The topological classification of one-dimensional symmetric quantum walks

We give a topological classification of quantum walks on an infinite 1D lattice, which obey one of the discrete symmetry groups of the tenfold way, have a gap around some eigenvalues at symmetry protected points, and satisfy a mild locality condition. No translation invariance is assumed. The classification is parameterized by three indices, taking values in a group, which is either trivial, the group of integers, or the group of integers modulo 2, depending on the type of symmetry. The classification is complete in the sense that two walks have the same indices if and only if they can be connected by a norm continuous path along which all the mentioned properties remain valid. Of the three indices, two are related to the asymptotic behaviour far to the right and far to the left, respectively. These are also stable under compact perturbations. The third index is sensitive to those compact perturbations which cannot be contracted to a trivial one. The results apply to the Hamiltonian case as well. In this case all compact perturbations can be contracted, so the third index is not defined. Our classification extends the one known in the translation invariant case, where the asymptotic right and left indices add up to zero, and the third one vanishes, leaving effectively only one independent index. When two translationally invariant bulks with distinct indices are joined, the left and right asymptotic indices of the joined walk are thereby fixed, and there must be eigenvalues at $1$ or $-1$ (bulk-boundary correspondence). Their location is governed by the third index. We also discuss how the theory applies to finite lattices, with suitable homogeneity assumptions.Comment: 36 pages, 7 figure

Propagation and spectral properties of quantum walks in electric fields

We study one-dimensional quantum walks in a homogeneous electric field. The field is given by a phase which depends linearly on position and is applied after each step. The long time propagation properties of this system, such as revivals, ballistic expansion and Anderson localization, depend very sensitively on the value of the electric field $\Phi$, e.g., on whether $\Phi/(2\pi)$ is rational or irrational. We relate these properties to the continued fraction expansion of the field. When the field is given only with finite accuracy, the beginning of the expansion allows analogous conclusions about the behavior on finite time scales.Comment: 7 pages, 4 figure

Random Aharonov-Bohm vortices and some funny families of integrals

A review of the random magnetic impurity model, introduced in the context of the integer Quantum Hall effect, is presented. It models an electron moving in a plane and coupled to random Aharonov-Bohm vortices carrying a fraction of the quantum of flux. Recent results on its perturbative expansion are given. In particular, some funny families of integrals show up to be related to the Riemann $\zeta(3)$ and $\zeta(2)$.Comment: 10 page

Hadronic Dijet Imbalance and Transverse-Momentum Dependent Parton Distributions

We compare several recent theoretical studies of the single transverse spin asymmetry in dijet-correlations at hadron colliders. We show that the results of these studies are all consistent. To establish this, we investigate in particular the two-gluon exchange contributions to the relevant initial and final state interactions in the context of a simplifying model. Overall, the results confirm that the dijet imbalance obeys at best a non-standard or "generalized" transverse-momentum-dependent factorization.Comment: 13 pages, 3 figure