derived. The secular equation allowing us to find the energy spectrum of the Dirac particles on periodic quantum graphs is obtained. Band spectra of the periodic quantum graphs of different topologies are calculated. Universality of the probability to be in the spectrum for certain graph topologies is observed.
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This work deals with quantum graphs, focusing on the transmission properties they engender. We first select two simple diamond graphs, and two hexagonal graphs in which the vertices are all of degree 3, and investigate their transmission coefficients. In particular, we identified regions in which the transmission is fully suppressed. We also considered the transmission coefficients of some series and parallel arrangements of the two basic graphs, with the vertices still preserving the degree 3 condition, and then identified specific series and parallel compositions that allow for windows of no transmission. Inside some of these windows, we found very narrow peaks of full transmission, which are consequences of constructive quantum interference. Possibilities of practical use as the experimental construction of devices of current interest to control and manipulate quantum transmission are also discussed.
We discuss some basic aspects of quantum fields on star graphs, focusing on boundary conditions, symmetries and scale invariance in particular. We investigate the four-fermion bulk interaction in detail. Using bosonization and vertex operators, we solve the model exactly for scale invariant boundary conditions, formulated in terms of the fermion current and without dissipation. The critical points are classified and determined explicitly. These results are applied for deriving the charge and spin transport, which have interesting physical features.
Kinetic theory of Dirac fermions is studied within the matrix valued differential forms method. It is based on the symplectic form derived by employing the semiclassical wave packet build of the positive energy solutions of the Dirac equation. A satisfactory definition of the distribution matrix elements imposes to work in the basis where the helicity is diagonal which is also needed to attain the massless limit. We show that the kinematic Thomas precession correction can be studied straightforwardly within this approach. It contributes on an equal footing with the Berry gauge fields. In fact in equations of motion it eliminates the terms arising from the Berry gauge fields.
The semiclassical kinetic theory of Dirac particles in the presence of external electromagnetic fields and global rotation is established. To provide the Hamiltonian formulation of Dirac particles a symplectic two-form which is a matrix in spin indices is proposed. The particle number and current densities for the Dirac particles are acquired in the helicity basis. Following a similar procedure, semiclassical kinetic theory of the Weyl particles is accomplished. It is shown that the phase-space dynamics of the Weyl and Dirac particles is directly linked. The anomalous chiral effects due to the external electromagnetic fields and angular velocity of the frame are calculated.
We consider quantum graphs with transparent branching points. To design such networks, the concept of transparent boundary conditions is applied to the derivation of the vertex boundary conditions for the linear Schrodinger equation on metric graphs. This allows to derive simple constraints, which use equivalent usual Kirchhoff-type boundary conditions at the vertex to the transparent ones. The approach is applied to quantum star and tree graphs. However, extension to more complicated graph topologies is rather straight forward.
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