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We develop a variational approximation to the entanglement entropy for scalar ?4?4 theory in 1+11+1, 2+12+1, and 3+13+1 dimensions, and then examine the entanglement entropy as a function of the coupling. We find that in 1+11+1 and 2+12+1 dimensions, the entanglement entropy of ?4?4 theory as a function of coupling is monotonically decreasing and convex. While ?4?4 theory with positive bare coupling in 3+13+1 dimensions is thought to lead to a trivial free theory, we analyze a version of ?4?4 with infinitesimal negative bare coupling, an asymptotically free theory known as precarious  ?4?4 theory, and explore the monotonicity and convexity of its entanglement entropy as a function of coupling. Within the variational approximation, the stability of precarious ?4?4 theory is related to the sign of the first and second derivatives of the entanglement entropy with respect to the coupling.  相似文献   

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A curve αα immersed in the three-dimensional sphere S3S3 is said to be a Bertrand curve if there exists another curve ββ and a one-to-one correspondence between αα and ββ such that both curves have common principal normal geodesics at corresponding points. The curves αα and ββ are said to be a pair of Bertrand curves in S3S3. One of our main results is a sort of theorem for Bertrand curves in S3S3 which formally agrees with the classical one: “Bertrand curves in S3S3 correspond to curves for which there exist two constants λ≠0λ0 and μμ such that λκ+μτ=1λκ+μτ=1”, where κκ and ττ stand for the curvature and torsion of the curve; in particular, general helices in the 3-sphere introduced by M. Barros are Bertrand curves. As an easy application of the main theorem, we characterize helices in S3S3 as the only twisted curves in S3S3 having infinite Bertrand conjugate curves. We also find several relationships between Bertrand curves in S3S3 and (1,3)-Bertrand curves in R4R4.  相似文献   

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In this paper we show that for a compact minimal hypersurface MM of constant scalar curvature in the unit sphere S6S6 with the shape operator AA satisfying ‖A‖2>5A2>5, there exists an eigenvalue λ>10λ>10 of the Laplace operator of the hypersurface MM such that ‖A‖2=λ−5A2=λ5. This gives the next discrete value of ‖A‖2A2 greater than 0 and 5.  相似文献   

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The effects associated to the length of stabilograms, a measure of the time dependence of the center of pressure of an individual standing up, are analyzed. The fractal characteristics of 27 signals with a length of 214214 points, each one corresponding to a different individual, are studied by using the Detrended Fluctuation Analysis technique. The properties of the complete signals are compared to those of various subsignals extracted from them. No differences have been found between the characteristic exponents found for xx and yy signals. The relation between the exponents of the position and velocity signals is accomplished by the 214214 point signals, while subsignals with up to 212212 points do not verify it. Using artificial signals with 214214 points, generated for αα values given, it has been demonstrated that the exponents obtained from these signals take values larger than expected for α<0.3α<0.3, while the exponents of the accumulated series are smaller than expected for 0.7<α0.7<α. For CoP trajectories this indicates that DFA-1 provides feasible exponents for the short ττ-end region of the velocity signal and the large ττ-end region of the accumulated (position) one. It has been found that the characteristic exponents vary along the series. A slightly larger persistence is found in the last part of the signal for large frequencies in the xx direction.  相似文献   

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An exact incompressible quantum liquid is constructed at the filling factor 1/m21/m2 in the square lattice. It supports deconfined fractionally charged excitation. At the filling factor 1/m21/m2, the excitation has fractional charge e/m2e/m2, where ee is the electric charge. This model can be easily generalized to the nn-dimensional square lattice (integer lattice), where the charge of excitations becomes e/mne/mn.  相似文献   

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We discuss space-time symmetric Hamiltonian operators of the form H=H0+igHH=H0+igH, where H0H0 is Hermitian and gg real. H0H0 is invariant under the unitary operations of a point group GG while HH is invariant under transformation by elements of a subgroup GG of GG. If GG exhibits irreducible representations of dimension greater than unity, then it is possible that HH has complex eigenvalues for sufficiently small nonzero values of gg. In the particular case that HH is parity-time symmetric then it appears to exhibit real eigenvalues for all 0<g<gc0<g<gc, where gcgc is the exceptional point closest to the origin. Point-group symmetry and perturbation theory enable one to predict whether HH may exhibit real or complex eigenvalues for g>0g>0. We illustrate the main theoretical results and conclusions of this paper by means of two- and three-dimensional Hamiltonians exhibiting a variety of different point-group symmetries.  相似文献   

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We discuss three Hamiltonians, each with a central-field part H0H0 and a PT-symmetric perturbation igzigz. When H0H0 is the isotropic Harmonic oscillator the spectrum is real for all gg because HH is isospectral to H0+g2/2H0+g2/2. When H0H0 is the Hydrogen atom then infinitely many eigenvalues are complex for all gg. If the potential in H0H0 is linear in the radial variable rr then the spectrum of HH exhibits real eigenvalues for 0<g<gc0<g<gc and a PT phase transition at gcgc.  相似文献   

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