Harpers equation (aka the almost Mathieu equation) famously describes the quantum dynamics of an electron on a one dimensional lattice in the presence of an incommensurate potential with magnitude $V$ and wave number $Q$. It has been proven that all
states are delocalized if $V$ is less than a critical value $V_c=2t$ and localized if $V> V_c$. Here, we show that this result (while correct) is highly misleading, at least in the small $Q$ limit. In particular, for $V<V_c$ there is an abrupt crossover akin to a mobility edge at an energy $E_c$; states with energy $|E|<E_c$ are robustly delocalized, but those in the tails of the density of states, with $|E|>E_c$, form a set of narrow bands with exponentially small bandwidths $ sim t exp[-(2pialpha/Q)]$ (where $alpha$ is an energy dependent number of order 1) separated by band-gaps $ sim t Q$. Thus, the states with $|E|> E_c$ are almost localized in that they have an exponentially large effective mass and are easily localized by small perturbations. We establish this both using exact numerical solution of the problem, and by exploiting the well known fact that the same eigenvalue problem arises in the Hofstadter problem of an electron moving on a 2D lattice in the presence of a magnetic field, $B=Q/2pi$. From the 2D perspective, the almost localized states are simply the Landau levels associated with semiclassical precession around closed contours of constant quasiparticle energy; that they are not truly localized reflects an extremely subtle form of magnetic breakdown.
Because a material with an incommensurate charge density wave (ICDW) is only quasi-periodic, Blochs theorem does not apply and there is no sharply defined Fermi surface. We will show that, as a consequence, there are no quantum oscillations which are
truly periodic functions of $1/B$ (where $ B$ is the magnitude of an applied magnetic field). For a weak ICDW, there exist broad ranges of $1/B$ in which approximately periodic variations occur, but with frequencies that vary inexorably in an unending cascade with increasing $1/B$. For a strong ICDW, e.g. in a quasi-crystal, no quantum oscillations survive at all. Rational and irrational numbers really are different.
Let $M$ be an $n$-dimensional complete Riemannian manifold with Ricci curvature $ge n-1$. In cite{colding1, colding2}, Tobias Colding, by developing some new techniques, proved that the following three condtions: 1) $d_{GH}(M, S^n)to 0$; 2) the volum
e of $M$ ${text{Vol}}(M)to{text{Vol}}(S^n)$; 3) the radius of $M$ ${text{rad}}(M)topi$ are equivalent. In cite{peter}, Peter Petersen, by developing a different technique, gave the 4-th equivalent condition, namely he proved that the $n+1$-th eigenvalue of $M$ $lambda_{n+1}(M)to n$ is also equivalent to the radius of $M$ ${text{rad}}(M)topi$, and hence the other two. In this note, we give a new proof of Petersens theorem by utilizing Coldings techniques.
Quantum systems in confined geometries are host to novel physical phenomena. Examples include quantum Hall systems in semiconductors and Dirac electrons in graphene. Interest in such systems has also been intensified by the recent discovery of a larg
e enhancement in photoluminescence quantum efficiency and a potential route to valleytronics in atomically thin layers of transition metal dichalcogenides, MX2 (M = Mo, W; X = S, Se, Te), which are closely related to the indirect to direct bandgap transition in monolayers. Here, we report the first direct observation of the transition from indirect to direct bandgap in monolayer samples by using angle resolved photoemission spectroscopy on high-quality thin films of MoSe2 with variable thickness, grown by molecular beam epitaxy. The band structure measured experimentally indicates a stronger tendency of monolayer MoSe2 towards a direct bandgap, as well as a larger gap size, than theoretically predicted. Moreover, our finding of a significant spin-splitting of 180 meV at the valence band maximum of a monolayer MoSe2 film could expand its possible application to spintronic devices.
In this paper, a nonlinear revision protocol is proposed and embedded into the traffic evolution equation of the classical proportional-switch adjustment process (PAP), developing the present nonlinear pairwise swapping dynamics (NPSD) to describe th
e selfish rerouting evolutionary game. It is demonstrated that i) NPSD and PAP require the same amount of network information acquisition in the route-swaps, ii) NPSD is able to prevent the over-swapping deficiency under a plausible behavior description; iii) NPSD can maintain the solution invariance, which makes the trial and error process to identify a feasible step-length in a NPSD-based swapping algorithm is unnecessary, and iv) NPSD is a rational behavior swapping process and the continuous-time NPSD is globally convergent. Using the day-to-day NPSD, a numerical example is conducted to explore the effects of the reaction sensitivity on traffic evolution and characterize the convergence of discrete-time NPSD.
Based on the reliability budget and percentile travel time (PTT) concept, a new travel time index named combined mean travel time (CMTT) under stochastic traffic network was proposed. CMTT here was defined as the convex combination of the conditional
expectations of PTT-below and PTT-excess travel times. The former was designed as a risk-optimistic travel time index, and the latter was a risk-pessimistic one. Hence, CMTT was able to describe various routing risk-attitudes. The central idea of CMTT was comprehensively illustrated and the difference among the existing travel time indices was analysed. The Wardropian combined mean traffic equilibrium (CMTE) model was formulated as a variational inequality and solved via an alternating direction algorithm nesting extra-gradient projection process. Some mathematical properties of CMTT and CMTE model were rigorously proved. In the end, a numerical example was performed to characterize the CMTE network. It is founded that that risk-pessimism is of more benefit to a modest (or low) congestion and risk network, however, it changes to be risk-optimism for a high congestion and risk network.
The scheduling utility plays a fundamental role in addressing the commuting travel behaviours. In this paper, a new scheduling utility, termed as DMRD-SU, was suggested based on some recent research findings in behavioural economics. DMRD-SU admitted
the existence of positive arrival-caused utility. In addition, besides the travel-time-caused utility and arrival-caused utility, DMRD-SU firstly took the departure utility into account. The necessity of the departure utility in trip scheduling was analysed comprehensively, and the corresponding individual trip scheduling model was presented. Based on a simple network, an analytical example was executed to characterize DMRD-SU. It can be found from the analytical example that: 1) DMRD-SU can predict the accumulation departure behaviors at NDT, which explains the formation of daily serious short-peak-hours in reality, while MRD-SU cannot; 2) compared with MRD-SU, DMRD-SU predicts that people tend to depart later and its gross utility also decrease faster. Therefore, the departure utility should be considered to describe the travelers scheduling behaviors better.
We study three dimensional insulators with inversion symmetry, in which other point group symmetries, such as time reversal, are generically absent. Their band topology is found to be classified by the parities of occupied states at time reversal inv
ariant momenta (TRIM parities), and by three Chern numbers. The TRIM parities of any insulator must satisfy a constraint: their product must be +1. The TRIM parities also constrain the Chern numbers modulo two. When the Chern numbers vanish, a magneto-electric response parameterized by theta is defined and is quantized to theta= 0, 2pi. Its value is entirely determined by the TRIM parities. These results may be useful in the search for magnetic topological insulators with large theta. A classification of inversion symmetric insulators is also given for general dimensions. An alternate geometrical derivation of our results is obtained by using the entanglement spectrum of the ground state wave-function.
Constraint-based modeling has been widely used on metabolic networks analysis, such as biosynthetic prediction and flux optimization. The linear constraints, like mass conservation constraint, reversibility constraint, biological capacity constraint,
can be imposed on linear algorithms. However, recently a non-linear constraint based on the second thermodynamic law, known as loop law, has emerged and challenged the existing algorithms. Proven to be unfeasible with linear solutions, this non-linear constraint has been successfully imposed on the sampling process. In this place, Monte - Carlo sampling with Metropolis criterion and Simulated Annealing has been introduced to optimize the Biomass synthesis of genome scale metabolic network of Helicobacter pylori (iIT341 GSM / GPR) under mass conservation constraint, biological capacity constraint, and thermodynamic constraints including reversibility and loop law. The sampling method has also been employed to optimize a non-linear objective function, the Biomass synthetic rate, which is unified by the total income number of reducible electrons. To verify whether a sample contains internal loops, an automatic solution has been developed based on solving a set of inequalities. In addition, a new type of pathway has been proposed here, the Futile Pathway, which has three properties: 1) its mass flow could be self-balanced; 2) it has exchange reactions; 3) it is independent to the biomass synthesis. To eliminate the fluxes of the Futile Pathways in the sampling results, a linear programming based method has been suggested and the results have showed improved correlations among the reaction fluxes in the pathways related to Biomass synthesis.
How do we uniquely identify a quantum phase, given its ground state wave-function? This is a key question for many body theory especially when we consider phases like topological insulators, that share the same symmetry but differ at the level of top
ology. The entanglement spectrum has been proposed as a ground state property that captures characteristic edge excitations. Here we study the entanglement spectrum for topological band insulators. We first show that insulators with topological surface states will necessarily also have protected modes in the entanglement spectrum. Surprisingly, however, the converse is not true. Protected entanglement modes can also appear for insulators without physical surface states, in which case they capture a more elusive property. This is illustrated by considering insulators with only inversion symmetry. Inversion is shown to act in an unusual way, as an antiunitary operator, on the entanglement spectrum, leading to this protection. The entanglement degeneracies indicate a variety of different phases in inversion symmetric insulators, and these phases are argued to be robust to the introduction of interactions.
