Bertrand curves in the three-dimensional sphere

A curve α$\alpha$ immersed in the three-dimensional sphere S³${\mathbb{S}}^3$ is said to be a Bertrand curve if there exists another curve β$\beta$ and a one-to-one correspondence between α$\alpha$ and β$\beta$ such that both curves have common principal normal geodesics at corresponding points. The curves α$\alpha$ and β$\beta$ are said to be a pair of Bertrand curves in S³${\mathbb{S}}^3$. One of our main results is a sort of theorem for Bertrand curves in S³${\mathbb{S}}^3$ which formally agrees with the classical one: “Bertrand curves in S³${\mathbb{S}}^3$ correspond to curves for which there exist two constants λ≠0$\lambda \ne 0$ and μ$\mu$ such that λκ+μτ=1$\lambda \kappa +\mu \tau =1$”, where κ$\kappa$ and τ$\tau$ 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 S³${\mathbb{S}}^3$ as the only twisted curves in S³${\mathbb{S}}^3$ having infinite Bertrand conjugate curves. We also find several relationships between Bertrand curves in S³${\mathbb{S}}^3$ and (1,3)-Bertrand curves in R⁴${\mathbb{R}}^4$.     

Is space-time symmetry a suitable generalization of parity-time symmetry?

We discuss space-time symmetric Hamiltonian operators of the form H=H₀+igH^′$H=H_0+ig{H}^{\prime }$, where H₀$H_0$ is Hermitian and g$g$ real. H₀$H_0$ is invariant under the unitary operations of a point group G$G$ while H^′$H^{\prime }$ is invariant under transformation by elements of a subgroup G^′$G^{\prime }$ of G$G$. If G$G$ exhibits irreducible representations of dimension greater than unity, then it is possible that H$H$ has complex eigenvalues for sufficiently small nonzero values of g$g$. In the particular case that H$H$ is parity-time symmetric then it appears to exhibit real eigenvalues for all 0<g<g_c$0<g<g_c$, where g_c$g_c$ is the exceptional point closest to the origin. Point-group symmetry and perturbation theory enable one to predict whether H$H$ may exhibit real or complex eigenvalues for g>0$g>0$. 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.     

Entanglement entropy and variational methods: Interacting scalar fields

We develop a variational approximation to the entanglement entropy for scalar ?⁴${?}^4$ theory in 1+1$1+1$, 2+1$2+1$, and 3+1$3+1$ dimensions, and then examine the entanglement entropy as a function of the coupling. We find that in 1+1$1+1$ and 2+1$2+1$ 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+1$3+1$ 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.     

The first Chevalley–Eilenberg Cohomology group of the Lie algebra on the transverse bundle of a decreasing family of foliations

In [L. Lebtahi, Lie algebra on the transverse bundle of a decreasing family of foliations, J. Geom. Phys. 60 (2010), 122–133], we defined the transverse bundle V^k$V^k$ to a decreasing family of k$k$ foliations _Fi$F_i$ on a manifold M$M$. We have shown that there exists a (1,1)$(1,1)$ tensor J$J$ of V^k$V^k$ such that J^k≠0$J^k\ne 0$, J^k+1=0$J^{k+1}=0$ and we defined by _LJ(V^k)$L_J\left(V^k\right)$ the Lie Algebra of vector fields X$X$ on V^k$V^k$ such that, for each vector field Y$Y$ on V^k$V^k$, [X,JY]=J[X,Y]$[X,JY]=J[X,Y]$.     

Three PT-symmetric Hamiltonians with completely different spectra

We discuss three Hamiltonians, each with a central-field part H₀$H_0$ and a PT-symmetric perturbation igz$igz$. When H₀$H_0$ is the isotropic Harmonic oscillator the spectrum is real for all g$g$ because H$H$ is isospectral to H₀+g²/2$H_0+g^2/2$. When H₀$H_0$ is the Hydrogen atom then infinitely many eigenvalues are complex for all g$g$. If the potential in H₀$H_0$ is linear in the radial variable r$r$ then the spectrum of H$H$ exhibits real eigenvalues for 0<g<g_c$0<g<g_c$ and a PT phase transition at g_c$g_c$.     

Minimal hypersurfaces in a nearly Kaehler 6-sphere

In this paper we show that for a compact minimal hypersurface M$M$ of constant scalar curvature in the unit sphere S⁶$S^6$ with the shape operator A$A$ satisfying ‖A‖²>5${\Vert A\Vert }^2>5$, there exists an eigenvalue λ>10$\lambda >10$ of the Laplace operator of the hypersurface M$M$ such that ‖A‖²=λ−5${\Vert A\Vert }^2=\lambda -5$. This gives the next discrete value of ‖A‖²${\Vert A\Vert }^2$ greater than 0 and 5.     

Deconfined fractionally charged excitation in any dimensions

An exact incompressible quantum liquid is constructed at the filling factor 1/m²$1/m^2$ in the square lattice. It supports deconfined fractionally charged excitation. At the filling factor 1/m²$1/m^2$, the excitation has fractional charge e/m²$e/m^2$, where e$e$ is the electric charge. This model can be easily generalized to the n$n$-dimensional square lattice (integer lattice), where the charge of excitations becomes e/mⁿ$e/m^n$.     

A separation property for magnetic Schrödinger operators on Riemannian manifolds

We consider a Schrödinger differential expression L=_ΔA+q$L={\Delta }_A+q$ on a complete Riemannian manifold (M,g)$(M,g)$ with metric g$g$, where _ΔA${\Delta }_A$ is the magnetic Laplacian on M$M$ and q≥0$q\ge 0$ is a locally square integrable function on M$M$. In the terminology of W.N. Everitt and M. Giertz, the differential expression L$L$ is said to be separated in L²(M)$L^2\left(M\right)$ if for all u∈L²(M)$u\in L^2\left(M\right)$ such that Lu∈L²(M)$Lu\in L^2\left(M\right)$, we have qu∈L²(M)$qu\in L^2\left(M\right)$. We give sufficient conditions for L$L$ to be separated in L²(M)$L^2\left(M\right)$.     

Weak mirror symmetry of complex symplectic Lie algebras

A complex symplectic structure on a Lie algebra h$\mathfrak{h}$ is an integrable complex structure J$J$ with a closed non-degenerate (2,0)$(2,0)$-form. It is determined by J$J$ and the real part Ω$\Omega$ of the (2,0)$(2,0)$-form. Suppose that h$\mathfrak{h}$ is a semi-direct product g?V$\mathfrak{g}?V$, and both g$\mathfrak{g}$ and V$V$ are Lagrangian with respect to Ω$\Omega$ and totally real with respect to J$J$. This note shows that g?V$\mathfrak{g}?V$ is its own weak mirror image in the sense that the associated differential Gerstenhaber algebras controlling the extended deformations of Ω$\Omega$ and J$J$ are isomorphic.     

On the length of stabilograms: A study performed with detrended fluctuation analysis

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 2¹⁴$2^{14}$ 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 x$x$ and y$y$ signals. The relation between the exponents of the position and velocity signals is accomplished by the 2¹⁴$2^{14}$ point signals, while subsignals with up to 2¹²$2^{12}$ points do not verify it. Using artificial signals with 2¹⁴$2^{14}$ points, generated for α$\alpha$ values given, it has been demonstrated that the exponents obtained from these signals take values larger than expected for α<0.3$\alpha <0.3$, while the exponents of the accumulated series are smaller than expected for 0.7<α$0.7<\alpha$. For CoP trajectories this indicates that DFA-1 provides feasible exponents for the short τ$\tau$-end region of the velocity signal and the large τ$\tau$-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 x$x$ direction.     

Zero finite-temperature charge stiffness within the half-filled 1D Hubbard model

Even though the one-dimensional (1D) Hubbard model is solvable by the Bethe ansatz, at half-filling its finite-temperature T>0$T>0$ transport properties remain poorly understood. In this paper we combine that solution with symmetry to show that within that prominent T=0$T=0$ 1D insulator the charge stiffness D(T)$D\left(T\right)$ vanishes for T>0$T>0$ and finite values of the on-site repulsion U$U$ in the thermodynamic limit. This result is exact and clarifies a long-standing open problem. It rules out that at half-filling the model is an ideal conductor in the thermodynamic limit. Whether at finite T$T$ and U>0$U>0$ it is an ideal insulator or a normal resistor remains an open question. That at half-filling the charge stiffness is finite at U=0$U=0$ and vanishes for U>0$U>0$ is found to result from a general transition from a conductor to an insulator or resistor occurring at U=U_c=0$U=U_c=0$ for all finite temperatures T>0$T>0$. (At T=0$T=0$ such a transition is the quantum metal to Mott-Hubbard-insulator transition.) The interplay of the η$\eta$-spin SU(2)$SU\left(2\right)$ symmetry with the hidden U(1)$U\left(1\right)$ symmetry beyond SO(4)$SO\left(4\right)$ is found to play a central role in the unusual finite-temperature charge transport properties of the 1D half-filled Hubbard model.     

Holomorphic Cartan geometries and Calabi–Yau manifolds

Let M$M$ be a connected complex projective manifold such that _c1(T^(1,0)M)=0$c_1\left(T^{(1,0)}M\right)=0$. If M$M$ admits a holomorphic Cartan geometry, then we show that M$M$ is holomorphically covered by an abelian variety.     

Half-monopole in the Weinberg–Salam model

We present new axially symmetric half-monopole configuration of the SU(2)×U(1) Weinberg–Salam model of electromagnetic and weak interactions. The half-monopole configuration possesses net magnetic charge 2π/e$2\pi /e$ which is half the magnetic charge of a Cho–Maison monopole. The electromagnetic gauge potential is singular along the negative z$z$-axis. However the total energy is finite and increases only logarithmically with increasing Higgs field self-coupling constant λ^1/2${\lambda }^{1/2}$ at sin²θ_W=0.2312${sin}^2{\theta }_W=0.2312$. In the U(1) magnetic field, the half-monopole is just a one dimensional finite length line magnetic charge extending from the origin r=0$r=0$ and lying along the negative z$z$-axis. In the SU(2) ’t Hooft magnetic field, it is a point magnetic charge located at r=0$r=0$. The half-monopole possesses magnetic dipole moment that decreases exponentially fast with increasing Higgs field self-coupling constant λ^1/2${\lambda }^{1/2}$ at sin²θ_W=0.2312${sin}^2{\theta }_W=0.2312$.     

Thermodynamics properties of an anisotropic Ising antiferromagnet in both external longitudinal and transverse fields

We have studied the anisotropic two-dimensional nearest-neighbor Ising model with competitive interactions in both uniform longitudinal field H$H$ and transverse magnetic field Ω$\Omega$. Using the effective-field theory (EFT) with correlation in cluster with N=1$N=1$ spin we calculate the thermodynamic properties as a function of temperature with values H$H$ and Ω$\Omega$ fixed. The model consists of ferromagnetic interaction _Jx$J_x$ in the x$x$ direction and antiferromagnetic interaction _Jy$J_y$ in the y$y$ direction, and it is found that for H/_Jy∈[0,2]$H/J_y\in [0,2]$ the system exhibits a second-order phase transition. The thermodynamic properties are obtained for the particular case of λ=_Jx/_Jy=1$\lambda =J_x/J_y=1$ (isotropic square lattice).     

On complete spacelike hypersurfaces with two distinct principal curvatures in Lorentz–Minkowski space

We investigate complete spacelike hypersurfaces in Lorentz–Minkowski space with two distinct principal curvatures and constant m$m$th mean curvature. By using Otsuki’s idea, we obtain the global classification result. As their applications, we obtain some characterizations for hyperbolic cylinders. We prove that the only complete spacelike hypersurfaces in Lorentz–Minkowski (n+1)$(n+1)$-spaces (n≥3$n\ge 3$) of nonzero constant m$m$th mean curvature (m≤n−1$m\le n-1$) with two distinct principal curvatures λ$\lambda$ and μ$\mu$ satisfying inf(λ−μ)²>0$inf{(\lambda -\mu )}^2>0$ are the hyperbolic cylinders. We also obtain a global characterization for hyperbolic cylinder Hⁿ⁻¹(c)×R${\mathbb{H}}^{n-1}\left(c\right)\times \mathbb{R}$ in terms of square length of the second fundamental form.     

Biminimal properly immersed submanifolds in the Euclidean spaces

We consider a complete nonnegative biminimal   submanifold M$M$ (that is, a complete biminimal submanifold with λ≥0$\lambda \ge 0$) in a Euclidean space E^N${\mathbb{E}}^N$. Assume that the immersion is proper  , that is, the preimage of every compact set in E^N${\mathbb{E}}^N$ is also compact in M$M$. Then, we prove that M$M$ is minimal. From this result, we give an affirmative partial answer to Chen’s conjecture. For the case of λ<0$\lambda <0$, we construct examples of biminimal submanifolds and curves.     

Non-Abelian fusion rules from an Abelian system

We demonstrate the emergence of non-Abelian fusion rules for excitations of a two dimensional lattice model built out of Abelian degrees of freedom. It can be considered as an extension of the usual toric code model on a two dimensional lattice augmented with matter fields. It consists of the usual C(Z_p)$\mathbb{C}\left({\mathbb{Z}}_p\right)$ gauge degrees of freedom living on the links together with matter degrees of freedom living on the vertices. The matter part is described by a n$n$ dimensional vector space which we call H_n$H_n$. The Z_p${\mathbb{Z}}_p$ gauge particles act on the vertex particles and thus H_n$H_n$ can be thought of as a C(Z_p)$\mathbb{C}\left({\mathbb{Z}}_p\right)$ module. An exactly solvable model is built with operators acting in this Hilbert space. The vertex excitations for this model are studied and shown to obey non-Abelian fusion rules. We will show this for specific values of n$n$ and p$p$, though we believe this feature holds for all n>p$n>p$. We will see that non-Abelian anyons of the quantum double of C(S₃)$\mathbb{C}\left(S_3\right)$ are obtained as part of the vertex excitations of the model with n=6$n=6$ and p=3$p=3$. Ising anyons are obtained in the model with n=4$n=4$ and p=2$p=2$. The n=3$n=3$ and p=2$p=2$ case is also worked out as this is the simplest model exhibiting non-Abelian fusion rules. Another common feature shared by these models is that the ground states have a higher symmetry than Z_p${\mathbb{Z}}_p$. This makes them possible candidates for realizing quantum computation.