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arXiv:2604.08937

Superconductivity and competing orders in honeycomb $t$-$J$ model: interplay of lattice geometry and next-nearest-neighbor hopping

Zhi Xu, Hong‐Chen Jiang, Yi‐Fan Jiang

Published Apr 10, 2026• 0 citations
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Abstract

We investigate the extended $t$-$J$ model on honeycomb lattices with next-nearest-neighbor (NNN) electron hopping $t'$ and superexchange coupling $J'=(t'/t)^2 J$ using large-scale density-matrix renormalization group (DMRG) simulations and slave-boson mean-field theory (SBMFT). By systematically varying $t'$ and cylinder geometries, our DMRG results reveal several competing phases with distinct charge and superconducting (SC) properties. On YC4-0 cylinders possessing bonds lying along $\vec{e}_y$ direction, the ground state of doped models exhibits pronounced quasi-long-range $d$-wave SC with coexisting armchair-oriented stripes (a-stripe) across a broad range of $t'$. Notably, the SC Luttinger exponent has a non-monotonic dependence on $t'$, showing an optimal $t'_{op}\sim0.4$ for dominant SC. Conversely, XC cylinders host a competing long-range zigzag stripes phase without SC for $t'>0.5$, highlighting the role of boundary geometry in stabilizing distinct competing phases in DMRG. To elucidate the stability of all these competing phases in 2D limit, we employ SBMFT and identify the a-stripe as the stable configuration across most of phase diagram, with a transition to uniform nematic $d$-wave SC at large $t'$ for $δ=1/8$. The combined results from two complementary approaches suggest a robust $t'$-induced SC phase that might remain stable in doped extended $t$-$J$ model on the honeycomb lattice.

DOI
10.48550/arxiv.2604.08937

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Recently, a flurry of works have found strong competition between charge density wave (CDW) and superconductivity (SC) in the doped Hubbard and $t\text{\ensuremath{-}}J$ models on the square lattice. Interestingly, some recent results suggest that the electron-phonon coupling may suppress CDW order and enhance SC. In this work, we consider the square-lattice Hubbard model with the Holstein or Su-Schrieffer-Heeger electron-phonon coupling at the large-UUU and antiadiabatic (infinite phonon frequency) limits, which gives an effective $t\text{\ensuremath{-}}J$ model with either a density attractive interaction VVV or a JP{J}_{P}JP​ term that contributes a larger spin exchange and a density repulsive interaction. To explore how these effective couplings may tune the competition between CDW and SC, we implement the density matrix renormalization group simulations on the $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}J$ model with VVV or JP{J}_{P}JP​ coupling. We focus on the six-leg cylinder system with the next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$, which hosts partially filled stripe and ddd-wave SC in phase diagram. By tuning ${t}^{\ensuremath{'}}/t>0$ and VVV or JP{J}_{P}JP​, we establish two quantum phase diagrams at 112\frac{1}{12}121​ doping ratio. In the SC phases, the increased VVV or JP{J}_{P}JP​ coupling can enhance the quasi-long-range SC order, consistent with some previous findings. Nonetheless, no SC emerges when the partially filled stripe phase disappears with increased VVV or JP{J}_{P}JP​. Instead, the system has a transition to either a phase-separation-like regime or a filled stripe phase. On the other hand, with increased ${t}^{\ensuremath{'}}/t$, not only the partially filled stripe but the phase separation and filled stripe can also be tuned to SC phase. Our results suggest that although VVV and JP{J}_{P}JP​ couplings may strengthen hole binding, the hole dynamics controlled by ${t}^{\ensuremath{'}}/t$ appears to play a more crucial role for obtaining a SC in the $t\text{\ensuremath{-}}J$ model.
Unravelling competing orders emergent in doped Mott insulators and their interplay with unconventional superconductivity is one of the major challenges in condensed matter physics. To explore the possible superconducting state in a doped Mott insulator, we study the square-lattice t-J model with both the nearest-neighbor and next-nearest-neighbor electron hoppings and spin interactions. By using the state-of-the-art density matrix renormalization group calculation with imposing charge U(1) and spin SU(2) symmetries on the six-leg cylinders, we establish a quantum phase diagram including three phases: a stripe charge density wave phase, a superconducting phase without static charge order, and a superconducting phase coexistent with a weak charge stripe order. Crucially, we demonstrate that the superconducting phase has a power-law pairing correlation that decays much slower than the charge density and spin correlations, which is a quasi-1D descendant of the uniform d-wave superconductor in two dimensions. These findings reveal that enhanced charge and spin fluctuations with optimal doping is able to produce robust d-wave superconductivity in doped Mott insulators, providing a foundation for connecting theories of superconductivity to models of strongly correlated systems.
The square-lattice Hubbard and closely related t-J models are considered as basic paradigms for understanding strong correlation effects and unconventional superconductivity (SC). Recent large-scale density matrix renormalization group simulations on the extended t-J model have identified d-wave SC on the electron-doped side (with the next-nearest-neighbor hopping t_{2}>0) but a dominant charge density wave (CDW) order on the hole-doped side (t_{2}<0), which is inconsistent with the SC of hole-doped cuprate compounds. We re-examine the ground-state phase diagram of the extended t-J model by employing the state-of-the-art density matrix renormalization group calculations with much enhanced bond dimensions, allowing more accurate determination of the ground state. On six-leg cylinders, while different CDW phases are identified on the hole-doped side for the doping range δ=1/16-1/8, a SC phase emerges at a lower doping regime, with algebraically decaying pairing correlations and d-wave symmetry. On the wider eight-leg systems, the d-wave SC also emerges on the hole-doped side at the optimal 1/8 doping, demonstrating the winning of SC over CDW by increasing the system width. Our results not only suggest a new path to SC in general t-J model through weakening the competing charge orders, but also provide a unified understanding on the SC of both hole- and electron-doped cuprate superconductors.
Inspired by the observation of a robust ddd-wave superconducting phase driven by tuning the next-nearest-neighbor (NNN) electron hopping in recent density matrix renormalization group (DMRG) studies of six- and eight-leg $t\text{\ensuremath{-}}J$ model, we systematically study the phase diagram of the two-leg $t\text{\ensuremath{-}}J$ ladder with the NNN couplings (including NNN hopping and spin interaction) in a large region of doping level, by means of the DMRG calculations. Upon doping from half filling, we identify the Luther-Emery liquid (LEL) phase, which can be distinguished as the pairing-dominant and charge density-dominant regime by comparing the Luttinger parameter ${K}_{\ensuremath{\rho}}$. With the growing NNN couplings, pairing correlations are enhanced and correspondingly ${K}_{\ensuremath{\rho}}$ increases, driving the system from the density-dominant to the pairing-dominant regime. In the Tomonaga-Luttinger liquid (TLL) phase in the larger doping region, we identify two TLL regimes with different features of charge density correlation. At the quarter filling (1/21/21/2 doping level), we find that the strong dimer orders of bond energy in the open system actually decay algebraically and thus do not indicate a spontaneous translational symmetry breaking. Our results show that in the LEL phase of the two-leg ladder, the NNN couplings seem to play the similar role as that on the wider $t\text{\ensuremath{-}}J$ cylinder, and studies on this more accessible system can be helpful toward understanding the emergence of the remarkable ddd-wave superconducting phase on the wider system.
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 Hubbard and closely related t-J models are exciting platforms for unconventional superconductivity (SC). Through state-of-the-art density matrix renormalization group calculations using the grand canonical ensemble, we address open issues regarding the ground-state phase diagram of the extended t-J model on a square lattice in the parameter regime relevant to cuprate superconductors. On large 8-leg cylinders, we demonstrate that the pure t-J model with only nearest-neighbor hoppings and superexchange interactions, for a wide range of doping ( δ = 0.1 to 0.2 ), hosts robust d-wave superconductivity possibly coexisting with weak unidirectional pair density wave. Furthermore, a small next nearest neighbor hopping t 2 suppresses pair and charge density waves, resulting in a uniform d-wave SC phase in both electron- and hole-doped cuprate model systems. Our work validates the t-J model as a proper minimum model for the emergence of superconductivity in cuprate superconductors.