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arXiv:1809.07775 · Proceedings of the National Academy of Sciences

Universal Fermi-surface anisotropy renormalization for interacting Dirac fermions with long-range interactions

Jia Ning Leaw, Ho-Kin Tang, Maxim Trushin, Fakher F. Assaad, Shaffique Adam

Published Dec 9, 2019• 8 citations
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Abstract

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.

DOI
10.1073/pnas.1913096116

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We perform density-matrix renormalization group studies of a two-dimensional electron gas in a high perpendicular magnetic field and with an anisotropic band mass. At half filling in the lowest Landau level, such a system is a Fermi liquid of composite fermions. By measuring the Fermi surface of these composite fermions, we determine a relationship between the anisotropy of composite fermion dispersion ${\ensuremath{\alpha}}_{CF}$ and the original anisotropy ${\ensuremath{\alpha}}_{F}$ of the fermion dispersion at zero magnetic field. For systems where the electrons interact via a Coulomb interaction, we find ${\ensuremath{\alpha}}_{CF}=\sqrt{{\ensuremath{\alpha}}_{F}}$ within our numerical accuracy. The same result has been found concurrently in recent experiments. We also find that the relationship between the anisotropies is dependent on the form of the electron-electron interaction.
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Both insulating and conducting electronic behaviors have been experimentally seen in clean bilayer graphene samples at low temperature, and there is still no consensus on the nature of the interacting ground state at half filling and in the absence of a magnetic field. Theoretically, several possibilities for the insulating ground states have been predicted for weak interaction strength. However, a recent renormalization-group calculation on a Hubbard model for charge-neutral bilayer graphene with short-range interactions suggests the emergence of low-energy Dirac fermions that would stabilize the metallic phase for weak interactions. Using a nonperturbative projective quantum Monte Carlo, we calculate the ground state for bilayer graphene using a realistic model for the Coulomb interaction that includes both short-range and long-range contributions. We find that a finite critical on-site interaction is needed to gap bilayer graphene and the transition belongs to the Gross-Neveu universality class, thereby confirming the Hubbard model expectations even in the presence of a long-range Coulomb potential, in agreement with our theoretical analysis. In addition, we also find that the critical on-site interactions necessary to destabilize the metallic ground state decrease with increasing interlayer coupling.
In a two-dimensional electron gas under a strong magnetic field, correlations generate emergent excitations distinct from electrons. It has been predicted that "composite fermions"--bound states of an electron with two magnetic flux quanta--can experience zero net magnetic field and form a Fermi sea. Using infinite-cylinder density matrix renormalization group numerical simulations, we verify the existence of this exotic Fermi sea, but find that the phase exhibits particle-hole symmetry. This is self-consistent only if composite fermions are massless Dirac particles, similar to the surface of a topological insulator. Exploiting this analogy, we observe the suppression of 2k(F) backscattering, a characteristic of Dirac particles. Thus, the phenomenology of Dirac fermions is also relevant to two-dimensional electron gases in the quantum Hall regime.
To understand correlated insulating and unconventional superconducting states in twisted bilayer graphene, we perform a systematic study of spin and pairing correlations in an effective two-orbital Hubbard model, by using the ground-state constrained-path quantum Monte Carlo method. Our numerical simulations reveal that when only the on-site Hubbard UUU is considered, the long-range magnetic order develops for $U\ensuremath{\ge}3.2t$ at half filling. Upon doping away from half filling, an analysis based on the pairing correlations and corresponding vertex contributions identifies that pairing with the d+idd+idd+id symmetry is the dominant pairing channel. As the on-site Coulomb interaction is increased, both spin correlation and effective pairing interaction are enhanced simultaneously, indicating the intimate relation between magnetism and superconductivity. An inclusion of nearest neighbor Coulomb interaction VVV results in a suppression of the d+idd+idd+id superconductivity, but it can survive in the small VVV regime. Our findings are useful for clarifying the ongoing controversy on the superconducting pairing symmetry in twisted bilayer graphene.
We present a nonperturbative study of the quantum many-body effects caused by the long-range Coulomb interaction in a two-dimensional semi-Dirac semimetal. This kind of semimetal may be realized in deformed graphene and a class of other realistic materials. In the noninteracting limit, the dispersion of semi-Dirac fermion is linear in one direction and quadratic in the other direction. When the impact of Coulomb interaction is taken into account, such a dispersion can be significantly modified. To reveal the correlation effects, we first obtain the exact self-consistent Dyson-Schwinger equation of the full fermion propagator and then extract the momentum dependence of the renormalized fermion dispersion from the numerical solutions. Our results show that the fermion dispersion becomes linear in two directions. These results are compared to previous theoretical works on semi-Dirac semimetals.