Stripe and superconducting order competing in the Hubbard model on a square lattice studied by a combined variational Monte Carlo and tensor network method
Stripe and superconducting order competing in the Hubbard model on a square lattice studied by a combined variational Monte Carlo and tensor network method
复制标题
DOI：
10.1103/physrevb.98.205132
复制
发表时间：
2018-08
影响因子：
3.7
通讯作者：
Andrew S. Darmawan;Y. Nomura;Y. Yamaji;M. Imada
中科院分区：
文献类型：
--
作者：
Andrew S. Darmawan;Y. Nomura;Y. Yamaji;M. Imada
长期研究的哈伯德模型是氧化铜超导体最简单的模型之一。然而，该模型与实验相图之间的联系仍存在争议，特别是关于d波超导相的存在及其范围。近期在提高数值求解器精度方面的快速进展为可靠地回答这个问题开辟了一条途径。在此，我们针对在位相互作用U和转移t，采用变分蒙特卡罗方法研究了在强耦合U/t = 10的正方晶格基态下哈伯德模型的空穴掺杂浓度（δ）依赖性。该方法在普凡菲安波函数基础上结合了张量网络和兰索斯方法，揭示了一个丰富的相图，其中许多序在0.01t的能量范围内剧烈竞争且简并。我们已经确定了不同的相，包括在0.17 ≲ δ ≲ 0.22时的均匀d波超导相，以及在δ ≲ 0.17时的条纹电荷/自旋有序相（条纹周期取决于δ），同时在δ ≲ 0.07时可能存在空间上共存的反铁磁和条纹序，在0.07 ≲ δ ≲ 0.17时存在条纹和d波超导共存。目前这种改进的方法揭示出比先前研究更宽的电荷均匀超导相区域，并且与铜酸盐超导体的相图在定性上相似。在基态下，超导序参量在掺杂约为δ = 0.17时最大，此时会发生从非均匀态到均匀态的相变。
The long-studied Hubbard model is one of the simplest models of copper-oxide superconductors. However, the connection between the model and the experimental phase diagram is still under debate, in particular regarding the existence and extent of the $d$-wave superconducting phase. Recent rapid progress in improving the accuracy of numerical solvers has opened a way to answer this question reliably. Here, we study the hole-doping concentration ($\delta$) dependence of the Hubbard model in the ground states on a square lattice at strong coupling $U/t=10$, for the on-site interaction $U$ and the transfer $t$, using a variational Monte Carlo method. The method, which combines tensor network and Lanczos methods on top of Pfaffian wave functions, reveals a rich phase diagram, in which many orders compete severely and degenerate within the energy range of 0.01$t$. We have identified distinct phases including a uniform $d$-wave superconducting phase for $0.17\lesssim \delta \lesssim0.22$ and a stripe charge/spin ordered phase for $\delta\lesssim0.17$ with the stripe period depending on $\delta$, together with presumable spatially coexisting antiferromagnetic and stripe order for $\delta\lesssim0.07$ and coexisting stripe and $d$-wave superconductivity for $0.07\lesssim\delta\lesssim0.17$. The present, improved method revealed a wider region of a charge uniform superconducting phase than the previous studies and shows a qualitative similarity to the phase diagram of the cuprate superconductors. The superconducting order parameter is largest at doping of around $\delta=0.17$ in the ground state, which undergoes phase transitions from an inhomogeneous to a uniform state.