Abstract:
Using effective field theories dictated by the symmetry of the system, as well as microscopic considerations, we map out the magnetic coupling-temperature phase diagrams of spin-1 Bose gases in both two- and three-dimensions. We also determine the nature of all phase boundaries, and critical properties in the case of 2nd order phase transitions at both zero and finite temperatures.

Abstract:
It is well known that bosons on an optical lattice undergo a second-order superfluid-insulator transition (SIT) when the lattice potential increases. In this paper we study SIT when fermions coexist with the bosons. We find that the critical properties of particle-hole symmetric SIT with dynamical exponent z=1 is modified when fermions are present; it either becomes a fluctuation-driven first order transition or a different second-order transition. On the other hand the more generic particle-hole asymmetric (with z=2) SIT is stable against coupling with fermions. We also discuss pairing interaction between fermions mediated by quantum critical fluctuations near SIT.

Abstract:
We propose a generalization of the chiral Luttinger liquid theory to allow for a unified description of quantum Hall edges with or without edge reconstruction. Within this description edge reconstruction is found to be a quantum phase transition in the universality class of one-dimensional dilute Bose gas transition, whose critical behavior can be obtained exactly. At principal filling factors $\nu=1/m$, we show the additional edge modes due to edge reconstruction modifies the point contact tunneling exponent in the low energy limit, by a small and non-universal amount.

Abstract:
Superfluidity in fermionic systems originates from pairing of fermions, and Bose condensation of these so-called Cooper pairs. The Cooper pairs are usually made of fermions of different species; for example in superconductors they are pairs of electrons with opposite spins. Thus the most favorable situation for pairing and superfluidity is when the two species of fermions that form pairs have the same density. This paper studies the possible superfluid states when the two pairing species have different densities, and show that the resultant states have remarkable similarities to the phases of liquid crystals. This enables us to provide a unified description of the possible pairing phases, and understand the phase transitions among them.

Abstract:
In a very interesting recent Letter\cite{machida}, the authors suggested the possibility of realizing the spatially modulated, or Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superfluid state in trapped atomic fermion systems. The authors this Letter used a 1D mean field solution as guidance to estimate the parameter range for the existence of the FFLO phase, and also discussed the possibility of its detection by imaging the atomic density of the system. In this comment I wish to make two points. (i) In 1D there exists an exact solution based on bosonization, which fully takes into account the important quantum fluctuation effects\cite{yang01}; the exact solution suggests a wider parameter range for the FFLO state than that obtained from the mean-field solution of the present Letter. (ii) One can detect the FFLO pairing (in which Cooper pairs carry finite momenta) more directly by extending the methods used to detect BCS pairing\cite{regal,altman,greiner}.

Abstract:
Haldane's geometrical description of fractional quantum Hall states is generalized to compressible states. It is shown that anisotropy in the composite fermion Fermi surface is a direct reflection of this intrinsic geometry. A simple model is introduced in which the geometric parameter can be obtained exactly from other parameters including electron mass anisotropy. Our results compare favorably with recent measurements of anisotropy in composite fermion Fermi surface [D. Kamburov, Y. Liu, M. Shayegan, L. N. Pfeiffer, K. W. West, and K. W. Baldwin, Phys. Rev. Lett. 110, 206801 (2013)]. Broader implications of our results are discussed.

Abstract:
We show that in the limit of zero temperature, double layer quantum Hall systems exhibit a novel phenomena called Hall drag, namely a current driven in one layer induces a voltage drop in the other layer, in the direction perpendicular to the driving current. The two-by-two Hall resistivity tensor is quantized and proportional to the ${\bf K}$ matrix that describes the topological order of the quantum Hall state, even when the ${\bf K}$ matrix contains a zero eigenvalue, in which case the Hall conductivity tensor does not exist. Relation between the present work and previous ones is also discussed.

Abstract:
In a very interesting recent Letter\cite{berg}, the authors suggested that a novel form of superconducting state is realized in La$_{2-x}$Ba$_x$CuO$_4$ with $x$ close to 1/8. This suggestion was based on experiments\cite{li} on this compound which found predominantly two-dimensional (2D) characters of the superconducting state, with extremely weak interplane coupling. Later this specific form of superconducting state was termed striped superconductors\cite{berg08}. The purpose of this note is to point out that the suggested form\cite{berg} of the superconducting order parameter can be detected directly using magnetic field modulated Josephson effect.

Abstract:
In this article we briefly review recent experimental and theoretical work on quantum Hall effect in graphene, and argue that some of the quantum Hall states exhibit spontaneous symmetry breaking that is driven by electron-electron interaction. We will also discuss how to experimentally determine the actual manner in which symmetry breaking occurs, and detect the collective charge and neutral excitations associated with symmetry breaking. Other issues will also be briefly mentioned.