We present a comprehensive study of vortex structures in $d$-wave superconductors from large-scale renormalized mean-field theory of the square-lattice $t\text{−}{t}^{\prime }\text{−}J$ model, which has been shown to provide a quantitative modeling for high-$T_c$ cuprate superconductors. With an efficient implementation of the kernel polynomial method for solving electronic structures, self-consistent calculations involving up to ${10}^5$ variational parameters are performed to investigate the vortex solutions on lattices of up to ${10}^4$ sites. By taking into account the strong correlation of the model, our calculations shed new light on two puzzling results that have emerged from recent scanning tunneling microscopy experiments. The first concerns the issue of the zero-biased-conductance peak (ZBCP) at the vortex core for a uniform $d$-wave superconducting state. Despite its theoretical prediction, the ZBCP was not observed in most doping range of cuprates except in heavily over-doped samples at low magnetic field. The second issue is the nature of the checkerboard charge-density waves (CDWs) with a period of about eight unit cells in the vortex halo at optimal doping. Although it has been suggested that such bipartite structure arises from low-energy quasiparticle interference, another intriguing scenario posits that the checkerboard CDWs originate from an underlying bidirectional pair-density wave (PDW) ordering with the same period. We present a coherent interpretation of these experimental results based on systematic studies of the doping and magnetic-field effects on vortex solutions with and without a checkerboard structure. Due to the small size of Cooper pairs, the vortex core has a radius of about three unit cells, which results in a strong spatial dependence on pairing fields. This may be an important mechanism for the formation of PDW states inside the vortex core.

• Received 20 December 2022
• Revised 3 May 2023
• Accepted 7 June 2023

DOI:https://doi.org/10.1103/PhysRevResearch.5.033028

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Published by the American Physical Society