We develop and apply the diagrammatic Monte Carlo technique to address the problem of the stability of the Dirac liquid state (in a graphene-type system) against the strong long-range part of the Coulomb interaction. So far, all attempts to deal with this problem in the field-theoretical framework were limited either to perturbative or random phase approximation and functional renormalization group treatments, with diametrically opposite conclusions. Our calculations aim at the approximation-free solution with controlled accuracy by computing vertex corrections from higher-order skeleton diagrams and establishing the renormalization group flow of the effective Coulomb coupling constant. We unambiguously show that with increasing the system size L (up to ln(L)∼40), the coupling constant always flows towards zero; i.e., the two-dimensional Dirac liquid is an asymptotically free T=0 state with divergent Fermi velocity.
We develop and apply the diagrammatic Monte Carlo technique to address the problem of the stability of the Dirac liquid state (in a graphene-type system) against the strong long-range part of the Coulomb interaction. So far, all attempts to deal with this problem in the field-theoretical framework were limited either to perturbative or random phase approximation and functional renormalization group treatments, with diametrically opposite conclusions. Our calculations aim at the approximation-free solution with controlled accuracy by computing vertex corrections from higher-order skeleton diagrams and establishing the renormalization group flow of the effective Coulomb coupling constant. We unambiguously show that with increasing the system size L (up to ln(L)∼40), the coupling constant always flows towards zero; i.e., the two-dimensional Dirac liquid is an asymptotically free T=0 state with divergent Fermi velocity.
Studying the strong correlation effects in interacting Dirac fermion systems is one of the most challenging problems in modern condensed matter physics. The long-range Coulomb interaction and the fermion-phonon interaction can lead to a variety of intriguing properties. In the strong-coupling regime, weak-coupling perturbation theory breaks down. The validity of 1/N expansion with N being the fermion flavor is also in doubt since N equals to 2 or 4 in realistic systems. Here, we investigate the interaction between (1+2)- and (1+3)-dimensional massless Dirac fermions and a generic scalar boson, and develop an efficient nonperturbative approach to access the strong-coupling regime. We first derive a number of self-consistently coupled Ward-Takahashi identities based on a careful symmetry analysis and then use these identities to show that the full fermion-boson vertex function is solely determined by the full fermion propagator. Making use of this result, we rigorously prove that the full fermion propagator satisfies an exact and self-closed Dyson-Schwinger integral equation, which can be solved by employing numerical methods. A major advantage of our nonperturbative approach is that there is no need to employ any small expansion parameter. Our approach provides a unified theoretical framework for studying strong Coulomb and fermion-phonon interactions. It may also be used to approximately handle the Yukawa coupling between fermions and order-parameter fluctuations around continuous quantum critical points. Our approach is applied to treat the Coulomb interaction in undoped graphene. We find that the renormalized fermion velocity exhibits a logarithmic momentum dependence but is nearly energy independent, and that no excitonic gap is generated by the Coulomb interaction. These theoretical results are consistent with experiments in graphene.
After its first isolation by Geim and Novoselov in 2004, graphene has grown quickly into an independent subfield of contemporary condensed matter physics, counting thousands of publications. This material has attracted so much attention not only because it is truly two-dimensional -- a feature long thought to be impossible to be realized in nature -- but also because it is the prime representative of a new class of condensed matter systems, the Dirac materials, whose electronic spectrum disperses linearly around isolated points in the Brillouin zone. An outstanding consequence of this linear energy-momentum dispersion is the collapse of the Fermi surface to isolated points at charge neutrality, accompanied by a vanishing single-particle density of states. While this feature of the noninteracting model is what separates graphene from most other, more conventional condensed matter systems, giving rise to many of its astounding physical properties, it leads to severe computational complications when extending the model to contain two-body interactions and/or disorder. In this thesis we consider both extensions separately. After rigorously deriving the low-energy quantum field theory from a tight-binding model and establishing the one-particle irreducible (1PI) vertex functions as the fundamental building blocks of correlation functions, we construct a nonperturbative formalism for the calculation of these vertex functions using the well-established functional renormalization group (fRG). By combining the fRG with the Keldysh formalism, accounting for possible external electromagnetic fields and nontrivial initial correlations, this framework is capable of handling thermal equilibrium and true nonequilibrium alike. To circumvent some of the technical problems of the standard fRG formalism when treating fermionic systems at finite density, we also explore a variant of the fRG, where the chemical potential is interpreted as a flow parameter. We obtain hierarchical sets of flow equations, which describe the change of the 1PI vertex functions upon varying an artificial cutoff scale or the chemical potential, respectively. These nonperturbative formalisms have been used to calculate the renormalization of the Fermi velocity and the static dielectric function at finite temperature and density in the strong coupling regime as an explicit demonstration of their capabilities. Since the Keldysh fRG is an exact formulation of quantum field theory, in particular when external magnetic fields are present, it should be possible to describe the fractional quantum Hall effect in this framework. However, the typical truncation schemes that are currently available to approximately solve the fRG flow equation fail to account for the nontrivial correlation effects necessary to access the fractional quantum Hall regime. To circumvent this problem we consider a modified field theory, where these correlations are implemented ``by hand'' via a Chern-Simons gauge field, coupling to the fermions in addition to the Coulomb interaction term. This modified field theory is analyzed in a stationary phase approximation including Gaussian fluctuations. The electromagnetic response tensor is calculated in the random phase approximation, yielding the Hall conductivities. In the second part of this thesis we consider disordered Dirac fermions in the absence of two-particle interactions. After sketching how the most general disorder potential for Dirac fermions arises from a tight-binding model, we first consider an explicit physical scenario, a disordered graphene pn junction in the presence of a quantizing magnetic field perpendicular to the graphene sheet. We derive an effective one-dimensional theory for the chiral states that propagate along the junction interface and calculate the full conductance distribution in the crossover between the clean and the strong-disorder limit, via an exact solution of a Fokker-Planck equation. Lastly, we develop a nonperturbative approach to calculate the disorder induced self-energy that is based on exact Schwinger-Dyson equations and Ward identities, not only for graphene, but also for other paradigmatic semimetals with a nodal point dispersion.
We report results of large-scale quantum Monte Carlo (QMC) simulations of graphene. Using cutting-edge algorithmic improvements, we are able to consider spatial volumes, corresponding to 20808 electrons, that allow us to access energy scales of direct relevance to experiments. Using constrained random phase approximation (cRPA) estimates of short-ranged interactions combined with a Coulomb tail, we are able to successfully confront numerical and experimental estimates of the Fermi velocity renormalization, thus fixing the parameters of microscopic Hamiltonian to exactly reproduce the experimental data. Subsequent comparison of the QMC results with perturbation theory not only show the non-Fermi liquid character of graphene, but also prove the importance of lattice-scale physics and higher-order perturbative corrections beyond RPA for the quantitative description of the experimental data for the Fermi velocity renormalization in suspended graphene. The same Monte Carlo technique is subsequently employed to investigate electrical transport by computing the renormalization of the optical conductivity in freestanding graphene. The large lattice sizes and low temperatures accessible in our QMC simulations enable a clear resolution of the Dirac plateau in the spectral function, even in the presence of interactions. We demonstrate that, despite the strong renormalization of the single-particle dispersion, the optical conductivity remains essentially constant, in agreement with current experimental observations. This finding points toward the possibility of extending the theorem on the absence of optical conductivity renormalization for the Hubbard model on the hexagonal lattice to the case of long-range interactions.
Recent experimental [I. Jo et al ., Phys. Rev. Lett. 119, 016402 (2017)] and numerical [M. Ippoliti, S. D. Geraedts, R. N. Bhatt, Phys. Rev. B 95, 201104 (2017)] evidence suggests an intriguing universal relationship between the Fermi surface anisotropy of the noninteracting parent 2-dimensional (2D) electron gas and the strongly correlated composite Fermi liquid formed in a strong magnetic field close to half-filling. Inspired by these observations, we explore more generally the question of anisotropy renormalization in interacting 2D Fermi systems. Using a recently developed [H. -K. Tang et al ., Science 361, 570 (2018)] nonperturbative and numerically exact projective quantum Monte Carlo simulation as well as other numerical and analytic techniques, only for Dirac fermions with long-range Coulomb interactions do we find a universal square-root decrease of the Fermi-surface anisotropy. For the ν = 1 / 2 composite Fermi liquid, this result is surprising since a Dirac fermion ground state was only recently proposed as an alternative to the usual Halperin–Lee–Read state. Our proposed universality can be tested in several anisotropic Dirac materials including graphene, topological insulators, organic conductors, and magic-angle twisted bilayer graphene.
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.
Within the infinite series of ring (or bubble) diagram approximation for the electronic self-energy as appropriate for the long-range Coulomb interaction, we calculate the density-dependent T=0 Fermi-liquid quasiparticle effective-mass renormalization in two-dimensional (2D) electron systems in the random-phase approximation. We find that the quasiparticle effective mass increases very strongly with increasing rs, exhibiting an unexpected divergence at a critical rs value ($\ensuremath{\sim}16$ in the ideal 2D system). We discuss the possibility for and the consequences of such an interaction-induced effective-mass divergence as the electron density decreases and critically comment on the likely theoretical scenarios which could lead to such a low-density effective-mass divergence in electron liquids.