Dynamical System of Scalar Field from 2-Dimension to 3-D and its Cosmological Implication

We give the three-dimensional dynamical autonomous systems for most of the popular scalar field dark energy models including (phantom) quintessence, (phantom) tachyon, k-essence and general non-canonical scalar field models, change the dynamical variables from variables $(x, y, \lambda)$ to observable related variables $(w_{\phi}, \Omega_{\phi}, \lambda)$, and show the intimate relationships between those scalar fields that the three-dimensional system of k-essence can reduce to (phantom) tachyon, general non-canonical scalar field can reduce to (phantom) quintessence and k-essence can also reduce to (phantom) quintessence for some special cases. For the applications of the three-dimensional dynamical systems, we investigate several special cases and give the exactly dynamical solutions in detail. In the end of this paper, we argue that, it is more convenient and also has more physical meaning to express the differential equations of dynamical systems in $(w_{\phi}, \Omega_{\phi}, \lambda)$ instead of variables $(x, y, \lambda)$ and to investigate the dynamical system in 3-Dimension instead of 2-Dimension. We also raise a question about the possibility of the chaotic behavior in the spatially flat single scalar field FRW cosmological models in the presence of ordinary matter.Comment: 20 pages, 8 figures,some references added. Minor changes according to the suggestions from referee

Criticality in Translation-Invariant Parafermion Chains

In this work we numerically study critical phases in translation-invariant $\mathbb{Z}_N$ parafermion chains with both nearest- and next-nearest-neighbor hopping terms. The model can be mapped to a $\mathbb{Z}_N$ spin model with nearest-neighbor couplings via a generalized Jordan-Wigner transformation and translation invariance ensures that the spin model is always self-dual. We first study the low-energy spectrum of chains with only nearest-neighbor coupling, which are mapped onto standard self-dual $\mathbb{Z}_N$ clock models. For $3\leq N\leq 6$ we match the numerical results to the known conformal field theory(CFT) identification. We then analyze in detail the phase diagram of a $N=3$ chain with both nearest and next-nearest neighbor hopping and six critical phases with central charges being $4/5$, 1 or 2 are found. We find continuous phase transitions between $c=1$ and $c=2$ phases, while the phase transition between $c=4/5$ and $c=1$ is conjectured to be of Kosterlitz-Thouless type.Comment: published versio

Phase diagram of the SO(n) bilinear-biquadratic chain from many-body entanglement

Here we investigate the phase diagram of the SO(n) bilinear-biquadratic quantum spin chain by studying the global quantum correlations of the ground state. We consider the cases of n=3,4 and 5 and focus on the geometric entanglement in the thermodynamic limit. Apart from capturing all the known phase transitions, our analysis shows a number of novel distinctive behaviors in the phase diagrams which we conjecture to be general and valid for arbitrary n. In particular, we provide an intuitive argument in favor of an infinite entanglement length in the system at a purely-biquadratic point. Our results are also compared to other methods, such as fidelity diagrams.Comment: 7 pages, 4 figures. Revised version. To appear in PR

Hexagon-singlet solid ansatz for the spin-1 kagome antiferromagnet

We perform a systematic investigation on the hexagon-singlet solid (HSS) states, which are a class of spin liquid candidates for the spin-1 kagome antiferromagnet. With the Schwinger boson representation, we show that all HSS states have exponentially decaying correlations and can be interpreted as a (special) subset of the resonating Affleck-Kennedy-Lieb-Tasaki (AKLT) loop states. We provide a compact tensor network representation of the HSS states, with which we are able to calculate physical observables efficiently. We find that the HSS states have vanishing topological entanglement entropy, suggesting the absence of intrinsic topological order. We also employ the HSS states to perform a variational study of the spin-1 kagome Heisenberg antiferromagnetic model. When we use a restricted HSS ansatz preserving lattice symmetry, the best variational energy per site is found to be $e_0 = -1.3600$. In contrast, when allowing lattice symmetry breaking, we find a trimerized simplex valence bond crystal with a lower energy, $e_0=-1.3871$.Comment: 14 pages, 12 figures, published versio

Universal Boundary Entropies in Conformal Field Theory: A Quantum Monte Carlo Study

Recently, entropy corrections on nonorientable manifolds such as the Klein bottle are proposed as a universal characterization of critical systems with an emergent conformal field theory (CFT). We show that entropy correction on the Klein bottle can be interpreted as a boundary effect via transforming the Klein bottle into an orientable manifold with nonlocal boundary interactions. The interpretation reveals the conceptual connection of the Klein bottle entropy with the celebrated Affleck-Ludwig entropy in boundary CFT. We propose a generic scheme to extract these universal boundary entropies from quantum Monte Carlo calculation of partition function ratios in lattice models. Our numerical results on the Affleck-Ludwig entropy and Klein bottle entropy for the $q$-state quantum Potts chains with $q=2,3$ show excellent agreement with the CFT predictions. For the quantum Potts chain with $q=4$, the Klein bottle entropy slightly deviates from the CFT prediction, which is possibly due to marginally irrelevant terms in the low-energy effective theory.Comment: 10 pages, 4 figures. Published versio

Topology and Criticality in Resonating Affleck-Kennedy-Lieb-Tasaki loop Spin Liquid States

We exploit a natural Projected Entangled-Pair State (PEPS) representation for the resonating Affleck-Kennedy-Lieb-Tasaki loop (RAL) state. By taking advantage of PEPS-based analytical and numerical methods, we characterize the RAL states on various two-dimensional lattices. On square and honeycomb lattices, these states are critical since the dimer-dimer correlations decay as a power law. On kagome lattice, the RAL state has exponentially decaying correlation functions, supporting the scenario of a gapped spin liquid. We provide further evidence that the RAL state on the kagome lattice is a $\mathbb{Z}_2$ spin liquid, by identifying the four topological sectors and computing the topological entropy. Furthermore, we construct a one-parameter family of PEPS states interpolating between the RAL state and a short-range Resonating Valence Bond state and find a critical point, consistent with the fact that the two states belong to two different phases. We also perform a variational study of the spin-1 kagome Heisenberg model using this one-parameter PEPS.Comment: 10 pages, 14 figures, published versio