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arXiv:2208.13681 · Physical review. B./Physical review. B

Competing orders in the honeycomb lattice t−J model

Zheng-Tao Xu, Zheng‐Cheng Gu, Shuo Yang

Published Jul 24, 2023• 14 citations
arXivHTMLPDF

Abstract

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 $d$-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.

DOI
10.1103/physrevb.108.035144

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We study the honeycomb lattice t\ensuremath-Jt\text{\ensuremath{-}}Jt\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 \ensuremathδ<0.05\ensuremath{\delta}<0.05\ensuremathδ<0.05, we find almost degenerate uniform ddd-wave superconducting ground states coexisting with antiferromagnetic order. While at larger doping \ensuremathδ>0.05\ensuremath{\delta}>0.05\ensuremathδ>0.05, the ground state is an approximately half-filled stripe-ordered state, where the stripe period decreases with increasing hole doping \ensuremathδ\ensuremath{\delta}\ensuremathδ. Furthermore, the stripe states with the lowest variational energy always display dx2\ensuremath−y2{d}_{{x}^{2}\ensuremath{-}{y}^{2}}dx2\ensuremath−y2​-wave pairing symmetry. The similarity between our results and those on the square lattice contributes to a more comprehensive understanding of doped Mott insulators.
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.
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The long-studied Hubbard model is one of the simplest models of copper-oxide superconductors. However, the connection between the model and the experimental phase diagram is still under debate, in particular regarding the existence and extent of the ddd-wave superconducting phase. Recent rapid progress in improving the accuracy of numerical solvers has opened a way to answer this question reliably. Here, we study the hole-doping concentration ($\ensuremath{\delta}$) dependence of the Hubbard model in the ground states on a square lattice at strong coupling U/t=10U/t=10U/t=10, for the on-site interaction UUU and the transfer ttt, using a variational Monte Carlo method. The method, which combines tensor network and Lanczos methods on top of Pfaffian wave functions, reveals a rich phase diagram, in which many orders compete severely and degenerate within the energy range of 0.01t0.01t0.01t. We have identified distinct phases including a uniform ddd-wave superconducting phase for $0.17\ensuremath{\lesssim}\ensuremath{\delta}\ensuremath{\lesssim}0.22$ and a stripe charge/spin ordered phase for $\ensuremath{\delta}\ensuremath{\lesssim}0.17$ with the stripe period depending on $\ensuremath{\delta}$, together with presumable spatially coexisting antiferromagnetic and stripe order for $\ensuremath{\delta}\ensuremath{\lesssim}0.07$ and coexisting stripe and ddd-wave superconductivity for $0.07\ensuremath{\lesssim}\ensuremath{\delta}\ensuremath{\lesssim}0.17$. The present, improved method revealed a wider region of a charge uniform superconducting phase than the previous studies and shows a qualitative similarity to the phase diagram of the cuprate superconductors. The superconducting order parameter is largest at doping of around $\ensuremath{\delta}=0.17$ in the ground state, which undergoes phase transitions from an inhomogeneous to a uniform state.
The emergence of superconductivity in doped Mott insulators has been debated for decades. In this paper, we report the theoretical discovery of a time-reversal symmetry breaking superconducting ground state in the doped Mott insulator (described by the well known ttt-JJJ model) on honeycomb lattice, based on a recently developed variational method: the Grassmann tensor product state approach. As a benchmark, we use exact diagonalization and density-matrix renormalization methods to check our results on small clusters. We find systematic consistency for the ground-state energy as well as other physical quantities, such as the staggered magnetization. At low doping, the superconductivity coexists with antiferromagnetic ordering.
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.
Abstract We investigate the hole-doped Hubbard model on a honeycomb lattice using a fermionic projected entangled pair states (fPEPS) method. Our study reveals the presence of quasi-long-range order of Cooper pairs, characterized by power-law decay of correlation functions with exponents K > 1. We further analyze the competing phases of superconductivity, specifically the antiferromagnetic (AFM) order and the charge density wave (CDW) order. Our results show that there are domain wall structures when the hole doping δ is small and the Coulomb parameter U is large. However, these structures disappear as we increase the hole doping δ or decrease U. Furthermore, for small hole doping, the system exhibits AFM order, which diminishes for δ > 0.05. Conversely, as the doping level increases, the CDW order gradually decreases. Notably, a considerable CDW order persists even at higher doping levels. These findings suggest a progressive suppression of the AFM order and a growing prominence of the CDW order with increasing δ.