The length operator in Loop Quantum Gravity

The dual picture of quantum geometry provided by a spin network state is discussed. From this perspective, we introduce a new operator in Loop Quantum Gravity—the length operator. We describe its quantum geometrical meaning and derive some of its properties. In particular we show that the operator has a discrete spectrum and is diagonalized by appropriate superpositions of spin network states. A series of eigenstates and eigenvalues is presented and an explicit check of its semiclassical properties is discussed.     

Entanglement of Area Quantums in Quantized Space

Employing the area quantum 1/2 excited by the Wilson loop, a possible origin of qubit is argued. At the same time, the existences of possible entanglement of area quantums, and non-local property of the entangled states are demonstrated in quantized space.     

On the Green’s function and iterative solutions of loop quantum cosmology

Here we shall find the Green’s function of the difference equation of loop quantum cosmology. To illustrate how to use it, we shall obtain an iterative solution for closed model and evaluate its corresponding Bohmian trajectory.     

Alternative quantization of the Hamiltonian in loop quantum cosmology

Since there are quantization ambiguities in constructing the Hamiltonian constraint operator in isotropic loop quantum cosmology, it is crucial to check whether the key features of loop quantum cosmology are robust against the ambiguities. In this Letter, we quantize the Lorentz term of the gravitational Hamiltonian constraint in the spatially flat FRW model by two approaches different from that of the Euclidean term. One of the approaches is very similar to the treatment of the Lorentz part of Hamiltonian in loop quantum gravity and hence inherits more features from the full theory. Two symmetric Hamiltonian constraint operators are constructed respectively in the improved scheme. Both of them are shown to have the correct classical limit by the semiclassical analysis. In the loop quantum cosmological model with a massless scalar field, the effective Hamiltonians and Friedmann equations are derived. It turns out that the classical big bang is again replaced by a quantum bounce in both cases. Moreover, there are still great possibilities for the expanding universe to recollapse due to the quantum gravity effect.     

Space–time dimensionality from brane collisions

Collisions and subsequent decays of higher dimensional branes leave behind three-dimensional branes and anti-branes, one of which could play the rôle of our universe. This process also leads to the production of one-dimensional branes and anti-branes, however their number is expected to be suppressed. Brane collisions may also lead to the formation of bound states of branes. Their existence does not alter this result, it just allows for the existence of one-dimensional branes captured within the three-dimensional ones.     

Differential geometry of the quantum 3-dimensional space

A Hopf algebra structure of the extended three-dimensional quantum space is defined. A differential algebra of the extended three-dimensional quantum space is introduced and its Hopf algebra structure is explicitly given. The (undeformed) Lie algebra of three dimensional quantum space is obtained.     

The emergence of background geometry from quantum fluctuations

We show how the quantization of two-dimensional gravity leads to an (Euclidean) quantum space–time where the average geometry is that of constant negative curvature and where the Hartle–Hawking boundary condition arises naturally.     

Note on Invariance of One-Dimensional Lattice-Boltzmann Equation

Invariance of the one-dimensional lattice Boltzmann model is proposed together with its rigorous theoretical background. It is demonstrated that the symmetry inherent in Navier-Stokes equations is not really recovered in the one-dimensional lattice Boltzmann equation （LBE）, especially for shock calculation. Symmetry breaking may be the inherent cause for the non-physical oscillations in the vicinity of the shock for LBE calculation.     

Spin geometry of entangled qubits under bilocal decoherence modes

The Lindblad generators of the master equation define which kind of decoherence happens in an open quantum system. We are working with a two qubit system and choose the generators to be projection operators on the eigenstates of the system and unitary bilocal rotations of them. The resulting decoherence modes are studied in detail. Besides the general solutions we investigate the special case of maximally entangled states—the Bell singlet states. The results are depicted in the so-called spin geometry picture which allows to illustrate the evolution of the (nonlocal) correlations stored in a certain state. The question for which conditions the path traced out in the geometric picture depends only on the relative angle between the bilocal rotations is addressed.     

Thermo-Hydrodynamic Lattice BGK Schemes with Lie Symmetry Preservation

A new class of lattice Bhatnagar-Gross-Krook （BGK） models is proposed, based on the Lie symmetry preservation ansatz for the local equilibria. This class extends the range of stability of previous models, especially for thermohydrodynamic lattice BGK schemes.     

A Characteristic Non-Reflecting Boundary Treatment in Lattice Boltzmann Method

In lattice Boltzmann methods, disturbances develop at the initial stages of the simulation, the decay characteristics depend mainly on boundary treatment methods; open boundary conditions such as equilibrium and bounce-back schemes potentially generate uncontrollable disturbances. Excessive disturbances originate from non-physical reflecting waves at boundaries. Characteristic boundary conditions utilizing the signs of waves at boundaries which suppress these reflecting waves, as well as their implementation in the lattice Boltzmann method, are introduced herein. The performance of our novel boundary treatment method to effectively suppress excessive disturbances is verified by three different numerical experiments.     

Quantum Regge Calculus of Einstein–Cartan theory

We study the Quantum Regge Calculus of Einstein–Cartan theory to describe quantum dynamics of Euclidean space–time discretized as a 4-simplices complex. Tetrad field eμ(x)$e_{\mu }(x)$ and spin-connection field ωμ(x)${\omega }_{\mu }(x)$ are assigned to each 1-simplex. Applying the torsion-free Cartan structure equation to each 2-simplex, we discuss parallel transports and construct a diffeomorphism and local   gauge-invariant Einstein–Cartan action. Invariant holonomies of tetrad and spin-connection fields along large loops are also given. Quantization is defined by a bounded partition function with the measure of SO(4)$\mathit{SO}(4)$-group valued ωμ(x)${\omega }_{\mu }(x)$ fields and Dirac-matrix valued eμ(x)$e_{\mu }(x)$ fields over 4-simplices complex.     

Anyonic loops in three-dimensional spin liquid and chiral spin liquid

We established a large class of exactly soluble spin liquids and chiral spin liquids on three-dimensional helix lattices by introducing Kitaev-type's spin coupling. In the chiral spin liquids, exact stable ground states with spontaneous breaking of the time reversal symmetry are found. The fractionalized loop excitations in both the spin and chiral spin liquids obey non-Abelian statistics. We characterize this kind of statistics by non-Abelian Berry phase and quantum algebra relation. The topological correlation of loops is independent of local order parameter and it measures the intrinsic global quantum entanglement of degenerate ground states.     

Ab initio calculation and analysis of the properties of digital magnetic heterostructures and diluted magnetic semiconductors of IV and III-V groups

We present the first-principles calculations of digital magnetic heterostructures Si/M, Ge/M. GaAs/M, GaSb/M, GaN/M and GaN/M (50%) with M=Cr, Mn, Fe, and Co. The interaction between magnetic dopants results in a wide spin-polarized two-dimensional band inside the gap. It is found that beginning occupation of the minority-spin band greatly increases the energy of the ferromagnetic (FM) state and leads, as a rule, to the antiferromagnetic (AFM) spin ordering. This mechanism causes transition to the AFM state, when interaction between magnetic atoms is too strong, and defines the optimum of Curie temperature as a function of transition element concentration in magnetic layers.     

Real-time gauge theory simulations from stochastic quantization with optimized updating

We investigate simulations for gauge theories on a Minkowskian space–time lattice. We employ stochastic quantization with optimized updating using stochastic reweighting or gauge fixing, respectively. These procedures do not affect the underlying theory but strongly improve the stability properties of the stochastic dynamics, such that simulations on larger real-time lattices can be performed.