After the theoretical prediction that SmB6 is a topological Kondo insulator, there has been an explosion of studies on the SmB6 surface. However, there is not yet an agreement on even the most basic quantities such as the surface carrier density and mobility. In this paper, we carefully revisit Corbino disk magnetotransport studies to find those surface transport parameters. We first show that subsurface cracks exist in the SmB6 crystals, arising both from surface preparation and during the crystal growth. We provide evidence that these hidden subsurface cracks are additional conduction channels, and the large disagreement between earlier surface SmB6 studies may originate from previous interpretations not taking this extra conduction path into account. We provide an update of a more reliable magnetotransport data than the previous one (Phys. Rev. B 92, 115110) and find that the orders-of-magnitude large disagreements in carrier density and mobility come from the surface preparation and the transport geometry rather than the intrinsic sample quality. From this magnetotransport study, we find an updated estimate of the carrier density and mobility of 2.71×10 13 (1/cm 2 ) and 104.5 (cm 2 /V·sec), respectively. We compare our results with other studies of the SmB6 surface. By this comparison, we provide insight into the disagreements and agreements of the previously reported angle-resolved photoemission spectroscopy, scanning tunneling microscopy, and magnetotorque quantum oscillations measurements.

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Focusing on the geometry of one hot spot in airframes, this paper discusses the onset of the interaction of two collinear cracks at adjacent holes and defines the onset as a criterion for multi-site fatigue damage failure. The finite element method is used to calculate the stress intensity factors at the tips of two collinear cracks at adjacent holes growing towards each other. The stress intensity factor is found to increase rapidly at the onset of interaction. Since a rapid increase in stress intensity factor results in a rapid and unstable growth of the crack, the onset of the interaction is proposed as the point where the multi-site fatigue damage starts. A criterion to avoid multi-site fatigue damage locally is then established based on the separation distance of two crack tips at the onset of the interaction. To speed up the simulation of crack growth under multi-site fatigue damage with the finite element method, a semi-empirical criterion is derived to determine the time at which the stress intensity factors at the tips of the cracks correlate. The numerical examples show that the proposed criterion saves simulation time while incurring negligible relative error in the computation of the final crack length.

We examine the dynamics of entanglement entropy of all parts in an open system consisting of a two-level dimer interacting with an environment of oscillators. The dimer-environment interaction is almost energy conserving. We find the precise link between decoherence and production of entanglement entropy. We show that not all environment oscillators carry significant entanglement entropy and we identify the oscillator frequency regions which contribute to the production of entanglement entropy. Our results hold for arbitrary strengths of the dimer-environment interaction, and they are mathematically rigorous. Lemma 6.2 (Reduced oscillators state) Let J ⊆ R + and denote by ρ J (t) the reduced density matrix of the oscillators having frequencies inside J, obtained from the total density matrix by tracing out all other oscillator degrees of freedom as well as those of the dimer. ThenDepending on time t, the matrix ρ J (t) has rank either one or two. It has rank one exactly whenApart from t = 0, condition (6.15) is difficult to satisfy, so generically, the rank of ρ J (t) is two.Proof. The reduced density matrix in question is obtained from (6.5),This shows the form of ρ J (t) as given in Lemma 6.2. Since the vectors Ψ ± J (t) are normalized, their span is one-dimensional exactly if | Ψ − J (t), Ψ + J (t) | = 1. This equation is equivalent to (c.f. (6.11), (6.13)) | ψ j , D j (2λg j (1 − e ıtω j t )/ω j )ψ j | = 1, ∀j ∈ J.Proof of Lemma 6.3. The entanglement entropy S(ρ J (t)) = −Trρ J (t) log ρ J (t) of the state ρ J (t), (6.14), is expressed purely through the two non-zero eigenvalues of ρ J (t). (The operator ρ J (t) acts on an infinite-dimensional Hilbert space of | J | independent harmonic oscillators, however, all but two of its eigenvalues are zero.)To find the two eigenvalues, we write ρ J = p|Ψ + J Ψ + J | + (1 − p)|Ψ − J Ψ − J | (not exhibiting the t dependence for simplicity of notation) and decompose Then we express ρ J in the ordered orthonormal basis {Ψ + J ,η J }, whereη J = η J / η J . Namely, it has the form (2.6). The nonzero eigenvalues of ρ J are then immediately found,