Single-Valued Hamiltonian via Legendre-Fenchel Transformation and Time Translation Symmetry

Under conventional Legendre transformation, systems with a non-convex Lagrangian will result in a multi-valued Hamiltonian as a function of conjugate momentum. This causes problems such as non-unitary time evolution of quantum state and non-determined motion of classical particles, and is physically unacceptable. In this work, we propose a new construction of single-valued Hamiltonian by applying Legendre-Fenchel transformation, which is a mathematically rigorous generalization of conventional Legendre transformation, valid for non-convex Lagrangian systems, but not yet widely known to the physics community. With the new single-valued Hamiltonian, we study spontaneous breaking of time translation symmetry and derive its vacuum state. Applications to theories of cosmology and gravitation are discussed.Comment: Journal Version, 16pp. All results + conclusions un-changed, only minor refinements to clarify the importance of our new LFT method and its physics applications; references adde

Twisted Gauge Theory Model of Topological Phases in Three Dimensions

We propose an exactly solvable lattice Hamiltonian model of topological phases in $3+1$ dimensions, based on a generic finite group $G$ and a $4$-cocycle $\omega$ over $G$. We show that our model has topologically protected degenerate ground states and obtain the formula of its ground state degeneracy on the $3$-torus. In particular, the ground state spectrum implies the existence of purely three-dimensional looplike quasi-excitations specified by two nontrivial flux indices and one charge index. We also construct other nontrivial topological observables of the model, namely the $SL(3,\mathbb{Z})$ generators as the modular $S$ and $T$ matrices of the ground states, which yield a set of topological quantum numbers classified by $\omega$ and quantities derived from $\omega$. Our model fulfills a Hamiltonian extension of the $3+1$-dimensional Dijkgraaf-Witten topological gauge theory with a gauge group $G$. This work is presented to be accessible for a wide range of physicists and mathematicians.Comment: 37 pages, 9 figures, 4 tables; revised to improve the clarity; references adde

Modular Matrices as Topological Order Parameter by Gauge Symmetry Preserved Tensor Renormalization Approach

Topological order has been proposed to go beyond Landau symmetry breaking theory for more than twenty years. But it is still a challenging problem to generally detect it in a generic many-body state. In this paper, we will introduce a systematic numerical method based on tensor network to calculate modular matrices in 2D systems, which can fully identify topological order with gapped edge. Moreover, it is shown numerically that modular matrices, including S and T matrices, are robust characterization to describe phase transitions between topologically ordered states and trivial states, which can work as topological order parameters. This method only requires local information of one ground state in the form of a tensor network, and directly provides the universal data (S and T matrices), without any non-universal contributions. Furthermore it is generalizable to higher dimensions. Unlike calculating topological entanglement entropy by extrapolating, which numerical complexity is exponentially high, this method extracts a much more complete set of topological data (modular matrices) with much lower numerical cost.Comment: 5+3 pages; 4+2 figures; One more appendix is adde

Design, Synthesis and Study of Functional Amphiphilic Polymers and Their Applications

Amphiphilic homopolymers with high densities of functional groups are synthetically challenging. Thiol-yne nucleophilic click reactions have been investigated to introduce multiple functional groups in polymers with high density. An electron deficient alkyne group bearing methacrylate monomer was polymerized using reversible addition−fragmentation chain-transfer (RAFT) polymerization. Subsequently, the electron deficient alkyne group on polymer side chain was readily reacted with a thiol reagent using triethylamine (TEA) as the organocatalyst. This reaction was found to be very efficient under mild conditions. The resultant homopolymer bearing thiol vinyl ether functional groups could perform a second thiol addition with a stronger base, such as triazabicyclodecene(TBD), to prepare multifunctional homopolymers. This stepwise addition process was monitored by different techniques. The fidelity of this method was demonstrated by attaching four different functionalities, including both hydrophobic and hydrophilic moieties. Also, a novel polymerization method was developed using the same thiol-yne nucleophilic addition strategy. Typically, two bi-functional thiol-based monomers with activated alkyne termini differencing in their linker length can be prepared and polymerized utilizing this strategy. Under mild conditions, monomers can be polymerized conveniently to yield linear structures with high molecular weights. The polymerization conditions were optimized by screening various catalysts, solvents, temperatures, reaction time, and monomer concentrations. This new methodology efficiently yields polymers with high molecular weight. Moreover, the resulting polymer contains alkene backbone that acts as a functional handle for further modifications. For example, it can be subjected to a second thiol species to achieve multi-functional capabilities. These post-modified polymers with dithioacetal units in their backbone degrade in the presence of reactive oxygen species, providing the opportunity to design functional materials with stimuli-responsive features

Empathy and Sympathy Learned from British and American Novels and Short Stories

Foreign literature courses, including both British and American literature, serve as compulsory and core courses for Chinese English Majors (CEMs). Literature courses make CEMs grasp aesthetic meanings as well as immerses themselves in a literary world, most importantly empower them to access different cultural experiences, and experience ups and downs of fictional characters. This article mainly delineates empathetic and sympathetic factors inherent in English novels and short stories included in literary textbooks used in foreign literature classes, such as A Guide to History and Anthology of American Literature, A Guide to History and Anthology of British Literature, Selected Readings in American Literature, and Selected Readings in British Literature. Some novice teaching staff usually find that the assiduous efforts they have to put into the teaching of foreign literature class is due to limited academic duration, numerous writers, countless literary works, difficult literary languages, complicated character maps, exotic life experiences, as well as a range variety of students’ demands and interests. This article aims to find out empathy- and sympathy-themed excerpts from the above-mentioned textbooks to help novice literature teachers to accustom themselves to the teaching of the courses, and even to realize a favorable teaching result. The effective pedagogy includes pre-teaching, during-teaching and after-teaching. The examples of literary texts contain both American and British writers. The exemplified text genre includes novels and short stories. The academics in literature courses can follow the suggestions and endeavors mentioned in this paper to better guide their teaching activities