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.     

Quantum spherical model with competing interactions

We analyse the phase diagram of a quantum mean spherical model in terms of the temperature T$T$, a quantum parameter g$g$, and the ratio p=−_J2/_J1$p=-J_2/J_1$, where _J1>0$J_1>0$ refers to ferromagnetic interactions between first-neighbour sites along the d$d$ directions of a hypercubic lattice, and _J2<0$J_2<0$ is associated with competing antiferromagnetic interactions between second neighbours along m≤d$m\le d$ directions. We regain a number of known results for the classical version of this model, including the topology of the critical line in the g=0$g=0$ space, with a Lifshitz point at p=1/4$p=1/4$, for d>2$d>2$, and closed-form expressions for the decay of the pair correlations in one dimension. In the T=0$T=0$ phase diagram, there is a critical border, _gc=_gc(p)$g_c=g_c\left(p\right)$ for d≥2$d\ge 2$, with a singularity at the Lifshitz point if d<(m+4)/2$d<(m+4)/2$. We also establish upper and lower critical dimensions, and analyse the quantum critical behavior in the neighborhood of p=1/4$p=1/4$.     

Non-convergent perturbation theory and misleading inferences about parameter relationships: The case of superexchange

We discuss the well-known three-centre cation–anion–cation model for superexchange in insulating transition-metal compounds using limiting expansions for the Anderson–Hubbard model. We find that due to the interfering energy scales in the model, a limiting expression for the superexchange J$J$ for the idealized Mott–Hubbard (M–H) case t?U?Δ$t?U?\Delta$ cannot be formally defined. We further show that the decomposition of the superexchange into range-dependent components is formally invalid. The well-known t⁴$t^4$ superexchange expression, obtained from path-dependent series expansions, is not unique to these systems as it can also be obtained with many other different expansions, in which either the d$d$–p$p$ energy difference Δ$\Delta$ or the d$d$-electron correlation U$U$ can actually be small. Particularly for milder relationships between the parameters, i.e.  t?U?Δ$t?U?\Delta$, the reverse from the usual form of the series expansions can yield better agreement with the exact results. This implies that the fitting of experimental data to the simple expressions derived from path-dependent series expansions can lead to qualitatively incorrect relationships between the parameters, fictitiously within the M–H regime.     

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.     

Pseudo-topological transitions in 2D gravity models coupled to massless scalar fields

We study the geometries generated by two-dimensional causal dynamical triangulations (CDT) coupled to d   massless scalar fields. Using methods similar to those used to study four-dimensional CDT we show that there exists a c=1$c=1$ “barrier”, analogous to the c=1$c=1$ barrier encountered in non-critical string theory, only the CDT transition is easier to be detected numerically. For d?1$d?1$ we observe time-translation invariance and geometries entirely governed by quantum fluctuations around the uniform toroidal topology put in by hand. For d>1$d>1$ the effective average geometry is no longer toroidal but “semiclassical” and spherical with Hausdorff dimension _dH=3$d_H=3$. In the d>1$d>1$ sector we study the time dependence of the semiclassical spatial volume distribution and show that the observed behavior is described by an effective mini-superspace action analogous to the actions found in the de Sitter phase of three- and four-dimensional pure CDT simulations and in the three-dimensional CDT-like Ho?ava–Lifshitz models.     

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.     

Intertwining operators for Sklyanin algebra and elliptic hypergeometric series

Intertwining operators for infinite-dimensional representations of the Sklyanin algebra with spins ?$?$ and −?−1$-?-1$ are constructed using the technique of intertwining vectors for elliptic L$L$-operator. They are expressed in terms of elliptic hypergeometric series with operator argument. The intertwining operators obtained (W$W$-operators) serve as building blocks for the elliptic R$R$-matrix which intertwines tensor product of two L$L$-operators taken in infinite-dimensional representations of the Sklyanin algebra with arbitrary spin. The Yang–Baxter equation for this R$R$-matrix follows from simpler equations of the star–triangle type for the W$W$-operators. A natural graphic representation of the objects and equations involved in the construction is used.     

Spin-1 Blume–Capel model with random crystal field effects

The random-crystal field spin-1 Blume–Capel model is investigated by the lowest approximation of the cluster-variation method which is identical to the mean-field approximation. The crystal field is either turned on randomly with probability p$p$ or turned off with q=1−p$q=1-p$ in a bimodal distribution. Then the phase diagrams are constructed on the crystal field (Δ$\Delta$)–temperature (kT/J)$(kT/J)$ planes for given values of p$p$ and on the (kT/J,p$kT/J,p$) planes for given Δ$\Delta$ by studying the thermal variations of the order parameters. In the latter, we only present the second-order phase transition lines, because of the existence of irregular wiggly phase transitions which are not good enough to construct lines. In addition to these phase transitions, the model also yields tricritical points for all values of p$p$ and the reentrant behavior at lower p$p$ values.     

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.     

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$g_c$.     

Lifshitz-point correlation length exponents from the large-n expansion

The large-n expansion is applied to the calculation of thermal critical exponents describing the critical behavior of spatially anisotropic d-dimensional systems at m  -axial Lifshitz points. We derive the leading non-trivial 1/n$1/n$ correction for the perpendicular correlation-length exponent _νL2${\nu }_{L2}$ and hence several related thermal exponents to order O(1/n)$O(1/n)$. The results are consistent with known large-n expansions for d  -dimensional critical points and isotropic Lifshitz points, as well as with the second-order epsilon expansion about the upper critical dimension d^?=4+m/2$d^{?}=4+m/2$ for generic m∈[0,d]$m\in [0,d]$. Analytical results are given for the special case d=4$d=4$, m=1$m=1$. For uniaxial Lifshitz points in three dimensions, 1/n$1/n$ coefficients are calculated numerically. The estimates of critical exponents at d=3$d=3$, m=1$m=1$ and n=3$n=3$ are discussed.     

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.     

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]$[X,JY]=J[X,Y]$.     

Pre-quantization of the moduli space of flat G-bundles over a surface

For a simply connected, compact, simple Lie group G$G$, the moduli space of flat G$G$-bundles over a closed surface Σ$\Sigma$ is known to be pre-quantizable at integer levels. For non-simply connected G$G$, however, integrality of the level is not sufficient for pre-quantization, and this paper determines the obstruction–namely a certain cohomology class in H³(G²;Z)$H^3(G^2;\mathbb{Z})$–that places further restrictions on the underlying level. The levels that admit a pre-quantization of the moduli space are determined explicitly for all non-simply connected, compact, simple Lie groups G$G$.     

Surface and bulk properties of ballistic deposition models with bond breaking

We introduce a new class of growth models, with a surface restructuring mechanism in which impinging particles may dislodge suspended particles, previously aggregated on the same column in the deposit. The flux of these particles is controlled through a probability p$p$. These systems present a crossover, for small values of p$p$, from random to correlated (KPZ) growth of surface roughness, which is studied through scaling arguments and Monte Carlo simulations on one- and two-dimensional substrates. We show that the crossover characteristic time _t×$t_{\times }$ scales with p$p$ according to _t×∼p^−y$t_{\times }\sim p^{-y}$ with y=(n+1)$y=(n+1)$ and that the interface width at saturation _Wsat$W_{sat}$ scales as _Wsat∼p^−δ$W_{sat}\sim p^{-\delta }$ with δ=(n+1)/2$\delta =(n+1)/2$, where n$n$ is either the maximal number of broken bonds or of dislodged suspended particles. This result shows that the sets of exponents y=1$y=1$ and δ=1/2$\delta =1/2$ or y=2$y=2$ and δ=1$\delta =1$ found in all previous works focusing on systems with this same type of crossover are not universal. Using scaling arguments, we show that the bulk porosity P$P$ of the deposits scales as P∼p^y−δ$P\sim p^{y-\delta }$ for small values of p$p$. This general scaling relation is confirmed by our numerical simulations and explains previous results present in literature.     

Post-breakthrough scaling in reservoir field simulation

We study the oil displacement and production behavior in an isothermal thin layered reservoir model subjected to water flooding. We use the CMG’s (Computer Modelling Group  ) numerical simulators to solve mass balance equations. The influences of the viscosity ratio (m≡_μoil/_μwater$m\equiv {\mu }_{oil}/{\mu }_{water}$) and the inter-well (injector-producer) distance r$r$ on the oil production rate C(t)$C\left(t\right)$ and the breakthrough time _tbr$t_{br}$ are investigated. Two types of reservoir configuration are used, namely one with random porosities and another with a percolation cluster structure. We observe that the breakthrough time follows a power-law of m$m$ and r$r$, _tbr∝r^αm^β$t_{br}\propto r^{\alpha }{m}^{\beta }$, with α=1.8$\alpha =1.8$ and β=−0.25$\beta =-0.25$ for the random porosity type, and α=1.0$\alpha =1.0$ and β=−0.2$\beta =-0.2$ for the percolation cluster type. Moreover, our results indicate that the oil production rate is a power law of time. In the percolation cluster type of reservoir, we observe that P(t)∝t^γ$P\left(t\right)\propto t^{\gamma }$, with γ=−1.81$\gamma =-1.81$, where P(t)$P\left(t\right)$ is the time derivative of C(t)$C\left(t\right)$. The curves related to different values of m$m$ and r$r$ may be collapsed suggesting a universal behavior for the oil production rate.     

If Gauss–Bonnet interaction plays the role of dark energy

A cosmological model has been constructed with Gauss–Bonnet-scalar interaction, where the Universe starts with exponential expansion but encounters infinite deceleration, q→∞$q\to \infty$ and infinite equation of state parameter, w→∞$w\to \infty$. During evolution it subsequently passes through the stiff fluid era, q=2$q=2$, w=1$w=1$, the radiation dominated era, q=1$q=1$, w=1/3$w=1/3$ and the matter dominated era, q=1/2$q=1/2$, w=0$w=0$. Finally, deceleration halts, q=0$q=0$, w=−1/3$w=-1/3$, and it then encounters a transition to the accelerating phase. Asymptotically the Universe reaches yet another inflationary phase q→−1$q\to -1$, w→−1$w\to -1$. Such evolution is independent of the form of the potential and the sign of the kinetic energy term, i.e., even a non-canonical kinetic energy is unable to phantomize (w<−1)$(w<-1)$ the model.