Density and Spin Linear Response of Atomic Fermi Superfluids with Population Imbalance in BCS-BEC Crossover

We present a theoretical study of the density and spin (representing the two components) linear response of Fermi superfluids with tunable attractive interactions and population imbalance. In both linear response theories, we find that the fluctuations of the order parameter must be treated on equal footing with the gauge transformations associated with the symmetries of the Hamiltonian so that important constraints including various sum rules can be satisfied. Both theories can be applied to the whole BCS-Bose-Einstein condensation crossover. The spin linear responses are qualitatively different with and without population imbalance because collective-mode effects from the fluctuations of the order parameter survive in the presence of population imbalance, even though the associated symmetry is not broken by the order parameter. Since a polarized superfluid becomes unstable at low temperatures in the weak and intermediate coupling regimes, we found that the density and spin susceptibilities diverge as the system approaches the unstable regime, but the emergence of phase separation preempts the divergence.Comment: 15 pages, 5 figure

BCS thermal vacuum of fermionic superfluids and its perturbation theory

The thermal field theory is applied to fermionic superfluids by doubling the degrees of freedom of the BCS theory. We construct the two-mode states and the corresponding Bogoliubov transformation to obtain the BCS thermal vacuum. The expectation values with respect to the BCS thermal vacuum produce the statistical average of the thermodynamic quantities. The BCS thermal vacuum allows a quantum-mechanical perturbation theory with the BCS theory serving as the unperturbed state. We evaluate the leading-order corrections to the order parameter and other physical quantities from the perturbation theory. A direct evaluation of the pairing correlation as a function of temperature shows the pseudogap phenomenon results from the perturbation theory. The BCS thermal vacuum is shown to be a generalized coherent and squeezed state. The correspondence between the thermal vacuum and purification of the density matrix allows a unitary transformation, and we found the geometric phase in the parameter space associated with the transformation.Comment: 14 pages, 2 figure

The strangeness form factors of the proton within nonrelativistic constituent quark model revisited

We reexamine, within the nonrelativistic constituent quark model (NRCQM), a recent claim that the current data on the strangeness form factors indicates that the uuds\bar{s} component in the proton is such that the uuds subsystem has the mixed spatial symmetry [31]_X and flavor spin symmetry [4]_{FS}[22]_F[22]_S, with \bar{s} in S state (configuration I). We find this claim to be invalid if corrected expressions for the contributions of the transition current to G_A^s and G_E^s are used. We show that, instead, it is the lowest-lying uuds\bar{s} configuration with uuds subsystem of completely symmetric spatial symmetry [4]_X and flavor spin symmetry [4]_{FS}[22]_F[22]_S, with \bar{s} in P state (configuration II), which could account for the empirical signs of all form factors G_E^s, G_M^s, and G_A^s. Further, we find that removing the center-of-mass motion of the clusters will considerably enhance the contributions of the transition current. We also demonstrate that it is possible to give a reasonable description of the existing form factors data with a tiny probability P_{s\bar{s}}=0.025% for the uuds\bar{s} component. We further see that with a small admixture of configuration I, the agreement of our prediction with data for G_A^s at low-q^2 region can be markedly improved. We find that without removing CM motion, P_{s\bar{s}} would be overestimated by about a factor of four in the case when transition current dominates. We also explore the consequence of a recent estimate reached from analyzing existing data on \bar{d} -\bar{u}, s +\bar{s}, and \bar{u} + \bar{d} - s -\bar{s}, that P_{s\bar{s}} lies between 2.4-2.9%. It would lead to a large size for the five-quark system and a small bump in both G_E^s+\eta G_M^s and G_E^s in the region of q^2<=0.1 GeV^2 within the considered model.Comment: 6 pages, 2 figure

Horocycle Decision Boundaries for Large Margin Classification in Hyperbolic Space

Hyperbolic spaces have been quite popular in the recent past for representing hierarchically organized data. Further, several classification algorithms for data in these spaces have been proposed in the literature. These algorithms mainly use either hyperplanes or geodesics for decision boundaries in a large margin classifiers setting leading to a non-convex optimization problem. In this paper, we propose a novel large margin classifier based on horocycle (horosphere) decision boundaries that leads to a geodesically convex optimization problem that can be optimized using any Riemannian gradient descent technique guaranteeing a globally optimal solution. We present several experiments depicting the performance of our classifier