Towards a sound massive cosmology

It is known that de Rham-Gabadadze-Tolley (dRGT) massive gravity does not permit a homogeneous and isotropic universe with flat or spherical spatial metrics. We demonstrate that a singular reference metric solves this problem in an economic and straightforward way. In the dRGT massive gravity with a singular reference metric, there are sound homogeneous and isotropic cosmological solutions. We investigate cosmologies with the static and dynamical singular reference metrics, respectively. The term like dark energy appears naturally and the universe accelerates itself in some late time evolution. The term simulating dark matter also naturally emerges. We make a preliminary constraint on the parameters in the dRGT massive gravity in frame of the present cosmological model by using the data of supernovae, cosmic microwave back ground radiations, and baryonic acoustic oscillations.Comment: 21 pages, 3 figures, version accepted by Physics of the Dark Univers

Conserving and Gapless Hartree-Fock-Bogoliubov theory for 3D dilute Bose gas at finite temperature

The energy spectrum for the three dimensional Bose gas in Bose-Einstein Condensation phase is calculated with Modified Hartree-Fock-Bogoliubov theory, which is both conserving and gapless. From Improved $\Phi -$% derivable theory, the diagrams needed to preserve Ward-Takahashi Identity are resummed in a systematic and nonperturbative way. The results show significant discrepancies with Popov theory at finite temperature. It is valid up to the critical temperature where the dispersion relation of the low energy excitation spectrum changes from linear to quadratic. Because of the repulsive interaction, the critical temperature has a positive shift from that of idea gas, which is in accordance with the result from the previous calculations in the uncondensed phase.Comment: 4pages, 5figure

Bridging Hubbard Model Physics and Quantum Hall Physics in Trilayer Graphene/h-BN moir\'e superlattice

The moir\'e superlattice formed by ABC stacked trilayer graphene aligned with a hexagonal boron nitride substrate (TG/h-BN) provides an interesting system where both the bandwidth and the topology can be tuned by an applied perpendicular electric field $D$ . Thus the TG/h-BN system can simulate both Hubbard model physics and nearly flat Chern band physics within one sample. We derive lattice models for both signs of $D$ (which controls the band topology) separately through explicit Wannier orbital construction and mapping of Coulomb interaction. When the bands are topologically trivial, we discuss possible candidates for Mott insulators at integer number of holes per site (labeled as $\nu_T$). These include both broken symmetry states and quantum spin liquid insulators which may be particularly favorable in the vicinity of the Mott transition. We propose feasible experiments to study carefully the bandwidth tuned and the doping tuned Mott metal-insulator transition at both $\nu_T=1$ and $\nu_T=2$. We discuss the interesting possibility of probing experimentally a bandwidth (or doping) controlled continuous Mott transition between a Fermi liquid metal and a quantum spin liquid insulator. Finally we also show that the system has a large valley Zeeman coupling to a small out-of-plane magnetic field, which can be used to control the valley degree of freedom.Comment: 20 pages, 13 figure

Misner-Sharp Mass and the Unified First Law in Massive Gravity

We obtain the Misner-Sharp mass in the massive gravity for a four dimensional spacetime with a two dimensional maximally symmetric subspace via the inverse unified first law method. Significantly, the stress energy is conserved in this case with a widely used reference metric. Based on this property we confirm the derived Misner-Sharp mass by the conserved charge method. We find that the existence of the Misner-sharp mass in this case does not lead to extra constraint for the massive gravity, which is notable in modified gravities. In addition, as a special case, we also investigate the Misner-Sharp mass in the static spacetime. Especially, we take the FRW universe into account for investigating the thermodynamics of the massive gravity. The result shows that the massive gravity can be in thermodynamic equilibrium, which fills in the gap in the previous studies of thermodynamics in the massive gravity.Comment: 11 pages, no figure, version published in PR

Tilings of convex polyhedral cones and topological properties of self-affine tiles

Let $\textbf{a}_1,\dots, \textbf{a}_r$ be vectors in a half-space of $\mathbb{R}^n$. We call $$C=\textbf{a}_1\mathbb{R}^++\cdots+\textbf{a}_r \mathbb{R}^+$$ a convex polyhedral cone, and call $\{\textbf{a}_1,\dots, \textbf{a}_r\}$ a generator set of $C$. A generator set with the minimal cardinality is called a frame. We investigate the translation tilings of convex polyhedral cones. Let $T\subset \mathbb{R}^n$ be a compact set such that $T$ is the closure of its interior, and $\mathcal{J}\subset \mathbb{R}^n$ be a discrete set. We say $(T,\mathcal{J})$ is a translation tiling of $C$ if $T+\mathcal{J}=C$ and any two translations of $T$ in $T+\mathcal{J}$ are disjoint in Lebesgue measure. We show that if the cardinality of a frame of $C$ is larger than $\dim C$, the dimension of $C$, then $C$ does not admit any translation tiling; if the cardinality of a frame of $C$ equals $\dim C$, then the translation tilings of $C$ can be reduced to the translation tilings of $(\mathbb{Z}^+)^n$. As an application, we characterize all the self-affine tiles possessing polyhedral corners, which generalizes a result of Odlyzko [A. M. Odlyzko, \textit{Non-negative digit sets in positional number systems}, Proc. London Math. Soc., \textbf{37}(1978), 213-229.]

Paired exciton condensate and topological charge-4e4e composite fermion pairing in half-filled quantum Hall bilayers

Half-filled Landau levels admit the theoretically powerful fermion-vortex duality but longstanding puzzles remain in their experimental realization as $\nu_T=1$ quantum Hall bilayers, further complicated by Zheng et al's recent numerical discovery of an unknown phase at intermediate layer spacing. Here we propose that half-filled quantum Hall bilayers ($\nu_T=1$) at intermediate values of the interlayer distance $d/\ell_B$ enter a phase with \textit{paired exciton condensation}. This phase shows signatures analogous to the condensate of interlayer excitons (electrons bound to opposite-layer holes) well-known for small $d$ but importantly condenses only exciton pairs. To study it theoretically we derive an effective Hamiltonian for bosonic excitons $b_k$ and show that the single-boson condensate suddenly vanishes for $d$ above a critical $d_{c1} \approx 0.95 l_B$. The nonzero condensation fraction $n_0=\langle b(0) \rangle ^2$ at $d_{c1}$ suggests that the phase stiffness remains nonzero for a range of $d>d_{c1}$ via an intermediate phase of paired-exciton condensation, exhibiting $\langle bb \rangle \neq 0$ while $\langle b \rangle =0$. Motivated by these results we derive a $K$-matrix description of the paired exciton condensate's topological properties from composite boson theory. The elementary charged excitation is a half meron with $\frac{1}{4}$ charge and fractional self-statistics $\theta_s=\frac{\pi}{16}$. Finally we argue for an equivalent description via the $d=\infty$ limit through topological charge-$4e$ pairing of composite fermions. We suggest graphene double layers should access this phase and propose various experimental signatures, including an Ising transition $T_{Ising}$ below the Berezinskii-Kosterlitz-Thouless transition $T_{BKT}$ at $d \sim d_{c1}$.Comment: 9 pages, 2 figure

Electrical detection of spin liquids in double moir\'e layers

Although spin is a fundamental quantum number, measuring spin transport in traditional solid state systems is extremely challenging. This poses a major obstacle to detecting interesting quantum states including certain spin liquids. In this paper we propose a platform that not only allows for the electrical measurement of spin transport, but in which a variety of exotic quantum phases may be stabilized. Our proposal involves two moir\'e superlattices, built from transition metal dichalcogenides (TMD) or graphene, separated from one another by a thin insulating layer. The two Coulomb coupled moir\'e layers, when suitably aligned, give rise to a layer pseudospin degree of freedom. The transport of pseudospin can be accessed from purely electrical measurements of counter-flow or Coulomb drag conductivity. Furthermore, these platforms naturally realize Hubbard models on the triangular lattice with $N = 4\, {\rm or}\, 8$ flavors. The flavor degeneracy motivates a large-N approximation from which we obtain the phase diagram of Mott insulators at different electron fillings and correlation strengths. In addition to conventional phases such as psuedospin superfluids and crystallized insulators, exotic phases including chiral spin liquids and a $U(1)$ spinon Fermi surface spin liquid are also found, all of which will show smoking gun electrical signatures in this setup.Comment: 5+8 pages, 7 figure

Separating Style and Content for Generalized Style Transfer

Neural style transfer has drawn broad attention in recent years. However, most existing methods aim to explicitly model the transformation between different styles, and the learned model is thus not generalizable to new styles. We here attempt to separate the representations for styles and contents, and propose a generalized style transfer network consisting of style encoder, content encoder, mixer and decoder. The style encoder and content encoder are used to extract the style and content factors from the style reference images and content reference images, respectively. The mixer employs a bilinear model to integrate the above two factors and finally feeds it into a decoder to generate images with target style and content. To separate the style features and content features, we leverage the conditional dependence of styles and contents given an image. During training, the encoder network learns to extract styles and contents from two sets of reference images in limited size, one with shared style and the other with shared content. This learning framework allows simultaneous style transfer among multiple styles and can be deemed as a special `multi-task' learning scenario. The encoders are expected to capture the underlying features for different styles and contents which is generalizable to new styles and contents. For validation, we applied the proposed algorithm to the Chinese Typeface transfer problem. Extensive experiment results on character generation have demonstrated the effectiveness and robustness of our method.Comment: Accepted by CVPR201

Domain-Invariant Adversarial Learning for Unsupervised Domain Adaption

Unsupervised domain adaption aims to learn a powerful classifier for the target domain given a labeled source data set and an unlabeled target data set. To alleviate the effect of `domain shift', the major challenge in domain adaptation, studies have attempted to align the distributions of the two domains. Recent research has suggested that generative adversarial network (GAN) has the capability of implicitly capturing data distribution. In this paper, we thus propose a simple but effective model for unsupervised domain adaption leveraging adversarial learning. The same encoder is shared between the source and target domains which is expected to extract domain-invariant representations with the help of an adversarial discriminator. With the labeled source data, we introduce the center loss to increase the discriminative power of feature learned. We further align the conditional distribution of the two domains to enforce the discrimination of the features in the target domain. Unlike previous studies where the source features are extracted with a fixed pre-trained encoder, our method jointly learns feature representations of two domains. Moreover, by sharing the encoder, the model does not need to know the source of images during testing and hence is more widely applicable. We evaluate the proposed method on several unsupervised domain adaption benchmarks and achieve superior or comparable performance to state-of-the-art results

On majorization of closed walks vector of trees with given degree sequences

Let $C_{v}(k;T)$ be the number of the closed walks of length $k$ starting at vertex $v$ in a tree $T$. We prove that for a given tree degree sequence $\pi$, then for any tree with degree sequence $\pi$, the sequence $C(k;T)\equiv(C_{v}(k;T), v\in V(T))$ is weakly majorized by the sequence $C(k, T_{\pi}^*)\equiv C(k, T_{\pi}^*, v\in V(T^*))$, where $T_{\pi}^*$ is the greedy tree corresponding to $\pi$. In addition, for two trees degree sequences $\pi,~\pi'$, if $\pi$ is majorized by $\pi'$, then $C(k;T_{\pi}^*)$ is weakly majorized by $C(k;T_{\pi'}^*)$.Comment: 16 pages 2 figures, Applied Mathematics and Computation, 201