Topological non-linear σ\sigma-model, higher gauge theory, and a realization of all 3+1D topological orders for boson systems

A discrete non-linear $\sigma$-model is obtained by triangulate both the space-time $M^{d+1}$ and the target space $K$. If the path integral is given by the sum of all the complex homomorphisms $\phi: M^{d+1} \to K$, with an partition function that is independent of space-time triangulation, then the corresponding non-linear $\sigma$-model will be called topological non-linear $\sigma$-model which is exactly soluble. Those exactly soluble models suggest that phase transitions induced by fluctuations with no topological defects (i.e. fluctuations described by homomorphisms $\phi$) usually produce a topologically ordered state and are topological phase transitions, while phase transitions induced by fluctuations with all the topological defects give rise to trivial product states and are not topological phase transitions. If $K$ is a space with only non-trivial first homotopy group $G$ which is finite, those topological non-linear $\sigma$-models can realize all 3+1D bosonic topological orders without emergent fermions, which are described by Dijkgraaf-Witten theory with gauge group $\pi_1(K)=G$. Here, we show that the 3+1D bosonic topological orders with emergent fermions can be realized by topological non-linear $\sigma$-models with $\pi_1(K)=$ finite groups, $\pi_2(K)=Z_2$, and $\pi_{n>2}(K)=0$. A subset of those topological non-linear $\sigma$-models corresponds to 2-gauge theories, which realize and classify bosonic topological orders with emergent fermions that have no emergent Majorana zero modes at triple string intersections. The classification of 3+1D bosonic topological orders may correspond to a classification of unitary fully dualizable fully extended topological quantum field theories in 4-dimensions.Comment: 32 pages, 14 figure

Advanced Post-Processing Techniques of Molecular Dynamics Simulations in Studying Strong Anharmonic Thermodynamics of Solids

While the vibrational thermodynamics of materials with small anharmonicity at low temperatures has been understood well based on the harmonic phonons approximation; at high temperatures, this understanding must accommodate how phonons interact with other phonons or with other excitations. We shall see that the phonon-phonon interactions give rise to interesting coupling problems, and essentially modify the equilibrium and non-equilibrium properties of materials, e.g., thermal expansion, thermodynamic stability, heat capacity, optical properties, thermal transport and other nonlinear properties ofmaterials. To date the anharmonic lattice dynamics is poorly understood despite its great importance, andmost studies on lattice dynamics still rely on the harmonic or quasiharmonicmodels. With recent developement of computational models, the anharmonic information can be extracted from the atomic trajectories of molecular dynamics simulations. For example, the vibrational energy spectra, the effective potential energy surface and the phonon-phonon interaction channels can be derived from these trajectories which appear stochastic. These inter-dependent methods are adopted to successfully uncover the strong anharmonic phenomena while the traditional harmonic models fail dramatically, e.g., the negative thermal expansion of cuprite and the high temperature thermal stability of rutile

Gapped Domain Walls, Gapped Boundaries and Topological Degeneracy

Gapped domain walls, as topological line defects between 2+1D topologically ordered states, are examined. We provide simple criteria to determine the existence of gapped domain walls, which apply to both Abelian and non-Abelian topological orders. Our criteria also determine which 2+1D topological orders must have gapless edge modes, namely which 1+1D global gravitational anomalies ensure gaplessness. Furthermore, we introduce a new mathematical object, the tunneling matrix $\mathcal W$, whose entries are the fusion-space dimensions $\mathcal W_{ia}$, to label different types of gapped domain walls. By studying many examples, we find evidence that the tunneling matrices are powerful quantities to classify different types of gapped domain walls. Since a gapped boundary is a gapped domain wall between a bulk topological order and the vacuum, regarded as the trivial topological order, our theory of gapped domain walls inclusively contains the theory of gapped boundaries. In addition, we derive a topological ground state degeneracy formula, applied to arbitrary orientable spatial 2-manifolds with gapped domain walls, including closed 2-manifolds and open 2-manifolds with gapped boundaries.Comment: 5+9 pages, 3 figures, updated references, fixed typos and refinements, added proof for equivalence to Lagrangian subgroups in Abelian case

Constraints on the Dark Side of the Universe and Observational Hubble Parameter Data

This paper is a review on the observational Hubble parameter data that have gained increasing attention in recent years for their illuminating power on the dark side of the universe --- the dark matter, dark energy, and the dark age. Currently, there are two major methods of independent observational H(z) measurement, which we summarize as the "differential age method" and the "radial BAO size method". Starting with fundamental cosmological notions such as the spacetime coordinates in an expanding universe, we present the basic principles behind the two methods. We further review the two methods in greater detail, including the source of errors. We show how the observational H(z) data presents itself as a useful tool in the study of cosmological models and parameter constraint, and we also discuss several issues associated with their applications. Finally, we point the reader to a future prospect of upcoming observation programs that will lead to some major improvements in the quality of observational H(z) data.Comment: 20 pages, 6 figures, and 1 table, uses REVTeX 4.1. Review article, accepted by Advances in Astronom

Topological quasiparticles and the holographic bulk-edge relation in 2+1D string-net models

String-net models allow us to systematically construct and classify 2+1D topologically ordered states which can have gapped boundaries. We can use a simple ideal string-net wavefunction, which is described by a set of F-matrices [or more precisely, a unitary fusion category (UFC)], to study all the universal properties of such a topological order. In this paper, we describe a finite computational method -- Q-algebra approach, that allows us to compute the non-Abelian statistics of the topological excitations [or more precisely, the unitary modular tensor category (UMTC)], from the string-net wavefunction (or the UFC). We discuss several examples, including the topological phases described by twisted gauge theory (i.e., twisted quantum double $D^\alpha(G)$). Our result can also be viewed from an angle of holographic bulk-boundary relation. The 1+1D anomalous topological orders, that can appear as edges of 2+1D topological states, are classified by UFCs which describe the fusion of quasiparticles in 1+1D. The 1+1D anomalous edge topological order uniquely determines the 2+1D bulk topological order (which are classified by UMTC). Our method allows us to compute this bulk topological order (i.e., the UMTC) from the anomalous edge topological order (i.e., the UFC).Comment: 32 pages, 8 figures, reference updated, some refinement

A biologically inspired computational model of the Block Copying Task

We present in this paper a biologically inspired model of the Basal Ganglia which deals with Block Copying as a sequence learning task. By breaking a relatively complex task into simpler operations with well-defined skills, an approach which is termed as a skill-based machine design is used in the device of computational models to complete such tasks. Basal Ganglia are critically involved in sensorimotor control. From the learning aspects, Actor-Critic architectures have been proposed to model the Basal Ganglia and Temporal difference has been proposed as a learning algorithm. The model is implemented and simulation results are presented to show the capability of our model to successfully complete the task