# Superconductivity, pair density wave, and Neel order in cuprates

## TL;DR

This paper explores the coexistence of superconductivity, pair density waves, and Neel order in underdoped cuprates, revealing different favored states at half filling and finite doping through variational quantum Monte Carlo simulations.

## Contribution

It introduces a comprehensive analysis of competing orders in cuprates, identifying the energetically favored states and their evolution with doping levels.

## Key findings

- At half filling, the $pS_{\bf Q}+pT_0$ state is more stable.
- At finite doping, the $dS_0+dT_{\bf Q}$ state becomes more favorable.
- Results suggest proximity to a $\mathrm{Z_2}$ quantum spin liquid with a spinon Fermi surface.

## Abstract

We investigate in underdoped cuprates possible coexistence of the superconducting (SC) order at zero momentum and pair density wave (PDW) at momentum ${\bf Q}=(\pi, \pi)$ in the presence of a Neel order. By symmetry, the $d$-wave uniform singlet pairing $dS_0$ can coexist with the $d$-wave triplet PDW $dT_{\bf Q}$, and the $p$-wave singlet PDW $pS_{\bf Q}$ can coexist with the $p$-wave uniform triplet $pT_0$. At half filling, we find the novel $pS_{\bf Q}+pT_0$ state is energetically more favorable than the $dS_0+dT_{\bf Q}$ state. At finite doping, however, the $dS_0+dT_{\bf Q}$ state is more favorable. In both types of states, the variational triplet parameters, $dT_{\bf Q}$ and $pT_0$, are of secondary significance. Our results point to a fully symmetric $\mathrm{Z_2}$ quantum spin liquid with spinon Fermi surface in proximity to the Neel order at zero doping, and to intertwined $d$-wave triplet PDW fluctuations and spin moment fluctuations along with the dominant $d$-wave singlet SC at finite doping. The results are obtained by variational quantum Monte Carlo simulations.