Competing Orders in a Dipolar Bose-Fermi Mixture on a Square Optical Lattice: Mean-Field Perspective

We consider a mixture of a two-component Fermi gas and a single-component dipolar Bose gas in a square optical lattice and reduce it into an effective Fermi system where the Fermi-Fermi interaction includes the attractive interaction induced by the phonons of a uniform dipolar Bose-Einstein condensate. Focusing on this effective Fermi system in the parameter regime that preserves the symmetry of $D_4$, the point group of a square, we explore, within the Hartree-Fock-Bogoliubov mean-field theory, the phase competition among density wave orderings and superfluid pairings. We construct the matrix representation of the linearized gap equation in the irreducible representations of $D_4$. We show that in the weak coupling regime, each matrix element, which is a four-dimensional (4D) integral in momentum space, can be put in a separable form involving a 1D integral, which is only a function of temperature and the chemical potential, and a pairing-specific "effective" interaction, which is an analytical function of the parameters that characterize the Fermi-Fermi interactions in our system. We analyze the critical temperatures of various competing orders as functions of different system parameters in both the absence and presence of the dipolar interaction. We find that close to half filling, the d_{x^{2}-y^{2}}-wave pairing with a critical temperature in the order of a fraction of Fermi energy (at half filling) may dominate all other phases, and at a higher filling factor, the p-wave pairing with a critical temperature in the order of a hundredth of Fermi energy may emerge as a winner. We find that tuning a dipolar interaction can dramatically enhance the pairings with $d_{xy}$- and g-wave symmetries but not enough for them to dominate other competing phases.Comment: 18 pages, 9 figure

Provable Deterministic Leverage Score Sampling

We explain theoretically a curious empirical phenomenon: "Approximating a matrix by deterministically selecting a subset of its columns with the corresponding largest leverage scores results in a good low-rank matrix surrogate". To obtain provable guarantees, previous work requires randomized sampling of the columns with probabilities proportional to their leverage scores. In this work, we provide a novel theoretical analysis of deterministic leverage score sampling. We show that such deterministic sampling can be provably as accurate as its randomized counterparts, if the leverage scores follow a moderately steep power-law decay. We support this power-law assumption by providing empirical evidence that such decay laws are abundant in real-world data sets. We then demonstrate empirically the performance of deterministic leverage score sampling, which many times matches or outperforms the state-of-the-art techniques.Comment: 20th ACM SIGKDD Conference on Knowledge Discovery and Data Minin

Spontaneous phase oscillation induced by inertia and time delay

We consider a system of coupled oscillators with finite inertia and time-delayed interaction, and investigate the interplay between inertia and delay both analytically and numerically. The phase velocity of the system is examined; revealed in numerical simulations is emergence of spontaneous phase oscillation without external driving, which turns out to be in good agreement with analytical results derived in the strong-coupling limit. Such self-oscillation is found to suppress synchronization and its frequency is observed to decrease with inertia and delay. We obtain the phase diagram, which displays oscillatory and stationary phases in the appropriate regions of the parameters.Comment: 5 pages, 6 figures, to pe published in PR

CP1CP^{1} model with Hopf term and fractional spin statistics

We reconsider the $CP^{1}$ model with the Hopf term by using the Batalin-Fradkin-Tyutin (BFT) scheme, which is an improved version of the Dirac quantization method. We also perform a semi-classical quantization of the topological charge Q sector by exploiting the collective coordinates to explicitly show the fractional spin statistics.Comment: 15 page

Collective phase synchronization in locally-coupled limit-cycle oscillators

We study collective behavior of locally-coupled limit-cycle oscillators with scattered intrinsic frequencies on $d$-dimensional lattices. A linear analysis shows that the system should be always desynchronized up to $d=4$. On the other hand, numerical investigation for $d= 5$ and 6 reveals the emergence of the synchronized (ordered) phase via a continuous transition from the fully random desynchronized phase. This demonstrates that the lower critical dimension for the phase synchronization in this system is $d_{l}=4

Topological current of point defects and its bifurcation

From the topological properties of a three dimensional vector order parameter, the topological current of point defects is obtained. One shows that the charge of point defects is determined by Hopf indices and Brouwer degrees. The evolution of point defects is also studied. One concludes that there exist crucial cases of branch processes in the evolution of point defects when the Jacobian $D(\frac \phi x)=0$.Comment: revtex,14 pages,no figur

Symmetries of SU(2) Skyrmion in Hamiltonian and Lagrangian approaches

We apply the Batalin-Fradkin-Tyutin (BFT) method to the SU(2) Skyrmion to study the full symmetry structure of the model at the first class Hamiltonian level. On the other hand, we also analyze the symmetry structure of the action having the WZ term, which corresponds to this Hamiltonian, in the framework of the Lagrangian approach. Furthermore, following the BFV formalism we derive the BRST invariant gauge fixed Lagrangian from the above extended action.Comment: 14 pages, final revised version, to appear in Mod. Phys. Lett.

Simultaneous eigenstates of the number-difference operator and a bilinear interaction Hamiltonian derived by solving a complex differential equation

As a continuum work of Bhaumik et al who derived the common eigenvector of the number-difference operator Q and pair-annihilation operator ab (J. Phys. A9 (1976) 1507) we search for the simultaneous eigenvector of Q and (ab-a^{+}b^{+}) by setting up a complex differential equation in the bipartite entangled state representation. The differential equation is then solved in terms of the two-variable Hermite polynomials and the formal hypergeometric functions. The work is also an addendum to Mod. Phys. Lett. A 9 (1994) 1291 by Fan and Klauder, in which the common eigenkets of Q and pair creators are discussed

Anisotropic Cosmological Models with Energy Density Dependent Bulk Viscosity

An analysis is presented of the Bianchi type I cosmological models with a bulk viscosity when the universe is filled with the stiff fluid $p = \epsilon$ while the viscosity is a power function of the energy density, such as $\eta = \alpha |\epsilon|^n$. Although the exact solutions are obtainable only when the $2n$ is an integer, the characteristics of evolution can be clarified for the models with arbitrary value of $n$. It is shown that, except for the $n = 0$ model that has solutions with infinite energy density at initial state, the anisotropic solutions that evolve to positive Hubble functions in the later stage will begin with Kasner-type curvature singularity and zero energy density at finite past for the $n> 1$ models, and with finite Hubble functions and finite negative energy density at infinite past for the $n < 1$ models. In the course of evolution, matters are created and the anisotropies of the universe are smoothed out. At the final stage, cosmologies are driven to infinite expansion state, de Sitter space-time, or Friedman universe asymptotically. However, the de Sitter space-time is the only attractor state for the $n <1/2$ models. The solutions that are free of cosmological singularity for any finite proper time are singled out. The extension to the higher-dimensional models is also discussed

Exact solution of gyration radius of individual's trajectory for a simplified human mobility model

Gyration radius of individual's trajectory plays a key role in quantifying human mobility patterns. Of particular interests, empirical analyses suggest that the growth of gyration radius is slow versus time except the very early stage and may eventually arrive to a steady value. However, up to now, the underlying mechanism leading to such a possibly steady value has not been well understood. In this Letter, we propose a simplified human mobility model to simulate individual's daily travel with three sequential activities: commuting to workplace, going to do leisure activities and returning home. With the assumption that individual has constant travel speed and inferior limit of time at home and work, we prove that the daily moving area of an individual is an ellipse, and finally get an exact solution of the gyration radius. The analytical solution well captures the empirical observation reported in [M. C. Gonz`alez et al., Nature, 453 (2008) 779]. We also find that, in spite of the heterogeneous displacement distribution in the population level, individuals in our model have characteristic displacements, indicating a completely different mechanism to the one proposed by Song et al. [Nat. Phys. 6 (2010) 818].Comment: 4 pages, 4 figure