Phase diagram and Fermi-liquid properties of the extended Hubbard model on the honeycomb lattice (1312.5728v1)

Abstract: The Hubbard model and extended Hubbard model on the honeycomb lattice can be seen as prototype models of 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 the Dynamical Cluster Approximation at finite temperatures. After determining the stability of the semi-metallic 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 semi-metallic 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 $\rho(\omega) = |\omega|$ type local density of states. We also investigate the impact of the correlation effects on the renormalization of the Fermi velocity $v_F$. We find that $v_F$ is not renormalized when only on-site repulsion $U$ is present, but that near-neighbor repulsion $V$ does renormalize $v_F$. This may explain the variations between different measurements of $v_F$ in graphene.

• The paper determines the phase diagram by identifying semi-metal, spin-density wave, and charge-density wave phases based on on-site and nearest-neighbor interactions.
• The paper shows that nearest-neighbor interactions significantly renormalize the Fermi velocity, offering insight into the variations observed in graphene experiments.
• The paper employs advanced numerical techniques, combining DCA with DQMC and CTQMC to simulate large clusters without the fermion sign problem at low temperatures.

This paper (1312.5728) investigates the phase diagram and electronic properties of interacting electrons on a honeycomb lattice using the extended Hubbard model. The model includes on-site repulsion ($U$) and nearest-neighbor repulsion ($V$), representing electron interactions in materials like single-layer graphene. The paper employs advanced numerical techniques, specifically the Dynamical Cluster Approximation (DCA) combined with large-scale Determinant Quantum Monte Carlo (DQMC) and Continuous Time Quantum Monte Carlo (CTQMC) as impurity solvers. A key advantage is the absence of the fermion sign problem at half-filling, allowing for simulations on relatively large clusters (up to 96 sites) at low temperatures ($T \sim 1/30$ in units where hopping $t=1$).

Practical Applications and Relevance:

The research directly addresses fundamental questions about the electronic behavior of graphene, a material with significant potential for future electronics. Understanding how electron interactions affect properties like phase stability and Fermi velocity is crucial for predicting and controlling graphene's performance in real-world devices. Specifically:

1. Predicting Material Phases: The phase diagrams calculated (showing semi-metal, spin-density wave, and charge-density wave phases) help predict the electronic state of graphene or similar materials depending on the strength of on-site and inter-site Coulomb interactions. This depends heavily on the surrounding dielectric environment (substrate), which screens the Coulomb interaction.
2. Explaining Experimental Variations in Fermi Velocity: The finding that nearest-neighbor interactions ($V$) strongly renormalize the Fermi velocity ($v_F$), while on-site interactions ($U$) do not, provides a theoretical explanation for why experimental measurements of $v_F$ in graphene vary significantly depending on the substrate. Graphene on a high-dielectric substrate has stronger screening (smaller effective $V$), leading to less $v_F$ renormalization, consistent with experiments showing weak $v_F$ enhancement on substrates like SiC compared to suspended graphene or graphene on hBN.
3. Understanding Electron Dynamics: The conclusion that the semi-metallic phase remains a stable Fermi liquid at low temperatures despite the pseudogap nature of the non-interacting density of states is important for understanding electron transport and excitation properties in graphene when interactions are present but not strong enough to cause symmetry breaking.

Implementation Insights:

Implementing research like this involves sophisticated numerical methods for strongly correlated systems. The paper highlights several key aspects relevant to practitioners:

• Model: The Hamiltonian used is a standard extended Hubbard model:

  $H = -t \sum_{<i,j>,\sigma} c^{\dagger}_{i\sigma} c^{\null}_{j\sigma} + U \sum_{i} n_{i \uparrow}n_{i \downarrow} + \sum_{i>j,\sigma , \bar{\sigma} } V_{ij} n_{i\sigma}n_{j \bar{\sigma} } -\mu \sum_{i} n_{i\sigma}$

  where $t$ is hopping, $U$ is on-site repulsion, and $V_{ij}$ is non-local repulsion (here restricted to nearest-neighbor, $V$, and potentially next-nearest-neighbor, $V'$). Half-filling condition is maintained by adjusting $\mu$.

• Methodology (DCA+DQMC/CTQMC):
  • DCA maps the lattice problem onto an effective cluster problem coupled to a non-interacting bath. This involves self-consistent iterations.
  • DQMC (Determinant Quantum Monte Carlo) is used as the impurity solver to solve the effective cluster problem. Its key feature for this work is the absence of the sign problem at half-filling on bipartite lattices like honeycomb, allowing access to lower temperatures and larger system sizes than typically possible for fermionic QMC.
  • CTQMC (Continuous Time Quantum Monte Carlo) is used for complementary calculations, especially at very low temperatures on smaller clusters.
  • Implementing DCA requires defining clusters (e.g., 24, 54, 96 sites), setting up bath sites, performing QMC simulations to calculate cluster Green's functions, and solving the self-consistency loop to update the bath parameters.
  • DQMC implementation involves Hubbard-Stratonovich transformations to decouple interaction terms, Monte Carlo sampling of auxiliary fields, and calculations involving determinants of matrices related to the Green's function. Large cluster sizes significantly increase computational cost ($O(\beta N_{cluster}^3)$ where $\beta = 1/T$ and $N_{cluster}$ is cluster size).
• Observable Calculation:
  • Phase transitions are identified by monitoring order parameters (e.g., staggered magnetization for SDW, sublattice density difference for CDW) or single-particle gaps in the self-energy.
  • Fermi liquid behavior is assessed by calculating the quasiparticle residue $$Z_T(k) = (1 - \frac{Im\Sigma(k, \pi T)}{\pi T})^{-1}$$ which should remain finite at low $T$ for a Fermi liquid.
  • The imaginary part of the self-energy $\Sigma(i\omega_n)$ at the Dirac point is checked for linear frequency dependence ($Im\Sigma(i\omega_n) \sim \omega_n$) at low Matsubara frequencies $\omega_n$, a signature of Fermi liquid.
  • Local Density of States (LDOS) $\rho(\omega)$ is obtained by analytically continuing imaginary-frequency Green's functions $G(i\omega_n)$ to real frequencies $\omega$. Techniques like Maximum Entropy Method (MEM) or Padé approximation are used, which are numerically challenging and sensitive to noise. $v_F$ renormalization is inferred from the slope of $\rho(\omega)$ near the Fermi level ($\omega=0$) using the relation $\rho(E) \sim |E|/v_F^2$.
• Computational Resources: The paper mentions the need for large-scale simulations using high-performance computing clusters (Compute Canada, Calcul Québec), indicating significant computational requirements for running such methods on large clusters and accumulating sufficient Monte Carlo statistics ($2 \times 10^5$ to $5 \times 10^5$ sweeps per DCA iteration).

Trade-offs and Limitations:

• DCA is a cluster approximation and its accuracy depends on the cluster size. Smaller clusters may not capture long-range correlations adequately, as noted in the comparison of $v_F$ results with smaller DCA clusters vs. CDMFT.
• DQMC is limited to certain interaction forms (like nearest-neighbor $V$) and lattice structures at half-filling due to the sign problem in other cases. Real Coulomb interactions in graphene are long-ranged.
• Analytical continuation methods (MEM, Padé) are ill-posed problems, making the resulting real-frequency spectral functions (like LDOS) sensitive to input data quality (statistical noise from QMC) and regularization parameters.

In summary, the paper provides valuable theoretical insights into interaction effects in graphene and serves as a strong example of applying advanced, computationally intensive numerical methods to address fundamental condensed matter physics problems with direct relevance to materials science and device physics. Implementing these methods requires significant expertise in quantum many-body theory and high-performance computing.