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arXiv:2104.14152 · Physical review. B./Physical review. B

Stripe order in the doped Hubbard model on the honeycomb lattice

Yang Xu, Hao Zheng, Mingpu Qin

Published Apr 7, 2021• 13 citations
arXivHTMLPDF

Abstract

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 $U$ 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.

DOI
10.1103/physrevb.103.155110

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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.
We study the ground state of the doped Hubbard model on the honeycomb lattice in the small-doping and strongly interacting region. The nature of the ground state after doping holes into antiferromagnetic Mott insulating states on the honeycomb lattice remains a long-standing unsolved issue, even though tremendous effort has been spent investigating this challenging problem. In this work, we employ two complementary, state-of-the-art, many-body computational methods: constrained path (CP) auxiliary-field quantum Monte Carlo (AFQMC) with self-consistent constraint and density matrix renormalization group (DMRG) methods. Systematic and detailed cross validations are performed between these two methods for narrow systems for which DMRG can produce reliable results. AFQMC is then utilized to study wider systems to investigate the thermodynamic limit properties. The ground state is found to be a half-filled stripe state in the small-doping and strongly interacting region. The pairing correlation shows ddd-wave symmetry locally but decays exponentially with the distance between two pairs.
We determine the spin and charge orders in the ground state of the doped two-dimensional (2D) Hubbard model in its simplest form, namely with only nearest-neighbor hopping and on-site repulsion. At half-filling, the ground state is known to be an antiferromagnetic Mott insulator. Doping Mott insulators is believed to be relevant to the superconductivity observed in cuprates. A variety of candidates have been proposed for the ground state of the doped 2D Hubbard model. A recent work employing a combination of several state-of-the-art numerical many-body methods established the stripe order as the ground state near 1/81/81/8 doping at strong interactions. In this paper, we apply one of these methods, the cutting-edge constrained-path auxiliary field quantum Monte Carlo (AFQMC) method with self-consistently optimized gauge constraints, to systematically study the model as a function of doping and interaction strength. With careful finite size scaling based on large-scale computations, we map out the ground state phase diagram in terms of its spin and charge order. We find that modulated antiferromagnetic order persists from near half-filling to about 1/51/51/5 doping. At lower interaction strengths or larger doping, these ordered states are best described as spin-density waves, with essentially delocalized holes and modest oscillations in charge correlations. When the charge correlations are stronger (large interaction or small doping), they are best described as stripe states, with the holes more localized near the node in the antiferromagnetic spin order. In both cases, we find that the wavelength in the charge correlations is consistent with so-called filled stripes.
We study the ground state of the doped Hubbard model on honeycomb lattice with both nearest (t)(t)(t) and next-nearest-neighbor hoppings $({t}^{\ensuremath{'}})$ in the small doping and strongly interacting region. Previous study on the model without ${t}^{\ensuremath{'}}$ showed the ground state is a half-filled stripe. We employ density matrix renormalization group and extrapolate the results with truncation errors in the converged region. In the ${t}^{\ensuremath{'}}<0$ side, we find the half-filled stripe phase at ${t}^{\ensuremath{'}}=0$ is stable against the frustration of ${t}^{\ensuremath{'}}$ until a critical point $\ensuremath{-}0.4<{t}_{c}^{\ensuremath{'}}<\ensuremath{-}0.3$, beyond which the ground state switches to antiferromagnetic N\'eel phase with charge modulation. With further increase of ${t}^{\ensuremath{'}}$ to $\ensuremath{-}0.7$, the ground state becomes paramagnetic. In the ${t}^{\ensuremath{'}}>0$ side, the half-filled stripe stretches to ${t}^{\ensuremath{'}}\ensuremath{\approx}0.7$. We do not find obvious enhancement of pairing for the range of ${t}^{\ensuremath{'}}$ studied. We study width-4 cylinders in this paper but the results for spin, charge, and pairing correlation agree qualitatively for periodic and antiperiodic boundary conditions in the half-filled stripe and antiferromagnetic N\'eel phases, suggesting the results are likely to be representative for true two-dimensional systems. The half-filled stripe to antiferromagnetic N\'eel phase transition can be realized on real materials or ultracold atom platform.
The cluster slave-spin method is implemented to research the ground state properties of the honeycomb lattice Hubbard model with doping $\ensuremath{\delta}$ and coupling UUU being its parameters. At half filling, a single direct and continuous phase transition between the semimetal and antiferromagnetic (AFM) insulator is found at UAFM=2.43t{U}_{\text{AFM}}=2.43tUAFM​=2.43t that is in the Gross-Neveu-Yukawa universality class, where a relation between the staggered magnetization MMM and the AFM energy gap ${\mathrm{\ensuremath{\Delta}}}_{\text{AFM}}$ is established as $M\ensuremath{\propto}{\mathrm{\ensuremath{\Delta}}}_{\text{AFM}}$, compared to $M\ensuremath{\propto}{\mathrm{\ensuremath{\Delta}}}_{\text{AFM}}{(ln{\mathrm{\ensuremath{\Delta}}}_{\text{AFM}})}^{2}$ in the square lattice case. A first-order semimetal to the underlying paramagnetic (PM) insulator Mott transition is corroborated at UMott=8.36t{U}_{\text{Mott}}=8.36tUMott​=8.36t, which is responsible for a broad crossover around Uc=5.4t{U}_{c}=5.4tUc​=5.4t between the weak- and strong-coupling regimes in the AFM state that increases with $\ensuremath{\delta}$, in contrast to the square lattice case. In the doped system, the compressibility $\ensuremath{\kappa}$ near the van Hove singularity at $\ensuremath{\delta}=1/4$ is suppressed substantially by the interaction before the semimetal to AFM transition occurs, whereas $\ensuremath{\kappa}$ near the Dirac points is very close to the noninteracting one, indicating that the Dirac cone structure of the energy dispersion is rather robust. An overall phase diagram in the $U\text{\ensuremath{-}}\ensuremath{\delta}$ plane is presented, consisting of four regimes: the AFM insulator at $\ensuremath{\delta}=0$ for U>UAFMU>{U}_{\text{AFM}}U>UAFM​, the AFM metal with compressibility $\ensuremath{\kappa}>0$ or $\ensuremath{\kappa}<0$, and the PM semimetal; the AFM metal with $\ensuremath{\kappa}<0$ only exists in an extremely small area near the phase boundary between the AFM and PM state.
We study the honeycomb lattice $t\text{\ensuremath{-}}J$ model using the fermionic tensor network approach. By examining the ansatz with various unit cells, we discover several different stripe states with different periods that compete strongly with uniform states. At very small doping $\ensuremath{\delta}<0.05$, we find almost degenerate uniform ddd-wave superconducting ground states coexisting with antiferromagnetic order. While at larger doping $\ensuremath{\delta}>0.05$, the ground state is an approximately half-filled stripe-ordered state, where the stripe period decreases with increasing hole doping $\ensuremath{\delta}$. Furthermore, the stripe states with the lowest variational energy always display ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave pairing symmetry. The similarity between our results and those on the square lattice contributes to a more comprehensive understanding of doped Mott insulators.
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.