[Paper Review] Emergent Superconductivity and Competing Charge Orders in Hole-Doped Square-Lattice $t$-$J$ Model

Xin Lu, Feng Chen+3 more|arXiv (Cornell University)|Apr 8, 2023

Physics of Superconductivity and Magnetism4 citations

TL;DR

This study employs large-scale density matrix renormalization group (DMRG) simulations with enhanced bond dimensions to re-express the ground state of the hole-doped square-lattice $t$-$J$ model. It reveals that $d$-wave superconductivity (SC) emerges at low doping ($\delta = 1/24 - 1/36$) on six-leg cylinders and becomes dominant at optimal $1/8$ doping on eight-leg cylinders, overcoming competing charge density wave (CDW) order, thus providing a unified mechanism for SC in both hole- and electron-doped cuprates.

ABSTRACT

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 (DMRG) 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 DMRG calculations with much enhanced bond dimensions, allowing more accurate determination of the ground state. On 6-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 8-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 $t$-$J$ models through weakening the competing charge orders, but also provide a unified understanding on the SC of both hole- and electron-doped cuprate superconductors.

Motivation & Objective

• To resolve the longstanding inconsistency between numerical simulations and experimental observations in hole-doped cuprates, where simulations previously predicted CDW dominance instead of superconductivity.
• To investigate the interplay between superconducting (SC) and charge density wave (CDW) orders in the extended $t$-$J$ model under hole doping.
• To determine whether $d$-wave superconductivity can emerge in hole-doped systems when using higher-accuracy DMRG simulations with increased bond dimensions.
• To establish a unified framework for understanding unconventional superconductivity in both hole- and electron-doped cuprates within the $t$-$J$ model.

Proposed method

• Employed state-of-the-art density matrix renormalization group (DMRG) calculations with significantly enhanced bond dimensions (up to 28,000 SU(2) multiplets) to improve ground-state accuracy.
• Used bond dimension extrapolation via quadratic polynomial fitting $\mathcal{F}(1/D) = \mathcal{F}(0) + \alpha/D + \beta/D^2$ to estimate infinite-bond-dimension limits of correlation functions.
• Analyzed pairing correlation functions $P_{yy}(r)$ and charge density profiles $n(x)$ on six- and eight-leg cylinders to detect SC and CDW order parameters.
• Systematically varied doping $\delta$ and next-nearest-neighbor hopping $t_2/t_1$ to map the phase diagram, including $t_2 < 0$ for hole-doped regime.
• Compared results with and without $J_2$ interactions to test robustness of SC phases, using $J_2 = 0$ and $J_2 = (t_2/t_1)^2$.
• Assessed $d$-wave symmetry and algebraic decay of pairing correlations via power-law fitting with exponent $K_{\rm sc}$.

Experimental results

Research questions

• RQ1 Can $d$-wave superconductivity emerge in the hole-doped $t$-$J$ model when using high-accuracy DMRG simulations with large bond dimensions?
• RQ2 How does the competition between superconducting and charge density wave orders evolve with increasing system width (from six- to eight-leg cylinders)?
• RQ3 Does the $t$-$J$ model support a unified description of superconductivity in both hole- and electron-doped cuprates?
• RQ4 How robust are the superconducting phases to variations in model parameters such as $t_2/t_1$ and $J_2$?

Key findings

• On six-leg cylinders, $d$-wave superconductivity with algebraically decaying pairing correlations ($K_{\rm sc} \lesssim 1$) emerges at low doping ($\delta = 1/24 - 1/36$), coexisting with weak or vanishing CDW order.
• At optimal $1/8$ doping on eight-leg cylinders, $d$-wave superconductivity becomes dominant, with pairing correlation exponent $K_{\rm sc} \simeq 1.37$ and $1.38$ for $t_2/t_1 = -0.1$ and $-0.2$, respectively.
• The superconducting phase is robust against variations in $t_2/t_1$ and $J_2$ interactions, with consistent power-law decay of pairing correlations across different parameters.
• On the electron-doped side ($t_2 > 0$), the study confirms the existence of a robust uniform $d$-wave SC, consistent with previous findings.
• The results resolve the prior inconsistency between simulations and experiments by showing that SC wins over CDW in wider systems, especially at optimal doping.
• The $t$-$J$ model with large bond-dimension DMRG simulations provides a unified framework for understanding both hole- and electron-doped cuprate superconductivity.