[Paper Review] Hybrid Monte-Carlo simulation of interacting tight-binding model of graphene

Dominik Smith, Lorenz von Smekal|arXiv (Cornell University)|Nov 5, 2013

Graphene research and applications12 references3 citations

TL;DR

This study presents Hybrid Monte Carlo simulations of the interacting tight-binding model of graphene using a Hubbard-Stratonovich transformation to model long-range Coulomb interactions. It identifies a conductor-insulator phase transition at an effective fine-structure constant α_C ≈ 4, consistent with prior work, and demonstrates that compact Hubbard fields and second-order discretization schemes yield equivalent continuum limits despite higher-order corrections.

ABSTRACT

In this work, results are presented of Hybrid-Monte-Carlo simulations of the tight-binding Hamiltonian of graphene, coupled to an instantaneous long-range two-body potential which is modeled by a Hubbard-Stratonovich auxiliary field. We present an investigation of the spontaneous breaking of the sublattice symmetry, which corresponds to a phase transition from a conducting to an insulating phase and which occurs when the effective fine-structure constant $α$ of the system crosses above a certain threshold $α_C$. Qualitative comparisons to earlier works on the subject (which used larger system sizes and higher statistics) are made and it is established that $α_C$ is of a plausible magnitude in our simulations. Also, we discuss differences between simulations using compact and non-compact variants of the Hubbard field and present a quantitative comparison of distinct discretization schemes of the Euclidean time-like dimension in the Fermion operator.

Motivation & Objective

• To investigate spontaneous sublattice symmetry breaking in graphene's interacting tight-binding model using non-perturbative lattice simulations.
• To determine the critical value α_C of the effective fine-structure constant at which a conductor-insulator phase transition occurs.
• To assess the impact of different discretization schemes and Hubbard field variants (compact vs. non-compact) on simulation accuracy and convergence.
• To validate the simulation framework against earlier results and prepare for large-scale studies with improved boundary conditions.

Proposed method

• The path integral formulation of the tight-binding Hamiltonian is derived, incorporating a long-range two-body potential via a Hubbard-Stratonovich auxiliary field.
• A non-local, piecewise potential is constructed using cRPA values for on-site, nearest-neighbor, next-nearest-neighbor, and hexagon-crossing terms, with an unscreened Coulomb tail.
• Simulations are performed on a 6×6 rectangular graphene lattice using GPU-accelerated Hybrid Monte Carlo with Euclidean time discretization.
• The order parameter ⟨Δ_N⟩ is measured across multiple configurations to detect spontaneous symmetry breaking and extrapolate to the chiral limit (m→0).
• Discretization errors are quantified by fitting ⟨Δ_N⟩ to 1/N_t and 1/N_t² terms for standard and improved second-order Fermion actions.
• Compact Hubbard fields are preferred over non-compact ones due to better convergence and non-zero order parameter detection.

Experimental results

Research questions

• RQ1 Does the interacting tight-binding model of graphene exhibit a phase transition from a conducting to an insulating state at a physically plausible α_C?
• RQ2 How do compact and non-compact Hubbard field formulations affect the simulation of sublattice symmetry breaking?
• RQ3 What is the quantitative impact of first- vs. second-order discretization schemes on the continuum limit of the Fermion action?
• RQ4 Can the simulation framework reproduce prior results on α_C when using consistent potential parameters and boundary conditions?

Key findings

• The simulation identifies a conductor-insulator phase transition at α_C ≈ 4, which is qualitatively consistent with previous studies reporting α_C ≈ 3.12.
• Non-compact Hubbard fields fail to produce a non-zero order parameter, indicating numerical instability and poor convergence.
• Both first- and second-order discretization schemes converge to the same continuum limit, with fitted coefficients c1 and c2 showing negligible differences.
• The second-order scheme does not yield measurable improvements over the standard scheme, suggesting no benefit from higher-order terms in this setup.
• The results validate the simulation framework as a reliable platform for future large-scale studies with improved boundary handling and potential matching.
• The use of FFT-based potential computation enables future simulations at larger volumes and with consistent boundary conditions.