Graphene has been subject to intense theoretical and experimental research since its discovery in 2005. The lowered dimensionality of graphene leads to strong many-body interactions. This casts doubt on applicability of traditional models in solid state physics. In this work we describe the low-energy excitations of graphene in terms interacting fermionic quasiparticles. We present results from Monte Carlo simulations of this effective field theory, using the tadpole improvement method to eliminate discretization artefacts present in previous simulations. The simulations demonstrate that graphene undergoes a quantum phase transition from its observed semimetallic phase to a Mott insulating phase. The critical coupling under which the transition takes place depends on the fermion velocity. We compute the excitation spectrum, from which we extract renormalized value of the fermion velocity. Our results indicate that the renormalization has a significant effect on the critical coupling.
Graphene has been subject to intense theoretical and experimental research since its discovery in 2005. The lowered dimensionality of graphene leads to strong many-body interactions. This casts doubt on applicability of traditional models in solid state physics. In this work we describe the low-energy excitations of graphene in terms interacting fermionic quasiparticles. We present results from Monte Carlo simulations of this effective field theory, using the tadpole improvement method to eliminate discretization artefacts present in previous simulations. The simulations demonstrate that graphene undergoes a quantum phase transition from its observed semimetallic phase to a Mott insulating phase. The critical coupling under which the transition takes place depends on the fermion velocity. We compute the excitation spectrum, from which we extract renormalized value of the fermion velocity. Our results indicate that the renormalization has a significant effect on the critical coupling.
Monte Carlo simulation of a 2+1 dimensional model of voltage-biased bilayer graphene, consisting of relativistic fermions with chemical potential $\ensuremath{\mu}$ coupled to charged excitations with opposite sign on each layer, has exposed noncanonical scaling of bulk observables near a quantum critical point found at strong coupling. We present a calculation of the quasiparticle dispersion relation E(k) as a function of exciton source j in the same system, employing partially twisted boundary conditions to boost the number of available momentum modes. The Fermi momentum kF and superfluid gap $\mathrm{\ensuremath{\Delta}}$ are extracted in the $j\ensuremath{\rightarrow}0$ limit for three different values of $\ensuremath{\mu}$, and support a strongly interacting scenario at the Fermi surface with $\mathrm{\ensuremath{\Delta}}\ensuremath{\sim}O(\ensuremath{\mu})$. We propose an explanation for the observation $\ensuremath{\mu}<{k}_{F}$ in terms of a dynamical critical exponent z<1.
Optical conductivity of graphene is studied using quantum Monte Carlo calculations. We start from a Euclidean current-current correlator and extract $\ensuremath{\sigma}(\ensuremath{\omega})$ from Green-Kubo relations using the Backus-Gilbert method. Calculations were performed both for long-range interactions and taking into account only the contact term. In both cases we vary interaction strength and study its influence on optical conductivity. We compare our results with previous theoretical calculations choosing $\ensuremath{\omega}\ensuremath{\approx}\ensuremath{\kappa}$, thus working in the region of the plateau in $\ensuremath{\sigma}(\ensuremath{\omega})$ which corresponds to optical conductivity of Dirac quasiparticles. No dependence of optical conductivity on interaction strength is observed unless we approach the antiferromagnetic phase transition in the case of an artificially enhanced contact term. Our results strongly support previous theoretical studies that claimed very weak regularization of graphene conductivity.
We study a number of quantum phase transitions, which are exotic in their nature and separates non-trivial phases of matter. Since quantum fluctuations, which drive these phase transitions, are stronger in low-dimensions, we concentrate on low-dimensional systems. We consider two different two-dimensional systems in this thesis and study their phase transition. \n\nFirst, we investigate a phase transition in graphene, one of the most famous two-dimensional systems in condensed matter. For a suspended bilayer graphene in ν = 0 quantum Hall regime, the conductivity data and mean-field analysis suggests a phase transition from an antiferromagnetic (AF) state to a valence bond solid (VBS) state, when perpendicular electric field is increased. This AF to VBS phase transition is reminiscent of deconfined criticality, which is a novel phase transition that cannot be explained by Landau’s theory of symmetry breaking. We show that in the strong coupling regime of bilayer graphene, the AF state is destabilized by the transverse electric field, likely resulting in a VBS state. We also consider monolayer and bilayer graphene in the large cyclotron gap limit and show that the effective action for the AF and VBS order parameters have a topological Wess-Zumino-Witten term, supporting that the phase transition observed in experiments is in the deconfined criticality class. \n\nSecond, we study the model systems of cuprate superconductor, which is effectively a two-dimensionalal system in the CuO_2 plane. The proposal that the pseudogap metal is a fractionalized Fermi liquid described by a quantum dimer model is extended using the density matrix renormalization group. Measuring the Friedel oscillations in the open boundaries reveals that the fermionic dimers have dispersion minima near (π/2,π/2), which is compatible with the Fermi arcs in photoemission. Moreover, investigating the entanglement entropy suggests that the dimer model with low fermion density is similar to the free fermion system above the Lifshitz transition. We also study the phase transition from a metal with SU(2) spin symmetry to an AF metal. By applying the functional renormalization group to the two-band spin-fermion model, we establish the existence of a strongly coupled fixed point and calculate critical exponents of the fixed point.
The Hubbard and extended Hubbard models on the honeycomb lattice can be seen as prototype models of a single-layer graphene placed in a high dielectric constant environment that screens the Coulomb interaction. Taking advantage of the absence of a sign problem at half-filling, we study this problem with clusters up to 96 sites with the determinant quantum Monte Carlo method as an impurity solver for the dynamical cluster approximation at finite temperatures. After determining the stability of the semimetallic phase to interaction-induced spin-density wave (SDW), charge-density wave (CDW), and Mott insulating phases, we study the single-particle dynamics of the Dirac fermions. We show that when spontaneous symmetry breaking is avoided, the semimetallic phase is a stable Fermi liquid in the presence of repulsive interactions and that Kondo screening dominates the low temperature regime, even though there is a $\ensuremath{\rho}(\ensuremath{\omega})=|\ensuremath{\omega}|$ type local density of states. We also investigate the impact of the correlation effects on the renormalization of the Fermi velocity vF. We find that vF is not renormalized when only on-site repulsion U is present, but that near-neighbor repulsion V does renormalize vF. This may explain the variations between different measurements of vF in graphene.
The many-body correlation between electrons is the origin of many fascinating phenomena in condensed matter systems, such as high temperature superconductivity, superfluidity, fractional quantum Hall effect, and Mott insulator. Strongly correlated systems have been an important subject of condensed matter physics for several decades, especially after the discovery of high temperature cuprate superconductors. In this thesis, we apply first principles quantum Monte Carlo (QMC) method to several representative systems to study the electron correlations in transition metal oxides (vanadium dioxide) and low dimensional electronic systems (graphene and graphene-like two dimensional systems). Vanadium dioxide (VO2) is a paradigmatic example of a strongly correlated system that undergoes a metal-insulator transition at a structural phase transition. To date, this transition has necessitated significant post-hoc adjustments to theory in order to be described properly. We apply first principles quantum Monte Carlo (QMC) to study the structural dependence of the properties of VO2. Using this technique, we simulate the interactions between electrons explicitly, which allows for the metal-insulator transition to naturally emerge, importantly without ad-hoc adjustments. The QMC calculations show that the structural transition directly causes the metal-insulator transition and a change in the coupling of vanadium spins. This change in the spin coupling results in a prediction of a momentum-independent magnetic excitation in the insulating state. While two-body correlations are important to set the stage for this transition, they do not change significantly when VO2 becomes an insulator. These results show that it is now possible to account for electron correlations in a quantitatively accurate way that is also specific to materials. Electron correlation in graphene is unique because of the interplay of the Dirac cone dispersion of pi electrons with long range Coulomb interaction. The random phase approximation predicts no metallic screening at long distances and low energies because of the zero density of states at Fermi level. It is thus interesting to see how screening takes place in graphene at different length scales. We addressed this problem by computing the structure factor S(q) and S(q, ω) of freestanding graphene using ab initio fixed-node diffusion Monte Carlo and the random phase approximation. The X-ray measured structure factor is reproduced very accurately using both techniques, provided that sigma-bonding electrons are included in the simulations. Strong dielectric screening from sigma electrons are observed, which redshifts the pi plasmons resonance frequency at long distance and reduces the effective interactions between pi electrons at short distance. The short distance screening makes suspended graphene a weakly correlated semimetal which otherwise would be an insulator. The third piece of works is dedicated to studying the low energy excitation of many-body systems using extended Koopmans' theorem (EKT). The EKT provides a straight forward way to compute charge excitation spectra, such as ionization potentials, electron affinities from any level of theory. We implemented the EKT within the QMC framework, and performed systematic benchmark studies of ionization potentials of the second- and third-row atoms, and closed- and open-shell molecules. We also applied it to compute the quasiparticle band structure of solids (graphene). For complex correlated systems, identifying relevant low energy physics degrees of freedom is extremely important to understanding the system's collective behavior at different length scales. In this sense, bridging the realistic systems to lower energy effective lattice models that involve fewer but important degrees of freedom is significant to understanding correlated systems. We have formulated three ab initio density matrix based downfolding (AIDMD) methods to downfold the ab initio systems into effective lattice models. We have demonstrated the successfulness of these methods by applying them to molecules (H2) and periodic systems (hydrogen chain and graphene).
We present the results of numerical simulations of a (2+1)-dimensional fermion field theory based on a recent proposal for a model of graphene consisting of Nf four-component Dirac fermions moving in the plane and interacting via an instantaneous Coulomb interaction. In the strong-coupling limit we identify a critical number of flavors Nfc=4.8(2) separating an insulating from a conducting phase. This transition corresponds to the location of a quantum critical point, and we use a fit to the equation of state for the chiral order parameter to estimate the critical exponents. Next we simulate Nf=2 corresponding to real graphene and approximately locate a transition from strong- to weak-coupling behavior. Strong correlations are evident in the weak-coupling regime.