The Hubbard Model on the Honeycomb Lattice with Hybrid Monte Carlo

Abstract

We take advantage of recent improvements in the grand canonical Hybrid Monte Carlo (HMC) algorithm, to perform a precision study of the single-particle gap in the hexagonal Hubbard model, with on-site electron-electron interactions. After carefully controlled analyses of the Trotter error, the thermodynamic limit, and finite-size scaling with inverse temperature, we find a critical coupling of $U_c/\kappa =3.835(14)$ and the critical exponent $\nu =1.181(43)$ for the semimetal-antiferromagnetic Mott insulator quantum phase transition in the hexagonal Hubbard Model. Based on these results, we provide a unified, comprehensive treatment of all operators that contribute to the anti-ferromagnetic, ferromagnetic, and charge-density-wave structure factors and order parameters of the hexagonal Hubbard Model. We expect our findings to improve the consistency of Monte Carlo determinations of critical exponents. We perform a data collapse analysis and determine the critical exponent $\beta =0.898(37)$. We consider our findings in view of the $SU(2)$ Gross-Neveu, or chiral Heisenberg, universality class. We also discuss the computational scaling of the HMC algorithm. Our methods are applicable to a wide range of lattice theories of strongly correlated electrons. The Ising model, a simple statistical model for ferromagnetism, is one such theory. There are analytic solutions for low dimensions and very efficient Monte Carlo methods, such as cluster algorithms, for simulating this model in special cases. However most approaches do not generalise to arbitrary lattices and couplings. We present a formalism that allows one to apply HMC simulations to the Ising model, demonstrating how a system with discrete degrees of freedom can be simulated with continuous variables.

arXiv commentCumulative PhD thesis (HISKP, Bonn University) based on the publications arXiv:1912.03278, arXiv:2004.10472, arXiv:2005.11112, arXiv:2105.06936

References in corpus (50)

• Many-Body Physics with Ultracold Gases
• Theory of interacting electrons on the honeycomb lattice
• Light hadrons from lattice QCD with light (u,d), strange and charm dynamical quarks
• Quantum criticality with two length scales
• Quantum critical point in graphene approached in the limit of infinitely strong Coulomb interaction
• Dynamical Twisted Mass Fermions with Light Quarks: Simulation and Analysis Details
• Fermionic quantum criticality in honeycomb and $\pi$-flux Hubbard models: Finite-size scaling of renormalization-group-invariant observables from quantum Monte Carlo
• Resonance Parameters of the rho-Meson from Lattice QCD
• Stochastic Estimation with $Z_2$ Noise
• Four-loop critical exponents for the Gross-Neveu-Yukawa models
• Machine Learning of Explicit Order Parameters: From the Ising Model to SU(2) Lattice Gauge Theory
• Quantum Critical Behaviour in a Graphene-like Model
• Lattice Effective Field Theory for Medium-Mass Nuclei
• Excitonic instability in two-dimensional degenerate semimetals
• Optimized Verlet-like algorithms for molecular dynamics simulations
• Dynamical overlap fermions, results with hybrid Monte-Carlo algorithm
• Importance of Positive Feedbacks and Over-confidence in a Self-Fulfilling Ising Model of Financial Markets
• Finite-size scaling at quantum transitions
• Structure of Lefschetz thimbles in simple fermionic systems
• Superconductivity from the Condensation of Topological Defects in a Quantum Spin-Hall Insulator
• Monte-Carlo study of the electron transport properties of monolayer graphene within the tight-binding model
• First Physics Results at the Physical Pion Mass from $N_f=2$ Wilson Twisted Mass Fermions at Maximal Twist
• Toward solving the sign problem with path optimization method
• Momentum Distribution and Contact of the Unitary Fermi gas
• A lattice theory for low energy fermions at finite chemical potential
• Charge-Density-Wave Transitions of Dirac Fermions Coupled to Phonons
• Revisiting the Hybrid Quantum Monte Carlo Method for Hubbard and Electron-Phonon Models
• Deep Learning Beyond Lefschetz Thimbles
• Interaction driven metal-insulator transition in strained graphene
• Large $N$ critical exponents for the chiral Heisenberg Gross-Neveu universality class
• Machine Learning as a universal tool for quantitative investigations of phase transition
• Tempered transitions between thimbles
• Critical (Chiral) Heisenberg Model with the Functional Renormalisation Group
• Lefschetz thimbles decomposition for the Hubbard model on the hexagonal lattice
• Isospin-0 $\pi \pi$ s-wave scattering length from twisted mass lattice QCD
• Flavor-singlet meson decay constants from $N_f=2+1+1$ twisted mass lattice QCD
• Lemon: an MPI parallel I/O library for data encapsulation using LIME
• Machine learning determination of dynamical parameters: The Ising model case
• Numerical Methods for the QCD Overlap Operator IV: Hybrid Monte Carlo
• η and η' mesons from Nf=2+1+1 twisted mass lattice QCD
• Dirac electrons in the square lattice Hubbard model with a $d$-wave pairing field: chiral Heisenberg universality class revisited
• Quantum critical scaling in magnetic field near the Dirac point in graphene
• Control the model sign problem via path optimization method: Monte-Carlo approach to QCD effective model with Polyakov loop
• Nucleon average quark momentum fraction with $N_{\mathrm{f}}=2+1$ Wilson fermions
• Expanding the Interpolator Basis in the Variational Method to Explicitly Account for Backward Running States
• Hybrid Monte Carlo Simulation of Graphene on the Hexagonal Lattice
• Extraction of multiple exponential signals from lattice correlation functions
• Automated label flows for excited states of correlation functions in lattice gauge theory
• Extracting the Single-Particle Gap in Carbon Nanotubes with Lattice Quantum Monte Carlo
• Data collapse in the critical region using finite-size scaling with subleading corrections