Majoranas with and without a 'character': hybridization, braiding and Majorana number

In this paper we demonstrate under what conditions a pseudo-spin degree of freedom or character can be ascribed to the Majorana bound states (MBS) which can be created at the end of one dimensional non-interacting systems, corresponding to D, DIII and BDI in the usual classification scheme. We have found that such a character is directly related to the class of the topological superconductor and its description by a $\mathbb{Z}$, rather than a $\mathbb{Z}_2$, invariant which corresponds to the BDI class. We have also found that the DIII case with mirror symmetry, which supports multiple MBS, is in fact equivalent to the BDI class with an additional time-reversal symmetry. In all cases where a character can be given to the Majorana states we show how to construct the appropriate operator explicitly in various examples. We also examine the consequences of the Majorana character by considering possible hybridization of MBS brought into proximity and find that two MBS with the same character do not hybridize. Finally, we show that having this character or not has no consequence on the braiding properties of MBS.Comment: 10 pages, 1 figur

Hamilton's Turns for the Lorentz Group

Hamilton in the course of his studies on quaternions came up with an elegant geometric picture for the group SU(2). In this picture the group elements are represented by ``turns'', which are equivalence classes of directed great circle arcs on the unit sphere $S^2$, in such a manner that the rule for composition of group elements takes the form of the familiar parallelogram law for the Euclidean translation group. It is only recently that this construction has been generalized to the simplest noncompact group $SU(1,1) = Sp(2, R) = SL(2,R)$, the double cover of SO(2,1). The present work develops a theory of turns for $SL(2,C)$, the double and universal cover of SO(3,1) and $SO(3,C)$, rendering a geometric representation in the spirit of Hamilton available for all low dimensional semisimple Lie groups of interest in physics. The geometric construction is illustrated through application to polar decomposition, and to the composition of Lorentz boosts and the resulting Wigner or Thomas rotation.Comment: 13 pages, Late

Bargmann invariants and off-diagonal geometric phases for multi-level quantum systems -- a unitary group approach

We investigate the geometric phases and the Bargmann invariants associated with a multi-level quantum systems. In particular, we show that a full set of `gauge-invariant' objects for an $n$-level system consists of $n$ geometric phases and ${1/2}(n-1)(n-2)$ algebraically independent 4-vertex Bargmann invariants. In the process of establishing this result we develop a canonical form for U(n) matrices which is useful in its own right. We show that the recently discovered `off-diagonal' geometric phases [N. Manini and F. Pistolesi, Phys. Rev. Lett. 8, 3067 (2000)] can be completely analysed in terms of the basic building blocks developed in this work. This result liberates the off-diagonal phases from the assumption of adiabaticity used in arriving at them.Comment: 13 pages, latex, no figure

Examining the relationship between pubertal stage, adolescent health behaviours and stress

Background. This paper examines the associations between puberty and three important health behavlours (smoking, food intake and exercise) and explores whether these associations are mediated by puberty's relationship to stress and psychological difficulties.Method. Data were taken from the first year of the ongoing, 5-year, Health and Behaviours in Teenagers Study (HABITS). This is a school-based study set in 36 schools in London. In the first year of the study, 4320 students (2578 boys, 1742 girls) in their first year of secondary education took part.Results. Among girls, being more pubertally advanced was associated with a greater likelihood of having tried smoking. Among boys, being more pubertally advanced was associated with a greater likelihood of having tried smoking, a higher intake of high-fat food and higher levels of exercise. More pubertally advanced girls experienced more stress but not more psychological difficulties. There were no associations between puberty and either stress or psychological difficulties in boys. Stress and psychological difficulties were associated with health behaviours in girls and boys, but neither of these factors mediated the relationship between pubertal stage and health behaviours found in girls.Conclusions. These results suggest that the onset of puberty has a marked effect on the development of health behaviours. Puberty was related to an acceleration of the development of unhealthy behaviours, except for exercise behaviour in boys, where advanced puberty was associated with more exercise. These changes were unrelated to adolescent issues of stress and a causal explanation for these associations must be sought elsewhere

Evolution of the rates of mass wasting and fluvial sediment transfer from the epicentral area of the 1999, Mw 7.6 earthquake

The 1999 Chichi earthquake (Mw=7.6) triggered more than 20,000 landslides in the epicentral area in central west Taiwan, and subsequent typhoons have caused an even larger number of slope failures. As a result, the suspended sediment load of the epi- central Choshui River has increased dramatically. Measurements of suspended sedi- ment at a downstream gauging station indicate that the unit sediment concentration increased about six times due to the earthquake, and decreased exponentially due to flushing by subsequent typhoons. The e-folding time scale of the seismic perturbation of sediment transfer in the Choshui River is 3-5 years. Based on this estimate of the de- cay of the erosional response to the earthquake, a mass balance can be calculated for the earthquake, including co-seismic uplift and subsidence, post-seismic relaxation, and erosion. This mass balance shows that the Chi-Chi earthquake has acted to build ridge topography in the hanging wall of the fault, but in the far field, some destruc- tion of topography has occurred. However, our estimate of seismically-driven erosion may be incomplete. A detailed analysis of landsliding in the Chenyoulan tributary of the Choshui River indicates that most co-and post seismic landslide debris remains on hillslopes within the catchment. Recent typhoons have continued to cause high rates of landsliding high in the landscape, but rates of mass wasting near the stream net- work have decreased. The full geomorphic response to the Chi-Chi earthquake may be much larger, and more protracted than indicated by river gauging data

A generalized Pancharatnam geometric phase formula for three level systems

We describe a generalisation of the well known Pancharatnam geometric phase formula for two level systems, to evolution of a three-level system along a geodesic triangle in state space. This is achieved by using a recently developed generalisation of the Poincare sphere method, to represent pure states of a three-level quantum system in a convenient geometrical manner. The construction depends on the properties of the group SU(3)\/ and its generators in the defining representation, and uses geometrical objects and operations in an eight dimensional real Euclidean space. Implications for an n-level system are also discussed.Comment: 12 pages, Revtex, one figure, epsf used for figure insertio

Resources Required for Topological Quantum Factoring

We consider a hypothetical topological quantum computer where the qubits are comprised of either Ising or Fibonacci anyons. For each case, we calculate the time and number of qubits (space) necessary to execute the most computationally expensive step of Shor's algorithm, modular exponentiation. For Ising anyons, we apply Bravyi's distillation method [S. Bravyi, Phys. Rev. A 73, 042313 (2006)] which combines topological and non-topological operations to allow for universal quantum computation. With reasonable restrictions on the physical parameters we find that factoring a 128 bit number requires approximately 10^3 Fibonacci anyons versus at least 3 x 10^9 Ising anyons. Other distillation algorithms could reduce the resources for Ising anyons substantially.Comment: 4+epsilon pages, 4 figure

HI Imaging of LGS 3 and an Apparently Interacting High-Velocity Cloud

We present a 93' by 93' map of the area near the Local Group dwarf galaxy LGS 3, centered on an HI cloud 30' away from the galaxy. Previous authors associated this cloud with LGS 3 but relied on observations made with a 36' beam. Our high-resolution (3.4'), wide-field Arecibo observations of the region reveal that the HI cloud is distinct from the galaxy and suggest an interaction between the two. We point out faint emission features in the map that may be gas that has been tidally removed from the HI cloud by LGS 3. We also derive the rotation curve of the cloud and find that it is in solid-body rotation out to a radius of 10', beyond which the rotation velocity begins to decline. Assuming a spherical geometry for the cloud, the implied mass is 2.8 x 10^7 (d/Mpc) M_{Sun}, where d is the distance in Mpc. The observed HI mass is 5.5 x 10^6 (d/Mpc)^2 M_{Sun}, implying that the cloud is dark-matter dominated unless its distance is at least 1.9 Mpc. We propose that the cloud is a high-velocity cloud that is undergoing a tidal interaction with LGS 3 and therefore is located roughly 700 kpc away from the Milky Way. The cloud then contains a total mass of ~2.0 x 10^7 M_{Sun}, 82% of which consists of dark matter.Comment: 5 pages, 2 color figures. Accepted for publication in ApJ Letter

Numerical Analysis of Quasiholes of the Moore-Read Wavefunction

We demonstrate numerically that non-Abelian quasihole excitations of the $\nu = 5/2$ fractional quantum Hall state have some of the key properties necessary to support quantum computation. We find that as the quasihole spacing is increased, the unitary transformation which describes winding two quasiholes around each other converges exponentially to its asymptotic limit and that the two orthogonal wavefunctions describing a system with four quasiholes become exponentially degenerate. We calculate the length scales for these two decays to be $\xi_{U} \approx 2.7 \ell_0$ and $\xi_{E} \approx 2.3 \ell_0$ respectively. Additionally we determine which fusion channel is lower in energy when two quasiholes are brought close together.Comment: 4 pages, 3 figure