Momentum-dependent Mass Research Articles

Symmetry and fermion degeneracy on a lattice

We consider the general form of the finite difference approximation to the Dirac (Weyl) Hamiltonian on a lattice and investigate systematically the dependence on symmetry of the number of particles described by it. To a lattice with given symmetry, expressed by its crystallographic space group, there corresponds a minimal number of particles, which are associated with prescribed points of momentum space (the unit cell of the reciprocal lattice). We show in tables, using the existing detailed descriptions of the space groups, our results for all the relevant (symmorphic) symmetry groups. Only for lattice Hamiltonians with a momentum dependent mass term can this degeneracy be reduced or eliminated without reducing the symmetry.

New Approach to the Renormalization Group

A new set of renormalization-group equations is presented. These equations are based on a renormalization procedure in which counterterms are calculated for zero unrenormalized mass. Unlike the Gell-Mann-Low and Callan-Symanzik equations, they can be solved for arbitrary momenta. The solutions involve a momentum-dependent effective mass as well as a momentum-dependent effective coupling constant. By studying these solutions at large momenta, it can be shown that the nonleading terms discarded by previous authors do, in fact, remain negligible when the perturbation series is summed to all orders if, and only if, the effective mass vanishes at large momentum, which will be the case if a certain anomalous dimension is less than unity, as it is in asymptotically free theories. In this case, the new renormalization-group equations can be used at large momentum to derive not only the leading term, but the first three terms in an asymptotic expansion of any Green's function. These results are also applied to Wilson coefficient functions, and an important cancellation of anomalous dimensions is noted.