多轨道Hubbard模型在$\mathcal{N}=3$下的VDAT规范约束算法
A gauge constrained algorithm of VDAT at $\mathcal{N}=3$ for the multi-orbital Hubbard model
AI总结：
本文提出一种规范约束算法，自动满足SCDA在$\mathcal{N}=3$时的自洽条件，使多轨道Hubbard模型的VDAT求解更高效稳定，并推导了闭合表达式，促进其在强关联电子材料中的应用。
摘要：
最近发展的变分离散作用理论（VDAT）为量子多体问题的基态提供了一种系统的变分方法，其解的质量由整数$\mathcal{N}$控制，增加$\mathcal{N}$可单调逼近精确解。利用自洽正则离散作用理论（SCDA），VDAT可以在$d=\infty$的多轨道Hubbard模型中精确求解，这要求整数时间格林函数满足自洽条件。先前的工作表明，$\mathcal{N}=3$能以类似于Gutzwiller近似的代价准确捕捉多轨道Mott/Hund物理。在此，我们采用规范约束来自动满足SCDA在$\mathcal{N}=3$时的自洽条件，从而产生一种数值稳定性增强的更高效算法。我们推导了具有一般密度-密度相互作用的多轨道Hubbard模型的规范约束算法的闭合形式表达式，使得$\mathcal{N}=3$的VDAT能够直接应用于七轨道Hubbard模型。我们展示了在$d=\infty$且$\textrm{N}_{\textrm{orb}}=2-8$的$\textrm{SU}(2\textrm{N}_{\textrm{orb}})$Hubbard模型中使用$\mathcal{N}=2$和$\mathcal{N}=3$的结果及性能分析，并在可能的情况下与数值精确的动力学平均场理论解进行了比较。本工作的进展将极大促进$\mathcal{N}=3$的VDAT在强关联电子材料中的应用。
英文摘要：
The recently developed variational discrete action theory (VDAT) provides a systematic variational approach to the ground state of the quantum many-body problem, where the quality of the solution is controlled by an integer $\mathcal{N}$, and increasing $\mathcal{N}$ monotonically approaches the exact solution. VDAT can be exactly evaluated in the $d=\infty$ multi-orbital Hubbard model using the self-consistent canonical discrete action theory (SCDA), which requires a self-consistency condition for the integer time Green's functions. Previous work demonstrates that $\mathcal{N}=3$ accurately captures multi-orbital Mott/Hund physics at a cost similar to the Gutzwiller approximation. Here we employ a gauge constraint to automatically satisfy the self-consistency condition of the SCDA at $\mathcal{N}=3$, yielding an even more efficient algorithm with enhanced numerical stability. We derive closed form expressions of the gauge constrained algorithm for the multi-orbital Hubbard model with general density-density interactions, allowing VDAT at $\mathcal{N}=3$ to be straightforwardly applied to the seven orbital Hubbard model. We present results and a performance analysis using $\mathcal{N}=2$ and $\mathcal{N}=3$ for the $\textrm{SU}(2\textrm{N}_{\textrm{orb}})$ Hubbard model in $d=\infty$ with $\textrm{N}_{\textrm{orb}}=2-8$, and compare to numerically exact dynamical mean-field theory solutions where available. The developments in this work will greatly facilitate the application of VDAT at $\mathcal{N}=3$ to strongly correlated electron materials.
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