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

Significance

Electrons in a 2D metal have a 1-dimensional contour in momentum space, called the Fermi surface, that separates the occupied and unoccupied energy levels. For realistic materials the Fermi surface is usually anisotropic, while theoretical models often assume rotationally symmetric Fermi surfaces. We address a simple but fundamental question: Given a specified anisotropic Fermi surface for a noninteracting 2D electron gas, what happens to this anisotropy in the presence of electron–electron interactions? Remarkably, only for Dirac electrons with long-range Coulomb interactions there is a universal square-root renormalization of the anisotropy, independent of the interaction strength. This prediction can be tested in a variety of experimental systems including graphene, topological insulators, organic-layered compounds, twisted bilayer graphene, and others.

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 $\nu =1/2$ 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.

A plethora of quantum Hall ground states are observed when the 2-dimensional (2D) electron gas is placed in a strong magnetic field. Unique among these states is when the system is tuned to half-filling; here rather than observing an insulator as is typical, instead a metallic state is observed. This compressible phase called a “composite Fermi liquid” is believed to emerge because at the mean-field level, the magnetic flux attached to each composite fermion exactly cancels the external magnetic field (1, 2). The Fermi surface properties of this metallic state have long been explored. A very recent experiment at Princeton measured how the anisotropy of this strongly correlated Fermi surface along the high-symmetry axis (parameterized by $\eta$, defined below) was related to that of the original Fermi surface ${\eta }_0$ of the noninteracting electrons in the absence of an external magnetic field (3). Before performing the experiment, the researchers surveyed leading theorists on what they expected to observe, obtaining a range of answers: Since the composite Fermi liquid was a universal strongly correlated ground state, in which the kinetic energy of the noninteracting bands was quenched, some theorists expected no Fermi surface anisotropy in the composite Fermi liquid and others expected the state to have the same anisotropy as the noninteracting bands from which the state emerged. Remarkably, both the experiment and DMRG numerics (4) observed that $\eta =\sqrt{{\eta }_0}$; i.e., the interacting state was always more isotropic and universally so.

This experiment motivated us to ask a more fundamental question: What is the relationship between the Fermi surface anisotropies of an interacting Fermi liquid given a fixed anisotropy of the noninteracting bands? I.e., we are interested not only in the effective composite Fermi liquid at $\nu =1/2$, but also in the interacting 2D Fermi liquid in general. To illustrate that the answer is not obvious, consider the following: One might expect that interactions enhance the anisotropy. Anisotropy can be thought of as broken rotational symmetry, and it was predicted long ago that the exchange interaction can enhance the splitting between broken symmetry states (5). In the quantum Hall context, interaction-enhanced Zeeman splitting has been seen experimentally for the graphene integer quantum Hall effect (6) and more recently for an interaction-induced spontaneous symmetry breaking of nematic phases (7).

In other contexts, interactions wash away nonuniversal particularities of the noninteracting model flowing to a universal interacting fixed point (the universal $\nu =1/3$ Laughlin state observed in different parent materials and with different confining potentials leading to different effective interactions is but one example). Experimentally, the observed sequence of fractional quantum Hall plateaus in graphene suggests that the strongly interacting ground state partially restores the spin and valley splitting of the noninteracting system (8). For the specific case of $\nu =1/2$, using a Gaussian approximation for the electron–electron interactions, ref. 9 found analytically that interactions always make the composite Fermi liquid more isotropic, but in a nonuniversal way where $\eta /{\eta }_0$ could take on values between 0 and 1, depending on the length scale of the Gaussian. Other calculations in specific models (10, 11) suggest no change in the Fermi surface anisotropy; while yet others show a nonuniversal decrease in anisotropy (12, 13).

We are not the first to ask the question about the many-body renormalization of anisotropic Fermi surfaces. Back in 1960, Kohn and Luttinger (14) argued that the standard diagrammatic perturbation theory to account for electron–electron interactions failed when considering anisotropic Fermi surfaces. Moreover, the effect of correlations was understood to be nonuniversal where the anisotropy renormalization depends on material-specific parameters like the effective mass, carrier density, and dielectric substrate. Below we reproduce the leading-order term for “Schrödinger electrons” where the bare band dispersion is of the usual parabolic energy form to illustrate how this nonuniversality arises for generic band structures. However, and remarkably, we find that for Dirac fermions (e.g., graphene) with bare Fermi velocity anisotropy, in the presence of a long-range Coulomb interaction, there is a universal relationship $\eta =\sqrt{{\eta }_0}$ that does not depend on any of the material-specific parameters mentioned above. We emphasize that the long-range nature of the Coulomb potential is essential: Our analytical and numerical results also suggest that Dirac fermions with only contact interactions retain the anisotropy of the original noninteracting system. We find that the chirality of the Dirac bands and the long-range interaction are both necessary to obtain the square-root anisotropy.

In the presence of Coulomb interactions, the properties of correlated Dirac fermions are governed by 2 very different fixed points (15). There is a stable “weak-coupling” fixed point determined entirely by the long-range Coulomb tail, which flows at low energies to a noninteracting Lorenz invariant theory (16) (where the electron Fermi velocity is equal to the speed of the light). Then there is also an unstable “strong-coupling” Gross–Neveu fixed point controlled by the contact part of the Coulomb interaction and associated with the transition to a Mott insulating antiferromagnet at half-filling. While ultimately unstable, the proximity to the strong-coupling fixed point is largely responsible for most experimentally observable properties of Dirac fermions. Paradoxically, the closer to the strong-coupling fixed point the flow of the Dirac fermion under interactions starts, the more it behaves like the noninteracting theory (with typical Fermi velocities about 2 or 3 orders of magnitude slower than the speed of light) (15). The dichotomy between the long-range and contact parts of the Coulomb potential for Dirac fermions was also discussed in ref. 17. Since this particular velocity renormalization influences all points on the Fermi surface equally, it drops out when calculating the anisotropy renormalization that we are concerned with here.

To build some intuition into our results, we begin by considering the Hartree–Fock approximation and first-order perturbation theory (for the case we consider, both these considerations give identical results; SI Appendix, sections A and B). This is the leading-order many-body result as expanded directly in the bare interaction. We first observe that for Dirac fermions, the contact interaction does not renormalize the Fermi velocity (e.g., ref. 18 and SI Appendix, section B). For the long-range Coulomb interaction, the dominant contribution comes from the filled-hole bands. Although the Fermi surface is vanishing at the Dirac point, the Fermi surface anisotropy is still well defined. By first setting a finite Fermi level (defined by a Fermi momentum $k_F$) and then taking the limit $k_F\to 0$, the Fermi surface anisotropy converges to the Fermi velocity anisotropy ${\eta }_0=v_x/v_y$, where the subscript denotes that this is the noninteracting system. When the 2D Coulomb interaction is turned on, the effective single-particle energy is corrected by the exchange correlation, which within the Hartree–Fock approximation is given by (19)

$$E^{HF}\left(\mathbf{k}\right)=-{\int }_{\mathrm{\Omega }}\frac{d^2\mathbf{k}\prime }{{\left(2\pi \right)}^2}\frac{2\pi e^2}{ϵ|\mathbf{k}-\mathbf{k}\prime |}{\sin }^2\left(\frac{{\phi }_k-{\phi }_k^{\prime }}{2}\right),$$ [1]

where $e$ is the electron charge, $ϵ$ is the dielectric relative permittivity, and $\tan {\phi }_k=\left(v_y{k}_y\right)/\left(v_x{k}_x\right)$. The ${\phi }_k$-dependent factor accounts for the chirality of the Dirac fermions. The Fermi velocity renormalizations along the $x$ and $y$ axes were calculated in ref. 20. Taking the ratio of the velocity along the $x$ axis to the $y$ axis, we find the anisotropy

$$\eta =\frac{\mathrm{E}\left(1-{\eta }_0^2\right)-\mathrm{K}\left(1-{\eta }_0^2\right)}{\mathrm{K}\left(1-{\eta }_0^{-2}\right)-\mathrm{E}\left(1-{\eta }_0^{-2}\right)}\approx \sqrt{{\eta }_0}.$$ [2]

Here $\mathrm{K}\left(z\right)$ and $\mathrm{E}\left(z\right)$ are the complete elliptic integrals of the first and the second kind. By plotting this analytic result as the red dashed curve in Fig. 1, we make a crucial observation: The renormalized anisotropy is remarkably close to the square root of ${\eta }_0$. Since within Hartree–Fock and first-order perturbation theory the exchange energy contribution dominates over the noninteracting contribution, it is not surprising that the result is universal; nonetheless this behavior is unique (SI Appendix, section A) to Dirac fermions in 2 dimensions with a long-range Coulomb interaction. Going beyond perturbation theory, we find that the random phase approximation is consistent with perturbation theory only at lowest order, and the renormalization group random phase approximation (RG-RPA) gives an isotropic Fermi surface in the long wave-length limit. This discrepancy between the RG-RPA and perturbation theory is not present in the isotropic case, and we discuss this issue further in SI Appendix, section B. We emphasize that chirality is essential for our result. The chiral eigenfunctions of the Dirac fermions are invariant to the Hartree–Fock interaction and neglecting these, or even changing the winding number from 1, breaks the universality. By contrast, particle–hole symmetry-breaking terms do not couple to the chirality and preserve the universality (see SI Appendix, section A for details).

Fig. 1.

Universal decrease of Fermi velocity anisotropy for long-range interacting Dirac fermions. Analytical perturbation theory and Hartree–Fock results for Dirac fermions (red dashed line) show a square-root decrease of the Fermi velocity anisotropy. Lattice perturbation theory on the honeycomb lattice (green diamonds), the nonperturbative projective quantum Monte Carlo on the honeycomb lattice (purple triangles), and the $\pi$-flux model (red circles) are all consistent with this square-root behavior. Also shown are recent experiments (black squares) and density matrix renormalization group calculations (blue circles) for the $\nu =1/2$ quantum Hall state. We find that both the chirality of the Dirac bands and a long-range interaction potential are necessary for this universal decrease in anisotropy.

While the perturbative approach is useful to understand qualitatively how the universality arises in the Dirac system, here we use a nonperturbative, numerically exact projective quantum Monte Carlo simulation appropriate for the strongly correlated nature of the ground state. We use a honeycomb lattice with nearest-neighbor hopping and at half-filling there is no fermion sign problem. We are able to separately tune the short-range or Hubbard $U$ and long-range tail $r_s/r$ components of the Coulomb interaction (SI Appendix, section C). The noninteracting system contains Dirac cones at the high-symmetry $K$, $K\prime$ points in the Brillouin zone that are isotropic in the low-energy limit, but have trigonal warping away from the Dirac point, naturally providing an anisotropic Fermi surface, whose renormalization with interactions we can explore. Appropriate for the 3-fold symmetry, we define ${\eta }_q$ as the ratio of the energy deviation ${\left(E-E_0\right)}_{K-K\prime }/{\left(E-E_0\right)}_{K-\mathrm{\Gamma }}$ at a fixed momentum, where $K-K\prime$ and $K-\mathrm{\Gamma }$ are the principal directions.

In addition to the honeycomb lattice, we also perform projective quantum Monte Carlo on a $\pi$-flux model. We consider on a square lattice the nearest-neighbor hopping $t_{ij}$ with a spatially varying phase, such that the product of phases of hopping integrals around a plaquette is $e^{i\pi }=-1$. Similar to the honeycomb lattice, the $\pi$-flux model also has 2 Dirac points, which are situated at $k=\left(\pi /2,\pm \pi /2\right)$ in the Brillouin zone. To get anisotropic Dirac cones, we set the magnitude of the hopping integral along the $x$ axis to 1, while varying $t_y$, the magnitude of the hopping integral along the $y$ axis. In contrast to the honeycomb lattice, the noninteracting dispersion remains anisotropic even as we approach the Dirac point.

Our results are obtained from a finite-size scaling of the numerical data on lattice sizes up to $32\times 32$ unit cells. Fig. 2 A, Left shows a typical case where short-range interactions are dominant ($r_s=0;U=2t$). In this case the correlation-induced velocity renormalization is very small, and the data are consistent with no renormalization of the anisotropy, ${\eta }_q={\eta }_0$ within the error bars. Fig. 2 A, Right is typical for when the long-range tail dominates ($r_s=1/3,U=t$). The data show that strongly correlated Dirac fermions with long-range Coulomb interactions have the universal ${\eta }_q=\sqrt{{\eta }_{q,0}}$. Similar results are also shown for $\pi$ flux at $t_y=0.5$. To complement the quantum Monte Carlo, we calculate first-order perturbation theory on the honeycomb lattice, allowing us to go to much bigger system sizes ($\mathrm{1,500}\times \mathrm{1,500}$ unit cells), showing again good agreement with the square-root renormalization of the anisotropy (see SI Appendix, section E for details). The various numerical and analytical approaches all confirm that the long-range Coulomb potential gives a universal square-root suppression of the bare anisotropy for interacting Dirac fermions, reminiscent of the experimental finding at the half-filled Landau level in ref. 3.

Fig. 2.

Nonperturbative quantum Monte Carlo simulations on a $\pi$-flux model and the honeycomb lattice both show universal square-root anisotropy renormalization with long-range Coulomb interactions, but not for Hubbard interactions. (A) Quantum Monte Carlo (QMC) data on honeycomb lattice with on-site Hubbard interaction (Left) and long-range Coulomb interaction (Right). The energy renormalization $E-E_0$ of the quasiparticle is obtained for the $K-\mathrm{\Gamma }$ direction (squares) and the $K-K\prime$ direction (triangles). Solid lines indicate what one would expect for a square-root renormalization, while dashed lines show no anisotropy renormalization. (B) The interacting theory anisotropy as a function of momentum away from the Dirac point. The $\pi$-flux model has an anisotropy that is almost constant for small momentum down to the Dirac point, while the honeycomb lattice is isotropic as momentum vanishes but becomes more anisotropic at larger momentum. Both the nonperturbative QMC and the lattice perturbation theory show a clear square-root anisotropy renormalization for long-range Coulomb interactions.

A natural interesting question is whether this is a mere coincidence or is an indicator for the predicted emergent Dirac fermion nature of the half-filled Landau level (22) rather than the usual HLR state (23). While answering this question definitively is well beyond the scope of the current work, which is entirely on the many-body renormalization of bare anisotropy in zero-field interacting 2D Dirac and Schrödinger systems, we mention one more tantalizing experimental finding in this context by Pan et al. (21) who established that the half-filled Landau-level conductivity in high-mobility modulation-doped 2D structures manifests a linear-in-carrier density dependence [similar to that observed in graphene (24)] in contrast to the expected quadratic density dependence observed and predicted in the corresponding zero-field 2D conductivity of the usual Schrödinger system (25). If our speculations about a possible connection between our result and the emergent Dirac nature of the composite fermion liquid at half-filling are correct, then it also follows that the long-range nature of Coulomb interaction is an essential part of the physics here since we find the square-root suppression of the bare anisotropy only for long-range Coulomb interaction and not for the short-range contact interaction.

Shown in Fig. 3 is the conductivity for 2 of the samples reported in ref. 21 both at $B=0$ (blue data) and close to half-filling (black data). For unscreened long-range interactions, the conductivity $\sigma \sim \left(e^2/h\right)\left(n/n_i\right)\left(1/r_s^2\right)$, where $r_s=e^2/\left(ϵ\hslash \sqrt{v_x{v}_y}\right)$ is density independent for Dirac fermions (26) but scales as $n^{-1/2}$ for Schrödinger fermions. Assuming that for each sample the impurity concentration $n_i$ is the same both at $B=0$ and at half-filling, this allows us to fit for the conductivity of the composite Fermi liquid with a single Dirac Fermi velocity $v=\sqrt{v_x{v}_y}=1.8\times 10^6$ cm/s. This corresponds to $r_s\sim 10$, which is more than an order of magnitude more strongly interacting than most other condensed-matter realizations of Dirac fermions. Finally, we emphasize that our proposed universality can be tested in several anisotropic Dirac materials including graphene, topological insulators (27), organic conductors (28), and magic-angle twisted bilayer graphene (29). We encourage experimentalists working with these materials to look for our proposed universal many-body renormalization-induced suppression of the anisotropy.

Fig. 3.

The bare electron and composite fermion conductivities, $\sigma$, vs. fermion concentration, $n$, obtained for a given sample at zero magnetic field ($B=0$) and near half-filling ($B=B_{1/2}$). The circles and squares are the experimental data by Pan et al. (21) for 2 different samples. The zero-field data show a quadratic trend fitted by the Boltzmann conductivity for Schrödinger fermions with impurity concentrations $n_i=1.9\times 10^7{\text{cm}}^{-2}$ and $9.6\times 10^6{\text{cm}}^{-2}$ for samples A and B, respectively. The conductivity at half-filling for both samples is then obtained by the Boltzmann conductivity for Dirac fermions assuming a single Fermi velocity $v=\sqrt{v_x{v}_y}=1.8\times 10^6$ cm/s.

Materials and Methods

The details of the analytic and numerical methods used in this work are provided in SI Appendix. SI Appendix, section A contains the analytical Hartree–Fock calculations using the continuum model for anisotropic Dirac fermions. SI Appendix, section B shows the perturbation theory results for the same model and giving the same results. The numerical projective quantum Monte Carlo formalism is discussed in SI Appendix, section C and applied to the $\pi$-flux model in SI Appendix, section D. SI Appendix, section E describes the numerical lattice perturbation theory for both the honeycomb lattice and the $\pi$-flux lattice, while the Boltzmann transport theory is presented in SI Appendix, section F.

The full collection of data presented in this work is available at https://figshare.com/articles/Source_data_of_Green_s_function_and_figure_files_/10062485. For these calculations, we used a projective version of the auxiliary-field quantum Monte Carlo approach. All of the data presented in this article can be reproduced using the Algorithms for Lattice Fermions (ALF) open-source general implementation of the finite-temperature auxiliary-field quantum Monte Carlo available at https://alf.physik.uni-wuerzburg.de. There, detailed documentation on input–output and error analysis can be found. The ALF implementation allows one to simulate very general lattice models, including the long-range Coulomb repulsion.