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Journal of Physics Conference Series

Quantum Monte Carlo study of the half-filled Hubbard model on the honeycomb lattice

Yuichi Otsuka, Seiji Yunoki, Sandro Sorella

Published Aug 12, 2013• 2 citations
Publisher

Abstract

The half-filled Hubbard model on the honeycomb lattice is investigated by numerically exact large-scale quantum Monte Carlo simulations for lattice sizes up to 2592 sites. By performing careful finite-size scaling for the spin-spin correlation functions, calculated with a high degree of accuracy, we find that the ground state is antiferromagnetically long-range ordered at U / t = 4, where U is the on-site Hubbard interaction and t is the nearest neighbor hopping. Our result is in sharp contrast to a recent report [Meng et al. , Nature 464, 847 (2010)], where instead strong evidence of spin liquid behavior is found.

DOI
10.1088/1742-6596/454/1/012045

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The half-filled Hubbard model on the honeycomb lattice is investigated by numerically exact large-scale quantum Monte Carlo simulations for lattice sizes up to 2592 sites. By performing careful finite-size scaling for the spin-spin correlation functions, calculated with a high degree of accuracy, we find that the ground state is antiferromagnetically long-range ordered at U / t = 4, where U is the on-site Hubbard interaction and t is the nearest neighbor hopping. Our result is in sharp contrast to a recent report [Meng et al. , Nature 464, 847 (2010)], where instead strong evidence of spin liquid behavior is found.
The Hubbard model encapsulates the physics of strongly correlated quantum systems in its most basic form. It has been studied intensively in the context of the high-temperature superconductivity. A number of novel phases were recently proposed for Hubbard-like models on the honeycomb lattice, the structure of graphene. We analyzed the Hubbard model of spin- 12\frac{1}{2}21​ fermions on the honeycomb lattice at half-filling using large-scale quantum Monte Carlo simulations. We find that the weak coupling semimetal and the antiferromagnetic Mott insulator at strong interaction are separated by an extended gapped phase in an intermediate coupling regime. Exploring excitation gaps, various correlation functions as well as probing for flux quantization, we conclude that a quantum spin liquid, lacking any conventional order, emerges with local charge and spin correlations, best described by a resonating valence bonds state.
Recent numerical work [Z. Y. Meng et al., Nature (London) 464, 847 (2010)] indicates the existence of a spin liquid (SL) phase that intervenes between the antiferromagnetic and semimetallic phases of the half filled Hubbard model on a honeycomb lattice. To better understand the nature of this exotic phase, we study the quantum J(1)-J(2) spin model on the honeycomb lattice, which provides an effective description of the Mott insulating region of the Hubbard model. Employing the variational Monte Carlo approach, we analyze the phase diagram of the model. We find three phases-antiferromagnetic, an unusual Z(2) SL state, and a dimerized state with spontaneously broken rotational symmetry. We identify the Z(2) SL state as the likely candidate for the SL phase of the Hubbard model.
Despite being relevant to better understand the properties of honeycomblike systems, as graphene-based compounds, the electron-phonon interaction is commonly disregarded in theoretical approaches. That is, the effects of phonon fields on interacting Dirac electrons is an open issue, in particular when investigating long-range ordering. Thus, here we perform unbiased quantum Monte Carlo simulations to examine the Hubbard-Holstein model (HHM) in the half-filled honeycomb lattice. By performing careful finite-size scaling analysis, we identify semimetal-to-insulator quantum critical points, and determine the behavior of the antiferromagnetic and charge-density wave phase transitions. We have, therefore, established the ground state phase diagram of the HHM for intermediate interaction strength, determining its behavior for different phonon frequencies. Our findings provide quantitative and qualitative descriptions of the model at intermediate coupling strengths, and may shed light on the emergence of many-body properties in honeycomblike systems.
We study the ground state properties of the doped Hubbard model with strong interactions on the honeycomb lattice by the density matrix renormalization group (DMRG) method. At half-filling, due to the absence of minus sign problem, it is now well established by large-scale quantum Monte Carlo calculations that a Dirac semimetal to antiferromagnetic Mott insulator transition occurs with the increase of the interaction strength UUU for the Hubbard model on the honeycomb lattice. However, an understanding of the fate of the antiferromagnetic Mott insulator when holes are doped into the system is still lacking. In this work, by calculating the local spin and charge density for width-4 cylinders with DMRG, we discover a half-filled stripe order in the doped Hubbard model on the honeycomb lattice. We also perform complementary large-scale mean-field calculations with renormalized interaction strength. We observe half-filled stripe order and find stripe states with filling close to one half are nearly degenerate in energy.
A spin liquid is a novel quantum state of matter with no conventional order parameter where a finite charge gap exists even though the band theory would predict metallic behavior. Finding a stable spin liquid in two or higher spatial dimensions is one of the most challenging and debated issues in condensed matter physics. Very recently, it has been reported that a model of graphene, i.e., the Hubbard model on the honeycomb lattice, can show a spin liquid ground state in a wide region of the phase diagram, between a semi-metal (SM) and an antiferromagnetic insulator (AFMI). Here, by performing numerically exact quantum Monte Carlo simulations, we extend the previous study to much larger clusters (containing up to 2592 sites), and find, if any, a very weak evidence of this spin liquid region. Instead, our calculations strongly indicate a direct and continuous quantum phase transition between SM and AFMI.
We study the magnetic properties of the two-dimensional Hubbard model with nearest-neighbor (t) and next-nearest-neighbor (t2{t}_{2}t2​) hopping and on-site repulsion U. We first calculate the mean-field phase diagram as a function of band filling and U. Because of the Van Hove singularity in the density of states, we find a ferromagnetic phase extending to zero U for certain band filling. For the half-filled band case, antiferromagnetism sets in at a finite value of U if t2{t}_{2}t2​\ensuremath{\ne}0. We study the behavior of spin-spin correlation functions for small lattices of up to N=64 atoms using Monte Carlo simulations, as well as exact diagonalization for N=4. Our results show enhanced ferromagnetic correlations in some regions, but apparently no ferromagnetic long-range order. In the half-filled case, our numerical results are consistent with a nonzero critical U. For a non-half-filled band our results suggest that there is no long-range order.