Low-energy electronic structure of ͑unbiased and undoped͒ bilayer graphene consists of two Fermi points with quadratic dispersions if trigonal warping is ignored. We show that short-range ͑or screened Coulomb͒ interactions are marginally relevant and use renormalization group to study their effects on low-energy properties of the system. We find that the two quadratic Fermi points spontaneously split into four Dirac points. This results in a nematic state that spontaneously breaks the sixfold lattice rotation symmetry ͑combined with layer permutation͒ down to a twofold one, with a finite transition temperature. Critical properties of the transition and effects of trigonal warping are also discussed. This Rapid Communication is motivated by the observation that in noninteracting systems with susceptibilities diverging as the temperature approaches zero, the inclusion of arbitrarily small interaction leads to a finite but also arbitrarily small transition temperature into an ordered state. The analytical method of choice in this case is the renormalization group ͑RG͒, which has the virtue of unbiased determination of the leading instability. 1 Here we apply the RG method to the bilayer graphene with A-B stacking. 2-5 While in general, the motion of the noninteracting electrons in such potential does not lead to diverging susceptibilities due to trigonal warping, 3,4 if only nearest-neighbor ͑nn͒ hopping is considered each set of four Dirac points merges into a single degenerate point with parabolic dispersion ͑see Fig. 1͒. As the nearest-neighbor hopping amplitudes are the largest, the latter is the natural starting point of theoretical analysis. 6,7 The finite density of states associated with the parabolic dispersion leads to screening that renders Coulomb interaction short ranged and to diverging susceptibilities in several channels. We find that the leading instability triggered by the run-away RG flow is in the nematic channel, which effectively makes hopping amplitudes stronger along preferred direction ͓see Eqs. ͑17͒ and ͑18͔͒, and leads to spontaneous splittings of the Fermi points and breaking of the lattice rotation symmetry. Among other effects, this should lead to anisotropic transport in sufficiently clean samples, as well as suppression of the low-energy density of state: an effect, in principle, observable in STM.We start with the tight-binding Hamiltonian for electrons hopping on the bilayer honeycomb lattice with Bernal stackingwhere, in the nn approximation, the ͑real͒ hopping amplitudes t connect the in-plane nn sites belonging to different sublattices and, for one of the sublattices, also the sites vertically above it with amplitude t Ќ . Since there are four sites in the unit cell, there are four bands whose dispersion for the above model comes from the solution of the eigenvalue problem,K 0 K t 2t 3t 0 t 2t 3t K K' a 1 t b 1 a 2 t t 0.95K 1.05K K 0.4 0.2 0 0.2 0.4 FIG. 1. ͑Color online͒ ͑Upper left inset͒ Honeycomb bilayer unit cell. Atoms in the lower layer ͑2͒ are marked as smaller ͑black͒ c...

Double layer quantum Hall systems have interesting properties associated with interlayer correlations. At ν = 1/m where m is an odd integer they exhibit spontaneous symmetry breaking equivalent to that of spin 1/2 easy-plane ferromagnets, with the layer degree of freedom playing the role of spin. We explore the rich variety of quantum and finite temperature phase transitions in these systems. In particular, we show that a magnetic field oriented parallel to the layers induces a highly collective commensurate-incommensurate phase transition in the magnetic order. 73.20.Dx, 64.60.Cn Recent technological progress has allowed production of double-layer quantum Hall systems of extremely high mobility. The separation d of the two 2D electron gases is so small (d ∼ 100Å) as to be comparable to the spacing between electrons in the same layer and quantum states with strong correlations between the layers have been observed experimentally and discussed theoretically [1][2][3]. Wen and Zee have pointed out that at Landau level filling factor ν = 1/m and in the absence of interlayer tunneling, this system exhibits a spontaneously broken U(1) gauge symmetry [4]. (m is an odd integer.) The corresponding Goldstone mode is a neutral density wave in which the densities in the two layers oscillate out of phase. A finite temperature Kosterlitz-Thouless (KT) phase transition is expected to be associated with this broken symmetry.In this paper we focus for convenience on the case ν = 1 and show that this system can be viewed as an easy-plane quantum itinerant ferromagnet. Following Ref.[5] (but with a rotated coordinate system) we will use an 'isospin' magnetic language in which isospin 'up' ('down') refers to an electron in the 'upper' ('lower') layer [6]. Using this language and building upon recent progress in understanding the case of single-layer systems at ν = 1 with real spin [7,8] we explore the consequences of the mixing of charge and isospin degrees of freedom and discuss the rich variety of phase transitions controlled by temperature, layer separation, tunneling between layers, layer charge imbalance and magnetic field tilt angles. In addition to the KT transition we find a 'commensurateincommensurate' phase transition as a function of B , the component of the magnetic field in the plane. Furthermore we demonstrate that the Meissner screening of the in-plane component of the magnetic field (B ) predicted by Ezawa and Iwazaki does not occur. A portion of this rich set of phenomena is captured in the schematic zerotemperature phase diagram illustrated in Fig. 1. The present paper will be devoted to explication of the physical picture underlying this phase diagram. Technical details of the microscopic calculations on which it is based will be presented elsewhere [9].It is helpful to begin by discussing the limit of zero temperature, zero tunneling amplitude between the layers and layer spacing d = 0. We work entirely in the lowest Landau level and take the unit of length to be l ≡ (hc/eB) 1/2 . Coulomb repulsion induces ...