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 ${\alpha }_g^{\mathrm{crit}}=1.11\pm 0.06$, where ${\alpha }_g\simeq 2.16$ in vacuum, and ${\alpha }_g\simeq 0.79$ on a ${\mathrm{SiO}}_2$ 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 $4<N_f^{\mathrm{crit}}<6$. Our data are consistent with a second-order transition.

We study, in a fully nonperturbative 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 are calculated and we also determine 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 quantum Monte Carlo method for spin-$\frac{1}{2}$ 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 $T_c$ 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 $-0.5⩽{\left(k_{F}a\right)}^{-1}⩽0.2$, where $a$ is the $s$-wave scattering length and $k_F$ is the Fermi momentum of a noninteracting gas at the same density. In all cases our data are consistent with normal Fermi gas behavior above a characteristic temperature $T_0>T_c$, which depends on the coupling and is obtained by studying the deviation of the caloric curve from that of a free Fermi gas. For $T_c<T<T_0$ 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 present a high-precision determination of the universal contact parameter in a strongly interacting Fermi gas. In a trapped gas at unitarity, we find the contact to be $3.06\pm 0.08$ at a temperature of 0.08 of the Fermi temperature in a harmonic trap. The contact governs the high-momentum (short-range) properties of these systems, and this low-temperature measurement provides a new benchmark for the zero-temperature homogeneous contact. The experimental measurement utilizes Bragg spectroscopy to obtain the dynamic and static structure factors of ultracold Fermi gases at high momentum in the unitarity and molecular Bose-Einstein condensate regimes. We have also performed quantum Monte Carlo calculations of the static properties, extending from the weakly coupled BCS regime to the strongly coupled Bose-Einstein condensate case, that show agreement with experiment at the level of a few percent.