Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2511.18644 · npj Quantum Materials

Competition between charge-density-wave and superconducting orders on eight-leg square Hubbard cylinders

Hong-Chen Jiang, Thomas P. Devereaux, Steven A. Kivelson

Published Jun 10, 2026• 0 citations
arXivHTMLPDF

Abstract

Abstract The issue of whether d -wave superconductivity (SC) occurs in the square-lattice Hubbard model with U of order of the bandwidth has been one of the most debated issues to emerge from the study of high temperature SC. Here, we report variational results on eight-leg cylinders with next-nearest-neighbor hopping in the range $$-0.5t\le t^{\prime} \le 0.25t$$ − 0.5 t ≤ t ′ ≤ 0.25 t with U = 8 t and 12 t and doped hole concentrations δ = 1/12 and 1/8. For $$t^{\prime} \le 0$$ t ′ ≤ 0 , the ground-state appears to be a charge-density wave (CDW) of one sort or another with SC correlations that are extremely short-ranged. In contrast, in some cases, the local magnetic order has a correlation length greater than half the cylinder width - suggestive that magnetic order might also arise in the 2D limit. For $$t^{\prime}\,>\,0$$ t ′ > 0 , our results depend more strongly on boundary conditions (periodic vs antiperiodic), making it still harder to correctly guess whether SC or CDW correlations dominate in the 2D limit. These results were obtained employing matrix-product states with bond dimensions large enough that energy differences as small as 10 −3 t per site can be resolved.

DOI
10.1038/s41535-026-00910-7

Related papers

Tuning competition between charge order and superconductivity in the square-lattice t − t ′ − J model

Xin Lu, Huaiming Guo, Wei-Qiang Chen +2 more

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-$U$ and antiadiabatic (infinite phonon frequency) limits, which gives an effective $t\text{\ensuremath{-}}J$ model with either a density attractive interaction $V$ or a ${J}_{P}$ 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 $V$ or ${J}_{P}$ coupling. We focus on the six-leg cylinder system with the next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$, which hosts partially filled stripe and $d$-wave SC in phase diagram. By tuning ${t}^{\ensuremath{'}}/t>0$ and $V$ or ${J}_{P}$, we establish two quantum phase diagrams at $\frac{1}{12}$ doping ratio. In the SC phases, the increased $V$ or ${J}_{P}$ 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 $V$ or ${J}_{P}$. 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 $V$ and ${J}_{P}$ 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.

arXiv:2409.15270•Physical review. B./Physical review. B•Jan 21, 2025•3 citations

Emergent Superconductivity and Competing Charge Orders in Hole-Doped Square-Lattice t − J Model

Xin Lu, Feng Chen, Wei Kang Zhu +2 more

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.

arXiv:2304.03963•Physical Review Letters•Feb 8, 2024•49 citations

Ground-state phase diagram and superconductivity of the doped Hubbard model on six-leg square cylinders

Yi‐Fan Jiang, Thomas Devereaux, Hong‐Chen Jiang

The authors employ the density-matrix renormalization group to investigate the doped Hubbard model on six-leg square cylinders. It uncovers a rich quantum phase diagram, intricately sensitive to next-nearest-neighbor electron hopping ($t\ensuremath{'}$). The positive-$t$\ensuremath{'} region shows a robust $d$-wave superconducting phase with intertwined superconducting and charge-density-wave orders. In contrast, the negative-$t$\ensuremath{'} side remains insulating, where doped holes form either long-range charge stripe order at small $t$\ensuremath{'} or a holon Wigner crystal with one doped hole per emergent unit cell and short-range spin correlations at larger $t$'.

arXiv:2303.15541•Physical review. B./Physical review. B•Feb 14, 2024•36 citations

Global phase diagram of D-wave superconductivity in the square-lattice t-J model

Feng Chen, F. D. M. Haldane, D. N. Sheng

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.

arXiv:2311.15092•Proceedings of the National Academy of Sciences•Mar 21, 2025•27 citations

Uniform-Charge-Density $d$-Wave Superconductivity in the Pure $t$-$J$ Model at $1/8$ Doping on an Infinite Cylinder

G. Yu, Zheng Zhu

The ground state of the square-lattice $t$-$J$ model at $1/8$ hole doping has been the subject of a long-standing controversy, with conflicting numerical reports of stripe order versus superconductivity. Here, we address this debate using the variational uniform matrix product state (VUMPS) method, which optimizes the wave function directly in the thermodynamic limit along the cylinder axis and eliminates the boundary pinning effects that can arise in finite-cylinder calculations. On an infinite cylinder of circumference $L_y = 8$, we find a uniform-charge-density $d$-wave superconducting ground state with quasi-long-range pairing correlations. The dominant pairing occurs at zero center-of-mass momentum, while no sizable finite-momentum pairing component is detected. Twisted-boundary-condition calculations reveal a finite superconducting phase stiffness, with the order-parameter phase winding smoothly to follow the applied flux, providing an independent probe of the robustness of the superconductivity. Moreover, both charge and spin correlations decay rapidly, with no signatures of long-range stripe or magnetic order, in contrast to an $L_y=6$ cylinder at $1/6$ doping where the same method readily identifies stripe order. Our results provide methodologically distinct evidence in the long-standing stripe--superconductivity debate in the pure $t$-$J$ model and reveal uniform $d$-wave superconductivity in a minimal model of doped Mott insulators.

arXiv:2609.37191•Sep 29, 2026•0 citations

Intertwined order in a frustrated four-leg t − J cylinder

John F. Dodaro, Hong‐Chen Jiang, Steven Allan Kivelson

We report a density-matrix renormalization group study of the $t\text{\ensuremath{-}}J$ model with nearest $({t}_{1}$ and ${J}_{1})$ and next-nearest $({t}_{2}$ and ${J}_{2})$ interactions on a four-leg cylinder with concentration $\ensuremath{\delta}=1/8$ of doped holes. We observe an astonishingly complex interplay between uniform $d$-wave superconductivity (SC) and strong spin and charge-density wave ordering tendencies (SDW and CDW). Depending on parameters, the CDWs can be commensurate with period 4 or 8. By comparing the charge ordering vectors with $2{k}_{F}$, we rule out Fermi surface nesting-induced density wave order in our model. Magnetic frustration (i.e., ${J}_{2}/{J}_{1}\ensuremath{\sim}1/2)$ significantly quenches SDW correlations with little effect on the CDW. Typically, the SC order is strongly modulated at the CDW ordering vector and exhibits $d$-wave symmetry around the cylinder. There is no evidence of a near-degenerate tendency to pair-density wave (PDW) ordering, charge $4e$ SC, or orbital current order.

arXiv:1610.04300•Physical review. B./Physical review. B•Apr 12, 2017•70 citations
Abstract The issue of whether d -wave superconductivity (SC) occurs in the square-lattice Hubbard model with U of order of the bandwidth has been one of the most debated issues to emerge from the study of high temperature SC. Here, we report variational results on eight-leg cylinders with next-nearest-neighbor hopping in the range −0.5t≤t′≤0.25t-0.5t\le t^{\prime} \le 0.25t−0.5t≤t′≤0.25t − 0.5 t ≤ t ′ ≤ 0.25 t with U = 8 t and 12 t and doped hole concentrations δ = 1/12 and 1/8. For t′≤0t^{\prime} \le 0t′≤0 t ′ ≤ 0 , the ground-state appears to be a charge-density wave (CDW) of one sort or another with SC correlations that are extremely short-ranged. In contrast, in some cases, the local magnetic order has a correlation length greater than half the cylinder width - suggestive that magnetic order might also arise in the 2D limit. For t′ > 0t^{\prime}\,>\,0t′>0 t ′ > 0 , our results depend more strongly on boundary conditions (periodic vs antiperiodic), making it still harder to correctly guess whether SC or CDW correlations dominate in the 2D limit. These results were obtained employing matrix-product states with bond dimensions large enough that energy differences as small as 10 −3 t per site can be resolved.
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.
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.
The authors employ the density-matrix renormalization group to investigate the doped Hubbard model on six-leg square cylinders. It uncovers a rich quantum phase diagram, intricately sensitive to next-nearest-neighbor electron hopping ($t\ensuremath{'}$). The positive-ttt\ensuremath{'} region shows a robust ddd-wave superconducting phase with intertwined superconducting and charge-density-wave orders. In contrast, the negative-ttt\ensuremath{'} side remains insulating, where doped holes form either long-range charge stripe order at small ttt\ensuremath{'} or a holon Wigner crystal with one doped hole per emergent unit cell and short-range spin correlations at larger ttt'.
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.
Uniform-Charge-Density ddd-Wave Superconductivity in the Pure ttt-JJJ Model at 1/81/81/8 Doping on an Infinite Cylinder
The ground state of the square-lattice ttt-JJJ model at 1/81/81/8 hole doping has been the subject of a long-standing controversy, with conflicting numerical reports of stripe order versus superconductivity. Here, we address this debate using the variational uniform matrix product state (VUMPS) method, which optimizes the wave function directly in the thermodynamic limit along the cylinder axis and eliminates the boundary pinning effects that can arise in finite-cylinder calculations. On an infinite cylinder of circumference Ly=8L_y = 8Ly​=8, we find a uniform-charge-density ddd-wave superconducting ground state with quasi-long-range pairing correlations. The dominant pairing occurs at zero center-of-mass momentum, while no sizable finite-momentum pairing component is detected. Twisted-boundary-condition calculations reveal a finite superconducting phase stiffness, with the order-parameter phase winding smoothly to follow the applied flux, providing an independent probe of the robustness of the superconductivity. Moreover, both charge and spin correlations decay rapidly, with no signatures of long-range stripe or magnetic order, in contrast to an Ly=6L_y=6Ly​=6 cylinder at 1/61/61/6 doping where the same method readily identifies stripe order. Our results provide methodologically distinct evidence in the long-standing stripe--superconductivity debate in the pure ttt-JJJ model and reveal uniform ddd-wave superconductivity in a minimal model of doped Mott insulators.
We report a density-matrix renormalization group study of the $t\text{\ensuremath{-}}J$ model with nearest (t1({t}_{1}(t1​ and J1){J}_{1})J1​) and next-nearest (t2({t}_{2}(t2​ and J2){J}_{2})J2​) interactions on a four-leg cylinder with concentration $\ensuremath{\delta}=1/8$ of doped holes. We observe an astonishingly complex interplay between uniform ddd-wave superconductivity (SC) and strong spin and charge-density wave ordering tendencies (SDW and CDW). Depending on parameters, the CDWs can be commensurate with period 4 or 8. By comparing the charge ordering vectors with 2kF2{k}_{F}2kF​, we rule out Fermi surface nesting-induced density wave order in our model. Magnetic frustration (i.e., ${J}_{2}/{J}_{1}\ensuremath{\sim}1/2)$ significantly quenches SDW correlations with little effect on the CDW. Typically, the SC order is strongly modulated at the CDW ordering vector and exhibits ddd-wave symmetry around the cylinder. There is no evidence of a near-degenerate tendency to pair-density wave (PDW) ordering, charge 4e4e4e SC, or orbital current order.