[Paper Review] Superconducting valence bond fluid in lightly doped 8-leg $t$-$J$ cylinders

Hongchen Jiang, Steven A. Kivelson+1 more|arXiv (Cornell University)|Feb 22, 2023

Physics of Superconductivity and Magnetism4 citations

TL;DR

This study uses density matrix renormalization group (DMRG) simulations on 8-leg t-J cylinders to investigate superconductivity in lightly doped quantum spin liquids and valence bond crystals. It finds robust d-wave superconducting order with quasi-long-range correlations and no competing charge-density-wave order, suggesting a nodeless d-wave superconducting ground state in the 2D thermodynamic limit for both doped quantum spin liquid and valence bond crystal phases at δ=1/12 and δ=1/8.

ABSTRACT

Superconductivity in doped quantum paramagnets has been a subject of long theoretical inquiry. In this work we report a density matrix renormalization group study of lightly doped $t$-$J$ models on the square lattice (doped hole densities $δ= 1/12$ and 1/8) with parameters for which previous studies have suggested that the undoped system in 2D is either a quantum spin liquid or a valence bond crystal. Our studies are performed on cylinders with width up to 8. Ground-state correlations are found to be nearly identical for the ``doped quantum spin liquid'' and ``doped valence bond crystal''. Upon increasing the cylinder width from 4 to 8, we observed a significant strengthening of the quasi-long-range superconducting correlations, and a dramatic suppression of any ``competing'' charge-density-wave order. Extrapolating from the observed behavior of the width 8 cylinders, we speculate that the system has a nodeless d-wave superconducting ground-state in the 2D limit.

Motivation & Objective

• To investigate the emergence of superconductivity in lightly doped t-J models on square lattices with parameters that favor either a quantum spin liquid or a valence bond crystal in the undoped limit.
• To determine whether superconducting order persists and dominates over competing orders such as charge-density-wave (CDW) or Néel antiferromagnetic order in doped systems.
• To assess the stability and nature of superconducting correlations as cylinder width increases to 8, extrapolating toward the 2D thermodynamic limit.
• To examine the isotropy and d-wave symmetry of superconducting order parameters in finite-width cylinders with broken rotational symmetry.
• To compare the behavior of superconducting and charge-density correlations across different doping levels and J2/J1 ratios, particularly at δ=1/12 and δ=1/8.

Proposed method

• Employed density matrix renormalization group (DMRG) method to compute ground-state properties of 8-leg t-J cylinders with hole doping δ=1/12 and δ=1/8.
• Used t1/J1=3 and J2/J1=0.5 and 0.55 to tune the system between a quantum spin liquid (QSL) and a valence bond crystal (VBC) at half-filling.
• Analyzed superconducting pair-field correlation functions Φαβ(r,r′) to extract the exponent Ksc governing power-law decay, indicating quasi-long-range order.
• Computed charge-density-density correlation functions D(r) to extract the Luttinger liquid parameter Kc, comparing it with Ksc to assess competing orders.
• Used finite-size scaling and extrapolation techniques to infer behavior in the 2D limit, particularly focusing on the width dependence of Ksc and correlation lengths.
• Applied boundary-induced oscillation models to interpret finite-size effects on CDW and superconducting components, using the form Γ(r) = a1 + a2A(r)cos(Q·r + φ(r)).

Experimental results

Research questions

• RQ1 Does superconducting order emerge in lightly doped t-J cylinders when the undoped system is a quantum spin liquid or a valence bond crystal?
• RQ2 How does the strength and symmetry of superconducting correlations evolve with increasing cylinder width (W=4 to W=8)?
• RQ3 What is the role of next-nearest-neighbor hopping t2 and the resulting J2 in suppressing competing orders like CDW and enhancing d-wave superconductivity?
• RQ4 Is the superconducting order in the doped system isotropic despite the broken 90-degree rotational symmetry of the cylindrical geometry?
• RQ5 Can the observed power-law decay of superconducting correlations and the absence of long-range CDW order support a nodeless d-wave superconducting ground state in the 2D limit?

Key findings

• For W=8 cylinders at δ=1/12 and δ=1/8, superconducting pair correlations exhibit quasi-long-range order with power-law decay characterized by exponent Ksc, which is roughly halved compared to W=4 and W=6, indicating enhanced superconducting correlations.
• The superconducting order parameter is d-wave in symmetry and exhibits surprising isotropy in the cylinder geometry, despite the absence of 90-degree rotational symmetry.
• Charge-density-wave (CDW) order is dramatically suppressed in the doped regime, with no evidence of long-range CDW order and CDW correlation functions decaying with exponent Kc > 2, consistent with Luttinger liquid behavior.
• The Luttinger liquid parameter Kc extracted from both D(r) and n(x) is consistently greater than Ksc, indicating that superconducting order dominates over charge-density fluctuations.
• The spin gap persists in the doped system, as indicated by exponentially decaying spin-spin and dimer-dimer correlations with correlation lengths < W for W≥6, suggesting a gapped spin liquid phase in the 2D limit.
• Extrapolation from W=8 results suggests the system develops a nodeless d-wave superconducting ground state in the 2D thermodynamic limit, regardless of whether the undoped state is a QSL or VBC.