We present evidence, from Lattice Monte Carlo simulations of the phase diagram of graphene as a function of the Coulomb coupling between quasiparticles, that graphene in vacuum is likely to be an insulator. We find a semimetal-insulator transition at α crit g = 1.11 ± 0.06, where α g ≃ 2.16 in vacuum, and α g ≃ 0.79 on a SiO 2 substrate. Our analysis uses the logarithmic derivative of the order parameter, supplemented by an equation of state. The insulating phase disappears above a critical number of four-component fermion flavors 4 < N crit f < 6. Our data are consistent with a second-order transition. Graphene, a carbon allotrope with a two-dimensional honeycomb structure, has become an important player at the forefront of condensed matter physics, drawing the attention of theorists and experimentalists alike due to its challenging nature as a many-body problem, its unusual electronic properties and possible technological applications (see Refs. [1, 2] and references therein). Graphene also belongs to a large class of planar condensed-matter systems, which includes other graphite-related materials as well as high-T c superconductors.A distinctive feature of graphene is that its band structure contains two degenerate 'Dirac points', in the vicinity of which the dispersion is linear, as in relativistic theories [3]. The low-energy excitations in graphene are thus Dirac quasiparticles of Fermi velocity v ≃ c/300, where c is the speed of light in vacuum. These are described by the Euclidean actionwhere g 2 = e 2 /ǫ 0 for graphene in vacuum, ψ a is a four-component Dirac field in 2+1 dimensions, A 0 is a Coulomb field in 3+1 dimensions, N f = 2 for real graphene, andwhere the Dirac matrices γ µ satisfy the Euclidean Clifford algebra {γ µ , γ ν } = 2δ µν . The strength of the Coulomb interaction is controlled (as can be shown by rescaling t and A 0 ) by α g = e 2 /(4πvǫ 0 ), which is the graphene analogue of the fine-structure constant α ≃

We have studied, in a fully non-perturbative calculation, a dilute system of spin 1/2 interacting fermions, characterized by an infinite scattering length at finite temperatures. Various thermodynamic properties and the condensate fraction were calculated and we have also determined the critical temperature for the superfluid-normal phase transition in this regime. The thermodynamic behavior appears as a rather surprising and unexpected mélange of fermionic and bosonic features. The thermal response of a spin 1/2 fermion at the BCS-BEC crossover should be classified as that of a new type of superfluid. The unitary regime is commonly referred to as the situation in which the scattering length a greatly exceeds the average inter-particle separation, thus n|a| 3 ≫ 1, where n is the particle number density [1,2]. It is widely accepted by theorists that at T = 0 these systems are superfluid and that in the unitary regime the coherence length is comparable in magnitude with the average interparticle separation. At T = 0 this problem has been considered by a number of authors [3] and the most accurate results so far have been reported in Refs. [4,5,6]. In 2002 it was shown experimentally that such systems are (meta)stable, and they have been studied extensively experimentally ever since [7,8].The typical theoretical treatment of such systems is based on the idea put forward by Eagles, Leggett and others [9], and used subsequently by most authors [10,11]. The form of the many-body wave function is as in the weak coupling BCS limit and is used for all values of the scattering length a. The particle number projected BCS wave function has the functional formwhere odd subscripts refer to spin-up particles and even subscripts to spin-down particles, A is the antisymmetrization operator, r 12 = |r 1 − r 2 | and φ(r) is either the Cooper pair wave function in the BCS limit, or the two-bound state wave function in the BEC limit. The main difficulty with this approach becomes evident when one tries to use this kind of wave function in the unitary regime, where n|a| 3 ≫ 1. In the extreme BEC limit, this wave function describes a state with all bosons (dimers) at rest, in the condensed state. The fraction of non-condensed bosons (dimers) is known to be small thenwhere n d = n/2 and a dd = 0.6a is the dimer-dimer scattering length [6,12]. When one approaches the unitary regime, the fraction of non-condensed bosons becomes of order one [13], which resembles qualitatively the situation in superfluid 4 He, and then a meanfield description (with or without fluctuations) becomes questionable. In order to calculate the thermal properties of a system of fermions in the unitary regime, we have placed them on a 3D-spatial lattice and used a path integral representation of the partition function. We start fromwhere β = 1/T = N τ τ andÔ is a quantity of interest. T stands for the temperature and µ for the chemical potential, andĤ andN are the Hamiltonian and the particle number operators respectively.Since the system under consideration is di...

The Quantum Monte Carlo method for spin 1/2 fermions at finite temperature is formulated for dilute systems with an s-wave interaction. The motivation and the formalism are discussed along with descriptions of the algorithm and various numerical issues. We report on results for the energy, entropy and chemical potential as a function of temperature. We give upper bounds on the critical temperature Tc for the onset of superfluidity, obtained by studying the finite size scaling of the condensate fraction. All of these quantities were computed for couplings around the unitary regime in the range −0.5 ≤ (kF a) −1 ≤ 0.2, where a is the s-wave scattering length and kF is the Fermi momentum of a non-interacting gas at the same density. In all cases our data is consistent with normal Fermi gas behavior above a characteristic temperature T0 > Tc, which depends on the coupling and is obtained by studying the deviation of the caloric curve from that of a free Fermi gas. For Tc < T < T0 we find deviations from normal Fermi gas behavior that can be attributed to pairing effects. Low temperature results for the energy and the pairing gap are shown and compared with Green Function Monte Carlo results by other groups.

We discuss the Monte Carlo method of simulating lattice field theories as a means of studying the low-energy effective theory of graphene. We also report on simulational results obtained using the Metropolis and Hybrid Monte Carlo methods for the chiral condensate, which is the order parameter for the semimetal-insulator transition in graphene, induced by the Coulomb interaction between the massless electronic quasiparticles. The critical coupling and the associated exponents of this transition are determined by means of the logarithmic derivative of the chiral condensate and an equation-of-state analysis. A thorough discussion of finite-size effects is given, along with several tests of our calculational framework. These results strengthen the case for an insulating phase in suspended graphene, and indicate that the semimetal-insulator transition is likely to be of second order, though exhibiting neither classical critical exponents, nor the predicted phenomenon of Miransky scaling.

We survey approaches to nonrelativistic density functional theory (DFT) for nuclei using progress toward ab initio DFT for Coulomb systems as a guide. Ab initio DFT starts with a microscopic Hamiltonian and is naturally formulated using orbital-based functionals, which generalize the conventional local-density-plus-gradients form. The orbitals satisfy single-particle equations with multiplicative (local) potentials. The DFT functionals can be developed starting from internucleon forces using wave-function based methods or by Legendre transform via effective actions. We describe known and unresolved issues for applying these formulations to the nuclear manybody problem and discuss how ab initio approaches can help improve empirical energy density functionals.